1979 April 18 20

A p r -i!

1 8 ^ § 0 , 1 9 7 9

C h € y m e a H c ! e a . y

A§pect$ ef Centyotted

Q)ovat #ecem# $#ean$ g!¥ania

— - !97 9 S h e r w o o d (Deet ing —

j, L.ABOr^’:*OH Y

’,-”-j’…’:,r !”r BY

’ .

’

COMMITTEE

r = -D. /D

L. Friedland and I. B. Bernstein

Yale University, New Haven, CT 06520

case, one can trace the rays by solving equations

This paper exploits a general approach to geometric optics in inhomogeneous

makes the formulation of the geometric optics equations different in an aniso

tropic plasma, where only one eigenvalue vanishes, from that in an isotropic

geometric optics approximation the determinant D-E^E^e^ vanishes giving in gen

eral three branches of the dispersion relation. The possibility of branching

(degenerate) plasma with more than one zero eignevalues. In the nondegenerate

in terms of its eigenvalues Ej and eigenvectors e^. Then to zero order in the

plasmas based on the properties of the local dielectric tensor E. We express E

These equations, however, are singular in the degenerate plasma. Here one can use

GEOMETRIC OPTICS IN INHOMOGENEOUS ISOTROPIC AND ANISOTROPIC PLASMAS AND ON THEIR BOUNDARIES*

The transition from (2) to (1) on the boundary between degenerate and nondegenerate

regions presents numerical difficulties, since D = D =D =0 on the boundary. It will

be shown that 1 ‘Hospital’s rule applied to (1) as one approaches the boundary from

and then follow each of the modes by solving (1). We will demonstrate this method

and finds the amplitudes, polarization and absorption of the waves along the rays,

in a case where radiation from a vacuum region enters an inhomogeneous magnetized

ferent branches of the dispersion relation. On using these derivatives, one can

plasma. The details of our general geometric optics code, which traces the rays

split the rays on the boundary, make a small step into the nondegenerate region,

the fact that the sum F = E ^ 2+ ^ E g + E 2E2 of the second order minors of E also van

the nondegenerate side, gives in general two values for k, corresponding to dif

Work supported by the U.S. Department of Energy, contract EG-77-S-02-4349.

ishes and, therefore, can be used in defining nonsingular ray equations

will also be reported.

; k = D /D jr w

r = -F, /D k w

k = F /F r w

k w

(2)

(1)

—

A. B. Hassam

APPLICATION TO TEMPERATURE GRADIENT-DRIVEN MODES*

HIGHER ORDER CHAPMAN-ENSKOG THEORY FOR ELECTRONS:

The Chapman-Enskog expansion is carried to second order

Department of Physics & Astronomy University of Maryland College Park, Maryland 20742

for the electron fluid. The resulting equations, an order more accurate than those of Braginskii, are employed to obtain a class of temperature gradient-driven modes. These modes were heretofore not derivable from fluid theory. In particular, the Collisional^’^ and semi—Collisional^ ver sions of the tearing mode are recovered. Likewise, the higher-m temperature gradient-driven drift and drift-tearing modes^ are also considered.

J. F. Drake and Y. C. Lee, Phys. Fluids ^(3, 1341 (1977).
N. T. Gladd, J. F. Drake, C. L. Chang, and C. S. Liu,
R. D. Hazeltine, D. Dobrott, and T. S. Wang, Phys. Fluids

^Research supported by a fellowship from the Center for

Theoretical Physics, University of Maryland.

18, 1778 (1975).

this meeting.

IN A SHEARED MAGNETIC FIELD*

DISSIPATIVE DRIFT MODES DRIVEN

BY THE ELECTRON TEMPERATURE GRADIENT

We have investigated the Collisional electrostatic

C. L. Chang, J. F. Drake, N. T. Gladd and C. S. Liu

= d In T /d In n. Collisions have been included with

Department of Physics & Astronomy University of Maryland College Park, Maryland 20742

drift wave in.slab geometry and have demonstrated both numerically and analytically that this mode can be driven unstable by a positive electron temperature gradient

a velocity-dependent Lorentz collision o p e r a t o r T h e temperature gradient produces “wells” on either side of the rational surface which localize the radial eigenmode. For a reasonable shear (1^/1 ^ 20), the dissipative drift waves has a threshold n % 3. As the collisionality is reduced, these n driven modes become stable, making a smooth transition to previous collisionless results.^ Analytic expressions for the growth rate have been obtained and are in good agreement with the numerical results.

^Research supported by the Department of Energy.

J. F. Drake and Y. C. Lee, Phys. Fluids _20, 1341 (1977).
N. T. Gladd and C. S. Liu, to be published in Phys. Fluids.

I B 14

A numerical and analytic study of the temperature-

MICROTEARING MODES AND ANOMALOUS TRANSPORT IN TOKAMAKS*

N. T. Gladd, J. F. Drake, C. S. Liu, and C. L. Chang Department of Physics & Astronomy University of Maryland College Park, Maryland 20742

gradient-driven tearing mode^ has been carried out which demonstrates that this mode is unstable in a slab model for realistic tokamak parameters. Collisions have been included with a velocity-dependent Lorentz collision operator. Modes with the growth rate peaking around growth rates, eigenfunctions, and the stability boundaries w^.ll be presented. Radial magnetic field fluctuations ]B)^/B^ associated with this instability lead to enhanced electron thermal transport by effectively converting Xejj toXei* In a nonlinear calculation, we have shown that’ this instability saturates by transferring energy from unstable long wavelengths to stable short wavelengths, a form of eddy viscous damping. We find that the resulting perpendicular transport Xgj_ scales as Tgl/2/n, which is consistent with Alcator scaling, and is of the same order of magnitude as the experimentally observed transport coefficient.

1341 (1977); D. D’Ippolito, J. F. Drake and Y. C. Lee, BAPS 23, 867 (1978).

^Research supported by the Department of Energy.

J. F. Drake and Y. C. Lee, Phys. Fluids
J. F. Drake and C. S. Liu, see separate abstract at this

are found to be unstable with

Details of

meeting.

Techniques for accurately measuring local values of heat transport

Mario Soler’ Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

Observation of Transport in Tokamaks of Arbitrary Shape and Approximate Numerical Description*

coefficients have been reported p r e v i o u s l y . T h e y involve the analysis of electron temperature perturbations produced by internal disruptions, made apparent in fluctuations of the soft x-ray Bremstrahlung emission. The advent of elaborate multi-channel x-ray diagnostics looking at the plasma cross section in two orthogonal directions will allow to obtain, by means of the quoted methods, local measurements of heat transport flowing in these two directions. For elliptical cross section plasmas such as those to be produced in ISX-B, values of local transport differing by a large (2 to 4) factor should be observed depending on the direction of observation, because of the influence of geometry. The descriptions of transport in noncylindrical geometry usually imply a surface averaging of transport, effectively effacing local features. The comparison of these descriptions with accurate local measurements does not seem very meaningful. In particular, the influence of transport on the evolution of geometry is not clear. Some preliminary work is presented on approximate computational methods being developed with the purpose of taking into account the effects of local values of transport.

^Corporation. ‘Visitor from Junta de Energia Nuclear, Madrid, Spain.

Callen, J. D. and Jahns, G. L., Phys. Rev. Lett., 33, 491
Soler, M. and Callen, J. D . , to be published in Nucl. Fus.
Soler, M., Callen, J. D., Navarro, A. P., Granetz, R., Seguin,

^Research sponsored by the Office of Fusion Energy, U. S. Department

Jahns, G. L., Soler, M . , Waddell, B. V., Callen, J. D., and

of Energy under contract W-7405-eng-26 with the Union Carbide

F., Petrasso, R., Bull. Am. Phys. Son., 23, 759 (1978).

Hicks, H. R., Nucl. Fus. 18, 609 (1978).

(1977).

Columbia University

A. Aydemir, C.K. Chu

EQUILIBRIUM NUMERICAL STUDY OF

THE FORMATION OF THE PLASMA IN TORMAC*

Results from our two-dimensional, single fluid resistive

magnetohydrodynamic simulation of TORMAC are presented. We find

the plasma during the implosion phase. With these modifications,

desired cusp equilibrium, and the overall performance, in terms

“start-up” problem. Starting with some reasonable initial condi

distribution and timing, which greatly improve the behavior of

Work supported by U.S. Department of Energy under contract

tions, the plasma fails to implode towards the desired min-B

the plasma is_ able to implode to a state which resembles the

that with the existing design, TORMAC suffers from a serious

cusp equilibrium. The reasons for this failure are presented.

of confinement and temperature, is better than the original

We also present some modifications to the external current

EY-76-S-02-2456.

TORMAC’s.

Nathaniel J. Fisch

Princeton, NJ 08544

of Lower-Hybrid Waves

Plasma Physics Laboratory, Princeton University

Two-Way Diffusion Equations and Diffuse Reflection

We consider the general two-way diffusion equation, h(6)3f/3x =

ditions are given where h is positive and final conditions where h is

(3/3e)(D3f/36), for 0 < x < L , which is well-posed when initial (in x) con

(in x) eigenfunction obtains completeness. This eigenfunction expansion

negative. Here separation of variables does not yield a complete set of

has been used in the special case of diffusion through a slab.*** Another

eigenfunctions; however, we prove that supplementing that set with a linear

“**H.A. Bethe, M.E. Rose, and L.P. Smith, Proc. Am. Philos. Soc. 78, 573 (1938).

as the inverse fluctuation thickness, i.e., -1/L . We use the eigenfunction

density fluctuations, which can be described by diffusion in perpendicular

special case of interest is the propagation of lower-hybrid waves through

expansion to numerically find the transmitted and reflected spectra.

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

^A. Sen and N.J. Fisch, M.I.T. PRR 78/16 (May, 1978).

The transmitted power falls off only algebraically

^E. Ott, Cornell U. LPS 253 (August, 1978).

(to B) velocity space. ’

< J ;

the zero-

specifically

becomes very impor

, the poloidal structure of

Our calculation is based on the bounce-averaged drift kinetic equation

There are two basic assumptions inherent in all neoclassical transport

EBT Neoclassical Ion Transport with Non-Maxwellian f^ and Higher Order Poloidal Expansions* *- R. B. Campbell, T. Kammash, The University of Michigan, Ann Arbor, Michigan 48109, and R. J. Kashuba, McDonnell Douglas Corporation, St. Louis, Missouri 63166

these assumptions are justified. However, in the lossy regions of velocity space, where J2. — a tant. Moreover, for a collisionless plasma order distribution exhibits non-maxwellian features a “loss cone” distribution.

calculations for bumpy tori reported on to date (1,2). The first is that the distribution function can be Fourier decomposed in poloidal angle 0 , and only terms up to the first harmonic (1, cos & , sin 9 ) need be retained. The second states that the Q independent distribution is Maxwellian, f =f . For a simple BGK collision model, relatively high collisionality , ) , and for a 9 independent poloidal drift frequency

for ions, in which we use a simple BGK collision model. The collision frequency is velocity dependent and modeled to crudely take into account the large velocity space derivatives near the /la: o model, and with the restriction tions of the neoclassical particle diffusion coefficient, D^. Two of the coefficients are computed using poloidal terms up to and including the first harmonic (cos 9 , sin 6 ) , one with f^=f^, and the other with f^=f^.. The other two coefficients are similar to these, but we have retained up to and including the second poloidal harmonic. Sample results are sketched below for typical EBT-S operating conditions. The loss cone results can be interpreted physically by noting that the regions where o are de populated, thus reducing the driving term in the neoclassical fluxes in this region. The plateau result in ref. 2 relys on this lossy area con tributing greatly to D’, as it should in that case, since the assumption made. This, however, implies that the bulk distribution neither ^o”^M sees nor responds to the loss cone. The introduction of non-maxwellian f^ driving terms seems rather important at low collisionalities since it appears to allow lisionality .

(2) . R. D. Hazeltine, X. A. Krall, H. H. Klein, “Neoclassical Transport

(1) C. L. Hedrick, D. A. Spong, L. W. Owen, Bull. Am. Phvs. Soc. 23_

Sept. 1973 p. 376 papers 8P1 through 8P3.

in EBT” March 1979 (to be published).

(2), we have calculated four varia

, decreasing with decreasing col

f =f-^ lst t 2nd harmonic

loss cone. With this

*wcrk suncarted bv DCE

— f =f.. 2nd harmonic

—f =f„ 1st harmonic

to scale like

o LC

o M

*7”

I B 19

Abstract

R. Magelssen

Argonne National Laboratory

Argonne’ Beam Propagation and Target Experimental Program for Proposed Heavy Ion Facility

Argonne National Laboratory is proposing to test the feasibility of heavy ion beam drivers for inertial confinement fusion. The first proposed facility would irradiate small targets, and study beam propagation in a background gas with Xe beams. The proposed program structure contains two phases. The Phase I facility could be used to create Xe beams with particle energies of 1.6 to 6.4 GeV and pulse lengths of 2 to 10 nsec. The total beam energy on target would be 1 to 8 kJ. increased to 40 to 50 kJ.

Me will discuss the direction and goals for the associated programs. Simple unclassified ion beam driver compression targets as well as foil targets will be described, and implosion calculations presented The important heavy ion background plasma beam instabilities will be reviewed. A discussion of diagnostics for the experimental programs will also be given.

For Phase II the total energy would be

Yale University-

New Haven, Ct. 06520

Bernstein, A. Weiser, S. Eisenstat, and M. Schultz

A Finite Element Solution of a Reduced Fokker-Planck Equation

In this paper we compare several methods for the numerical

solution of a reduced Fokker-Planck equation, and conclude that

one method, the Rayleigh-Ritz finite element method, has innate

which needs no special treatment, and a singularity in the

yields approximations to the flux across the boundary, the

advantages over the others. In particular, a minimum principle

numerical results comparing the flux estimates obtained using

the Rayleigh-Ritz method to previous analytic and numerical flux

curved portion of the domain has a natural boundary condition

describe an efficient implementation of the Rayleigh-Ritz method

using tensor products of one-dimensional piecewise-polynomial

solution can be handled easily using nonuniform meshes. We

Work supported by Department of Energy contract EG-77-S-02-4349

basis functions and numerical quadrature rules.

We present

estimates.

Princeton, New Jersey 08544

Alpha-Particle Heating in Tokamaks

A Monte Carlo alpha-particle heating routine for use in 1-D

the time delay in the heating due to the finite slowing-down

order guiding-center drift equations for axisymmetric tokamaks.

time. The alpha-particles follow orbits prescribed by the first-

D. R. Mikkelsen and D. E. Post Plasma Physics Laboratory, Princeton University

tokamak transport codes has recently been developed. The algorithm models first-orbit losses, spatial spreading of the plasma heating profile which is due to wide banana orbits, and

survey in which we varied the aspect ratio, the current profile, the density profile, the Tg profile, the alpha birth profile, and the position of the wall which acts as the alpha-particle

results to previous calculations of the influence of the plasma current and of changes in the alpha-particle orbits on the heating

The average drag caused by Coulomb scattering is used to compute the bulk heating rates for the electrons and ions and self-

profiles. We then present results from an extensive parameter

consistently slow down the sample alpha particles; changes in

the orbits caused by the drag are included. We compare our

Work supported by U. S. DoE Contract No. EY-76-C-02-3073.

limiter.

Princeton, NJ 08544

P.H. Kaw, E.J. Valeo, and P.H. Rutherford

Tearing Modes in a Braided Magnetic Field

Plasma Physics Laboratory, Princeton University

(constant-^) m>_2 tearing mode.”** These rates scale as p ^ and p ^ ,

Magnetic braiding, together with large electron mobility parallel to B ,

mal conductivity observed in toroidal confinement experiments. Simple esti

has been suggested as an explanation for the anomalously large electron ther

mates demonstrate that the anomalous electron viscosity p that should be an

additional consequence of the magnetic braiding can yield tearing mode growth

We calculate the growth rate of the m = 1 tearing mode as well as that of the

rates much larger than those determined by the (observed) classical resistivity.

Work jointly supported by U.S. AFOSR Contract No. F 44620-75-C-0037 and U.S. DoE Contract No. EY-76-C-02-3073.

with a previous calculation for the resistive problem.

charge radius for plausible values of p and could be responsible for triggering

when its size is a small fraction of the discharge radius; at later times, the

cate that this should typically occur during the initial growth of the island,

as the island width exceeds a critical value. We have modeled this effect by

Recent calculations suggest that field line stochasticity may onset abruptly

tron viscosity of this type could play a role in the disruptive instability.

increase, w(t) accelerates, followed by a return of w(t) to the resistive

“**H.P. Furth, P.H. Rutherford, and H. Selberg, Phys. Fluids 1(5, 1054 (1973).

usual resistive growth dominates. We also examine the possibility that elec

than the constant rate determined by resistivity alone. Our estimates indi

rate. The impulsive increment in w can be a moderate fraction of the dis

rapidly increasing p(t) during a short interval. Immediately upon this

The nonlinear behavior of the constant-^ mode is calculated in analogy

inates, the island width w increases in time as t

^P.H. Rutherford, Phys. Fluids 16, 1903 (1973).

at a rate much faster

When viscosity dom-

respectively.

disruptions.

V ?

lower

Princeton, NJ 08544

E.J. Valeo and Liu Chen

Coupling of Lower Hybrid to Acoustic Modes

stantially reduce the convective growth factor.

Plasma Physics Laboratory, Princeton University

We calculate the convective amplification factor and absolute

instability thresholds and growth rates for decay of lower hybrid

the thresholds by limiting the spatial extent over which appreciable ion

response occurs. Relative pump bandwidths greater than ( m / M ) c a n sub

hybrid + acoustic quasimode. Density and temperature gradients establish

Work supported by U.S. DoE Contract No. EY-76-C-02-3073 and by U.S. AFOSR Contract No. F 44620-75-C-0037.

lA,

and

ABSTRACT

Y. M. Treve, La Jolla Institute

P. Manley, Office of Fusion Energy

A POSSIBLE STRANGE ATTRACTOR IN MHD CONVECTIVE INSTABILITIES

We discuss the possibility that in a tokamak for sufficiently large

temperature gradients, the convective motion driven by those gradients

a theorem from which the level of turbulent fluctuations may be estimated^”^.

Y. M. Treve, in “Topics in Nonlinear Dynamics,” AlP-Conf. Proc., No. 46,

in fluid mechanics. Next, motivated by physical considerations, we present

E. K. Maschke, R. B. Paris, and B. Saramito, Calculs Non Lineaires de

motion is intrinsically stochastic, similar to that previously encountered

evolves into a strange attractor ’ . Such dissipative (non-Hamiltonian)

We begin by introducing the concept of a strange attractor as it arises

Finally, in the context of a simple model, we discuss the possible impli

D. Ruelle and F. Takens, Comm. Math. Phys., 20^ (1971) 120.
Y. M. Treve, J. Math. Phys. (submitted for publication).

Stabilite MHD, Eur-Cea-FC-938, Fontenay-Aux-Roses (France).

. This model of MHD turbulence is expected to lead to

American Institute of Physics, NY, 1978, Ed. S. Jorna.

E. N. Lorenz, J. Atm. Sci., _20 (1963) 130.

enhanced heat transport.

cations for tokamaks.

by E. N. Lorenz

References

(2 3)

(4)

Cubic Turbulence*

by D. R. Nicholson and D. F. DuBois**

and a model for the nonlinear interactions of drift cyclotron modes

Hie nonlinear interaction of a wave with itself is often described

include the nonlinear Schrodinger equation model of Langmuir turbulence,

by a cubiciy nonlinear partial differential equation. Important examples

Department of Physics and Astronomy The University of Iowa Iowa City, Iowa 52242

V. E. Zakharov, Sov. Phys. - JETP 35, 90S* (1972).
A. K. Nekrasov, Nucl. Fusion 14, 865 (1975); R. E. Aamodt, Y. C. Lee, C. S. Liu, M. N. Rosenbluth, B. 1. Cohen, and D. R. Nicholson, to be published.

Work supported by U.S. D.O.E. and KSF Atmospheric Research Section. Address: Los Alamos Scientific Laboratorv

theory is analogous to Kraichnan’s direct interaction approximation for

the results are implicit in the elegant formalism of Martin et al. Our

of this approximate theory. For example, when applied to the nonlinear

P. C. Martin, H. A. Rose, and E. D. Siggia, Phys. Rev. AS, 423 (1973).

Schrodinger equation, the theory yields the oscillating two stream in

statistical theory for such equations. While our development is new,

We present a progress report of an investigation of the properties

which has been applied to mirror plasmas. We develop an approximate

stability as an almost trivial consequence.

quadratically nonlinear equations.

ABSTRACT

Cheng Chu

General Atomic Company

San Diego, California 92138

THE SLOW ION CYCLOTRON WAVE IN TOKAMAKS

numerically in the ion cyclotron frequency

The propagation properties of electromagnetic waves in a bounded

plasma imbedded in a tokamak type nonuniform magnetic field is studied

slow ion cyclotron mode in a tokamak type device.

^Y. Yasaka, S. Komori and R. Itatani, Paper C3-1, Third Topical Conference

coupled wave equations, it is found that, in addition to the global fast

increase of loading at M = 0 . 8 $2. seems to indicate the excitation of this

on Radio Frequency Plasma Heating, Pasadena, California, Jan. 11-13, 1978.

dominant left-hand polarized electric field and can propagate within the

gives a more efficient heating and better energy deposition pattern than

is primarily from the left-hand polarized electric field, this slow mode

poloidal mode number with respect to its center. Since the ion heating

that from the fast mode, which has a small and usually ill-placed left-

be a global mode provided that the parallel wavelength is sufficiently

plasma. A unique feature of this mode is that it always has a small

Supported by the Electric Power Research Institute, EPRI Contract

hand polarized component. A recent experimental observation of the

this mode will be discussed in the context of RST and PLT.

Lo) > (Vp^/(ML - LO )^. This slow mode has a

the slow ion cyclotron wave with M

range. By solving the

The ramifications of

cavity mode with (D >

No. RP 323-3.

short ^k^C

/ 2 2 2

can also

2 \

Di:

GUIDING

Lu Jniversiu*.*

Maryland, and 3. Mcncrcmerv

Glenn uoyce (Universiuy of ..cva, ,

The nest common feature of the two-dimensional guiding cenuer

field gradients, or more simply, a uniform gravitational field. The

scale vortices and for exhibiting associated enhanced transport. The

of the charge. A slab geometry is considered, with an electric field

Here we point out another, essentially universal, mechanism: magnetic

essential feature is a guiding-center drift which depends upon the sign

plasma model has been its propensity for the nonlinear formation of large-

external electric fields, or microscopic instabilities driven by gradients.

large vortices can result from a variety of stimuli: high initial energies,

E = -Vd>, <j)=<j)(x,y,t), B = B e , g = -ge^, with 3/3z = 0. Simulations

involving 10,000 particles, with periodic boundary conditions in x and

conditions. Clamping the potential between the two faces normal to g at the

the most interesting feature of the analysis is that it appears possible to

which reach maxima above the random loading value (the threshold value for

where a finite potential jump is permitted. This suggests the possibility

pass from negative to positive temperatures by changing only the boundary

the onset of the negative temperature regime in the g = 0 case), and then

same value can enhance the vortex formation in contrast to the situation

static energy by randomly loading pairs. Electrostatic energies develop

execute large fluctuations. A maximum in the electrostatic energy’ as a

function of g is observed, and is unexplained. A most-probable-states

<{) = 0 on the y boundaries, are started from a condition of zero electro

which has no_ spatially uniform solution for either- sign of 0.

priate potential differences between the current-carrying

rermauion m multipcies by applying appro

analysis leads to the Poisson equation

= -47fe{n^exp[-(etp + nugy)/e]

-n ^expt-(-e^) + m gy)/Gj)

ods and the walls.

;enuiaa i; Lan:

sunnressinr

Probably

o z —

—

’

FIELD REVERSED PLASMA ROTATION AND TRANSPORT

Field Reversed Plasmas (FRP) have been observed to spin up and sub

Loren C. Steinhauer Mathematical Sciences [Northwest, Inc. Bellevue, Washington 98009

sequently develop a destructive m=2 stability. While the stability limit for spin velocity has been characterized approximately, the cause of rota tion remains unclear although several mechanisms have been proposed. We present theoretical evidence for one of these, i.e., the shorted, rotating, open field line plasma causes plasma on unshorted closed lines to spin up by viscous friction. This model seems to explain several observations; the time delay in onset of spin, the rapid spin up rate, and the achieve ment of rotation’ rates faster than the diamagnetic drift frequency.

diffusion from closed to open field lines followed by convective stream ing along field lines. Since the latter process is typically much faster, it will rapidly deplete the open line plasma, leading to steep density gradients near the separatrix; this will accelerate the cross-field dif fusion and possibly excite anomalous processes. We describe a model based on a quasi static open field line plasma. The results indicate that the global confinement is. a geometric combination of the diffusive and streaming timescales (the geometric mean for the case of classical transport). In addition there is an extremely sensitive dependence on aspect ratio; low. aspect ratio plasmas have much faster instantaneous decay rates.

The decay of stable FRPs is necessarily a two-step process; particle

This work was supported by USDOE contract no EY-76-C-06-2319

1 B 2 9

New Y o r k U n i v e r s i t y , N e w York, N.Y. 10012

F. Ba u e r , 0. B e t a n c o u r t and P. G a r a b e d i a n

N O N L I N E A R S A T U R A T I O N OF B A L L O O N I N G M O D E S F O R T O K A M A K S

A CRAY v e r s i o n of our e q u i l i b r i u m and s t a b i l i t y code for

CDC6600 (cf. F. Bauer, 0. B e t a n c o u r t and P. G a r a b e d i a n , “A

that runs 30 times f a s t e r tha n the p u b l i s h e d v e r s i o n for the

p l a s m a s in t h r e e - d i m e n s i o n a l t o r o i d a l g e o m e t r y has b e e n w r i t t e n

o r d e r a c c u r a t e e s t i m a t e s of the en e r g y l a n d s c a p e can be calcul a t e d .

C o m p u t a t i o n a l P h y s i c s , S p r i n g e r - V e r l a g , N e w York, 1978). F o u r t h

of i n s t a b i l i t i e s for s c r e w p i n c h e s w i t h r e a l i s t i c d i s t r i b u t i o n s

The i m p r o v e d code has e n a b l e d us to study n o n l i n e a r s a t u r a t i o n

l e a r n e d fro m l i n e a r s t a b i l i t y theory. An i n v e s t i g a t i o n of the

s t e m m i n g fro m t r u n c a t i o n e r r o r in the v a r i a t i o n a l m e t h o d are

C o m p u t a t i o n a l M e t h o d in P l a s m a P h y s i c s , ” S p r i n g e r Series in

o v e r c o m e , results are o b t a i n e d that go b e y o n d w h a t has b e e n

s a t u r a t i o n cf b a l l o o n i n g m o d e s for T o k a m a k s is in pro g r e s s .

of p r e s s u r e and r o t a t i o n a l trans f o r m .

A f t e r d i f f i c u l t i e s

Princeton, New Jersey

Plasma Physics Laboratory,Princeton University

OKADA**, S. DALHED, J. DELUCIAand M. OKABAYASHI

Free Boundary Equilibria with Multipole Expansion of External Field in Noncircular Tokamaks

The external magnetic field for noncircular equilibrium is studied in terms of multipole moment coefficients, and these co efficients are expressed in terms of geometrical parameters and the plasma properties. The noncircularity for a given external field configuration is compared with the exact numerical calcul ation.

The Grad-Shafranov equation is solved for arbitrary noncir cular tokamaks. Noncircularity, e.g., b-a/b+a for ellipticity, is assumed to be of the order of the inverse-aspect-ratio; therefore, the elongation ratio of up to about 2 is covered with in the framework of the present study. The external poloidalflux is expressed in terms of multipole moment coefficients. By ex pressing the external field requirement with the moment coefficients, we can directly invert the problem, namely, the geometrical para meters such as a major radius, ellipticity, triangularity, etc., are obtained from the given external fields.

calculation by the Princeton Equilibrium Code, which gives a satisfactory agreement.

This work is supported in part by United States Department of

The analytic result is compared with the result of numerical

On leave from Central Research Laboratory, Hitachi Ltd. ,

Energy Contract No. EY-76-C-02-3073.

Tokyo, Japan.

WITH STOCHASTIC RAY TRAJECTORIES*

Steven W. McDonald and Allan N. Kaufman

SPECTRUM AND EIGENFUNCTIONS FOR A FIELD EQUATION

separable geometry, we investigate the two-dimensional Helmholtz equation^

As a model for linear wave equations arising in plasma physics, with non-

ratio, the ray trajectories (of geometrical optics) are stochastic, except for

(V + k )^(x) = 0. For a racetrack boundary, whose only parameter is its aspect

Physics Department and Lawrence Berkeley Laboratory University of California, Berkeley, California 94720

Work supported by the Office of Fusion Energy of the U.S. Department of Energy under contract No. W-7405-ENG-48.

the limiting case of a circular boundary. We examine the eigenvalue spectrum (for

nodal curves, which almost never cross. Also, they fill the whole area uniformly,

the racetrack and the circle: the racetrack’s eigenfunctions have random-looking

accidental near-degeneracies. The eigenfunctions are qualitatively different for

in contrast to the Bessel eigenfunctions of the circle. The sensitivity of these

a given parity), and find the eigenvalues well-spaced, with no near degeneracies;

in contrast, for the circle, the eigenvalues are highly clustered, with frequent

S. McDonald and A. Kaufman, LBL-8587, submitted to Phys. Rev. Letters.

features to the aspect ratio will be presented.

A B S T R A C T

IN S H E A R E D S Y S T E M S

Princeton, New Jersey 08544

Shoichi Yoshikawa and Roscoe White

Plasma Physics Laboratory, Princeton University,

M A G N E T O H Y D R O D Y N A M I C A L I N T E R C H A N G E I N S T A B I L I T Y IN L O W - S P L A S M A S

The stability criterion of a magnetized plasma with respect

ing the finite growth rate, the singularity at the magnetic surface,

Suydam condition is both a necessary and a sufficient condition for

is a small positive number (less than 0.03), which presumably can

to interchange in sheared configuration was reexamined. By retain

be made arbitrarily small by improving the numerical approximation.

removed. The resultant wave equation yields the result that the

Here 8^ is the maximum 8 for the original Suydam criterion and A

where the perturbation is constant along the magnetic line, is

stability at least for the plasma pressure less than (1 - A) 8^..

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

** Princeton University, Princeton, N.J., U.S.A.

We have made an analytical and numerical analysis of the

Department of Physics, University of Tokyo, Tokyo,Japan

drift-wave instability in a sheared magnetic field by taking both the current and the finite-p effects into consideration.

Current-Driven Drift-Wave Instability of a Finite-6 Plasma in a Sheared-Magnetic Field T.Tange, C.Ueno, H.Irid, T.Watanabe,

S.Inoue*, K.Itoh*, Kyoji Nishikawa and S.Yoshikawa** Institute for Fusion Theory and Faculty of Science, Hiroshima University, Hiroshima, Japan

Recent analysis has shown the stability of the collisionless drift wave in a sheared magnetic field. The plasma current which produces the magnetic shear can, however, drive an instability. In low-p plasmas where the drift wave can be treated as electro static, the stability criterion is obtained numerically as shown in Fig.l. Here u is the electron drift velocity along the mag netic field and the other notation is standard. 0-2

For 6>m/M, the electron-to-ion mass ratio, the drift wave couples to the Alfven wave. We found that al though this coupling tends to stabilize the short wavelength modes, as shown previously,1) it destabilizes a long wavelength mode. Our method is to first derive a coupled set of equa tions for the electromagnetic drift wave in the presence of both the current and the finite-6 value, and then apply the technique similar to that of Antonsen for the electro static mode to the coupled equa tions. The result indicates the existence of a new-type of insta bility driven by the combined effect of the current and the coupling to the Alfven mode. We then obtained numerical solutions for the local ized mode of the coupled equations and derived the stability criterion which is shown in Fig.2. In the numerical analysis, we developed a new technique, i.e. the Newton method, in order to obtain the eigenvalue by the shooting method. This technique has considerably improved the convergence of the iterative solution. The result shows that the drift wave, when coupled to the Alfven mode, can be driven more unstable by ,a cur rent of substantially smaller electron drift velocity than the electrostatic case.

Stability criterion for trostatic drift wave of plasma(T„=T^, M/m=1836, L

Stability criterion for elec tromagnetic drift wave (T =T., M/m=1836, L /L =32).

Kyoji Nishikawa and S.Yoshikawa, IAEA-CN-37/W-3 (1978).

_ _ [_ _ i_ _ ’ 1 ’_ _ !_ ! ) ! 0-5 k pi

S.Inoue, K.Itoh, T.Tange,

[T =T., /L =32)^ ^

t !— )— !— r— r* — !— !— -

elec- l o w - ;

B C*/. 3

U N S T A B L E

Fig.2

Fig.l

e i

UNSTABLE

… . ” I

I—.

S T A B L E

s n

STABLE

0.05

o-i

0.6

0 0

1.2

& >

1-0

0.1

R. Marchand, G. Rewoldt, a n d W. M, Tang

T w o - D i m e n s i o n a l E i g e n m c d e A n a l y s i s o f t h e T r a p p e d - I o n I n s t a b i l i t y ^

An analysis of the two-dimensional eigenmode structure of the trapped-

Plasma Physics Laboratory, Princeton University, Princeton, N. J, 0.8544

energy and pitch-angle-dependent Krook operator. The perturbed electro

the analysis of eigenfrequencies both less than and of the order of the

governing equations take into account the spatial variations in the equi

librium profiles (e.g., density, temperature, etc.). They also allow for

the poloidal structure and each harmonic <j)^ is expressed as a truncated

static potential i is expanded in a Fourier series in 9 to account for

Taylor series in the minor radius to account for the radial structure. The

is based on the drift-kinetic equation in which collisions are modeled by an

ion instability in axisymmetric toroidal geometry is presented- The approach

frequency. Our approach, assumes the large aspect ratio limit with circular,

whole plasma cross section. A comparison of a full two-dimensional calcula

which radial derivatives of the perturbed potential are ignored, and to the

The main original contribution of this work, however, is that it is capable

assumed to be nearly flutelike. Results corresponding to these two special

average ion transit frequency. In addition to the two-dimensional problem,

of treating the full two-dimensional structure of the instability over the

length perturbations for which ^r^bi^ ^ ^ well-satisfied, where k^ and

tion with its corresponding one-dimensional counterpart shows significant

quantitative and qualitative differences in the mode structure and eigen-

concentric magnetic surfaces. It is limited to electrostatic, long wave

Our basic analysis is applied to the familiar radially local problem, in

one-dimensional radial analysis of Gladd and Ross,^ in which the mode is

cases are found to be in reasonable agreement with previous calculations.

Work supported by U.S. Department of Energy Contract #EY-76-C-02-3073,

p . are the typical mode radial wavelength, and the ion hanana width.,

“**N. T. Gladd, D. W. Ross, Phys. Fluids 16, 1706 CL973).

b i *

ABSTRACT

L. Sulton, M. Cotsaftis, ^ and H. H. Klein

Science Applications, Inc., La Jolla, California 92037

ANALYSIS OF PLT DISCHARGES WITH HIGH NEUTRAL INJECTION*

We have analyzed a PLT discharge with high neutral injection (P = 2.4 MW) with the 2-D tokamak transport code, G2M, including the previously developed global transport model. ^ The model accounts for the anomalous electron and ion heat and particle transport arising from drift-tearing islands in the outer region, from trapped electron modes in the central region, and from sawtooth modes inside the region, q = 1. Also, the programmed input rate of neutral gas was taken in order to study the density evolution. We find that the build up of the electron and ion temperatures, T^ and T., and the density as well as the displacement of the flux surfaces is in close agreement in space and time with experimental results. During the heating process where the ion temperature jumps from 1 keV up to 5. 7 keV, the transport improves in the center due to the particle loading increasing the density, though the safety factor remains below 1. As a consequence, the energy lifetime, after a deterioration at the beginning of the turn on of the neutral beam, shows a slow recovery, varying from 37 ms down to 18. 5 ms, and then back to 23. 2 ms. Various properties of the system during its time evolution are likewise discussed.

*M. Cotsaftis and H. H. Klein, APS Meeting, Colorado Springs, November 1978.

Work supported by the U. S. Dept, of Energy.

^Permanent address Fontenay-aux-Roses.

CONDUCTING SHELL STABILIZATION OF FCT EQUILIBRIA*

The computer code ERATO^* has been used in an ongoing study of the

equilibria were found with g in excess of 10% and stability for values

(about half the aspect ratio) and a q ratio (safety factor at the edge/

n=l,2,3 and 4 of the toroidal mode number. These equilibria had 3^=2.5

with aspect ratio of 4 and elongation of 1.65. As reported previously ,

stability properties of D-shaped flux conserving tokamak (FCT) equilibria

safety factor at the axis) of 2.0. A conducting shell was assumed, having

L. A. Charlton, R. A. Dory, Y-K. M. Peng D. J. Strickler, S. J. Lynch and D. K. Lee Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

^Research sponsored by the Office of Fusion Energy (ETM), U. S. Department of Energy under contract W-eng-26 with the Union Carbide Corporation. ^D. Berger, R. Gruber and F. Troyon, Second European Conference on Computa tional Physics, Garching, Germany (1976). ^L. A. Charlton, R. A. Dory, Y-K. M. Peng, D. J. Strickler, S. J. Lynch and D. K. Lee, Bull. Am. Phys. Soc. 23 897 (1978).

been completed with the conducting shell removed to infinity. The resulting

change in stability is about as one might expect on the basis of conducting

minimize the higher modes and maximize [3, these equilibria have relatively

but varies only slightly for the higher modes whose radial wavelengths are

broad current profiles; they rely on the D-shapedness of the cross section

a radius 20% larger than the plasma radius. Additional studies have now

small enough to provide isolation of plasma effects from the shell. To

conclude that wall stabilization is not required to obtain stability at

to keep q above the Mercier criterion limit at the axis and above the

Because plasma stability is determined by the most unstable model, we

the critical i3 value falls from 16% to 10% for n=l,

finite n for 3 values as high as 10%.

empirical limit q surface H 2-3.

wall stabilization:

ABSTRACT

Thomas B. Kaiser

A technique is presented for optimizing the coil design with respect

Lawrence Livermore Laboratory, University of California Livermore, California 94550

OPTIMIZATION OF TRANSITION COIL DESIGN IN TANDEM MIRROR SYSTEMS FROM THE POINT OF VIEW OF INTERCHANGE STABILITY

Tandem mirror systems are potentially subject to curvature-driven ballooning-interchange modes because of unfavorable field-line curvature in the transition regions between the minimum-B end cells and the axisymmetric central cell. One way to alleviate this problem is to shape the transition coils in a way that optimizes the stability of the system with respect to these modes.

to a particular subclass of curvature-driven instabilities, viz . , those with a purely interchange character. The calculation uses a variational procedure to find the transition field configuration that maximizes the value of the quantity ^da(K^/B) ) 3 ^ + p„)/3^[, whose positivity is a sufficient condition for stability. curvature, p^,„ the plasma pressures perpendicular and parallel to the magnetic field, and ^ the magnetic flux.

and anisotropic pressure one based on the low-g, isotropic pressure stability criterion Vp - VjfdR/B >_ 0 used by Riordan et of the Multiple Mirror Experiment.

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under Contract number W-7405-Eng-48.

This approach to improving interchange stability generalizes to finite S

^J. C. Riordan, A. J. Lichtenberg and M. A. Lieberman, Nuc. Fusion 1J3,

is the normal component of

in the design

21 (1979).

and

H. Okuda

A. T. Lin and J. M. Dawson

Princeton Plasma Physics Laboratory

Princeton University, Princeton, New Jersey 08540

CROSS-FIELD ELECTRON TRANSPORT DUE TO THERMAL ELECTROMAGNETIC FLUCTUATIONS*

Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024

The cross-field electron particle and energy transport for a high 3 thermal plasma is investigated using a magnetostatic plasma model. The diffusion process due to magnetic fluctuations alone is observed to have a Bohm-like scaling, whereas in the presence of both electrostatic convec tive cells and magnetic fluctuations, the diffusion, because of the strong interaction between these static modes, becomes smaller than the diffusion- assuming these modes are independent.

*h!ork supported by USDOE.

(l)

(2 3)

Current

to the

Kyoto University

Gokasho, Uji, Kyoto

heliotron configuration.

Plasma Physics Laboratory

configuration with 3 = 2 short pitch helical coils

investigated by applying the Stellarator expansion ’

For the low 3 current carrying heliotron configuration, the

A rotational transform exceeding unity at -+*he plasma surface,

ip(0) = n/m in the (i^(a), i^(a)) stability diagram. The m = 1

Magnetohydrodynamic Instabilities in a High Shear helical System

driven kink and tearing modes and pressure driven low m modes are

kink and tearing modes become unstable along the line of 1 ^ ( 0 ) +

M. Wakatani, T . Yoshioka, K. Hanatani, 0. Motojima, A. Iiyoshi, K . Uo

i-(a) > 1, and a high shear can be produced in a helical heliotron ^

is examined by comparing our results with the initial and boundary

K. Matsuoka, K. Miyamoto, K. Ohasa and M. Wakatani, Nucl. Fusion

Eigenfunctions of kink modes are fairly localized inside the plasma

modes with n > 2 give wide unstable regions and large growth rates.

pressure driven modes is estimated and g > 5 % may be expected by

column when the resonant surface exists outside the plasma column.

m modes and give larger growth rates and wider unstable regions

than the low g case. The validity of the Stellarator expansion

value problem of linearized MHD equations for the heliotron con

tailoring the pressure profile according to the shear parameter.

Comparison is made with the results of Heliotron-D experiments

For finite g plasmas, the pressure terms destabilize the low

J.L. Johnson, C.R. Oberman, R.M. Kulsrud and E.A. Frieman,
K. Uo et.al., in 7th Int. Coni, on Plasma Phys. and Contr.

For current-less finite g plasmas, g limit due to low m

K. Uo et.al., Phys. Rev. Lett. 3_1_ (1973) 986.

Nucl. Fusion Res. (Innsbruck, 1978) CN-37/L-1.

K. Uo, Nucl. Fusion 1_3 (1973) 661.

Phys. Fluids 1 (1958) 281.

17 (1977) 1123.

figuration^^.

(4)

H .C . Lui

New York, N.Y. 10027

Columbia University

Plasma Physics Laboratory

NONLINEAR KINK INSTABILITIES IN FORCE-FREE FIELDS*

For fixed force-free currents, nonlinear kink stability is always

mine the nonlinear kink stability is derived for long wave length

current. In general, the optimum value of the force-free current

current strength in the force-free region up to an optimum value.

vacuum is extended to include force-free and distributed plasma

increases as the plasma current increases. A condition to deter

currents. Nonlinear stabilization of the external kink (m = 1)

decreased by increasing the strength of the distributed plasma

unstable kink modes in a sharp boundary plasma surrounded by a

instability by force-free fields is enhanced by increasing the

The theory of P a o ^ for the nonlinear behavior of linearly

Work supported by U.S. Department of Energy under contract

Y.P. Pao, Phys. Fluids 2j^, 765 (1978).

EY-76-S-02-24S6.

perturbations.

ABSTRACT

Using GIM/HYBRED code, which features self-consistent

S. Hamasaki Science Applications, Inc., La Jolla, California 92037

ANOMALOUS DIFFUSION AND PLASMA LEAKAGE THROUGH OPEN FIELD LINES IN FIELD REVERSAL CONFIGURATIONS*

anomalous resistivities due to various microinstabilities, we have studied anomalous diffusion in reversed field configurations like field reversal experiments (FRX) of LASL. In the FRX, the open field lines at outer regions support the inner closed reversed field configuration. The plasma contained in the closed field region gets diffused out into the open field region due to the lower hybrid drift or drift ion cyclotron instability. The diffused-out plasma leaks out through open field lines, which enables the plasma edge near the open field lines to sustain, in spite of large diffusion, a finite gradient which feeds the microinstabilities. In this manner, the microinstabilities pump out the plasma from the closed field region to the open field region. Our calculation of these leakage rates due to the lower hybrid drift instability and the drift ion cyclotron instability showed a rough agreement, in order of mag nitude, with the experimental data of the FRX even though the FRX usually develops the rotational m = 2 MHD instability which terminates the containments.

*Work supported by the U. S. Dept, of Energy.

WITH TENSOR PRESSURE”

EQUILIBRIUM AND STABILITY OF TOKAMAKS

T . Kammash University of Michigan Ann Arbor, Michigan 48105

A. Cooper, D. B. Nelson, Glenn Bateman Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

Research sponsored by the Office of Fusion Energy (ETM) , U.S. Department of Energy under contract W*7405eng26 with the Union Carbide Corporation.

guiding center fluid energy principle. This second order ordinary differential

guiding center plasma depending on whether the double adiabatic contribution

The stability of these equilibria to ballooning modes of large toroidal mode

number are examined by numerically solving an Euler equation derived from a

criteria obtained are either necessary or sufficient for the stability of a

equation is similar in form to the corresponding ideal MHD equation. The

of arbitrary cross section. The perpendicular and parallel pressures are

evaluated from a distribution function that models neutral beam injection.

Tensor pressure equilibria are computed for small aspect ratio tokamaks

to the Euler equation is kept or ignored.

and

2—5

Atlanta, Georgia 30332

IMPURITY CONTROL BY NEUTRAL BEAM INJECTION

in response to beam injection and external drags.

Stacey showed that a general external momentum

We have used a recently developed generalization of neoclassical theory,^

W. M. Stacey, Jr. School of Nuclear Engineering Georgia Institute of Technology

D. J. Sigmar Fusion Energy Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

The use of momentum transfer due to neutral beam infection to drive impurities out of a plasma was apparently first suggested by Ohkawa and has subsequently been

examined by several workers. source could drive radial particle fluxes, in the Collisional regime. While the

direct effect of external momentum transfer was treated in these earlier works, two important indirect effects were omitted: the modification of the first-order flows in the flux surface and the adjustment of the radial, ambipolar potential gradient

Ohkawa, T . , General Atomic Report GA-A12926 (1974).
Connor, J. W., Cardey, J. C., Nucl. Fusion 14, 185 (1974).
Callen, J. D . , private communication.
El-Derini, Z., Earnert, G. A., Nucl. Fusion 1 6 , 342 (1976).
Fomenko, V. V., Sov. J. Plas. Phys. 3, 775 (1977).
Stacey, W. M., Jr., Phys. Fluids 21, 1404 (1978).
Stacey, W. N . , Jr., Sigmar, D. J . , ORNL/TM-6575 (1978); submitted to

self-consistent calculation of particle transport in a tokamak plasma in the presence of neutral beam injection and external drags. Our formalism is valid in

formalism is developed for a general momentum input, it is generally applicable for the analysis of impurity control by other forms of toroidal momentum input (e.g.

As an auxiliary result, we have extended the theory ’ for impurity control Finally, we have

S. Burrell, K. H., Phys. Fluids 19, 401 (1976). 9. Wong, J. K . , Phys. Fluids 2 1 , 299 (1978).

worked out the general theory for particle transport (subject to the Lorentz form

which is valid when external momentum sources and drags are present, to make a

specialize to the low- ^limit for clarity. We establish the conditions for which

effects could be observed in PLT and ISX-B, for example, and that beam injection

all collisionality regimes and for arbitrary geometry and beta, although we

might be a feasible means of impurity control in a reactor-type plasma.

Three important subsidiary results are contained in our work.

by asymmetric particle sources to all collisionality regimes.

coinjection drives impurities out of a plasma.

Acknowledgement: This work was sponsored by USDOE.

of the friction) in a two-species plasma.

We estimate that order-unity

Phys. Fluids.

References

Since the

rf).

8 9

ABSTRACT

TO BALLOONING MODES

R. W. Moore, R. L. Miller, and R. E. Waltz

STABILITY OF NEUTRAL BEAM HEATED EQUILIBRIA

General Atomic Company San Diego, California 92138

Neutral beam heated PLT equilibria are modeled using the General Atomic

1-1/2-D transport code.* The stability of these equilibria to localized

density, and neutral beam injection parameters. Special attention is paid

ballooning modes is studied as a function of time during the neutral beam

current profile is illustrated by changing transport coefficients, plasma

heating. The ballooning stability dependence on the transport generated

to current broadening caused by poor beam penetration and its effect on

Work supported by the Department of Energy, Contract No. EY-76-C-03-

*R. L. Miller, “Shape Control of Doublets,” General Atomic Company

Report, GA-A15186 (November 1978), submitted to Nucl. Fusion.

0167, Project Agreement No. 38.

ballooning stability.

and

D. P. Tewari*

Drexel University

RADIATION IN A PLASMA

Philadelphia, Pennsylvania 19104

Department of Electrical Engineering

In the vicinity of upper hybrid resonance (in the higher

V. K. Tripathi** Department of Physics and Astronomy University of Maryland College Park, Maryland 20742

density regime) an upper hybrid EM wave resonantly excites a second harmonic upper hybrid radiation. The efficiency of har monic generation is limited by the wave number mismatch created by the density gradient. However, the yield of harmonic conversion for typical laser produced plasmas could be as high as 5% which is in order of magnitude agreement with the experimental observations. This mechanism of harmonic generation is operative in tokamak type plasmas also.

On leave from Physics Department, Indian Institute of Technology, New Delhi, and supported by Indo-American Exchange Program.

** Supported by the Department of Energy.

T. Uckan

chosen to be

Oak Ridge, Tennessee 37830

Oak Ridge National Laboratory

ELECTRON CYCLOTRON RESONANCE HEATING RATE IN EET PLASMA*

extraordinary microwave field in EBT is calculated by means of the

The perpendicular energy gain, AtV^, of electrons from the applied

stochastical model^ for the field-plasma cyclotron resonance interac

tions. In these calculations, the inhomogeneous external bumpy field is

Research sponsored by the Office of Fusion Energy, U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

B(z) = B„ ^ ( r ’)

mirror ratio on the trapped and untrapped electrons in the bumpy field

are discussed. Then the heating rate AW^/At, At being one reflection

M is the mirror ratio and L is the distance between the mirror sectors.

The effects of the initial energy of the electrons as well as the

^D. A. Spong et al., Oak Ridge National Laboratory Report ORNL-TM

which simulates the field strength reasonably well for the EBT.

time from the mirror, is estimated for the trapped electrons.

^H. Grawe, Plasma Phys.

6215 (1978).

151 (1969).

Here

ABSTRACT

FINITE TEMPERATURE EFFECTS ON MICROWAVE PROPAGATION IN EBT

A three dimensional ray tracing code, RAYS, has been developed as

a part of the ongoing theoretical study of microwave heating in the

the effects of finite temperature on the ray paths as well as cyclotron

and higher harmonic resonances. In particular, we observe conversion of

Elmo Bumpy Torus device. Recently this code has been improved to include

damping. Our dispersion relation allows us to study effects at the second

B. Batchelor and R. C. Goldfinger Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

has upon total absorption (/ ds k. * V ). It was found that when the *”**- absorption is weak jk^l << jk^) the total absorption is virtually indep en-

Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation,

are given for ordinary and extraordinary mode waves at first and second

jected parameters for EBT-11. We have investigated the influence which

hybrid resonance and second harmonic resonance. Total absorption rates

harmonic resonances for plasma parameters appropriate for EBT-I and pro

the extraordinary mode to electrostatic Bernstein waves at the upper

the choice of direction for the imaginary part of the refractive index

dent of the direction of k..

” i

-L

A Simple Annulus Power Balance in EBT-1

An essential feature of the ELMO Bumpy Torus (EBT) concept is the

S. K. Borowski,’ M. A. Uckan, E. F. Jaeger 1 T. Kammash Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

presence of a relativistic electron annulus in each of the toroidal mirror sections. These high-beta annuli are formed and sustained by microwave heating and are of sufficient density and temperature that diamagnetic currents produce the necessary minimum in the magnetic field required for MHD stability of the toroidal core plasma. Since the electron rings play an important role in the confinement characteristics and performance of EBT, the trade off between the formation of the rings and the power required to sustain them represents an important problem in a fusion reactor. A power balance for the rings which includes drag cooling in addition to the radiation (synchrotron and bremsstrahlung) and annulus electron-electron scattering losses, indicates that drag dominates the annulus energy balance in EBT-1. Drag cooling of the relativistic annulus electrons on the toroidal core plasma appears to provide a reasonable explanation for the decrease in the annulus electron temperature in going from ELMO to EBT-1, Theoretical estimates of the microwave power required to sustain the annulus are found to be within a factor of 2 of the experimentally determined value. Scaling projections are shown for EBT-S and parametric study results for ELMO, EBT-1 and EBT-S are presented for various microwave power levels. The Maxwellian distribution function is assumed in these calculations. The results are found to be sensitive to the details of the hot electron distribution function as well as geometric and scaling parameters. Improvements to the model are underway in order to increase its capability and accuracy in assessing the overall power balance.

Research sponsored by the U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

^University of Michigan, Ann Arbor, MI.

1C 1

ABSTRACT

IN LOWER-HYBRID HEATING ’

Kyoko Matsuda, Y. Matsuda,f G. E. Guest, and T. Ohkawa

RESONANCE WAVE-WAVE COUPLING AND PONDEROMOTIVE EFFECTS

General Atomic Company San Diego, California 92138

study plasma response to high-power, lower-hybrid heating. In general,

The electrostatic particle simulation code “EZOHAR”^ has been applied to

strong edge heating of electrons and ions, and tail heating of electrons in

^Y. Matsuda, W. M. Nevins, and M. Gerver, in Proc. 8th Conf. Numerical Simulation of Plasma, Monterey, CA, June 1978.

harmonic of the external frequency. The electric field associated by these

76-C-03-0167, Project Agreement No. 38, and in part under Contract W-7405-

tion of Langmuir waves, have been found where the Langmuir frequency is a

depending on the density profile. New phenomena, such as resonance excita

higher harmonics may become comparable with the fundamental modes at the

the interior have been observed. Spatial density modulation by pondero-

motive forces does not occur at the edge due to large velocities of par

ticles;^ but such ponderomotive effects may appear inside the plasma,

Work supported by Department of Energy in part under Contract No. EY-

tPermanent address: Lawrence Livermore Laboratory, Livermore,

^A. Bers, Bull. Am. Phys. Soc. 15 (1978) 765.

CA 94550.

interior.

ENG-48.

Abstract

and R.D. Hazeltine

Fusion Research Center

X.S. Lee, Swadesh M. Mahajan,

Effects of Ion Dynamics on Tearing Modes

Using a simple self-consistent derivation of ion

The University of Texas at Austin Austin, Texas 78712

dynamics, we find two terms contributed by ions to the parallel electrical conductivity. One corresponds to the familiar ion acoustic term, and the other, which we call the ‘frictional term’, is due to the ion-electron

that most known unstable tearing modes are, in fact, the manifestations of the same mode in different regimes of

frictional term is also investigated, and is found to be destabilizing in one case. Quantitative expressions for

collisions. The tearing mode equations are modified by adding these terms to the parallel conductivity, and are

the change in mode frequency are given. We further show

tendency of the ion acoustic termt , earlier found on the

semi-collisional drift-tearing mode, is found to be true

This work is supported by the U.S. Department of Energy

for several other tearing modes. The effect of the ion

then solved to analyze their effects. The stabilizing

^Bussac, et. al., Phys. Rev. Lett. 40, 1500(1978).

Contract DE-AC05-79ET53036.

plasma parameters.

E. A, Frieman and Lin Chen

Nonlinear Interactions of Drift-Alfven Waves*

tively, by the FILR and diamagnetic-drift effects.

We present a general gyro-kinetic formalism for the nonlinear

derived include full finite ion-Larmor radius (FILR) effects and are

parametric decays- of kinetic drift-Alfven waves, it is found that the

valid in the strong-turbulence regime. Applying this formalism to the

Plasma Physics Laboratory, Princeton “University, Princeton, N. j. Q8544

nonlinear decay processes are modified, both qualitatively and quantita

interactions of kinetic drift-Alfven waves. The nonlinear equations thus

Work supported by U.S. Department of Energy Contract #EY-76-C-02-3073.

Control of a reactor-grade plasma may be lost if a sizable thermal

is a promising means of inhibiting such a thermal runaway.^ However,

excursion occurs after ignition is achieved. Because of its strong tem

which fixed ripple values lead to a satisfactory burn, i.e., a stable,

BURN CONTROL VIA REGULATED RIPPLE APPLIED TO REACTOR-GRADE PLASMAS?

achieving this ripple via a conventional TF-coil configuration provides

little flexibility for subsequent modification of startup operation. Fur

thermore, the degradation of confinement associated with ripple transport

the plasma, the magnetic ripple produced by the toroidal field (TF) coils

perature dependence and its finite value in the critical center region of

renders heating to ignition more difficult. Although there is a range over

J. M. Rawls, T. W. Petrie and W. Chen General Atomic Company San Diego, California

siderably relaxed, resulting in a much larger effective operating “window”.

by a small number of large superconducting TF-coils supplemented by copper

finement and auxiliary heating demands, and to reduce the current in these

Furthermore, the small number of superconducting coils provides a spatial

distribution of ripple more suitable for burn control. This approach may

sufficient for burn control. In this way, the ignition requirement is con

coils when ignition is achieved, thus enhancing ripple losses to a level

ignited plasma within prescribed beta limits, the size of this “window”

coils during startup to minimize the impact of ripple on both plasma con

R. W. Conn, gt al., Trans, of Third Topical Meeting on the Technology

^C. Baker and T. Ohkawa, Kakuyugo-Kenkyu, Vol. 35, No. 3, 224 (March

^T. W. Petrie and J. M. Rawls, “Burn Control Resulting From Toroidal

pull-back coils, a TF-array of the type employed in the recent NUMAK

These difficulties can be relieved by a TF-coil network characterized

provide the flexibility needed to track the plasma and to control the

Work supported by Department of Energy, Contract EY-76-C-03-0167,

of Controlled Nuclear Fusion, Santa Fe, New Mexico, 351 (May 1978).

The dynamical scenario proposed is to activate the “correction”

Field Ripple,” General Atomic Report GA-A15218, March 1979.

reactor power level.

suggests a somewhat restrictive operating mode.

Project Agreement No. 38.

q design.

1976).

—

M. J. Gerver

Short, Pat, Field-Reversed Ion Rings*

Electron Landau Damping of Instabilities in

The integral equation for the most dangerous (oj - v^/L) normal

allowing practical numerical solution, for an arbitrary (axisym-

in a cool, high density background plasma is simplified to a form

modes of a high energy, low density field-reversed ion ring immersed

Laboratory of Plasma Studies Cornell University Ithaca, New York 14850

& This work supported under U.S. Department of EY-76-S-02-3170.

v » v., there are far fewer electrons with v - v, than with Vi . - v,. e

V] j < Vj_ are trapped; for trapped electrons the resonance condition is

metric) geometry. Che qualitatively different feature of the short,

far fewer resonant electrons. In a long layer or bicycle tire, the

electron Landau damping is much less effective, because there are

magnetic field is uniform along a given field line, and resonance

fat ring, as opposed to the long layer or bicycle tire, is that

magnetic field varies along a field line, and electrons with

This result does not apply to ion damping since v^ < v^.

v/L), i.e., v - v^. For moderately large p,

v^. In a short, fat ring, the

requires M = kj jV

Energy Contract

i.e. Vi

(where

a) =

I I

J. M. Finn

Ithaca, New York 14853

with a Toroidal Magnetic Field

Kink Instabilities of a Field Reversed Ion Ring

Laboratory of Plasma Studies, Cornell University

treats’ the background plasma by fluid equations and the beam by

kinetic theory is employed. The major effects upon stability are

The low frequency MHD stability of an axisymmetric field reversed

ion ring in a current-carrying background plasma with a toroidal magne

tic field is studied. A generalization of the energy principle^ which

lR. N. Sudan and M. N. Rosenbluth, Phys. Fluids 22_, 282 (1979). 2j. M. Finn and R. N. Sudan, Phys. Rev. Lett. 41, 695 (1978). 3R. A. Meger, Ph.D. Thesis, Cornell University, 1977.

*Work supported under U.S. Department of Energy Contract EY-76-S-02-3170.

plasma currents are roughly equal, and if the exterior region contains

the MHD response of the plasma and beam, a collective reaction of the

a cold but highly conducting plasma, there is a window of stability

for kinks with safety factor q as low as 1/2. The results may also

explain the anomalous trapped current losses in the RECE-Christa

electron ring experiment. *

It is found that, if the beam and

beam, and betatron resonances.

Harold Weitzner

New York University

Courant Institute of Mathematical Sciences

STABILITY OF LOW BETA AXISYMMETRIC MIRROR MACHINES

With the use of an energy principle of W. Newcomb (LLL Report UCID

Unpublished results of W. Newcomb on a wide class of unstable systems are

pressure is studied.* The analysis is particularly simple in the low beta

recovered. Other unstable configurations are also described. In the appro

limit. For a mirror machine the low beta limit may be taken in several ways.

17182 (1976)) the stability of an axisymmetric mirror machine with anistropic

unstable only on the outer edge, where line tying may stabilize the system.

priate limit a stable configuration is given for a plasma with a very weak

singularity in the magnetic field, B is smooth but VB tends to infinity at

a point. If such a singularity is smoothed out, then the plasma will be

Some numerical examples will be given and extensions to high beta systems

will be considered.

and

ABSTRACT

Potsdam, Nek York

Bell Laboratories

SPECTRUM CASCADE IN

DRIFT WAVE TURBULENCE

Murray Hill, New Jersey 07974

Yuji Kodama Clarkson College

Akira Hasegawa and Carol G. Maclennan

The energy spectrum of drift wave turbulence with lnk/no)>^k/^ci is shown to be divided into two regions. One is the region with wavenumbers larger than a criti cal wave number k , in which the spectrum obeys the two dimensional hydrodynamic dual-cascade law where the energy spectrum cascades into smaller wavenumbers. is broad and the unidirectional k spectrum obeys k”8/3 or k”4 inertia range spectrum.2’3

The other region has wavenumbers smaller than the critical wavenumber kg and the spectrum obeys the resonant wave interaction law of the weak turbulence theory. Here the n) spectrum is narrow and peaked near the frequencies at which the resonant condition, for k = k’ + k” and the spectrum cascades to lower frequen cies.4 The critical wavenumber is decided roughly by the condition kcPg = (K/k)^(e]^k)/Tg)’^ where ps = Cg/aici’ x = [dinnQ/dx[.

R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory
D. Fyte and D. Montgomery, Phys. Fluids 22^ 246 (1978).
A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 41y 1470

The situation is analogous to Rossby wave turbulence.^

R. H. Kraichnan, Phys. Fluids 10_, 1417 (1967).
P. B. Rhines, J. Fluid Mech. 69, 417 (1975).

(Benjamin, New York, 1969), p. 103.

= aik’ + ^k” is satisfied

Here, the

REFERENCES

(1978).

ABSTRACT

THERMAL FLUCTUATION LEVELS

AND CONVECTIVE AMPLIFICATION

R. R. Dominguez, R. E. Waltz, and W. Pfeiffer

General Atomic Company San Diego, California 92138

It is well known that the thermal level of fluctuations in a stable

fluctuation spectrum for electrostatic drift waves in a sheared magnetic

regions of instability. Using the method of Kent and Taylor,^ the thermal

nonuniform plasma can greatly exceed the uniform plasma level due to local

forward) . We find that the thermal level spectral function, over a wide range

field is calculated (the extension to magnetic perturbations is straight

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

of parameters, is far below the experimentally measured level.

^A. Kent and J. B. Taylor, Phys. Fluids

Project Agreement No. 38.

(1969) 209.

(jJ

1 10.

“pe’ ce

ABSTRACT

), cold-fluid,

Simulations of DCLC Modes Near Linear Marginal Stability* Bruce 1. Cohen and Neil Maron Lawrence Livermore Laboratory

A particle-fluid hybrid simulation code has been used to study nonlinear properties of the drift-cyclotron-loss-cone (DCLC) instability near marginal stability. We are using a simple local model focusing on ion nonlinearities. The simulation allows only electrostatic perturbations and adopts the usual one-dimensional slab configuration for drift wave^. The electron response is taken to be the low frequency (tn « linear response with E X B and polarization drift effects retained.

We have investigated Maxwellian, subtracted-Maxwellian and delta function f.(v^) for nonuniform plasmas near linear marginal stability, i.e., at low densities 2 1 ^ / ^ 1 10. Drift-cyclotron and DCLC instabilities were observed which exhibited frequencies iRe and growth rates 0 jy Im theory over the range 0.2 jy a^/L^ wavelengths are characterized by k a - ^ ( m g / m ^ + These grew from an initial small level to a large amplitude in all cases. Nonlinear saturation was accomplished by ion trapping as evidenced by particle orbits of selected particles, vortex formation in phase space, and amplitude oscillations of the electrostatic potential at the trapping frequency was accompanied by spreading and filling in of the velocity distribution function, and by slowing of the average v^. The implied relaxation of the density gradient increased dramatically with the onset of trapping but remained insignificant, < 1 or 2%. The saturation of a single dominant DCLC mode was in excellent qualitative agreement and in good quantitative agreement with the trapping theory of Aamodt and Bodner (1969), [e^/T.[ ^ (Mpj/tDgj)/k a^ at saturation. A good fit to the data for subtracted Maxwellian f^(v^) was also given by je^/T^ saturation. Our simulation results near marginal stability exhibited a general insensitivity of the saturation levels with respect to a^/L^, in disagreement with some recent theoretical calculations. This, as well as the feedback influence of a self-consistent density profile, L^ L„(t} will be described in further detail.

“Work performed under the auspices of the U S . Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48.”

= Cf(l)k[e4-/m^[””^ % ka.]e<i)/T..]I/^. Trapping

— 0.2 in good agreement with linear

The most unstable

$ 1 at

0.4.

^ c r

Abstract

and W. Horton, Jr.

Austin, Texas 78712

Stability Analysis of

Fusion Research Center

Duk-In Choi, J.C. Wiley,

Runaway Distribution Function

The University of Texas at Austin

scheme for velocities u = v/v < 10. The tail of the

The steady state electron runaway distribution function

has been computed numerically using a spline collocation

boundary is calculated in terms of parameters m /hi , k„ and k,.

previously derived theoretical formulas for the distribution

resonance is then investigated using the analytical formula

to extrapolate the distribution function to high velocities

where numerical computation is inefficient. The stability

This work is supported by the U.S. Department of Energy

simple analytical formula which is parameterized in terms

frequency electrostatic mode due to the R = -1 cyclotron

in the runaway regime. The stability against the high

distribution function (5 < u < 10) has been fit by a

and E/Ep. The analytical formula is compared to

Contract DE-AC05-79ET53036.

pe’ ce

E/Ep,

The resulting

ion self collision time. A study of

these ion anisotropy reducing modes is thus

THE NONLINEAR EVOLUTION OF THE ION MIRROR INSTABILITY*

plasma primarily in the directions perpendicular to the field.

endloss rate becomes large on timescales of the order of the implosion time.

Collisionless shocks propagating normal to an ambient magnetic field heat a

necessary to the understanding of particle endloss from such pinches, since the

greater than classical electron self collision time, but much less than the classical

anisotropic distribution function is unstable to modes which would reduce this

anisotropy. In many fast 6 pinch experiments, the shock propagation time is much;

A. G. Sgro, D. W. Hewett, and T. C. Cayton Los Alamos Scientific Laboratory, Los Alamos, New Mexico

effects, this mode is studied with a hybrid (Vlasov ions, fluid electrons) simulation;

nonlinear evolution of the instability will be presented. The saturation mechanism

code. The early growth of the wave will be compared with linear theory and the

two-dimensional (r and z) model. In order to include finite ion Larmor radius

The ion mirror mode, having unstable m = 0 waves, may be addressed by s

*Work performed under the auspices of the U. S. Department of Energy.

will be discussed.

Princeton, NJ 08544

Plasma Physics Laboratory, Princeton University

P.NJ Guzdar, Liu Chen, W.M. Tang, and P.H. Rutherford

Ion-Temperature-Gradient Instability in Toroidal Plasmas

The stability of the ion-temperature-gradient mode in a toroidal plasma

studies show that for q. = d l n T . / d l n n > l toroidal effects further desta-

reduces to an ordinary differential equation. Analytic and computational

bilize the mode and hence the corresponding growth rates far exceed those

effects give rise to higher q^ threshold compared to the slab case. Exten

full kinetic effects. The equation is examined in, various limits where it

we have derived an ordinary difference-differential equation which includes

has been investigated. Using the newly developed ballooning mode formalism,

obtained from the slab calculations. However, it is also found that toroidal

out to delineate the regions of instability. Unlike the universal drift waves,

which require weak shear (s = rq’/q< % ) for the nullification of shear damping

sive numerical calculations over a wide range of parameters have been carried

by toroidal effects, the present q^ instability persists even for s = l and

hence is of relevance to present-day beam-heated tokamaks. *

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

Michael J. Schmidt

C O U R A N T I N S T I T U T E OF M A T H E M A T I C A L S C I E N C E S

HIGH BETA STELLARATOR STABILITY THEORY*

New York University New York, New York 10012

a sufficient condition for stability. The results of the calculation suggests that if the equilibrium model is valid,

presented. It is found that a solvability condition for the equilibrium of such a plasma is intimately related to

high beta Stellarator with arbitrary wall corrugation is

*This work was supported by the Department of Energy

An analytic study of the stability of a diffuse

then all high beta stellarators are unstable.

Contract Number EY-76-C-02-3077.

Princeton, NJ 08544

H. Okuda and 0.Z. Cheng

Plasma Physics Laboratory, Princeton University

Plasma Diffusion in the Presence of Strong Turbulence

Plasma diffusion in the presence of electrostatic turbulence has been

ion inertia. It is shown that the mode-coupling equations for both models

long-wavelength fluctuations and, hence, the presence of strong turbulence

the spreading of electrostatic energy toward long-wavelength modes (inverse

even for modest level of fluctuation. Numerical simulations reveal the spread

are essentially the same which indicate large mode-coupling coefficients for

studied analytically and numerically. First, the two-dimensional convective

ing of localized plasma density through vortex formation and at the same time

cell turbulence is studied in the guiding-center limit and keeping the finite

Both observations are interpreted in terms of small mode-coupling coefficients

in a steady state using a quasineutral simulation model in which the electrons

follow Boltzmann distribution. Numerical simulation reveals the diffusion in

tuations do not easily cascade toward long-wavelength fluctuations (kp. < 1).

vective cells using fluid theory. For drift turbulence, the coupling to con

this case is much smaller than the previous case and the electrostatic fluc

Drift wave turbulence and the associated particle diffusion are studied

vective cells is much more important than the drift wave nonlinearity.

Finally, a coupled set of equations are derived for drift wave and con

Numerical solutions of the mode-coupling equations will be presented.

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

for long-wavelength fluctuations in this model.

cascades).

and

1 C 1 6

Ernesto Canobbio

ON THE CYLINDRICAL LIMIT OF VARIOUS MHD PHENOMENA

In dealing with toroidal systems, cylindrical coordinates are used either

Department of Physics, University of California Los Angeles, California 90024, USA*

Association EURATOM-CEA, Departement de Physique du Plasma et de la Fusion Controlee Centre d’Etudes Nucleaires** Grenoble, 38041, France

essentially one-dimensional topics are discussed: 1. The nature of singu larities and cutoffs of the Hain-Lust equation. 2. The existence of toroidal surface MHD-modes. 3. The simplest possible form of a ballooning-like insta bility. 4. The qualitative equilibrium-pressure profile versus major radius in finite-beta toruses.

with z along the straightened toroidal direction, as in most stability and wave problems, or with z along the vertical symmetry axis of the system, as customary in equilibrium studies.

In this paper, assuming the appropriate axis orientation, the following

*Partially supported by USDOE **Perma’nent address.

D. Dobrott, J.A. Tataronis,^ and R.W. Moore

These codes are time consuming and may be imprac

Magnetohydrodynamic Stability Analysis Using Approximate Codes

General Atomic Company San Diego, California 92138

Linearized magnetohydrodynamic (MHD) stability of tokamak equilibria of

An example of such a code is that developed from the Mercier criterion.3

arbitrary cross section presently may be examined by large numerical codes such as PEST^ and ERATO. tical for comparison or optimization studies involving many equilibria. How ever, such studies may be done with the aid of smaller, faster, special- purpose, or approximate codes. All of the codes discussed here are based Upon the Lagrangian formulation of the linearized MHD equations of motion. Approx imate forms of the Lagrangian may be derived from geometrical considerations such as aspect ratio and vertical elongation or from the limit of high toroidal mode number.

The Mercier code is used to study interchange modes, radially localized to magnetic flux surfaces within the plasma. Another such code is that developed for the high toroidal mode number ballooning mode.’* This code examines the stability of modes that are radially extended and predominate toward the out side of the plasma toroidal cross section. Strictly speaking, these analyses are not independent, but must be treated so in practice. Stability information gained from these two economical codes is practically sufficient for the study of “pressure-driven” modes in the interior of the plasma.

Long wavelength (small toroidal mode number) “current-driven” modes, such as kinks and axially-symmetric displacements, also may be studied by an approx imate code for equilibria with vertically elongated cross section.^ A greatly simplified Lagrangian is obtained using a double expansion in inverse aspect ratio and elongation. A new analysis has been carried out to second-order in elongation to properly describe the vacuum region surrounding the plasma.

with simply connected flux surfaces and comparison of results has been made with ERATO. These approximate codes proved to be economical and may be used to study stability of equilibria with internal separatrices, such as doublets.

D. Berger, et al., Proc. of the 6th Conf. on Plasma Physics & Controlled Nuclear Fusion Research (IAEA, 1977), Vol. II, p. 411.

Work supported by the U.S. DoE Contract No. EY-76-C-03-0167, Project Agreement No. 38.

***R.C. Grimm, J.M. Greene, and J.L. Johnson, Methods in Computational Physics

^W. Grossman, J.A. Tataronis, and H. Weitzner, Phys. Fluids 20, 239 (1977).

These three codes have been used to examine the stability of equilibria

Present address Courant Institute, New York University, New York 10012.

^D. Dobrott, et al., Phys. Rev. Letts. _39, 943 (1977).

C. Mercier, Nucl. Fusion _1, 47 (1960).

16, 253 (1976).

HEARED MAGNETIC

JAVE TURBULENCE IN A S

C. Whitson Laboratory

S. P. Hirshman, J. Oak Ridge National Oak Ridge, TH

We have developed a self-consistent nonlinear resonance broadening

and electrons decorrelate at a rate ^ = [(Vjjkjp^D]^. in tokamaks,

theory for electrons in a drift-wave turbulent, sheared magnetic field.

combines with rapid parallel motion to induce random poloidal motion

The phase space islands overlap at very low fluctuation levels

resulting in stochastic electron orbits. With shear, radial diffusion

Kim Molvig Massachusetts Institute of Technology Cambridge, MA 02139

Department of Energy under contract W-7405-eng-26 with the Union

Research sponsored by the Office of Fusion Energy (ETM), U.S.

Nonlinear stabilization at modest saturation levels occurs when the

broadened inverse electron Landau resonance balances the turbulently

Apg PgCg/Lg, where Apg = (Lg/L^)^(mg/m^), is required to saturate

this exceeds the decorrelation rate kj*D for ions in a uniform magnetic

destabilized for ^ > oj, which occurs at very low levels of turbulence.

Carbide Corporation and U.S. ERDA Grant No. EG-77-G-01-4108.

is shown that linearly stable drift waves can be

electrostatic drift modes for which 8(Lg/L^)3 < i .

A turbulent diffusion coefficient, Dg ‘v. 15

enhanced shear damping.

field.

3/2 2

Jae Koo Lee and C. K. Birdsall

PARTICLE SIMULATION OF DRtFT-CYCLOTRON INSTABILITY-

The drift-cyclotron instability is a collisionless instability of a Max

Electronics Research Laboratory University of California Berkeley, California 94720

wellian magnetized plasma driven by the free energy associated with a spa tial density gradient normal to B, requiring k_^ and v^ only. To study the linear and nonlinear behavior of this instability we use an electrostatic particle code^ with no restrictions on the dynamics of electrons as well as ions; namely, both species are magnetized and treated fully nonlinearly. During the linear stage our simulations show exponential growth in time over a few ion .cyclotron periods, with the growth rates in quantitative agreement with the predictions of a linear nonlocal theory, while the real frequencies match only qualitatively with the predictions of a linear theory. The latter discrepancy might be related to the generation of a zero-frequency collisionless vortex mode. At saturation, the total elec trostatic field energy reaches a few percent of the initial ion kinetic energy for most runs, with mass ratios ranging from 25 to 200. These simulation saturation levels are compared with some nonlinear theories; namely, simulations produce the scaling of the nonlinear frequency shift theory, ^ = <6? [where ^ ( e ^ / T } ) ^ ^ ^ ^ ^ , <=(ai/n)(dn/dx), 5=me/mj+^j/J,] and that of the trapping,^ <XK*3/4gi/4, jn various parameter regimes. By the saturation time, the phase space pictures of electrons and ions reveal vortex-like structure and appreciable density-profile modification in some cases. The details of the comparison with linear and nonlinear theories will be presented.

^A. B. Langdon and B. F. Lasinski, Methods in Computational Physics, Vol. 16 pp. 327-366; Academic Press, N.Y., N.Y. (1976); Y. Matsuda, W. M. Nevins, and M. J. Gerver (in preparation).

^R. E. Aamodt et al., Phys. Rev. Lett. 39., 1660 (1977); B. ). Cohen, private

-Work supported by U.S. DOE Contract EY-76-S-03-0034-PA128.

3R. E. Aamodt, Phys. Fluids 20^ 960 (1977).

commun i cat i on.

J. Nuhrenberg

Federal Republic of Germany

= 1 field).The elliptical plasma cross-

Hedium-B, Medium Aspect-Ratio Stellarators

Max-Planck-Institut fur Plasmaphysik, 8046 Garching

meters. The magnetic axes are a set of closed curves

Within the framework of the expansion a three-dimensional

a class of stable stellarators without ohmic heating is con

described by two parameters, the number of periods and the

sidered which may be characterized by five independent para

MHD equilibrium around its magnetic axis (see, e.g. [1 , 2])

helical amplitude -section ( ^ = 2 field) turns at the constant rate - 7T/Lp

(Lp period of the axis) with respect to the normal of the axis. For a suitable range pressure gradients, given by the

serves to satisfy a stability criterion. B-values of about 10 % together with an aspect ratio of about 20 have been

^2] Lortz,D., Nuhrenberg,J., Proceedings of 7 ^ Int.Conf.on

association between the Max-Planck-Institut fur Plasmaphysik

“This work was performed under the terms of the agreement on

surfaces) these configurations have magnetic surfaces which

angularity parameters ( ^ = 3 fields) are used, one of which

[3J Lortz, D., Nuhrenberg,J., to be published in Z. Natur-

parameter dp/dV on the magnetic axis (V volume inside flux

found taking into account the necessary stability criterion.

are nearly centered []3j. In addition two stabilizing tri

Plasma Physics and Contr.Nucl.Fus.Res., IAEA-CN-37-H-5.

Lortz, D., Nuhrenberg,J., in Theoretical and

Computational Plasma Physics/ IAEA 1978, 305

and EURAT0M.”

forschung

Garching

Abstract

H. Welter, D. Biskamp*

Anomalous Reconnection in

Max-Planck-Institut fur Plasmaphysik

Disruptive Processes in Tokamak Like Plasmas

Interpreting the results of numerical simulations a

picture is given of the mechanisms that lead to rapid

expansion of magnetic flux during disruptive processes in

tokamaks. The competing role of resistivity and electron viscosity in the rapid growth of a (m,n) = (3,2) tearing mode in the presence of large (2,1) magnetic islands is studied quantitatively. When viscosity is dominating, the growth rate scales as y <x ^ , the process being very insensitive to the actual value of p. Finally certain shortcomings of the (3,2)-(2,1) mode interaction theory in explaining major tokamak disruptions are outlined. While interaction of any low m-number modes of different helicity may lead to an explosive transport of flux and electron energy across a certain region, a model of the major disruption seems to require a coupling to the (1,1) mode.

The University of Texas at Austin Austin, Texas 78712

*Present address: The Fusion Research Center

Physics Department

and

U s i n g MACSYMA*

Haifa, Israel

J. L. Schwarzmeier

p. Rosenau TECHNION

Courant Institute of Mathematical Sciences

Similarity Solutions of Partial Differential Equations

The use of the MACSYMA algebraic computing system to aid in the construction of exact (nonlinear) similarity solutions

larity form of the solutions of the partial differential equa tions can be obtained. Finally, the (hopefully nontrivial)

Supported by th^^U.S. Department of Energy, Contract No. EY-76-C-02-3077.

group under which the considered equations are invariant. Once

ary conditions of the problem is found. The use of MACSYMA in

of systems of second-order, quasi-linear partial differential

equations is discussed. Specifically, MACSYMA is used to cal

subgroup which leaves invariant the boundary curves and bound

obtaining similarity solutions is illustrated by an example

the group is known, its invariants and consequently the simi

culate systematically the generators of the infinitesimal

New York, N.Y. 10012

from fluid mechanics.

New York University

(

Long plasma columns can be heated economically by low frequency

compressive pumping with an axial wavelength, A , that is of the

a second order perturbation method and by numerical simulation,

velocity). It is shown by numerical simulation that in spite of

order of or long compared to the mean free path, A. Heating rates

Axial Collisional Heating of Linear Magnetic Fusion Systems

in the heating rate is found near the resonance between the pumping

using Collisional viscosity and heat flow co-efficients. A maximum

phase velocity, A^/a^, and the axial magneto acoustic velocity (cusp

are calculated as a function of A/A^, 8, and pumping frequency, 0)^, by

P. McKent;’ .;’ R. Morse and G. Sowers The University of Arizona Tucson, Arizona 85721

temperature changes, increases of plasma temperature by a factor

the detuning of the wave velocities that occurs as the plasma

of ten are quite plausible.

ABSTRACT

P. J. Morrison

Application to the Electrostatic Double Layer

Electron Stability Analysis of the Inhomogeneous Beam Plasma System—

University of California, San Diego La Jolla, California 92093

The electrostatic double layer has been proposed as a mechanism for particle acceleration. Such a B.G.K. type structure (related to the collisionless shock) has been observed in double and triple plasma de vices, and has been the subject of several computer simulations. We model this structure in terms of two interpenetrating cold beams (one electron and one ion), together with background populations. The sta bility analysis, facilitated by employing the localizing approximation, is performed. A classical perturbation expansion using the wave ampli tude and the ratio of the wavelength to the characteristic scale length of the medium as small parameters, yields the energy flux conserving In approximation. Such an expansion breaks down at transition points. order to understand the nature of these transition points we reformulate the problem in terms of a system of five first order differential equa tions. This system is then subjected to a reductive perturbation pro cedure which reduces it to tractable independent systems of lower order. We observe that in addition to the four traditional modes observed in the homogenous theory there exists an additional singular mode. The behavior of this additional mode is discussed.

Work supported by National Science Foundation under grant ATM77-12866.

“Pinch-Tormac”- A New Fusion Device. T. Hatori

and A.K. Sen*, Institute of Plasma Physics, Nagoya U., Japan. We propose a new figure 8 fusion device whose

lines, we use the Braginskii transport equations with friction and thermal forces and momentum terms correspond

linear segments are Theta pinches and whose curved ends are tormac like sectors. To derive the sheath thickness of the transition layer between the closed and open field

sheath confinement time and thickness. However, the sheath thickness is between one and two gyroradius and the sheath confinement time is typically <^10*4 s. For high p

ing to particle losses from the sheath. We find that the transit time of ions in the linear segment is a new

parameter which is critical in the determination of the

fusion type parameters and length of the linear segment

*Permanent address: Columbia University, New York, N.Y.

between 100 to 150 m one can obtain ni^lO***”’.

S. Migliuoloand B. Coppi

’ FOR TOROIDAL BALLOONING MODES

TWO DIMENSIONAL STRUCTURE AND VARIATIONAL PRINCIPLES

the resulting trial function for the perturbing displacement.

A comparison with the marginal stable eigenvalues and eigen

representation is that the topological and physical properties

functions,obtained by a well-known infinite series representa tion^, vields good agreement. The advantage of our direct

flux surfaces are still circular, though with shifted centers. We obtain marginal stability curves as well as a picture of

We emoloy a direct two dimensional representation of ballooning modes and evaluate their stability, in axysimmetric

toroidal devices, for the following equilibrium configurations a low-g equilibrium, where the flux surfaces can be described by concentric circles and a finite-^ equilibrium in which the

B. Coppi, J. Filreis, F. Pegoraro, MIT Report PRR 78/22,

Massachusetts Institute of Technology, Cambridge, Ma. 02139

(Cambridge, Ma., 1978) to be published in Ann. Phys.

of the considered modes are immediately evident.

The Trapped-Untrapped Electron Boundary Layer in Tokamak Geometry

Boundary layer effects are found to be important when solving the drift

boundary conditions at the trapped-untrapped electron boundary. Electron

for all of velocity space in a radially local analysis, applying no special

kinetic equation in toroidal geometry. A Lorentz operator is used for collisions

Landau resonances and trapped particle effects are thereby consistently connected

J.F. Santarius Nuclear Engineering Department, University of Wisconsin, Madison, Wisconsin, 53706 F.L. Hinton and D.W. Ross Fusion Research Center, University of Texas, Austin, Texas, 78712

functions showing boundary contributions will be exhibited. The results are

in the transition region. This improves upon analyses using a Krook model

used, with simple assumptions for the ion physics, to examine the trapped

for collisions which solve in the resonance and trapped particle regions

The drift kinetic equation is here numerically solved, and distribution

studied. Evidence indicating that this may be due to a collisionally

electron mode, which is found to be damped for most parameter regions

broadened Landau resonance at the boundary layer will be presented.

separately, then seek a plausible connection criterion.

STABILITY OF FIELD REVERSED THETA PINCHES*

In field reversed theta pinch experiments at Los Alamos and elsewhere,

modes. In these simulations, toroidal mode numbers of n = 0,1,2, and 3 are resolved

with little effect from finite grid resolution. Higher n modes are attenuated on

stationary plasma configurations which are stable for many Alfven transit times havg

been produced. Three-dimensional, initial value calculations have been performed

with the MALICE code to study the stability of such a configuration to ideal MH#

D. V. Anderson Lawrence Livermore Laboratory Livermore, California 94550

D. C. Barnes and C. E. Seyler Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

plasma pressure on open field lines outside the separatrix. The stability of n = !

unstable. Comparison of these results with the experimental observations and with

the equilibrium and the rotational velocity the n = 2 or the n = 3 mode may be most

Configurations which are completely stable to low n modes are found by allowing

Rotationally driven modes are also examined using the simulation. Depending ot

modes is shown to depend on the details of the open field pressure profile.

*Work performed under the auspices of U. S. Department of Energy.

other theoretical work indicates qualitative agreement.

totally supressed.

v’

F. L. Cochran, P. McKenty, R, Morse and G. Sowers

The University of Arizona Tucson, Arizona 85721

Studies of solid material end plugging of linear open ended

can eliminate plasma end loss and considerably reduce the plasma

confinement systems have shown that plugs with Z(atomic number) >1

column length needed to limit the energy end loss rate to a given

value. These studies include line and continuum radiation energy

Solid Material End Plugging cf Linear Magnetic Fusion Svgtens

model which includes Monte Carlo alpha particle transport, and with

loss rates and have been done with a time dependent, numerical, MHD

Improvements in performance are obtained from use of multi material

a quasi-stationary analytic model. Calculations done with reactor

parameters indicate system lengths less than ten kilometers.

layered end plugs.

ABSTRACT

Princeton, New Jersey, 08544

M. Katsurai”** and D^ L^ Jassby

CATALYZED DEUTERIUM TOKAMAK PLASMAS*

CHARACTERISTICS OF IGNITED, HIGH-WALL-LOADING

Plasma Physics Laboratory, Princeton Univarsity

high power loading. Neoclassical scaling for the ion

Scoping studies for ignited catalyzed-deuterium tokamak

energy confinement time and various empirical scalings for

limit is determined by MHD ballooning mode theory. Because

plasmas are carried out with emphasis on attaining medium to

electron energy confinement time are used. The critical beta

about 13.5 T In the extreme case of 8 (0) - 1-0 with appropriate radial profiles, B^ ^, as small as 12 T is sufficient.

and <P> = 0.15. If T <x T ^ the required B e

The results indicate that to achieve a total wall power loading

no tritium breeding is required, the plasma can be surrounded

tPermanent address. Dept, of Electronic Engn., Univ. of Tokyo.

the high densities (n^> 3 x 1 0 ^ cm”^) and medium temperatures

Cyclotron radiation is found to be a minor loss mechanism at

equilibrium ignition, taking into account radial profiles of

plasma parameters. In the numerical solutions, the relative

<T> - 25 KeV used here. <T^> is always at least 0.80 <T^>.

by a thick conducting wall which raises the limiting beta.

The thermal stability of practical operating regimes is

ofv3 MW/m , the reactor dimensions can be kept reasonable

magnitudes of T^ and T^ are calculated self-consistently.

Both analytical and numerical solutions are found for

*Supported by U.S. D.O.E. Contract EY-76-C-02-3073

(at the conductor) is 15 to 16 T

(R=8.4 m, a=3 m), if

under investigation.

is reduced to

max

BY MAGNETIC RIPPLE EFFECTS*

ALPHA PARTICLE “PUMPING” IN A TOROIDAL FUSION REACTOR

It is generally considered that the alpha particles generated by D-T fusions in

In this work we estimate the fraction of the alpha particles that are lost through

D. Callen, R. H. Fowler, and J. A. Rome Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

a fusion relator must be pumped away as neutral helium at the plasma edge. A corollary is that divertors are necessary to pump the charged alpha particles into a remote chamber where hopefully a means can be found to exhaust helium sufficiently and thus prevent quenching of the burn by an accumulation of alpha particles in the plasma. However, as has been pointed out previously,^ a possible difficulty with this scenario is that in toroidal plasmas with any reasonable amount of magnetic field ripple most of the alpha particles should pitch angle scatter into the ripple loss region, become ripple trapped, and drift vertically out of the machine by B x VB drifting along [B] contours. Thus, the problem of handling alpha particles is probably not primarily one of handling a diffusing thermal component, but instead becomes one of providing “channels” at the top or bottom of the tokamak through which the energetic (t 0.3-3 MeV) alpha particles can drift out of and be removed from the plasma chamber.

the ripple trapping process, and the fraction of their energy that is deposited in the plasma before they are lost. Since D-T fusion alpha particles are produced with a monoenergetic (3.5 MeV), but isotropic pitch angle distribution, a small fraction of them (Vjj/v^ $ /<5, where 6 is the ripple depth) are b o m in the* ripple loss region and are lost immediately without depositing any of their energy in the plasma. The remaining alpha particles slow down by Collisional drag on the background plasma in a manner similar^ to neutral beam injected fast ions. During the slowing down process they pitch angle scatter, with the scattering becoming most significant for alpha particle energies at or below the critical energy (Eg v. 33Tg for alpha particles in a 50:50 D-T plasma) at which energy is transferred equally to plasma ions and electrons. Thus, the alpha particles should deposit a large fraction of their energy in the plasma before being scattered into the ripple loss region and drifting out of the machine with a remaining energy of Eg ^ 300-600 keV. That is, the ripple effects may provide the primary “pump” for removing alpha particles from the plasma. . More detailed estimates of these processes and of other ramifications of magnetic field ripple effects will be presented.

N. A. Uckan, K. T. Tsang, and J. D. Callen, “Toroidal Field Ripple Effects in Large Tokamaks,” Proc. 6th Symp. of Eng. Prob. of.Fusion Research, San Diego, CA, Nov. 18-21, 1975 (IEEE, New York, 1976), p. 1105 (IEEE Prob. No. 75CH1097-5-NPS).

“Neutral Beam Injection into Tokamaks,” Plasma Physics and Controlled Nuclear Fusion Research, 1974 (IAEA, Vienna, 1975), Yol. I, p. 101.

Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

J. D. Callen, R. J. Colchin, R. H. Fowler, D. G. McAlees, and J. A. Rome,

References

fin

the

ting.

;ucv th

rrent in

Of? percent

port code t- 0

sr.d L. E. Nelson

current may result.

BCU1DAPY E^yTLIB^m*

reasonable adjustment of the vertical field and the

windings the 3 of the ecuilibriUm can be raised ft

We have used an accurate, efficient 1-1/2 D transport code t

The equilibrium module ’ of the code employs a combination of a

semi-fixed boundary Buneman solver with an efficient surface Green’s

reversed current regions or undesirable shape changes. However, unless the

primary current is adjusted, unexpected plasma compression or surface

three percent without separatrix formation at the plasma ed<ae, appearance of

1^. Cak Hidge national Laborstcry Caw Ridge, Tennessee egc-n

^Research sponsored by the Office of Fusion Energy (ETM), U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation. **

whereas, for peaked current, the plasma edge is almost circular even at 3 ^

increasing 6 and scales as 3*^- for high 3; 1^ rises linearly at low 3 but

increasing 3 because of increasing volume near the axis due to the outward

plasma edge becomes elliptical at high 3 in a uniform vertical field,

begins to saturate at higher 3; the required vertical field also begins to

they show that nonintuitive changes in the poloida.1 field coil currents may

function for the plasma current to reduce the computation time. Feedback of

destabilizing skin currents. These same methods will be used to adopt the

saturate at high 3 and the primary flux must be decreased. Evolution of the

the vertical field and primary flux is used to position the plasma against a.

fixed limiter and. to adjust the plasma volume to avoid skin current

required to preserve the plasma shape and volume and to avoid

In agreement with previous thatory,’* we find that 3j saturates with

These results have implications for rapid heating of a tokamak because

Interestingly, the density at the magnetic axis decreases with

1j.F. Clarke and D. J.Sigmar, Phys. Rev. Lett. ^8, 70 (1977).

diffusive 1-1/2 D transport code to study free boundary plasmas.

plasma shape depends upon the current profile.

Visitor from JAERI, Tokai, Japan.

With broad current, the

diamagnetic shift.

formation.

20%.

and

J . Denavit

Evolution of Drift Waves’*

C. 0. Beasley, Jr., and W. 1. van Rij

TEDI - A Numerical Simulation of the Time

Northwestern University Evanston, Illinois 60201

In order to study drift wave instabilities, we have developed a

Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

numerical model - TEDI - to study the time evolution of drift waves. The

used for these early tests. Landau damping is shown to be correctly included

Research sponsored by the Office of Fusion Energy (ETM), U. S. Department of

first calculations testing the model are shown here. These include 1) the

equation, and the electron equation being a drift-kinetic equation. These

results obtained by a well-known eigenmode solver. Kinetic equations are

are solved on a grid in velocity space and “radius.” A slab geometry is

linear evolution of a drift mode in a cylinder, and 2) a reproduction of

used for both ions and electrons, the ion equation being a gyro-kinetic

Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

ON TRAPPED-ELECTRON MODES*

L. Crystal and J. Denavit

CURVATURE DRIFT RESONANCE EFFECTS

Northwestern University, Evanston, Illinois 60201

These simulations are based on the linearized electron drift-kinetic

Computer simulations of dissipative trapped-electron modes in toroidal

plasmas, including curvature and gradient drift effects^ are presented.

equation, Fourier transformed with respect to the poloidal and toroidal

angles. No a priori distinction is made between trapped and circulating

particles, and collisions are represented by a Lorentz model giving pitch-

^J. C. Adam et al., Phys. Fluids 1J9, 561 (1976).

modifies significantly the mode structure of the dissipative trapped-electron

damping due to resonant circulating electrons reduces the growth rates, and

used “effective” collision frequency, based on an assumed trapped-electron

angle diffusion, which does not necessitate the introduction of the often-

dissipative trapped-electron instability occurs. In this regime, Landau

regime, v/u)^ ^ 0.1, curvature drift resonance effects are strong but are

destabilizing only for p^/r below a critical value. These growth rates

A series of computations shows the dependence of the growth rates on

J. Denavit and C. E, Rathmann, Phys. Fluids 21, 1533 (1978).

(p^: electron gyroradius, r: flux surface minor radius,

decrease rapidly for collision frequencies V ^ 0.1

drift wave frequency). In the weak collision

Work supported by DOE contract EY-76-S02.2200

V: collision frequency,

distribution.

and for v =*

p^/r and on

instability.

the

ABSTRACT

New York University

Gudmundur Vigfusson

New York, New York 10012

ADIABATIC COMPRESSION OF PLASMA^

MATHEMATICAL PROBLEMS ARISING IN

Courant Institute of Mathematical Sciences

We present some results on the “Generalized Differential

Equations” (GDE) of adiabatically evolving plasma equilibria.

These are non-linear differential-functional equations of the form Aijj = F(V,^,^‘,i^”), where V = V(i^) is the volume (area)

existence and uniqueness theorem is given for the associated . linearized problem, which is also a functional-differential

formulation will be used to study examples of bifurcation, exchange of stability and transfer into more complicated

equation. Finally we will mention an isoperimetric problem

derivatives and simple cases for the nonlinear problem. An

on the right hand side are with respect to the dependent

We describe so-called microcanonical averages and their

related to the geometry of GDE’s, and this variational

Work supported by U.S. D O E contract No. E Y - 7 6 - C - 0 2 - 3 0 7 7 .

a constant, and the derivatives

inside the levelsets ^(r) =

variable V.

geometries.

8 0

H ^f GIT

desired).

c Labe ra tcry e?

K. kai er .icge Lat icna Ridge, Tenne

We obtained the Hamiltonian for a charged particle in

r. TCf * FT K’ELEC TilTV f)M crT7 r ElEL p a

It is advantageous tc have a Hamiltonian formulation of the

toroidal geometry through the use of Poisson brackets. This procedure

is simpler than the generating function approach, especially for

consistent (i.e., they conserve energy exactly to whatever order

guiding center equations so that the equations are internally self

Department of Energy under contract W-?405-eng-26 with the Union

angle, and hence posseses an invariant momentum corresponding to the

geometry as opposed to systems which require knowledge of the length

this was not a perturbative problem, Lie transforms were inapplicable.

The Hamiltonian obtained has no explicit dependence on the gyro

“Research sponsored by the Office of Fusion Energy (ETM), U.

poloidal and toroidal angles were obtained together with

In addition, coordinates corresponding to the

Thus, the coordinates are directly related to the

For axisymmetric configurations, the momentum

conjugate to the toroidal angle is conserved.

obtaining higher order terms in 1/M

being the gyrofreauency). Since

along a field line.

conjugate momenta.

Carbide Corporation

magnetic

moment.

their

REDUCED SET OF RESISTIVE MHD EQUATIONS IN TOROIDAL GEOMETRY* *

code, Lcbeto, is used to numerically advance this set of equations.^

A detailed analysis in the linear regime has been performed to

Me have studied the evolution of tearing modes using a reduced set of

toroidal geometry of the equations employed in Ref. 1. A three-dimensional

investigate the toroidal effects on the linear growth rate and eigenfunctions

of resistivity and toroidicity. These equations are the generalization tc

low 8, three dimensional, resistive MHD equations which includes the effects

E. Carreras**, H. R. Hicks, arc J. A. Holmes Oak Ridge National Laboratory Oak Ridge, Tennessee 37820

‘B. V. Waddell, B. C=rrer=s, H. R. Hicks, J. A. Holmes and D. K. _Lec, Phys. Rev. Lett. VL. 1*8c (1978). *H. R. Hicks, B. Carreras, and S. J. Lynch, abstract submitted tc this

^Research sponsored by the office of Fusion Energy (ETM), U.S. Department .of Energy under contract W-7i)05-eng-26 with the Union Carbide Corporation.

nonlinear phase has also been studied. Me have found that the 2/1 tearing

modes are wisely chosen. A scheme for performing such calculations has been

different values for the aspect ratio. These results show that semianalytic

calculations based on the coupling of only two modes can give a reasonable

of the tearing modes. We have considered several safety factor profiles and

understanding of the toroidal effects in the large aspect ratio limit, if the

The toroidal coupling between the 1/1 and 2/1 tearing modes during the

mode can be destabilized by the V I mode through the toroidal coupling.

‘Visitor from J.E.N., Madrid, Spain.

developed.

FREE AND FORCED m = 0 OSCILLATIONS OF A SHARP-BOUNDARY VLASOV-FLUID SCREW PINCH*

Thomas E. Cayton and H. Ralph Lewis University of California LoS Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

A dispersion differential equation has been derived for the study of finite ioti gyroradius effects in free and forced oscillations of a high-8, sharp-boundary seres pinch. The dispersion differential equation is derived from the equations of th{ Vlasov-fluid model. An approximate solution of the linearized ion Vlasov equatiot for the sharp-boundary pinch is obtained analytically via assumption of fast gyratiot and small gyroradius. This approximate solution together with the Ampere equatiop generates the dispersion equation, which explicitly exhibits two length scales: at MHD length scale (the pinch radius) and a microscopic scale (the ion gyroradius). It the limit of vanishingly small ion gyroradii, the equation and two of the boundary conditions reduce to the ones derived from the guiding center plasma model; witi finite ion gyroradii, boundary-layer phenomena are obtained.

In the case of forced oscillations, we compute the coil impedance as a function of real frequency. Two separate resonances are noted: (relatively high frequency magnetoacoustic resonances and a (relatively low frequency) sloshing resonance magnetoacoustic resonance involves primarily radial plasma motion; whereas t’*- sloshing resonance in the guiding center plasma description exhibits substantial flo’^ along the magnetic field lines between the high- and low-pressure regions. As tb axial wavelength of the excitation increases, the magnetoacoustic resonance frequent approaches a low-frequency cutoff, while the sloshing resonance frequency scales witij} the axial wavenumber. The important implications for rf heating in high-p systen^ a r e a s follows:

The dispersion differential equation has been applied to study free and force axisymmetric oscillations of a sharp-boundary screw pinch. In the case of fre oscillations, we examine the effects of finite ion gyroradii on the eigenfrequencie and eigenfunctions of two types of modes: magnetoacoustic modes and mirror modes. A the ion gyroradius increases, the eigenfrequencies are modified; however, tb eigenfunctions become considerably distorted from the guiding center plasma solution this probably signals the breakdown of the approximation.

Absorption is due to ion transit-time damping: it depends critically up4 the value of (*)/kv^, the ratio of the resonance frequency to the thermal ion transl frequency.
The use of the magnetoacoustic resonance requires high frequencies.
Arbitrarily low frequencies may be achieved for the sloshing rsonances b
Comparable resistances can be achieved with each type of resonance.
Finite ion gyroradius effects modify the solutions in ways similar to thO=

*Wo,rk performed under the auspices of the U. S. Department of Energy.

using arbitrarily long wavelength excitations.

described above for free oscillations.

D. A. Larrabee and R. V. Lovelace, Cornell University.

An analytic and numerical study has been made of the single

PARTICLE ORBITS IN FIELD-REVERSING ION RINGS: ERGODIC OR NOT?

to the Hamiltonian and the canonical angular momentum. In one

analysis of possible effects of a third constant of the motion

with about 10% of the particles in the ring being ergodic. An

indicating the existence of a constant of the motion in addition

the cases studied the numerically computed orbits are non-ergodic

compressed ring equilibrium limited stochastic behavior was found

particle orbits in self-consistent ion ring equilibria. In most of

is obtained from an analysis of the orbital stability of the class

case, is shown below. The dotted line is the poloidal projection

An example of a stable mid-plane orbit, which is the typical

of particles which have orbits near the mid-plane of the ring.

An understanding of the ergodic to non-ergodic transistion

of the orbit, and the solid line }$

constant curve of the effec

has been begun.

tive potential.

Yale University

New Haven, Ct. 06520

Fokker-Planck Equation in Two Velocity Coordinates

D. Fyfe, S. Eisenstat, M. Schultz, and 1. Bernstein

Numerical Approaches to a Time-dependent Non-linear

distribution function is assumed to be independent of gyration

resulting ordinary differential equations in time are solved

product B-splines in the Galerkin form of the equation. A

using a fast Poisson solver and then these potentials are

using a stiff ODE package. A fully implicit method (fixed step

phase. The Poisson equations for the Rosenbluth potentials are

comparison with a finite difference discretization is made. The

solved in the perpendicular and parallel velocity coordinates

size backward Euler) and a time centered scheme (fixed step size

Work supported by Department of Energy contract EG-77-S-02-4349

Some numerical approaches to a Fokker-Planck equation for a

Fokker-Planck equation itself is discretized using

of particle trapped in a joint magnetic

trapezoidal rule) are also described.

electrostatic square-well

Fokker-Planck

coefficients.

differenced

discussed.

potential

species

tensor

single

The

are

for

the

Princeton, NJ 08544

Vlasov Turbulence ;

The formalism employs

R.V. Jensen and J.A. Krommes

Plasma Physics Laboratory, Princeton University

Renormalized Dispersion Tensor for Electromagnetic

The nonlinear dispersion tensor for electromagnetic fluctuations in

straightforward but powerful functional techniques to express the disper

sion function entirely in terms of observable quantities such as the elec

a turbulent Vlasov plasma is derived. The calculation extends the electro-

demonstrated. The application of the results to the turbulence theory of

Interaction Approximation; the relation to weak turbulence theory is then

tric field fluctuation spectrum. Explicit formulas are given in the Direct

static results given recently by Krommes and Kleva.

Work jointly supported by U.S. AFOSR Contract No. F 44620-75-C-0037 and U.S. DoE Contract No. EY-76-C-02-3073.

^*J.A. Krommes and R.G. Kleva, Princeton Plasma Physics Lab. Rept. PPPL-1522

finite-g drift waves is discussed.

(1979).

WAVE. PARTICLE TRANSPORT FROM ELECTROSTATIC INSTABILITIES: AN OVERVIEW*

Wave-particle transport from short wavelength electrostatic instabilities driven

momentum and energy for the lower hybrid drift, ion cyclotron electron drift,

this model, resistivities and heating frequencies of the universal drift instability

are substantially greater than those due to the other drift modes. And wave-particle

universal drift, ion acoustic current and ion cyclotron current instabilities. In

procedure is used to evaluate and compare wave-particle exchange frequencies of

plasma is reviewed. Both electrons and ions are taken as magnetized, and propagation

is in the plane defined by the drift velocities and the magnetic field. A consistent

by currents both across and parallel to a unidirectional magnetic field in a Vlasov

S. Peter Gary University of California Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

problem of thermal flux inhibition vs. enhanced radial diffusion in a linear theta

transport due to the ion cyclotron electron drift instability is larger than that of

*Work perfomed under the auspices of the U. S. Department of Energy.

the lower hybrid drift instability at

The results are applied to the

pinch.

> TL.

K. D . Marx

In a tokamak, a slow wave antenna of finite length will radiate waves

It is found that radiation at parallel phase velocities in excess of the

field. For a range of plasma parameters corresponding to the Doublet-IIA

within a broad range of phase velocities parallel to the toroidal magnetic

is used to examine penetration of the full spectrum excited by the antenna.

experiment, a quasilinear-collisional theory of lower hybrid wave absorption

STUDIES OF CURRENT DUE TO RF INDUCED RUNAWAY IN THE DIIA LOWER HYBRID EXPERIMENT*

R. W. Harvey, J. C. Riordan and J. L. Luxon General Atomic Company San Diego, California

National Magnetic Fusion Energy Computer Center Lawrence Livermore Laboratory Livermore, California

also been undertaken using a 2-D velocity space Fokker-Planck code including

able runaway current (but no substantial heating in these cases). The most

terms describing an applied dc electric field, QL electron diffusion due to

diffusion, even though the nominal phase velocities excited by the antenna

density discharges with n^ = 11. Simple numerical estimates give rise to

Dreicer velocity Vp can penetrate to the plasma center. The quasilinear

experiment provides evidence of quasilinear diffusion in the tail of the

pronounced effects occur in accordance with the experiment for the lower

waves nominally characterized by n^ = 11 and n^ = 14 may induce an observ

As a result, it is predicted that antennas in Doublet-IIA that radiate

For completeness, a more comprehensive analysis of runaway current has

diffusion of this component of the wave spectrum dominates Collisional

runaway currents consistent with experimental observation. Hence the

Work supported by Department of Energy, Contract EY-76-C-03-0167,

rf, and braided magnetic field induced transport.

Project Agreement No. 38.

are of the order Vp/2.

electron distribution.

Princeton, New Jersey 08544

W. M. Nevins, Liu Chen, and C. Z. Cheng

not assumed to be small, and we have found no

Plasma Physics Laboratory, Princeton University

The universal drift instability as an initial value problem

space (x, v ^ ) , where x is the inhomogeneous coordinate, and

Vjj is the component of the velocity parallel to the magnetic

is studied in the slab geometry. The linearized drift kinetic

Convective Drift Wave Instability In A Sheared Magnetic Field*

equation is integrated numerically in the two-dimensional phase

field. ^x^i absolute instabilities associated with finite ion gyroradius

This work was supported by US Department of Energy Contract No. EY-76-C-02-3073.

2 3 stable” drift wave eignemodes ’ after substantial convective

growth of the perturbation. Energy amplification factors of 0(10 )

effects.^ In the weak shear limit perturbations that are local

in both space and time are found to excite the “marginally

^Y. C. Lee, Liu Chen, and W. M. Nevins, to be published.

equilibrium fluctuation spectrum will be discussed.

^D. W. Ross and S. M. Mahajan, RPL 40_, 324 (1978).

^Y. C. Lee and Liu Chen, PRL 42, 708 (1979).

The contribution of these modes to the

^Liu Chen et al., PRL 41, 649 (1978).

have been obtained.

Y. Y. iau

Transient Amplification of Shear Alfven Waves

It is shown that a current-carrying plasma, such as that in a

the ideal MHD equations. For parameters typical of tokamak geometry,

tokamak or in a pinch, could be subject to transient amplification of

based on a preliminary study of a slab model of a plasma described by

magnetic field fluctuations of shear Alfven waves. This conclusion was

Department of Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 02139

the amplitude of a shearing wavelet may gain by a factor of 50-100 in a

trigger other instabilities if these fluctuations attain a sufficiently

speculated that these magnetic fluctuations, while “ever-present”, may

enhance the energy loss in a plasma, and in the worst case, may even

time scale of order 5-10 psec before it eventually decays. It is

Work supported in part by the National Science Foundation.

^*Y. Y. Lau, Phys. Rev. Lett. 42, 779 (19/9).

high level.

and

J . Denavit

reversed ion ring.

A. Mankofsky and R. N. Sudan

Heating by Field-Reversed Ion Rings”

Numerical Simulation of Plasma Confinement and

The RINGA code^ has been used to study confinement and heating

of a finite-8 plasma on closed field lines produced by a field-

Plasma is described by a Grad-Shafranov term in the field equation

Laboratory of Plasma Studies, Cornell University Ithaca, New York 14853

Department of Mechanical Engineering, Northwestern University Evanston, Illinois 60201

on Numerical Simulation of Plasmas, (Monterey, CA, 1978), Paper #PE-4; A. Mankofsky, A. Friedman, and R. N. Sudan, Cornell Univ. LPS #245 (1978).

is nonzero only on closed field lines.

E^, and the ring halfwidths Ar and Az increase, while the total particle

pressure, the field reversal factor t , the total magnetic field energy

energy Ep decreases. Most importantly, the innermost flux surfaces of

*Work supported under U.S. Department of Energy-Contract EY-76-S-02-3170.

^A. Mankofsky, A. Friedman, R. N. Sudan, and J. Denavit, Eighth Conf.

the fieId-reversed region become stable to the interchange mode, as

which is in addition to the term representing the current contributed

loss of particle energy is balanced by increases in the plasma energy

by the ion ring. The plasma pressure p(p) is a given function which

^A. Mankofsky, R. N. Sudan, and J. Denavit, Bull. Am. Phys. Soc. 23,

pg = -Vp^mrVg, where p^ is the ion canonical angular momentum. The

and pressure. Recent results from these studies will be presented.

We have also included the fast ion-electron drag term to describe

the slowing down of the energetic ions by means of the equation

As we increase the plasma

determined by ^d%/[B[.

842 (1978).

An important question regarding the stability of plasmas

Department of Physics and Astronomy University of Maryland College Park, Maryland 20742

with reversal magnetic fields is whether microscopic processes exist in the vicinity of the reversal point; and if so, what are their effects. Particle simulation is a convenient method of studying the development of instabilities in such a con figuration, since complicated particle orbits make analytic theory intractable. The simulations are done in two dimensions perpendicular to the magnetic field, so that’the resulting tur bulence is due to a cross-field instability, instead of the collisionless tearing mode. Away from the null point where strong density gradients exist, the lower hybrid drift insta bility is excited, giving rise to electrostatic and electro magnetic fluctuations of comparable size. At the null point the fluctuations are larger and primarily electromagnetic. The fluctuations are not simply an extension of the lower hybrid drift instability, since nonlocal linear theory shows that this mode is stabilized at the null point by finite beta effects;-*- rather, they are parametrically driven by the lower hybrid drift waves. The turbulence at the field reversal point produces strong electron heating. Both linear and non linear behavior will be described.

^*J. D. Huba, J. F. Drake and N. T. Gladd, “Nonlocal Theory of the Lower Hybrid Drift Instability in a Reversed Field Con figuration”, presented at this meeting. ^Research supported by U. S. Department of Energy.

1C 48

N. Sharkv, B. Coppi and T. Antonsen

Massachusetts Institute of Technology

NUMERICAL SIMULATION OF IMPURITY TRANSPORT AND

PLASMA DECONTAMINATION BY IMPURITY DRIVEN MODES

The effects of impurity driven modes are analyzed with a

one-dimensional impurity transport model which includes both neoclas

into account the different collisionality regimes of the main ions. A

temperatures are proportional. We then study the time evolution of the

single impurity species is included in the model, and it is assumed that the magnetic field is constant in time and the electron and ion

sical and anomalous transport. The expressions of the anomalous fluxes contain the quasi-linear effects of both Collisional and collisionless impurity driven modes”**. The evaluation of the neoclassical fluxes takes

outward flow of impurity ions until the impurity density profile is peaked at the edge of the plasma. In this case the neoclassical terms

anomalous trans varied. We find that when l i ^ H c (with r^t=l) ’ port occurs and collisions cause an accumulation of impurities at the

contaminated by an impurity ion in a single ionization state, most of

main ion density, the total impurity density and the ion temperature.

exceed the neoclassical ones. Furthermore, the peak impurity density

do not affect the results considerably, because the anomalous fluxes

in steady state is typically 24-5 times larger than the value at the

the computations are done for this simple case. However, the effects

Furthermore, since the results of ref. (1) are derived for a plasma

value of the relative ion temperature gradient (^^=dlnT^/dlnn^) is

on the results of the ^Z/^r terms, in the neoclassical fluxes, are

^S. Coppi, G. Rewoldt and T. Schep, Phys. Fluids 19_ (1976) 1144.

Parameters typical of the Alcator device are used, and the

examined by using the coronal equilibrium model.

, the impurity driven modes produce an

center. However, when

center.

3,4,5 ’

appears.

1 2 magnetic shear. ’

SELF-HEALING OF BALLOONING MODES

A first threshold for instability is reached

The growth rates and the stability limits of ideal M.H.D.

In the vicinity of the magnetic axis, where the magnetic

poloidal angle dependence of the poloidal field and the rate of

of the pressure gradient has a stabilizing effect because of the

ballooning modes have been obtained for models which include the

“stiffening” of the poloidal field lines on the outer side of the

torus and the poloidal angle dependence of the shear. This occurs

surfaces can be described accurately by shifted circles, all the

when there is a sufficiently large pressure gradient acting against

on the pressure gradient. As a result, a second stability region

the curvature of the magnetic field lines. However, further increase

because the general governing equation exhibits non-linear dependence

A. Ferreira, B. Coppi, J. W-K. Mark, J.J. Ramos, L. Sugiyama Massachusetts Institute of Technology, Cambridge, Ma.

B. Coppi, J. Filreis and J. W-K. Mark, 7th International Conference on Plasma Phvsics and Controlled Nuclear Fusion Research, Innsbruck Austria (1978) IAEA-CN-37-W-4.
J.J. Ramos, B. Coppi, A. Ferreira and J. W-K. Mark, 20tk Annual Meeting of the Division of Plasma Physics (A.P.S.) Colorado, Nov. 1978. Bull. An. Soc. 2_3, p 785 (1978).
B. Coppi, A. Ferreira, J. Filreis, J. W-K. Mark and J. Ramos, Annual Controlled Fusion Theory Conference, Gattlinburc,Ten’n. (April 1978).

the rate of magnetic shear and to the pressure gradient.

B. Coppi, A. Ferreira, J. W-K. Mark and J.J. Ramos, to be oublished

6 . B. Coppi, A. Ferreira, J. W-K. Mark and L. Sugiyama, M.I.T. Reocrr

pressure gradient which define the two boundaries of the instability

B. Coppi*, in Proceedings of the Finite Beta Theory Workshop held

finite-beta configurations, we have tested the stability of a sequence

solutions of the general ballooning mode equation based on this exact

equilibria again demonstrates the existence of a second stability

limit, we can determine the critical values of magnetic shear and

equilibrium parameters of the model are related in a simple way to

In order to confirm the predictions of the model with more realist

of flux-conserving Tokamak equilibria generated numerically. The

PRR 78/43 (Cambridge, Ma. 1978).

in Varenna, Italy, Sept. 1977.

in Nuclear Fusion.

In this

domain.

region.

ABSTPACT

(Euratom/UKAEA Fusion Association)

ing mode. This result is first obtained from a

It is shown that the shear in the magnetic field

GRADIENT INSTABILITIES IN A SHEARED MAGNETIC FIELD

frit. COUPLING OF THE RLSISTJOu—g AND iON tEHPER-O-’.‘tE

couples together the resistive-g and ion temperature

gradient instabilities to form a single strongly grow

J G Ccrdey, E M Jones and D F H Start Culham Laboratory, Abingdon, Oxon. 0X14 3DB, U.K.

of this instability and the low frequency fluctuations

shear, curvature and collisionality etc will be given.

solution of the full radial eigenvalue problem. The

dependence of the growth rate of this instability on

consideration of the structure of the turning points

Comparisons will be made between the characteristics

of the two modes and then confirmed by a numerical

observed on the Culham Levitron.

In the first version of the code we treated the case of constant resistivity and in

The work described here concerns investigations of resistive instabilities in the re

Cambridge, Massachusetts 02139 D. Hewett Los Alamos Scientific Laboratory Los Alamos, New Mexico 87544

RESISTIVE INSTABILITIES IN THE REVERSE FIELD PINCH J.P. Freidberg Massachusetts Institute of Technology

verse field pinch configuration. We have developed a new numerical coce which solves the full set of linearized resistive instability equations for one dimensional equilibria with B^(r), B^(r) and 8 arbitrary.

The novel feature of our approach is that the equations are solved as an eigenvalue problem. Those codes in existence, treating similar problems, utilize an initial value (1) The numerical approach. Two advantages of the eigenvalue approach are as follows. procedure is inherently very fast. A converged eigenvalue (to three significant figures) using a grid with 500 points is obtained in about 10-15 sec. on a PDP-10. (2) the eigen value approach allows us to examine the behavior, not only of the fastest growing mode, but of any mode. This feature is important in understanding the complete spectrum of instab ilities.

transition from one mode to another; that is, for these values of Reynolds number, we do not see clearly distinct regions where say y T ^ S ’ 3 / 5 as predicted by analytic theory. For the who le range o f uns ta b le k, the gro wt h rate of the slow mode s c a l e s ’ i n v e r s e l y w ith Reynolds number, yi.n,i/s. sistive diffusion m o t io n ^ away from our only app ro x im a te initial equilibrium. If the fast and slow growth rates are plotted s im ul t a n e o u s l y vs k for fixed S; these curves intersect at two d if f e r e n t k values. 4A For values of k outside this range, it is e x t re m el y likley that unstable modes exist, but w ith compl e x eigenvalues. Finally, we point out that the space between the inters ec t in g k values is a f u n c tion of S. action between d i f f u s i o n . a n d r e s is t iv e instabilities. a bout two and three dimensional MHD s im ul ations which are often forced to operate at rel at i ve l y low Reynolds numbers because of c omputer limitations.

for the m=l mode are as follows:

In general, for profiles satisying the Suydam criterion, there are two unstable modes for any given values of k and $ (Reynolds number). One of these is called the fast mode, the other one, the slow mode. For fixed S, the fast mode smoothly transforms from the ideal MHD mode, to a tearing mode, to a resistive interchange mode as k is varied.

compressible displacements. Even though there is no rigorous proof, substantial theo retical and numerical work indicates that under these assumptions, unstable modes are purely growing and not overstable. This feature, which-is sometimes not true, was also incorporated in the code. None of these assumptions, however, is at all critical to the numerical method.

We have run the code for RFP like profiles for a wide range of parameters. .The results

3, Curves of growth rate y vs S (for $< 10^) with k as a parameter show a smooth

Typically, for S ^ 1 5 0 - 2 0 0 these values c o a le s ce i nd icating a strong i n t e r

We int er p re t this, not as instability, but as a r e

This result raises que st i on s

F^s-r

Daniel C. Barnes and Charles E. Seyler Los Alamos Scientific Laboratory Los Alamos, New Mexico 87545

Numerical Calculations of Necessary and Sufficient Conditions for MHD Stability of a Stationary Field Reversed Mirror Plasma*

by David V. Anderson and William A. Newcomb Lawrence Livermore Laboratory Livermore, California 94550

Stationary plasma configurations have been observed in the field reversed theta pinches (FRX) at Los Alamos* and elsewhere which are observed to persist for many MHD Alven transit periods. Recent 3D time dependent computer simulations of these profiles with the MALICE code have also failed to show MHD instabilities. It is not clear that these results really indicate MHD stability because of other effects present In the experiment finite Larmor in the experiment or in the code. radius stabilization may reduce or eliminate the instabilities. The code, on the other hand, can only evolve long wavelength modes without dissipation while shorter wavelengths suffer numerical dissipation and those on the sub—grid scale are absent altogether. Hence stability of shorter wavelength modes is not assessed. We have turned to the MHD energy principle to get a better understanding of the field reversed plasma stability. Given equilibrium configurations calculated from a 2D r,z code (CYLEQ) for a scalar pressure profile P = P(i/’) we use our stability code (STABCR1T) to evaluate the MHD energy principle. A generalized Sturm-L*iouville problem is solved on each field line to sufficient conditions , for stability. evaluate the Criteria for interchange and co-interchange (ballooning) displacements Methods, first proposed by Johnson^, for calculating are found. marginally stable pressure

J. L. Johnson, R. M. Kulsrud, and K. E. Weimer, Plasma Phys., 11, 463, 1969.
R. K. Linford, Proc. 7th Inti. Conference on Plasma Physics and Controlled Fusion Research, Innsbruck, Austria, August 23-30, 1978.

Work performed under the Auspices of the Z. &. B. <9. B. by Lawrence Livermore Laboratory and Los Aiamos Scientific Laboratory under contract numbers W-7405-ENC-48 and W-7405-ENG-36 respectiveiy.

are also being investigated.

References:

necessary

profiles

and

Tueor tea.

G. P.ewclct,

‘larchand, and W. y. Tanc

Plasma Physics Laboratory, Princeton University, Princeton, N. J. 08544

Recent experimental measurements of density fluctuations in the neutral-

results. Taking into account conditions appropriate to the experiment, a

has been carried out. It is found that instead of a single type of insta

hybrids of the trapped-electron mode, trapped-ion mode, and the ion-tempera

theory of such modes are in apparent qualitative agreement with experimental

comprehensive calculation of electrostatic drift waves in a tokamak geometry

presence of low-frequency drift-type microinstabilities. In this paper it is

beam-heated PLT by means of microwave scattering”*” have suggested the possible

pointed out that a number of physical characteristics predicted by the linear

bility (such as the trapped-ion modes), the dominant drift modes are actually

ture-gradient-driven drift instability. These linear eigenmode calculations 2 3 employ one- and two-dimensional codes ’ embodying all features known to be

”*“V. Arunasalam, P. Efthimion, B. Gaulke, J. Hosea, E. Mazzucato, and M. Yamada, Bull. Am. Phvs. Soc. 23^, 901 (1978) .

poloidal or toroidal mode number) are found to be close to the observed peaks

experimentally for a large increase in the fluctuation level; and (.3) the two

corresponds to the ballooning character of the calculated eigenfunctions; (2)

the linear instability thresholds on the ion temperature and the ion tempera

rium profiles are obtained from transport code calculations which model well

R. Marchand, G. Rewoldt, and k. H. Tang, Bull. Am. Phys. Soc. 23, 785 (1978),

the experimental results. Among the particular points of qualitative agree

ment between linear theory predictions and the microwave scattering results

are: (1) the strong enhancement of fluctuations at the outside of the torus

important to the stability of toroidal drift waves. The necessary equilib

maxima in the computed linear growth rate curve (as a function of either

G. Rewoldt, W. M. Tang, and E. A. Frieman, Phys. Fluids 21, 1513 (1978).

*Work supported by U.S. Department of Energy Contract #EY-76-C-Q2-3073.

ture .gradient fall roughly within factors of two of those observed

in the k-spectrum of the density fluctuations.

Abstract

Austin, Texas 78712

W. Horton, Duk-In Choi,

Fusion Research Center

D. Biskamp, and P. Terry

The Trapped Ion Mode in the

The University of Texas at Austin

Presence of Drift Wave Fluctuations

The influence of a spectrum of drift wave fluctuations

on the trapped ion mode is investigated. It is shown that a broad spectrum of drift wave fluctuations with a monotonically decreasing k^-spectrum leads to a stochastic damping of the trapped ion modes. For the drift wave spectrum given by a previous theoretical model the nonlinear turbulent damping

equations are derived and reduced with the properties of the trapped ion-drift wave interactions. In the second approach the separation of the space-time scales is used at the outset

is sufficient to inhibit the onset of the usual trapped ion mode. In contrast, the analysis predicts that a peaked kj_- spectrum of drift modes results in the stimulation of trapped

ion mode dispersion relation renormalized by the presence of drift wave fluctuations. In one case general mode coupling

to write a wave-kinetic equation for the drift modes propagating

Two theoretical approaches are used to derive the trapped

This work is supported by the U.S. Department of Energy

in the presence of the slowly varying trapped ion mode.

Contract DE-AC05-79ET53036.

ion modes.

Sheared Magnetic Field

Ion Temperature Drift Instabilities in a

W. W. Lee, W. M. Tang, W. M. Nevins, and H . okuda

Results from the first particle code simulations of ion-

experiment exhibits characteristically large ion temperature

temperature-gradient-driven drift instabilities in a sheared

gradients. The purpose of this investigation is to verify the

Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544

code in a sheared slab geometry, where exact dynamics are kept for the ions, while the electron response is assumed to be

magnetic field are reported. This type of instability^ has received renewed interest recently because the beam-heated PLT

linear theory of the instability and to study its nonlinear consequences. The simulation has been carried out using a 21/2-D

accordance with the usual theoretical model and has the advantage of suppressing the unnecessary discrete particle noise associated with the electron motion. In the linear stage of the instability,

profile and the Doppler frequency shift resulting from the build up of the ambipolar potential. Details will be reported along

edited by M. A. Leontovich (Consultants Bureau, NY, 1970) Vol. 5, p. 303.

simulations, the dominant nonlinear saturation mechanisms are found to be the quasilinear diffusion of the ion temperature

the simulation results agree very well with the WKB and shooting

B. B. Kadomtsev and 0. P. Pogutse, in Reviews of Plasma Physics,

adiabatic, i.e., n^/n^ - e^/T^. The latter approximation is in

the spatial structure. In the nonlinear stage, a large amount

code calculations of the mode frequency, the growth rate, and

of ion energy transport has been observed. For the present

This work was supported by the United States Department of

with preliminary results using 3-D models.

Energy Contract No. EY-76-C-02-3073.

BY A LOWER HYBRID PUMP IN A Q MACHINE*

SUPPRESSION OF CURRENT DRIVEN ION CYCLOTRON WAVES

In this paper we have explained the experimental results

C . S. Liu and V. K. Tripathi Department of Physics and Astronomy University of Maryland College Park, Maryland 20742

of Lashinsky et al on the suppression of current driven ion cyclotron waves by a lower hybrid pump in a Q machine. The lower hybrid pump (of finite wave number) interacts with the cyclotron wave to produce high frequency sidebands, which in turn couple (through the dominant _E x B electron nonlinearity) to the pump to produce a low frequency ponderomotive force. When the ponderomotive potential is out of phase with the wave potential (as is the case with Q machine parameters), the wave frequency suffers a downward shift and the ion cyclotron damping is greatly enhanced, thus stabilizing the instability. This effect is shown to arise when the electron oscillatory velocity exceeds the acoustic speed. *This work supported by the Department of Energy.

occurs in EBT.

(1 + cx + ox”). The solution

E = Ep (e^ cos a)t + e ^ sin ait) and B = *e^.

As a first step we use a non-relativistic calculation in

ECRF Absorption Related to EBT*^. J. F. Pipkins and R. L. rlickok,

slab geometry and solve the equation m dv/dt = q E + g (v x B) where

Rensselaer Polytechnic Institute. — A single particle model has been

used to study ECRF absorption in a non-uniform magnetic field such as

also generates a positive potential barrier and a suppression of the mag

of this non-linear equation shows that the magnetic field gradient sets

an upper limit on the runaway energy of the resonant electrons, but the

energy fluctuates between zero and this maximum level. The resonant layer

netic field, but they also fluctuate with the runaway energy. The response

curve for this resonance resembles that of a “soft” oscillator — i.e. the

Experimentally it is observed that EBT operates at a drive frequency which

in the field direction (as it is in EBT) there will be a return flux that

upper hybrid resonance also occurs at the ring location and may be respon

and the maximum, but will be restricted to small fluctuations about the

characteristics, decreasing the density will trap the rings — i.e. the

energy of the resonant particles will no longer fluctuate between zero

the results are in qualitative agreement with experimentaT measurements.

rings- the response curve will change to resemble a “hard” oscillator.

If typical parameters for EBT are substituted into the model,

^Supported by DOE under Contract EY-76-S-02-2229.*000

corresponds to walking along the response curve.

must be included in the self-consistent field.

If this is true then varying the density

restoring force decreases with amplitude.

If the resonance zone is limited

sible for the energy absorption.

At the operating density the

For a “hard” oscillator

For appropriate length

at the ring location.

corresponds to 2

maximum.

K. Evans, Jr. and E. M. Gelbard

MAGNETIC FIELD DIFFUSION THROUGH A MAGNETIC CONDUCTING WALL*

time is T - L/R - pcaA, which is much longer if the shell is thin.

Two time scales enter in magnetic field diffusion problems where the

diffusion is through a shell of radius a and thickness A, as for example

in the poloidal field control of a tokamak plasma. The time scale of the

diffusion equation, A^B = po(9B/3t), is T * gcA^, and the eddy current rise

Applied Physics Division Argonne National Laboratory Argonne, Illinois 60439

were to experience a perturbation, an increase in its pressure, for example,

through a cylindrical shell, which could be magnetic, as well as conducting,

with copper shell tokamaks, the conducting shell also tends to retain the

penetrating and restoring the desired equilibrium. On the other hand, as

the conducting wall would tend to keep the corrected external field from

in order to represent the use of ferritic materials in wall and blanket/

plasma in its original equilibrium. The net result of these competing

A model of a plasma in such a shell is also presented. If the plasma

This paper presents numerical calculations of the field diffusion

shield design. The appearance of the two time scales is examined.

Work supported by the U. S. Department of Energy.

effects is examined.

Princeton, N. J. 08544

A. Bhattacharjee and R. L. Dewar

Plasma Physics Laboratory, Princeton University

A variational principle is given for constructing magnetohydro

Variational Principle for Magnetohydrodynamic Equilibrium States

complete for axisymmetric systems and are generalized versions of the

set of constraints. These constraints are global, may be shown to be

conventional class which conserve only the number of particles, entropy

dynamic equilibria, with arbitrary pressure, subject to a generalized

sion in the integral constraints of a complete set of weight functions

reversed field pinches may be generated. By considering the second

realistic pressure and density profiles of interest in tokamaks and

and magnetic helicity. This generalization is achieved by the inclu

variation of the generalized thermodynamic energy criteria for the

the energy are derived and are direct generalizations of the results

of the totoidal flux T. The Euler-Lagrange equations for minima of

stability of the equilibrium states to ideal MHD and a class of

Work supported by U. S. DoE Contract No. EY-76-C-02-3073.

Taylor, J. B., Phys. Rev. Letters, 13 (1974), 1139.

By a suitable choice of the basic functions,

dissipative perturbations are obtained.

f obtained by Taylor.

k(x) E Vijj(x).

ponderomotive Hamiltonian is (m = c = e = 1)

Celso Grebogi, Allan N. Kaufman, and Robert G. Littlejohn

In terms of the gyromomentum (or generalized magnetic moment) g ,

The ponderomotive Hamiltonian is derived for an electromagnetic wave of

guiding-center position X, and parallel momentum P^ , the result for the

The magnetostatic field is nonuniform and has arbitrary geometry . The pertur

PONDEROMOTIVE EFFECTS OF AN ELECTROMAGNETIC WAVE IN A NONUNIFORM MAGNETIC FIELD”

bation vector potential is represented as A(x)exp i[^(x) - mt] + c.c., where

Physics Department and Lawrence Berkeley Laboratory University of California, Berkeley, California 94720

1 2 arbitrary polarization, and with spatial variation of wavevector and amplitude ’

Work supported by the Office of Fusion Energy of the U.S. Department of Energy under contract No. W-7405-ENG-48.

tation of the physical meaning of each term is presented. The ponderomotive

expressions for the displacement of the turning point and the shifts of the

effects on the containment of particles in a mirror field are analyzed. The

turbed gyro-frequency. The equations of motion are derived, and the interpre

J.R. Cary and A.N. Kaufman, Phys. Rev. Lett. 7^, 402 (1977).

gyro-, bounce- and drift-frequencies are obtained.

J.R. Cary, Ph.D. Thesis, LBL-8185 (1979).

is the Fourier component of the perturbation:

k^(X) /2p/Q(X)’, and Q(X) is the unper

R.G. Littlejohn, paper at this meeting.

is the Bessel function of argument

II - 3Pj,/uj-.E.0(X)-k„(X)P

E (X) = k (X)-E(X);

B(X) = ik(X)xA(X),

where H (X;P,,;p)

E(X) = imA(X)

lH„(X;P,,;p)l =

K^(X;P,^;p) =

M(X)E, (X)

2iQ(X)pB,,(X)

P,,E„(X)

tHyX;F

--- J, +

x- * I:

mk^(X)

%=-°°

kf(X)

}E(X)

,3p

!l *

+CO

k ^ _ -

’

Robert G, Littlejohn

H(X,U,p) = B(X)p + y U ^ + O(e^)

A GUIDING CENTER HAMILTONIAN USING PHYSICAL VARIABLES*

of a systematic ordering scheme. The result to lowest two orders is

A guiding center Hamiltonian is rigorously derived within the framework

ables being expressed directly in terms of locally measurable quantities such as

potentials or Euler potentials is avoided, the Hamiltonian and all associated vari

where e is the ratio of gyroradius to scale length. The use of magnetic vector

where X and U are the position and parallel velocity of the guiding center and

Physics Department and Lawrence Berkeley Laboratory University of California, Berkeley, California 94720

Work supported by the Office of Fusion Energy of the U.S. Department of Energy under contract No. W-7405-ENG-48.

X = [X,H] = [X,X].^ + [X,U]^L = bU + ^ b*[pVB + U^b-Vb] + O(s^).

velocities and magnetic fields. Effects beyond lowest order in gyroradius, such as

second order drifts, are relatively easy to study. Perturbations such as small

where all field quantities are evaluated at the guiding center position. Hence

amplitude electromagnetic waves^ can be treated within a Hamiltonian framework.

A mathematical novelty of the method is the use of noncanonical coordinates

C. Grebogi, A.N. Kaufman, and R.G. Littlejohn, paper at this meeting.
R.G. Littlejohn, LBL-8917 (1979), submitted to J. Math. Phys.

U = [U,H] = [U,X].^. = - p b . V B + 0 ( e ) ;

[X,U] = b + - ^ bx(b-Vb) + O(e^)

in phase space.

[X,X].=#-b*

0(^)

Thus

”

—

Multipole Equilibria with Beta Equal to One*

R.L. SPENCER, University of Wisconsin-Madison— Hemholtz’s free boundary conformal mapping technique is used to find {3=1 sharp boundary equilibria for linear multipoles with conducting walls. It seems to be always possible to find such equilibria, indicating that there is no equilibrium beta limit in multipoles. Octupole equilibria are studied for all values of the fluid pressure. At low pressures, cusp equilibria are obtained. As the pressure is increased, the fluid closes on itself in the bridge region. At the instant of closure, the vacuum is split into separate regions, and for yet higher pressures, more parameters are needed to uniquely determine an equilibrium. Thus, there is a kind of bifurcation when the pressure exceeds the value for closure in the bridge. Although they are unstable, these sharp boundary {3=1 equilibria are useful because they provide an opposite extreme from the zero beta equilibria obtained from vacuum flux plots. They also provide information on the parameter ranges for which high beta equilibria exist, and on the number of parameters required to determine equilibrium uniquely.

*Work supported by USDOE.

LH-Quasimode Parametric Excitation at the Edge of a T o k a m a k Plasma.? E. VILLALON,?? MIT-Parametric excitations via quasimode decay of a lower-hybrid pump w a v e are shown to be strong near the edge of the piasma. Catenations in the s h a d o w of the limiters for Alcator A heating experiment wit! be presented. This region is characterized by a big drop in the eiectron temperature which makes pump and sideband be c o m e strongly coupled. The linear theory predicts that the rf-power is mainly distributed between the w a v e numbers

(e.g. 7 or 8) is significant, and may lead to a shift of the initial n power spectrum toward higher n’s. These wav e s m a y transfer energy to the electrons, through tinear Landau damping, as they get inside the plasma. A nonlinear analysis of the steady-state evolution has also been carried out, showing that the most powerful sideband fields are created in this region of the plasma.

(ET78-S-02-4682). ??Supported by Grant PFP) (MEC, Spain).

^ Work supported by U. S. Department of Energy Contract

’ ck^/ta). The excitation of fietds with high values of

= 1 to 3 (where

New York University

REVERSED.FIELD PLASMAS’

New York, New York 10012

William Grossmann and Eliezer Hameiri

Courant Institute of Mathematical Sciences

A numerical simulation of adiabatic compression in various

RF configuration is produced numerically by tying the field

with and without toroidol field has been investigated in two

lines at the two ends of the device. Specifically, RF plasma

using the “1-1/2 D” methods originally proposed by Grad”**. The

reversed field (RF) plasma configurations has been carried out

stable plasmas remain stable; conversely interchange unstable plasmas remain unstable. Numerical results for typical experi

Simple arguments show that under adiabatic evolution interchange

this contradicts previous predictions based on “ID” simulations.

plasma is accompanied by a strong axial contraction, the axial

require the imposition of different boundary conditions at the

liners the initial elongation is increased during compression.

In both cases the plasma beta is increased during compression;

mental plasmas will be shown which illustrate the interesting

kinds of current experiments, theta-pinches and liners, which

the plasma tends to move to a bicycle tire shape whereas in

H. Grad, P.N. Hu, D.C. Stevens, PNAS, Vol. 72, 10, pp. 3789-

Numerical results show evidence of similarity like solutions.

contraction being strongest for the theta-pinch case where

’ Work supported by U.S. DOE Contract No. EY-76-C-02-3077.

For both physical cases the radial compression of the

features of plasma compression.

3793 (1975).

wall.

ABSTRACT

V. S. Chan, S. C. Chiu, and T. Ohkawa

IN THE PRESENCE OF ANOMALOUS TRANSPORT

ELECTRON HEATING BY LONER HYBRID WAVES

General Atomic Company San Diego, California 92138

We consider the effect of anomalous transport on electron heating by

assume stochastic magnetic fluctuations as the mechanism for anomalous trans

lower hybrid waves in a low-to-medium density plasma. For definiteness, we

the electron distribution increases anomalous losses resulting in less power

regime, anomalous transport can eventually reduce the RF damping rate. For

enhance the electron heating rate with a concomitant increase in anomalous

examined both in the weak RF and strong RF limits. Physical pictures are

losses for example by direct heat convection; and (2) the modification of

port. The modification in rhe quasistationary electron distribution is

available for heating. The relevance of present study to current lower

(1) heating is shifted toward the plasma periphery thus increasing heat

presented to explain the distinction in the two cases. In the weak RF

high power heating experiments, anomalous transport can significantly

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

loss rate. This can reduce the efficiency of RF heating in two ways:

hybrid heating experiments will be discussed.

Project Agreement No. 38.

POSTER SESSION

As a r e s ul t of this o ve r

P revious mod el s *

C. W h i t s o n and K. T. Tsang

P. J. Cat to and M. N. Ros en b lu t h

FINITE 6 T R A P P E D ELE CT R ON INSTABILITIES*^

Science A pp l i c a t i o n s , Inc., Boulder, Col or a do 80302

Oak Ridge National L ab or a to r y, Oak Ridge, T e n n e s s e e 37830

The effects of t r a pp e d e l e ct r on s on d ri f t - A l f v ^ n waves has been stu di e d have failed to r e c og n iz e that by a num be r of authors. ’ the trapp e d e lectrons becau s e of t heir bounce m o t i o n are unable to respond to the p e r tu r be d parallel m a g n e t i c v e c to r potential A,. . sight, the coupled radial e i g e n v a l u e equ at i on s for A„ and the e le c t r o s t a t i c potential $ a n d /o r the p e r tu r be d d i s t r i b u t i o n fun ct i on for the electrons e x h i b i t a non-physical s i n g u l a r i t y at the m o d e rational surface.

In o rder to rem ov e the p r e c e d i n g dif fi c ul t y, a m or e careful der iv a ti o n of the per tu r be d trapped e l e ct r on d i s t r i b u t i o n fun ct i on f^. is p e r fo r me d by s e p ar a ti n g f^. into portions even and odd in the parallel velocity. res ul t in g radial e i g e n v a l u e e q u at i on s for $ and A„ are well behaved at the rational surface. and unt ra p pe d ele ct r on s can be ignored, this set of radial differential e q u a P r e l i m in a ry results tions is solved n um e r i c a l l y to d e t e r m i n e the eigenvalue. show that the e l e c t r o s t a t i c d r i ft branch of the trapped e l e ct r on mode is w e a k l y aff ec t ed by finite 8, w h i ch in m o s t cases is a d es t a b i l i z i n g influence. In addition, the trapped e l e ct r on tea ri n g mode (§ odd, A,, even about the rational surface) is found to have a growth rate small e r than the twisting m ode (§ even, A„ odd a bout the rational surface).

For the s m a ll e r c o l l i s i o n a l i t i e s a vel oc i ty space b oundary layer exists between the trapp e d and u n t r a p p e d e l e ct r on d is tr i b u t i o n f un ctions so that a Krook model is inappropriate. a ngle sca tt e ri n g c ol lision o p e r a t o r to treat this bou nd a ry layer.

W. M. Tang, C. S. Liu, M. N. R os en bluth, P. J. Catto, and J. D. Callen Nucl. Fusion 1(5, 191 ( 1976); and K. T. Tsang, J, C. Whitson, J. D. Callen, P. J. Catto, and J. Smith, Phys. Rev. Lett. 4T, 557 (1978). L. Chen, P. H. R u t he r fo r d and W. M. Tang, Phys. Rev. Lett. 39, 460 (1977); and S. M. Mahajan, U n i v e r s i t y of Texas, FRCR =179, A u g us t (1978).

tWork sup po r te d by U. S. D e p a r t m e n t of Energy under c o n tr a ct E Y - 7 6 - 0 3 - 1 0 1 8 at Science Applic at i on s, Inc. and under con tr a ct with Union Carbi d e C o r p o r at i on at Oak Ridge National Laboratory. 1.

In the limit in w h i c h the b o u nd a ry layer between the trapped

A model will be p resented whi ch employs a pitch

The

Princeton, New Jersey 08544

Plasma Physics Laboratory, Princeton University

Stability of High 3eta Tokamaks to Ballooning Modes*

D. A. Monticello, H. R. Strauss,+ W. Park, R. B. White,

S. C. Jardin, M. S. Chance, A. M. M. Todd, and A. H. Glasser

*Work supported by U. S. DoE Contract No. EY-76-C-02-3073.

^Permanent address: Fusion Research Center, University

particular, on shear, which can be strongly

the details of the equilibrium, and in

a second stable regime for high beta. The

range of unstable beta values depends on

Ballooning modes are found to possess

of Texas, Austin, Texas.

stabilizing.

P l a s m a Fusion Center

C a m b r i d g e , Massachusetts 0 2 1 3 9

V l a d i m i r K t a p c h e v a n d A b h a y R A H )

Massachusetts Institute of T e c h n o l o g y

A Nonlinear M o d ? B e l o w the Electron P t a s m a Frequenc y

a n d the p o n d e r o m o t i v e a n d ambipolar potentials are balanced to produce c h a r g e neutrality. T h e

potential w d l lead to a n o m a l o u s propagation in plasmas. T h e effect of the ions has been neglected

p r o b l e m . A large amplitude, high frequency, spatially m o d u l a t e d w a v e E(x)cos(raf - R.x) l a u n c h e d

wav e-p a r t i c l e interaction w e find the nonlinear dispersion relation to all orders in the electric field

T h e existence of the nonlinear m o d e implies that the velocity d e p e n d e n t p o n d e r o m o t i v e

W e find the exact V l a s o v distribution function for a one dimensiona] b o u n d a r y v a l u e

nonlinear m o d e , other than the ordinary p l a s m a wave, exists. Its range of frequencies is

n o r m a l m o d e s w e considered are orders of m a g n i t u d e abo v e the ion acoustic w a v e .

“Work supported by National Science Foundation (Grant ENG77-00340).

b y an external source cha nges significantly the p l a s m a equilibrium.

the frequency of the L a n g m u i r wave.

a m p l i t u d e a n d second order in

B y a s s u m i n g n o n r e s o n a n t

A b o v e a certain critical

an u n d a m p e d

Abstract.

*/^) of

a n d

plasmas with D-

and Plasma Fusion Center

J.H. Schultz , L. Bromberg and D.R. Cohn

Ignited Plasmas with Advanced Fuel Cycles

Francis Bitter National Magnet Laboratory

and catalized-D cycles. Present confinement

Equilibrium and Thermal Stability Properties of

Massachusetts Institute of Technology, Cambridge, Ma.

T^/i^«l which facilitates ion-electron decoupling. This

studies indicate that these plasmas may be characterized by

Ignition requirements are determined self-consistently for

decoupling would be enhanced by anomalous slowing down of the

anomalous slowing down has been recently suggested by Molvig.^

fusion products; in this case all of the energy of the fusion products is transferred to the plasma ions. The possibility of

Westinghouse Co., Pittsburg Penn. Kim Molvig, Ignition Experiment Design Meeting, M.I.T. Cambridge

slowing down. Thermal stability properties are studied using a simplified Fokker-Planck model of the fusion products. It is

at the temperature that results in the minimum size or in the maximum power density.

The anomalous slowing down results in a factor of two increase in

Work supported by U.S.D.O.E. Contract No. EG-77-S-02-4183.A002

slowing down. Similarly the minor radius of the ignited plasma

can be reduced significantly (— 30%) in the case of anomalous

the fusion power density relative to the case with classical

found that the ratio between i

and T … is

(Jan 1979)

runaway

^global

global

< o 5

e l

and neoclassical for

and Plasma Fusion Center

Features of Ignited Operation’

are used to project the features of

L. Bromberg, D.R. Cohn and J. Fisher

Francis Bitter National Magnet Laboratory

Massachusetts Institute of Technology, Cambridge Ma.

Regimes of ignited operation in D-T plasmas are explored

of electron-ion decoupling is calculated as a function of the

ignited operation in recent next step tokamak reactor designs.

in terms of a general requirement on nr , ni., T and T.. The

conditions under which T. >T and T. <T are found. The amount

ion temperature and the ratio of T /i. . Thermal stability **

characteristics are determined in the context of the four dimensional ignition requirement. An empirical scaling for

very short (-T ) until ion temperatures approaching 50 keV are reached . The effectiveness of gas control and compression-

‘Supported by U.S. D.O.E. Contract No. EG-77-S-02-4183.AOO2 * C.S. Draper Laboratory

L. Bromberg, D.R. Cohn and J. Fisher , MIT Plasma Fusion Center

temperature can significantly reduce the value of nr at ignition

J.F. Clarke, Ignition Experiment Design Meeting, MIT Cambridge

and stability properties. Operation at T. >T and at high ion

and therefore leads to a reduction in the beam energy required

m full size full density startup . Thermal runaway times are

decompression as means of plasma control are discussed.

Devices with similar values of

Report RR-79-3 (March 1979)

have similar equilibrium

(Jan 1979)

ABSTRACT

M. S. Chu and C. Chu

ELECTRON TRANSPORT IN RANDOM MAGNETIC FIELDS

General Atomic Company San Diego, California 92138

Electron transport in a random magnetic field has been studied taking

into account the effect of the perpendicular wavelength d of the perturbing

ticle dissociates itself from the field line before it diffuses a distance d,

dissociate itself after it diffuses a distance d, the diffusion law given

by Kadomtsev and Pogutse^ is obtained. In a high-g plasma, such that the

the diffusion coefficient is given by the Rechester-Rosenbluth^ formula.

fields. If the perturbing fields have low amplitudes and the typical par

Whereas in large amplitude stochastic fields, the typical particle can

random magnetic field results from excitation of magnetostatic modes,^

the relationship of the diffusion law to Ohkawa’s formula* ** (which fits

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

^C. Chu, M. S. Chu, and T. Ohkawa, Phys. Rev. Lett. 41 (1978) 853.

^B. B. Kadomtsev and 0. P. Pogutse, IAEA Innsbruck (1978).

*A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett.

**T. Ohkawa, Phys. Lett. t?7A (1978) 35.

Alcator scaling) is discussed.

Project Agreement No. 38.

(1978) 38.

ABSTRACT

COUPLING OF DRIFT MODES IN A TORUS

R. E. Waltz, W. Pfeiffer, and R. R. Dominguez

perturbed electrostatic potential written as $ = Z d)-(x)

General Atomic Company San Diego, California 92138

The coupling of electrostatic drift modes due to ion, magnetic curva

we reduce the general system of coupled differential equations for the

poloidal harmonics (j)^(x) to a single equation by considering a class of

ture drift in a torus is examined analytically and numerically. With the

half of the shear damping is nullified. However, when nonadiabatic electrons

are considered, numerical solutions show that, discounting trapped electrons,

mode coupling is insufficient to destabilize electron drift modes within the

Numerical and perturbative analyses with adiabatic electrons show that about

mode (which is least stable at outer radii with strong shear: s = rdUnq/dr

practical ranges of shear, temperature gradients, and current drive for a

In contrast to previous work we emphasize that only the inward ballooning

from the q = m/i rational surface and A is the spacing of the surfaces.

Solutions with K = 0(7r) correspond* to outward (inward) ballooning modes.

> 1/2) is physically realizable within the constraints cf the theory.

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

(x ** jA) 0 < K < 2?r. x is the radial distance

Project Agreement No. 38.

solutions

tokanak.

= e ^ ^

^ ^

J = - C O T J

D. Lortz, J. Nuhrenberg

F e d e r a l R e p u b l i c of G e r m a n y

ballooning Stable Profiles in Circular Tokamaks

c h a r a c t e r i z e d b y a b a l l o o n i n g u n s t a b l e b a n d in the p l a n e

R e c ently, b a l l o o n i n g i n s t a b i l i t i e s , as o b t a i n e d f r o m the

p r o p e r t i e s of s e l f - c o n s i s t e n t a x i s y m m e t r i c e q u i l i b r i a are

b a l l o o n i n g i n s t a b i l i t y e q u a t i o n (ij, h a v e gained m u c h i n t e r

est a n d it has b e e n s hown j[2j t h a t the b a l l o o n i n g s t a b i l i t y

Max-Planck-Institut fur Plasmaphysik, 8046 Garching

are a p p r o x i m a t e l y given (to w i t h i n 10 % accuracy) by Bp - 0.45 “V S* , Bp = V ? , in the range 5 < S < 40. W e hav e

to determine, for g iven p r e s s u r e p r o f i l e s , b a l l o o n i n g m a r g i n a l

n o w w r i t t e n a code w h i c h e v a l u a t e s the b a l l o o n i n g i n s t a b i l i t y

a s s o c i a t i o n b e t w e e n the M a x - P l a n c k - I n s t i t u t fur P l a s m a p h y s i k

“This w o r k w a s p e r f o r m e d u n d e r the terms of the a g r e e m e n t on

a p p l i e d to n o n l i n e a r e q u i l i b r i a w i t h c i r c u l a r c r o s s - s e c t i o n

c r i t e r i o n o v e r the w h o l e p l a s m a c r o s s - s e c t i o n of any g i v e n

a x i s y m m e t r i c e q u i l i b r i u m . In p a r t i c u l a r , this code w i l l be

[2j Lo r t z , D . , N u h r e n b e r g , J . , s u b m i t t e d for p u b l i c a t i o n

p o l o i d a l 6 vs. shear. D e f i n i n g Bp = 1- j[i(^/J^J(0),

[jj C o n n o r , J . , Hastie,R. , T a y l o r , J . B . ,

the b o u n d a r i e s of the u n s t a b l e b a n d

p r o f i l e s of the t o r o i d a l c u r r e n t J.

Phys. Rev. Lett. 40 (1978) 396

a n d E U R A T O M . ”

Princeton, NJ 08544

E.J. Caramana and F.W. Perkins

Plasma Physics Laboratory, Princeton University

Magnetic Field Profiles in the Reversed Field Pinch

A Numerical Study of the Effect of Impurities on Plasma and

We have developed a one-dimensional MHD simulation code including both

effects may be separated since impurities radiate energy out of the plasma

plasma transport and impurity effects that follows the time evolution of a

motion of plasma across magnetic flux surfaces. The full equations are thus

Two codes were developed and linked together to solve the full problem. The

causing an adiabatic change to a new equilibrium but do hot contribute to the

Reversed Field Pinch (RFP) through a series of hydrostatic equilibria. These

split into two sets, one which contains plasma transport and another radiation.

Work jointly supported by U.S. DoE Contract No. EY-76-C-02-3073 and U.S. AFOSR Contract No. F 49620-76-C-0005.

ties at present operating densities and impurity levels. The strong dependence

separate fluid, that contain energy loss terms due to radiation. When written

transport code has been described earlier.^* The radiation code is essentially

and reduce these gradients. In addition, we find that the plasma is usually

conductivity is not large enough to strongly couple neighboring plasma radii

those of ZT-S and ZT-40 at Los Alamos. These results show that the electron

in a Lagrangian coordinate system based on the poloidal flux, these become a

of radiation loss on density for a fixed relative impurity concentration is

through a radiation barrier in only one region. Classical electron thermal

temperature in the ZT-S experiment is radiation limited due to oxygen impuri

a set of ideal MHD equations, with each impurity charge state treated as a

also shown. Using classical transport coefficients for ZT-40, we find that

large electron temperature gradients can be created when the plasma burns

Results are presented for several RFP operating parameters, including

*^E.J. Caramana and F.W. Perkins, Bull. Am. Phys. Soc. _23, 811 (1978).

Suydam unstable in the outer third of the discharge.

simple set of ordinary differential equations.

IN A TANDEM MIRROR

ION CYCLOTRON RESONANCE HEATING

J. E. Howard and J. Kesner University of Wisconsin

First and second harmonic ion cyclotron resonance heating is being

studied as a possible alternative to neutral beam heating of the end plugs

of a tandem mirror. As a simple model we consider midplane and off-midplane

which is evaluated asymptotically in cases of interest. Various applications

to the local magnetic field. This analysis differs from previous theoretical

the single particle equations of motion for an RF wave travelling obliquely

treatments in that arbitrary harmonics and large doppler shifts are allowed.

heating in a parabolic well. The energy gain per pass is calculated from

The resulting Av^ is given in terms of a generalized Airy function,

to the Phaedrus experiment will be discussed.

Eliezer Hameiri

New York, N.Y. 10012

The Continuous Spectrum and Ballooning Modes

The appearance of a continuous spectrum in the ideal MHD

Alfven and Cusp continue arise from the presence of pressure

New York University Courant Institute of Mathematical Sciences

equations is related to the non-ellipticity of the time-inde pendent equations. Every family of characteristic surfaces gives rise to part of the continuous spectrum. The familiar

librium state involves mass flow. Ballooning modes are related to the existence of a second family of magnetic flux surfaces.

minimization of 6W and the use of eikonal forms but by direct derivation from the differential equations, in a way similar to

for closed field line systems. It will be demonstrated that in a mirror configuration, the outcome is equivalent to the modes

approach based on these ideas will help resolve questions con cerning boundary conditions for ballooning modes in sheared

This approach enables one to treat ballooning modes not through

Bernstein, Frieman, Kruskal and Kulsrud, Proc. Roy. Soc. A,

surfaces which are characteristic surfaces even when the equi

The equations determining ballooning modes will be derived

Work supported by U.S. DOE Contract No. EY-76-C-02-3077.

the traditional treatment of the Alfven continuum.

obtained in Ref. 1 in the limit m

°°. We anticipate that an

224, p. 17 (1958).

systems.

1-3

4 5

WKB analyses are

Princeton, NJ 08544

Liu Chen and C.Z. Cheng

Drift-Wave Eigenmodes in Toroidal Plasmas

Plasma Physics Laboratory, Princeton University

modes are studied using the ballooning-mode formalism.

rates. Both analytical and numerical results will be presented.

Effects of toroidal couplings on the shear damping of drift-wave eigen-

then carried out for the drift-ballooning eigenmode equation ’ in the cold-

ion limit. It is found that two types of eigenmodes exist. One is slab-like

of the eigenmodes, toroidicity can either enhance or reduce the shear-damping

and the other is toroidicity—induced. Depending on the parameters and the type

*^J.W. Connor, R.J. Hastie, and J.B. Taylor, Culham Rept. CLM-P537 (1978).

K.W. Hesketh, R.J. Hastie, and J^B. Taylor, Workshop on Drift Waves (Trieste, Italy, 1978).

^D.I. Choi, W. Horton, and R. Estes, University of Texas Rept. FRCR-184 (1978).

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

^Y.C. Lee and J.W. Van Dam, UCLA Rept. PPG-337 (1978).

*^*A. Glasser, et al. (to be published).

(Vlasov ions

in collisionless

From hybrid-kinetic theory

equations by an expansion in

and guiding-center electrons)^ an

eigenvalue equation for electro-magnetic perturbations with M

6-pinches with anisotropic ion energy was recently derived.^ This equation is

supplemented by appropriate boundary conditions for the case when the plasma is

presently reduced^ to two coupled, ordinary second order linear differential

CHERENKOV RESONANCE AS AN FLR EFFECT OK THE ALFVEN-ION-CYCLOTRON MODE*

J. Goedert Universidade Federal da Paraiba, Paraiba, Brazil

J. P. Mondt University of California Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

Conclusively, Cherenkov resonance as an FLR effect influences stability of the

Alfven-Ion-Cyclotron wave and might change the amount of anamolous transport

parallel to the instantaneous magnetic field. This electric field is annihilated by

electric field disturbance parallel to the direction of wave propagation. For high-P

resonance is due to the existence of a radial ion pressure gradient: the unperturbed

surrounded by a cylindrical, perfectly conducting wall. For weak inhomogeneities a

the rapid electron motions parallel, to the magnetic field, thereby inducing an

local dispersion equation is obtained that can be solved using standard numerical

electric field associated with this gradient induces an electric field component

correction terms contain Cherenkov resonances absent in the homogeneous case. This

J. Goedert and J. P. Mondt, to be published in J. Plasma Phys. (GB).
D. A. D’Ippolito and R. C. Davidson, Phys. Fluids _18^ 1507 (1975).

*Work performed under the auspices of the U. S. Department of Energy.

the phase-vleocity of the wave is comparable to the thermal

R. C. Davidson and J. M. Ogden, Phys. Fluids _1_8, 1045 (1975).

global and local analysis, the leading order

(Til ** Tin) associated with it.^

Both within the context of

These equations are

Ph.D. Dissertation,

Netherlands, 1977.

J. P. Mondt,

ion velocity.

thermal ion

University,

gyroradius.

procedures.

Eindhoven,

Eindhoven

The

J. L. Shohet

THE MICROWAVE SPHEROMAK*

The University of Wisconsin, Madison, Wisconsin 53706

The device is set up in a cylindrical microwave cavity, along the axis

A recent design proposed to construct a “spherical” tokamak by inducing

of which a uniform d.c. magnetic field is imposed. The required toroidal current is presumed to be driven by r.f. fields that are oriented in the azimuthal direction. This can be done by driving the cavity in a mode of the form T E ^ p , where a,m,n 7^ 0. The electric fields of such a mode are of the form:

a ring current perpendicular to a uniform d.c. magnetic field, producing a separatrix along the axis of the configuration and closed toroidal magnetic surfaces, i.e., the Spheromak.^ It is the purpose of this paper to propose a scheme to generate this configuration with the use of electromagnetic current drive.

of E is a standing wave which can be broken into two travelling waves. By suitably tuning the cavity, a single travelling wave can be excited. If a » 1, then conditions for rf current d r i v e d can be satisfied. A similar configuration may be set up in a spherical cavity.

4J. R. Hamann, A. J. Hatch and J. L. Shohet, IEEE Transactions on Plasma Science, PS-2, 241 (1974).

^M. N. Bussac, H. P. Furth, M. Okabayashi, M. N. Rosenbluth and A. M. Todd, Proc. IAEA Innsbruck Meeting (1978), paper X-l.

*Mork supported by tqe National Science Foundation under Grant ENG 77-14820.

Note that if a 7^ 0, the azimuthal variation of the azimuthal component

Plasma rings carrying such currents have previously been excited.^

3c. F. F. Karney and N. J. Fisch, PPPL MATT Report 1506 (1979).

2,N. J. Fisch, Phys. Rev. Lett. 4j, 373 (1978).

Eg = *3^(k^r) cos ae sin k^z

si” 4 S’” ^

Er = ’

T = °

(2)

(1)

(3)

ABSTRACT

INITIAL RESULTS OF TANDEM MIRROR TRANSPORT CALCULATIONS

Initial runs have been made with one central cell ion species,

A code previously used for mirror radial buildup s t u d i e s ^ ) has

James M. Gilmore, Department of Nuclear Engineering, U n i v e r s i t y o f Wisconsin, Madison, Wisconsin, and Ronald H. Cohen, Lawrence Livermore Laboratory, University of California, Livermore, California 94550

been modified to study tandem-mirror radial transport. The revised code includes modified endloss terms, extra (non-ambipolar) transport coefficients to describe enhanced diffusion caused by the quadrupole field, a procedure to solve for the radial potential profiles in the solenoid and plugs from requirements of charge neutrality, and bounce-averaged (over plugs and solenoid) equations for electrons. Atomic physics and finite gyroradius effects were included in the original code and have been retained. deuterium, a constant (in radius and time) ion source, and a deuterium plug with fixed density and temperature profiles. In these runs, quadrupole (21 field effects were treated by using order-of-magnitude approximations^ ^ to resonant i o n ^ ) and neoclassical electron transport coefficients.

Trial runs have been made with no transport and various combinations of classical and quadrupole enhanced electron and ion particle and energy transport, using TMX parameters. The results obtained indicate that resonant transport depresses the mid-solenoid density and raises the ion temperature, by a few percent, when compared with the results obtained using solely classical transport. A substantially higher (20%) mid-solenoid density and lower (40%) ion temperature are obtained in runs In with all transport coefficients as compared to runs with no transport. runs with particle transport but no energy transport, the equilibrium density profiles are lower than those obtained with no transport, and the depression is due mostly to ion resonant transport.

R. P; Fries, Lawrence Livermore Lab. CTR Annual Report UCRL-50002-96, p!08 (1976
R. H. Cohen, Comments Plasma Phys. Cont. Fusion 4, No. 5 (1979).
D. D. Ryutov and G. V. Stupakov, Dokl. Akad. Nauk SSSR 2 4 0 , 1086 (1978).

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contracts W-7405-ENG-48 and ET-78-5-02-4636.

Lawrence Livermore Laboratory

and energies are azimuthal averages.

A.A. M i r i n , R.E. Ccnen, M.E. Eensink and J. Killeen

^----‘/rr^pY RESULTS OF 0. TANDEM MIRROR TRFT’SFOFT CODE”

An arbitrary number of central cell ion species described by density

A radial transport code for tandem mirror devices has been developed.

profiles n^(r,t) and temperature profiles T^(r,t), plug ions of density

Tg(r,t) are considered. The quantity r is the radius of a magnetic flux

surface at the midplane of the central s.olenoid, and the above densities

n (r,t) and energy E (r,t), and electrons of density n (r,t) and temperature

iterative finite difference algorithm. Spatial gradients are centered, with

resonant transport models are included. Axial loss rates are computed using

exchange, charge exchange, ionization, end-loss and acceleration due to the

central solenoid and plugs consistent with charge neutrality are determined.

radial electric field are m o d e l e d . ’ Empirical, classical, neoclassical and

*Work-performed “under the auspices of the U.S. Department of Energy by the

a general Pastukhov formula. Radial ambipolar potential profiles in the

the exception of the convection term, which uses upwind differencing.

Preliminary results of applications to the Tandem Mirror Experiment

Particle and energy conservation up to roundoff error is maintained.

Lawrence Livermore laboratory under, contract number W-7205-ENG-18.

Particle diffusion, heat conduction, heat convection, energy

The transport equations are time-advanced using an implicit,

(TMX) are presented.

ABSTRACT

TRANSPORT EQUATIONS FOR TANDEM MIRROR MACHINES

We have developed an analytic approximation to the resonant plateau

-We have derived a set of one-dimensional transport equations for /

Ronald H. Cohen, Marvin E. Rensink and James H. Foote Lawrence Livermore Laboratory, University of California Livermore, California 94550

tandem mirror machines. This set of equations includes endloss and classical as well as quadrupole-field enhanced radial transport, and correctly describes transients due to time-varying magnetic and electric fields. The equations can be derived from either the drift-kinetic equation or the Boltzmann equation.

diffusion coefficients of Ryutov and S t u p a k o v ^ ^ . neglecting the azimuthal VB drift compared to the azimuthal E x B drift, and by adopting a semi-empirical model for the pitch-angle dependence of the radial displacement per bounce (obtained by fitting numerical drift calculations). The resulting expression is used to obtain numerical r e s u l t s ^ and analytic estimates for resonant transport in TMX, a scaled’-up tandem mirror experiment (MFTF-B), and tandem mirror reactors.

*Mork performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48.

D. D. Ryutov and G. V. Stupakov, Dokl. Akad. Nauk SSSR 2 4 0 , 1086 (1978).
A. A. Mirin, R. H. Cohen, M. E. Rensink and J. Killeen, Paper at this

It is derived by

meeting.

ABSTRACT

PARTICLE MOTION IN A CYCLOTRON RESONANT FIELD

The time averaged equations for a particle’s motion in a mirror field

Y. Matsuda and H. L. Berk Lawrence Livermore Laboratory, University of California Livermore, California 94550

with electric field r.f. present near the central cyclotron frequency (or its harmonic) is derived. The averaging method is an extension of the technique used by Aamodt and B o d n e r ^ who studied such particle motion in a uniform magnetic field. We obtain four coupled nonlinear equations for the axial position, axial velocity, perpendicular velocity and the particle’s relative gyrophase with respect to the wave phase. When the rate of change of relative gyrophase, M - M^(s), is larger than the bounce period, Mg, we derive a ponderomotive force for axial motion. the opposite limit, M - a^(s) < Mg, we obtain an analytic description of superadiabatic motion. energy particles that are trapped at the center of the mirror even in the presence of a repelling ambipolar well. This may have a stabilizing effect on the amplitude of loss cone modes. The intermediate limit, which is more difficult to analyze analytically, is the regime of stochastic motion. Numerical solutions will be compared to the analytic theory.

“Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG-48.”

lAamodt and Bodner, Phys. Fluids 12, 1971 (1969)

In both limits we show that there exists low

L n

/TV

2 B 2 9

Madison, Wisconin

= (c + 7 c + 4 ^ ^ )/2, c =

It is found that the lower hybrid

INTERACTION OF LOWER HYBRID FIELDS WITH

Ker-Chung Shaing, Robert W. Conn, and Jay Kesner

THE DRIFT-CYCLOTRON LOSS-CONE MIRROR INSTABILITY

Department of Nuclear Engineering University of Wisconsin

The effect of an externally applied large amplitude,spatially uniform electric field at the lower hybrid frequency on the drift-cyclotron loss- cone mirror instability is investigated. field has a stabilizing effect on the drift-cyclotron loss-cone mode if

u < ^ < M*, where uj, ^ is the lower hybrid frequency, m is the applied wave Ln frequency, and otherwise, the lower-hybrid field has a destabilizing effect on the drift- cyclotron loss-cone mode.

c / ^ ^ k , a n d c =

i ^

Abstract

and W. Horton

Austin, Texas 78712

D. Biskamp, R. Estes

Fusion Research Center

Three Dimensional Fluid

Simulations of Drift Waves

The University of Texas at Austin

The ion pressure gradient driven drift instability

has recently been invoked as a possible explanation for

the rather high frequency, large amplitude density fluctuations observed ih strongly beam heated PLT plasmas. A 3D code is developed to investigate the nonlinear behavior of this instability. A simple set of fluid equations is

used for the electrostatic potential, the parallel ion velocity and the ion pressure, neglecting quasilinear relaxation of the density. Besides giving saturation levels of this type of drift instability, these model

basic nonlinear dynamics of electrostatic drift waves, in particular the coupling of parallel phase-velocities

computations are of more general value to understand the

This work is supported by the U.S. Department of Energy

generating convective cells.

Contract DE-AC05-79ET53036.

F. Brunei,^ J. N. Leboeuf, T. Tajima, and J. M. Dawson

MAGNETOHYDRODYNAMIC PARTICLE CODE WITH THE LAX-MENDROFF METHOD*

A significant improvement of the particle MUD code^ is achieved by im

Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024

plementing the Lax-Mendroff method for advnacing the magnetic field in a way analogous to Makino et a l .2 Sharper mode spectra than are obtained by the Lax method have been observed with the present code, as the numerical diffu sion of the magnetic fields is reduced to the order of ( k A ) \ Thanks to the low magnetic diffusion with this algorithm along with the particle nature of the code, we are able to simulate problems with sharp plasma boundaries and large density ratio. The Adam-Bashferth method (exptrapolated leapfrog method), accurate also up to ( k A ) \ has been tried and compared to the Lax- Mendroff code: we find tha Adam-Bashferth code is more susceptible to numer ical difficulties in the case of handling sharp boundaries. Applications of the 2-1/2 D code have been started with studies of flute and ballooning instabilities and the area wave propagation in a high P plasma column. For a sharp boundary plasma in a gravitational field with finite P, we see a critical ratio of k„/k, (k„ parallel to B^) below which the ballooning mode is unstable.3

ij. N. Leboeuf, T. Tajima, and J. M. Dawson, to be published in J. Compt. Phys. ^M. Makino and T. Kamimura, private communication. 3p. L. Pritchett, C. C. Mu, and J. M. Dawson, Phys. Fluids 2^, 1543 (1978).

*Mork supported by USDOE and NSF. Supported by a D.G.E.S. Fellowship.

J. T. Woo and K. A. Connor

LOWER HYBRID HEATING IN TANDEM MIRROR GEOMETRY

The application of lower hybrid range of requencies (LHRF) to

Rensselaer Polytechnic Institute Troy, New York 12181

meters required for this application. The condition for efficient

for end plugging of a fusion plasma. We have considered the wave para

because it allows a significant relaxation of the ion energy required

tandem mirror geometry for direct heating of electrons, is of interest

absorption of wave energy by electron Landau damping is consistent with

in M-k space for effective application of LHRF waves is technologically

much more attainable than either ICRH supplementary heating of ions in

sion layer. By proper choice of wave frequency, electrons at the loss

which the potential barrier is amplified can therefore be very energy

both the accessibility condition and the avoidance of the mode conver

the end plugs or the application of ECRH that are presently being con

boundary can be selectively heated and driven out. This process by

For parameters typical of tandem mirror reactor, the window

efficient.

sidered.

Simulation of Multi Impurity Species Transport in Tokamaks*

We continue to upgrade our numerical simulation of transport of

E. C. Crume Jr. and D. E. Arnurius Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

multiple species of impurities in tokamaks.^ Our present emphasis is to develop a calculations! module treating impurity transport and atomic physics that can be utilized generally in tokamak transport simulation codes. As part of our review of neoclassical and classical ion diffusion coefficients we have developed some approximations for Pfirsch-Schluter regime coefficients that significantly reduce the complexity of the expressions while maintaining high accuracy. We present some comparisons, both of individual coefficients and of complete simulations, in which exact and approximate forms have been used.

of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

T. Amano and E. C. Crume, “Simulation of Multispecies Impurity

“Research sponsored by the Office of Fusion Energy, U. S. Department

Transport in Tokamaks,” ORNL/TM-&363 (June 1973)-

ABSTRACT

T. S. Wang

ON A HIGH-SPEED ARRAY PROCESSOR

REAL-TIME MHD COMPUTATIONS TOR NONCIRCULAR TOKAMAKS

General Atomic Company San Diego, California 92138

One of the most important tasks in high-8 noncircular tokamak experi

cross section throughout a discharge. Sequences of time-resolved MHD

ments, such as Doublet III, is to shape and maintain a desirable plasma

satisfy the stringent requirement on the computation time, we have interfaced

be presented. Initial timing comparison between A,17600, CRAY*-1, and DEC-10-

data corresponding to a single instant within a particular plasma shot. To

a high-speed array processor AP-190L capable of performing several million

MHD equilibrium code takes approximately 0.43 sec/step on the MFECC A-7600

of these noncircular plasma experiments. It would be especially useful if

equilibrium analyses, produced by fitting experimentally-measured magnetic

machine and typically, 100 steps are needed to produce one set of analyzed

data, greatly enhance prospects for the successful operation and diagnosis

system at General Atomic based on the USC DEC System-10 computer. The GA

The computational processing involved is significant: the General Atomic

free boundary MHD equilibrium code has also been converted to run on the

floating-point operations per second with the existing data acquisition

analyzed results returned within the 3-10 minute interval between shots.

Detailed structures of both the computer system and the MHD code will

the experimental data could be processed on a real-time basis and the

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

AP-190L will be presented as well.

Project Agreement No. 38.

DEC-10-AP-190L system.

P. Satyanarayana and P. Bakshi

Chestnut Hill, Massachusetts 02167

Department of Physics, Boston College

EFFECTS OF SHEAR ON DRIFT-CYCLOTRON INSTABILITY

essentially modifies the potential in the Weber equation.^

We have studied the effects of magnetic shear on the Drift-

The growth rates and the critical shear needed to trigger the

a sheared magnetic field. The main effect of including the exact

Cyclotron (DC) Instability by including the exact particle orbits in

stabilising process were calculated and compared with the conventional

partcile orbits is to introduce the Shear Kinematic Drift (SKD)l which

calculations and the results of the numerical study on the more general

the resonance factor.) When a’ is greater than a critical value a’, we

second order differential, equation with the full potential will also

“‘P. Satyanarayana and P. Bakshi, Bull. Am. Phys. Soc. 23, 891 (1978).

(y) for the very short wavelength, the kp >> 1 modes, now depends on

theory which uses uniform field orbits. We find that the growth rate

^**W. Bellew and P. Bakshi, Bull. Am. Phys. Soc. _22, 1089 (1977).

ion-cyclotron frequency; S is the inverse shear length, and m

characteristic SKD frequency; p^ is the Larmor radius;

less than tT, Y ^ is slightly greater than

is significantly less than Ycnventional’

the parameter ct’=a/(kp^)^, where a =

(m^ = p^Sp^k

be presented.

When a’ is

find that

Detailed

is the

c i ’

Field-Reversed Mirrors — MCFRM

A Monte Carlo Model of Particle Motion in

Fusion products (fps) are found to have a significant effect on both

D. E. Driemeyer, G. H. Mi ley, and W. C. Condit’ Fusion Studies Laboratory Nuclear Engineering Program University of Illinois Urbana, Illinois 61801

the steady-state particle and energy balances in Field-Reversed Mirror (FRM) plasmas in spite of the small size of the plasma (radius equal to In fact, over 40% of the fp energy is only a few fuel ion gyroradii)J retained in the closed-field region of a D- He FRM with S (the ratio of the plasma radius to the fuel ion gyroradius) equal to 5. This results in an energy multiplication factor (Q) of 12 as compared to the Q of 2.0 assoc iated with the same system without fp heating. Unfortunately in a steady- state system, the desirable contribution of fp heating is necessarily accompanied by an increase in the fp ash buildup which reduces the actual Q value of the above system to 4.5. An accurate calculation of fp ash depo sition is therefore seen to be an important consideration in evaluating steady-state FRM design concepts such as SAFFIRE.

To facilitate this, a Monte Carlo particle code, MCFRM, has been developed. It couples the Hill’s spherical vortex representation of a field-reversed equilibrium, with a Monte Carlo treatment of Coulomb scat tering; thus providing a complete picture of fp thermalization in the FRM, even at lower energies where pitch angle scattering becomes important, The basic algorithm will be discussed, along with results from several test cases which were run to establish the validity of the model. Additional results will also be presented which summarize the affect of fp heating and ash deposition on the SAFFIRE reactor, and illustrate several possible means of ash control.

J. ‘Lawrence Livermore Laboratory, Livermore, CA. *This work supported by Department of Energy Contract EY-76-S-02-2218.

D. E. Driemeyer, G. H. Mi ley, M. Y. Wang, and W. C. Condi t, Prccgga!fng’s Ann^aZ P o m t r o Z ZgdFMsfon T%gory Ponygrgneg_, D3, Gatlinburg, TN, 1978.
G. H. Miley, J. G. Gilligan, and D. Driemeyer, Prana. Am. Fnc. Foe.

3C, 47, (1978)

1 2

electric field (Ohmic heating) and Lorentz collisions.

I. B. Bernstein Yale University, New Haven, Conn.

Recent papers * on high frequency instabilities in tokamak discharges

inhomogeneous magnetic field (trapped and untrapped particles), the applied

The electron dynamics are described by the drift-kinetic equation expressed

charge. The numerical code which describes the particle dynamics includes an

function has been determined numerically as it evolves in time in a tokamak dis

order to improve the calculations, the shape of the full electron distribution

relied on simple models of.the anisotropic electron distribution function. In

W. H. Miner* Science Applications Inc., McLean, Va.

N. K. Winsor Naval Research Laboratory, Washington, D.C.

A Numerical Investigation of the Evolution of the Electron Distribution Function in Tokamaks*

V. V. Parail and 0. P. Pogutse , Nucl.. Fusion l8, 303 <)1978). D. 1. Choi and W. Horton, Jr., Plasma Physics 20, 903 (1978). V. s. n U C l . V. V. 1, 30

V ..M . Leonov, 5 0 9 (1975). A. Razumova and Y . A. Sokoi.ov, Sov. J. Plasma Physics

tribution function which in turn determine the runaway electron instability seen

away electrons have been calculated. This information should aid in determining

tric field strengths and varying degrees of magnetic field inhomogenity. Also

numerical procedure is employed. The differential equations are written in con

trapped electrons, 2) co-streaming passing electrons, and 3) counterstreaming

whether the changes in the tokamak discharge parameter alters the electron dis

in this region of parameter space the plasma resistivity and production of run

passing electrons, are then transformed to a compact domain where the actual

in energy and magnetic moment variables. The resulting three equations: l)

The electron distribution function has been determined for a range of elec

or whether the electron distribution function remains unchanged

Vlasenkov, Fusion 13, Alikaev, K. 3 (1975).

Work supported by U. S. Department of Energy.

V. G. Merezhkin and V. S. Muknovatov,

and some other physics is responsible.

servation form.

in tokamaks

3 1

J. A. Derr and J. L. Shohet

ALPHA PARTICLE ORBITS IN STELLARATORS AND TORSATRONS*

The University of Wisconsin, Nadison, Wisconsin 53705

Several orbit types have been studied in a comparison between

A code has been written at Wisconsin for numerical simulation of

alpha particle motion in vacuum fields of Stellarator type geometry. This code implements the Lorentz force equation for the magnetostatic case, avoiding problems encountered in guiding center methods which rely on the existence of adiabatic invariants of motion. This permits the code to be used in the study of the longitudinal invariant of motion for localized particles, which is of importance in the theory of superbanana diffusion.^

torsatron and Stellarator reactor configurations (R major - 30 M; R minor = 4 M; BT = 5 Tesla; 20 field periods). These simulations involve 3.5 MeV alpha particles launched under identical initial conditions in the reference Stellarator and torsatron of the a=3 type. The reference machines are designed to match flux surfaces, ripple profiles, mod-B surfaces and transform profiles within the regions enclosed by their separatrices.

rates, action, magnetic moment, and turning points. The absence of a strong longitudinal invariant of motion has been observed. of the action occurs on a piece-wise basis in general, with discontin uities due to particle detrapping at helical mirror boundaries. The computed Stellarator orbits are qualitatively similar to earlier simu lations for lower energy particles.^ However, for the torsatron, sig nificant differences in the localized orbit types are seen. In partic ular, the conditions for superbanana orbits, and their apparent absence in the torsatron geometry, are found to be related to the topological differences between the Stellarator and torsatron designs used for these calculations.

*This work was supported in part by the National Science Foundation under grant ENG 77-14820 and in part by the U.S. Department of Energy under Contract No. ET-78-S-02-5069.

iA. A. Galeev, R. S. Sagdeev, H. P. Furth, and M. N. Rosenbluth, Phys. Rev. Lett. 22, 511 (1969).

The orbits obtained have been compared in terms of their precession

^A. Gibson and J. 3. Taylor, Physics of Fluids 10, 2653 (1967).

Conservation

Modes in Hishiv Elongated Tokamaks*

Computational and Analytic Study of Ballooning

Computational results for high n ballooning modes in highly

C. H. An The University of Tennessee Knoxville, Tennessee 37916

Glenn Bateman’ Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

elongated elliptic plasma is presented. The effects of elongation, toroidicity, shear, and pressure profile are studied computationally using the ORNL BALOON code and these results are compared with and clearly understood through the analytic calculation of high n ballooning modes in highly elongated elliptic plasmas. The marginal 8’s are plotted as functions of 8 for different elongations, shears, pressure profiles, and aspect ratios. The analytic results predict and the computational results verify that high elongation, low aspect ratio, and broad pressure profile enhance the marginal beta value for 8p less than unity but severely reduce 8 for 8p larger than unity with pressure p(L) = A ^ and safety factor q(ijj) = B’- C^ . Stability sensitively depends on shear, q(^), as a function of ooloidal tjj, even when q(^) at the magnetic axis and at the plasma edge are fixed, while equilibrium is insensitive to q(^) in chis case. Detailed comparisons between computational results and analytic results are given for a variety of cases.

*Tnis research was sponsored jointly by the University of Tennessee under contract DOE EY-76-S-052593 and the Office of Fusion Energy (ETM), U. S. Department of Energy under contract W-7405-sng-26 with the Union Carbide Corporation.

4- ‘Present address: School of Nuclear Engineering, Georgia Institute of Technology, Atlanta, Georgia, 30332.

existing initial boundary value codes.

Hcnl inear Magnetohydrodynamics in Three Dimensions,

We describe a simple numerical method for nonlinear mag

and low speed flows, and can be incorporated easily into

The. method is similar to that of Jardin et al [1] in that

netohydrodynamics in three dimensions that is designed to be

the terms corresponding to the fastest time scale are system

efficient in any problem characterized by long thin geometries

J.U. Brackbill, Courant Institute of Mathematical Sciences, New York University.

ferenced, it is conservative in the low speed flow limit, and

and potentially even faster. Further, because the magnetohy-

terms corresponding to the fastest time scale are made fully

culation of an initial shear flow discontinuity in one dimen

atically identified. However, it is different in that the

sion, and helical equilibria in three dimensions will be pre

implicit. This selectively or semi-implicit formulation is

demonstrably twice as fast as an earlier implicit code [2 ],

drodynamic equations rather than derived equations are dif

terms are not then formally isolated. Rather, only those

The analysis, the formulation, and the results of the cal

S.C. Jardin et al, J. Comp. Phys. 29_, 101 (1978}.
J.U. Brackbill, Meth. Comp. Phys. 16_, 1, (1976).

incorporates resistive transport.

sented.

B. Coppi * and E . Mazzucato **

** Plasma Physics Laboratory, Princeton, N.J.

TRANSPORT OF ELECTRON THERMAL ENERGY IN CONFINED PLASMAS

Massachusetts Institute of Technology, Cambridge, Ma.

The nature of the anomalous transport of electron thermal

energy in existing experiments on magnetically confined toroidal

conductivity, that is consistent with the observed temperature pro

experiments are obtained. In the presence of ohmic heating alone

files, is presented. In particular, scalings of the energy replace

can be derived. The appropriate diffusion coefficient can be written

plasmas is discussed and a new form of the relevant electron thermal

ment time and the applied loop voltages that are consistent with the

a simple analytical form of the relevant electron temperature profile

B. Coppi and E. Mazzucato, Report PRR-78/40, R.L.E., Massachusetts Institute of Technology, (Cambridge, Ma., 1978) to appear in Phys. Letters A

This diffusion coefficient has been incorporated in the transport model

the Frascati FT device in which it has been possible to vary the plasma

simulate a variety of plasma discharges. The set of experiments for

which Eq. (1) aopear to be most appropriate have been performed on

B. Coppi and A. Taroni, Report PRR-79/7,R.L.E., Massachusetts

and code that are described in Ref. 2 and have been utilized to

_ and the other quantities have well known definitions.

Institute of Technology, (Cambridge, Ma., 1979)

current by a significant factor, up to 600 kA.

is the resistive diffusion coefficient^

is the local poloidal field

V: ‘/X”; ’

7 /c- X

” K o

^ ” 4TT

where

771” U J

M __

’ s

L-* V

(1 )

6.’

kev to

ABSTRACT

TOWARDS A C O M P L E T E THE OR Y OF FIELD R E V E R S E D E QU I L I B R I A

Experiments under construction at LLL propose to make magnetized

B. M cNamara, J.K. Boyd, H.L. Berk Lawrence Livermore Laboratory, University of California Livermore, California 94550

plasma rings by gun injection into a guide field leading to a quadrupole mirror trap. These rings are then to be heated from their initial temperatures around 20 kev by neutral injection. A number of theoretical models have been developed to describe plasma rings, with or without embedded toroidal fields and with poloidal and toroidal plasma flows. Single particle studies help to connect simple multifluid models to kinetic theory op these various equilibria. Numerical solutions of the equations enable the initial and final plasma states to be calculated and the accessibility of the transitions to be assessed.

These fluid equations do not describe the contribution of large gyroradius ions to the fields. Orbit studies in axisymmetric reversed fields, such as Hill’s vortex, show that a third invariant exists which is easily destroyed by collisions or quadrupole fields. High energy ions are therefore included in the studies via distribution functions, f, of energy c and canonical angular momentum p^.

balance equations for a rotating fluid plasma species. This can be reduced to a Bernoulli type equation on each magnetic surface, relating pressure, elecectric field 3, and rotation speed, and a pressure balance equation across the surfaces, relating centrifugal forces and pressure gradients. Entering the solutions into Amperes law gives a modified Grad Shafranov equation for a multifluid equilibrium:

where H.(3j) is an arbitrary profile function for each species, depending on the total flux 3j = 3^+ej-1 ai)j r potential is determined by the quasineutrality equation: e.

“Work performed under the auspices of* U.S. Department of Energy by the Lawrt! Livermore Laboratory under contract nun^ W-740i-ENG-48.”

As an example of the more general theory, we consider the pressure

A transport theory is required to determine Hj and f(e,pg) self

consistently, so present studies use model profiles.

. 2 ^ OJ J 33j

exp(H.- - 4 (3+1-

i- 2 n .c. J j*

e x p ( H .

A * ^ = -

2 3H.

Z n

2 33

The

3H.

4r?

‘j

-))

2 B 4 3

ABSTRACT

GENERALIZED WKB METHOD IN ONE DIMENSION

R. Dominguez General Atomic Corporation

A generalized WKB method is developed for calculating eigenvalues and

H. L. Berk Lawrence Livermore Laboratory, University of California Livermore, California 94550

eigenfunctions of electromagnetic waves governed by integral equations or, equivalently, differential equations of arbitrary order in a medium ythat is inhomogeneous in one dimension. The method extends previous work with one component of polarization^*^ (e.g., electrostatic waves) to the three components of polarization of an arbitrary electromagnetic wave. The wave amplitudes are expressed in terms of a superposition of eikonel solutions. Me find that the amplitude variation in space has a compact explicit solution when the kernal is symmetric with respect to position in the inhomogeneous direction x, and in the components of polarization. With a suitable choice of components, this symmetry applies to a wide class of problems and includes Landau damping.

k(x,tu) merge in the complex x-plane. the merging waves satisfy an Airy equation, which allows for the determination of reflected amplitudes and phases away from the turning point. By mapping the wave trajectories and demanding single valuedness of the solution, a general phase integral dispersion relation can be determined.

dispersion relation” to zero. The rules for obtaining the appropriate local dispersion relation are given and they contain corrections from what one usually expects. A specific example will be given.

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48.

The local wave number, k(x,o)), is determined by setting the “local

The WKB solutions fail near turning points, where two solutions,

3-H. L. Berk and D. Book, Phys. Fluids 12, 649, (1969)

In the vicinity of the turning point,

George Vahala (William and Mary)

STABILITY AND FORCE-FREE FIELDS IN AN ELLIPTICAL CYLINDER

Force-free fields in an elliptic cylinder are generated by finding

the eigenfunctions and point spectrum of the curl operator V x § = AB with $ - n = 0 at the vail. It is found that the point system is completely det ermined by the geometric boundary condition and consists of discrete eigen values on the real line. The minimum (non-zero) eigenvalue is the first zero of the radial Mathieu function.

is a singular limit^*. In the limit of zero eccentricity, the point spectrum remains completely determined and discrete, but.for the circular cross section itself, the point spectrum consists of a continuous eigenvalue (extending to the origin) together vith discrete eigenvalues bounded avay from the origin.

theory^, on linear MUD stability of the Lundquist solution^ and the discre pancy vith nonlinear stability results^ vill be considered.

3j. Kruger, J. Plasma Phys. 15_, 15, 31 (1976). ^D. Montgomery, L. Turner and G. Vahala, Phys. Fluids 21_, 757 (1978).

2j. B. Taylor, in Pulsed High Beta Plasmas ed. D. E. Evans (Pergamon, Oxford,

Thus the circular cylinder is a singular limit much like zero shear

The implications of this singular limit on Taylor’s reversed field

Grad, Proc. Natl. Acad. Sci. 70, 3277 (1973).

1976), p. 59.

and

Y.C. Lee

Princeton, NJ 08544

Liu Chen and W.M. Nevins

in Sheared Magnetic Field

Los Angeles, California 90024

Stability of Drift and Drift-Alfven Waves

Plasma Physics Laboratory, Princeton University

Department of Physics, University of California

Using Antonsen’s technique, we first show that the collisionless drift

powerful theory for analyzing the stability of drift-wave eigenmodes. The

that Antonsen’s technique is valid only for cold ions, we have developed a more

and drift-Alfven eigenmodes are stable in a sheared slab magnetic field. Noting

Work jointly supported by NSF Grant No. PHY-77-12873 and U.S. DoE Contract No. EY-76-C-02-3073.

arbitrary radial wavenumbers. Here, due to the finite-ion-^Larmor-radius effects,

exhibits wave-flux conservation. Applying this technique, we prove that, with

theory employs the S-matrix technique in the complex plane and is based on the

observation that the physical system, after a suitable complex transformation,

We further demonstrate that this theory can also be applied to the case with

the usual differential equation is replaced by an integral eigenmode equation.

full kinetic-ion effects, the collisionless drift-wave eigenmode is stable.

The universal drift-wave eigenmodes is found to remain absolutely stable.

VORTICES IN 2-D GUIDING ENTER PLASMA WITH GRAVITY^

H. H. Chen, Y. C. Lee, C. S. Liu, and D. Montgomery Department of Physics & Astronomy University of Maryland College Park, Maryland 20742

To study the convection cells in multipoles geometry, the equilibrium state of a two-dimensional guiding center plasma under gravity which simulates curvature effect is considered. The most probable state of this system can be described by a Poisson equation with Boltzman density distribution for both electrons and ions. We found a. con formal mapping which can transform away the gravity and reduce the equation to the nonlinear sinh- Poisson equation. Exact solutions are found and the effect of the gravity is to introduce a natural period independent of boundary condition in the direction perpendicular to the gravity. The guiding center plasma is thus quantized in this direc tion. The simplest solution shows a somewhat stochastic distribution of vortices of various sizes with the smaller ones at the bottom and the larger ones forming coherent, structure near the top. The implications of these highly complicated vortex structures to the convective-cell transport in multipoles devices will be discussed. *

*Research supported by National Science Foundation, Office

of Naval Research, and Department of Energy.

ABSTRACT

LOCALIZED MHD MODES

SHAPE OPTIMIZATION OF TOKAMAK PLASMAS TO

R. L. Miller, R. W. Moore, and L. C. Bernard

General Atomic Company San Diego, California 92138

We employ a numerical technique to optimize the shape of tokamak plasmas

to achieve maximum stable volume average beta, 8, with respect to localized

interchanges and localized ballooning modes. A free boundary equilibrium is

plasma. The maximum stable beta exceeds 14% for a moderately-peaked current

profile with beta poloidal = 1, aspect ratio = 2.76 and b/a, the height-to-

Optimization is then accomplished by automatically varying the equilibrium

width ratio of the rectangular limiter, = 3.0. Optimal doublet shapes are

Research (Proc. 6th Int. Conf., Berchtesgaden, 1976), Vol. 2, IAEA, Vienna

boundary conditions in the direction of increasing maximum stable 8. For

plasmas without internal separatrices, we find the optimal shape to be a

strongly modified dee with a large indentation on the inside edge of the

calculated numerically and its stability to internal modes is assessed.*

also presented. MHD stability to external modes is evaluated for the in

Work supported by the Department of Energy, Contract No. EY-76-C-03-

^D. Dobrott, gf aZ., 7th Int. Conf. on Plasma Physics and Controlled

^D. Berger, gf aZ., in Plasma Physics and Controlled Nuclear Fusion

Nuclear Fusion, IAEA, CN-37-P-4 (Innsbruck, 1978).

0167, Project Agreement No. 38.

dented dees using ERATO.^

(1977) 411.

Z B 48

MODELLING OF STAGED LASER HEATING*

The laser solenoid Is a linear magnetic fusion concept which e m

David Qulmby and Loren Stelnhauer Mathematical Sciences Northwest, Inc. Bellevue, Washington 98009

ploys heating by an axially propagated laser beam and adiabatic magnetic compression. A number of experiments on this concept has been carried out Including a Targe on-going experiment at Mathematical Sciences North west, Inc., (MSNW). Laser heating and adiabatic compression overlap In time and occur on comparable timescales. The objective of this work Is to characterize the plasma conditions which may be achieved by this method.

an analytical modet to give approximate scaling and a one-dimensional magnetohydrodynamlc code for precise solutions. Previous analytical models have assumed the laser heating time to be much less than the compression time or else have held “the electron density constant. Our analytical model accounts for staged heating on comparable timescales as well as Including two temperatures and crude radial structure. Results will be presented characterizing the plasma temperature and plasma radius as a function of laser energy and filling pressure for conditions rdlevent:to the current MSNW experiment and hypothetical reactors. Results of the ID MHD code DYNASOR will also be presented and compared with experimental data. The code Includes additional effects of Importance such as radial dynamics, classical radial thermal and field diffusion, Ionization, and Impurity radiation loss.

Staged laser heating Is treated using a dual approach which Includes

Abstract

Daniel A. Hitchcock

Austin, Texas 78712

Fusion Research Center

The University of Texas at Austin

We have constructed a quasilinear theory for the

effects of low frequency electromagnetic turbulence in a toroidal plasma, ignoring terms 0(E^ r/R). The qualitative

out the similarities of this theory to our earlier slab model theory^ and indicate the future applications which are planned.

behavior of the diffusion tensor and the special role played by

This work is supported by the U.S. Department of Energy

^D.A. Hitchcock, R.D. Hazeltine, and S.M. Mahajan, APS

will be discussed. In addition we shall point

Bulletin 2_3, September 1978.

Contract DE-AC05-79ET53036.

ABSTRACT

S. C. Chiu, V. S. Chan, and G. E. Guest

ON MODE CONVERSION OF LOWER HYBRID WAVES

General Atomic Company San Diego, California 92138

The’ problem of mode conversion of lower hybrid waves in the absence of

a reflection layer is considered. It is found that partial mode conversion

takes place for some ranges of plasma parameters. Nonetheless, the RF power

version to be observed. In such cases, mode conversion can become possible

tron Landau damping sets a power dependence to mode conversion. For small

ing of the electron distribution function with the resultant weakening of

if the RF power exceeds some critical level because of quasilinear flatten

incident lower hybrid powers, Landau damping may be too large for mode con

can penetrate to the plasma center. The quasilinear behavior of the elec

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

Project Agreement No. 38.

electron Landau damping.

Abstract

Austin, Texas 78712

R.D. Hazeltine, and H.R. Strauss

.David W. Ross, Swadesh M. Mahajan

Instabilities by Temperature Gradient

Stabilization of Trapped-Electron Shear-Alfven

1 2 Localized shear-Alfven modes with large m-numbers ’

Fusion Research Center The University of Texas at Austin

are shown, numerically, to be strongly damped by the collisionless electron response in the presence of a

critical value, typically between 0.1 and 0.2. Analytical models demonstrate the scaling of these results with

temperature gradient. The trapped-electron drift-tearing instability*** is stabilized by this effect in a tokamak*

“**L. Chen, P.H. Rutherford, and W.M. Tang, Phys. Rev. Lett.

This work is supported by the U.S. Department of Energy

unless the local inverse aspect ratio, r/R, exceeds a

K.T. Tsang, J.C. Whitson, J.D. Callen, P.J. Catto, and

J. Smith, Phys. Rev. Lett. _41, 557(1978).

Contract DE-AC05-79ET53036.

plasma parameters.

39, 460(1977).

Selberg and A. Glasser

Princeton, New Jersey 08544

Stable Spheromak Current Profiles*

Plasma Physics Laboratory, Princeton University

The original spheromak concept was the small aspect ratio

to consider the large aspect ratio limit of the spheromak,

found to be Mercier unstable at very low 8 unless a small hole

of a spheromak is the absence of external toroidal field coils

modeled by cylindrical fields B (r) and B (r), with B = 0 for

rhe middle and no external toroidal field coils. When that was

rather than the spherical shape.^ It is therefore worth while

limit of a toroidal magnetic confinement system, with no hole in

was put in the middle, it became clear that the essential feature

x E r/a > 1. We have studied a variety of smooth current profiles of the form J = cB, with c = a ^ d - x^l)^2 for x < 1 and a = 0 for x > 1, and found stability to all kink and tearing modes for

decrease of the current towards the edge of the plasma is more realistic than previous step-function models. Stability with the wall removed from the edge of the plasma is important for

Work supported by the United States Department of Energy Contract No. EY-76-C-02-3073.

Fusion Research (Proc. 7th International Conference, Innsbruck, 1978) IAEA-CN-37-X-1

^M. N. Bussac, et al., in Plasma Physics and Controlled Nuclear

practical reasons such as impurity control and thermal isolation.

p^ = 15, P 2 = 2, and with a wall at x = 1.0475. The smooth

IN VERSATOR II*

described elsewhere.

LOWER HYBRID HEATING AND CURRENT GENERATION

RF energy deposition into the bulk plasma is

parallel electron and perpendicular ion Landau absorption. A

included through appropriate quasilinear equations which describe

prepared for Versator II. Some of the features of the code are

modified version of Fisch’s theory has been used to obtain the

distribution function. Provisions for inductive effects have been

R. Englade, T. Antonsen, and M. Porkolab Massachusetts Institute of Technology, Cambridge, Ma. 02139

quasilinear corrections to the linear damping and the generation of RF current via plateau formation in the tail of the electron

We have used a one-dimensional transport code to model the time dependent plasma heating and current generation that we expect to result from the lower hybrid RF heating experiment being

combinations of waveguide array configuration and initial (pre-heating) plasma state, assuming the generation of a Brambilla

T. Antonsen, B. Coppi, and R. Englade, MIT Report PRR-78/29

power spectrum. We have attempted in this manner to estimate

optimum operating parameters for the Versator II RF experiment.

and RF current both during and after an RF pulse for various

We have followed the time evolution of electron temperature

N. Fisch, Phys. Rev. Lett. 4_1, No. 13, p 873 (1978).

(1978) submitted to Nuclear Fusion.

Work supported to USDOE

included in the code.

$4720

NY 14853

A. Friedman

University of California Berkeley

Department of Electrical Engineering and Computer Sciences

REFINEMENTS AND APPLICATIONS OF THE RINGHYBRID CODE

R. N. Sudan Laboratory of Plasma Studies Cornel 1 Un i vers i ty Ithaca

J. Denavit Department of Mechanical Engineering Northwestern University Evanston

strength, background density, and external field gradient has been carried out. We observe stability of the MHD precessional mode when the external magnetic field increases strongly with radius, and instability in the oppo site limit; near zero field gradient, however, results are inconclusive. We observe a regime of decreasing growth rate as the layer strength increases, as suggested by theory [2,3]. We have also examined the effects of various boundary conditions applied to the field equations at the outer wall and on axis.

nent, in addition to the electron and hot-ion components, because the present field solver requires a small At for convergence when regions of small total ion density exist. In order to model inhomogeneous hot mirror plasmas more efficiently, modifications of the fieldsolving algorithm to remove this limi tation have been proposed. We have also begun to consider models which match the usual equations in the plasma region to another set of equations assumed to hold in a surrounding vacuum region.

The linearized, 3**D hybrid code RINGHYBRID [1] was developed at Cornell University for the purpose of studying the low-frequency stability of field- reversed ion rings in a background plasma. With minor modification the pro gram is capable of examining the stability of axisymmetric field-reversed mirror equilibria.

plasma (in the absence of any ring) in greater detail than previously pre sented, and find finite cell-size effects to be of the sign and approximate magnitude expected.

A preliminary study of infinite-layer stability as a function of layer

The current plasma model requires the cold fluid background ion compo

We have examined the dispersion properties of waves in the background

[1] A. Friedman, R. N. Sudan, J. Denavit, P-tocAAcDcngA

eg P&timaA, Monterey CA, June 1 9 7 8 .

‘Work supported by U.S. DOE.

ovelace (to aooear

and 3. N. Sudan,

K . V . r !

Chance, on

(1971’/ u .

!”<! t-3 j

Coti^

i 2!

IN A THERMONUCLEAR PLASMA*

THE DISTRIBUTION OF AND CLASSICAL TRANSPORT BY ALPHA PARTICLES

analytically from the Fokker-Planck equation. The time-asymptotic

rate by thermonuclear reactions in a Maxwellian plasma is obtained

The velocity distribution of alpha particles produced at a constant

J. D. Gaffey, Jr. and R. S. Schneider Instituto de Fisica Universidade Federal do Rio Grande do Sul 90000 Porto Alegre, RS, Brasil

Research supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) and Financiadora de Estudos e Projetos (FINEP).

and for the high-energy tail. The time evolution of the density, momentum,

distribution can be divided into three regions: a thermalized region with

a nearly Maxwellian distribution, a slowing-down region with a power law

evolution, loss term and a weak parallel electric field is given for the

slowing-down region, which contains the majority of the alpha particles,

kinetic energy and heat flux in calculated. The electron and background

exponential distribution. A more detailed treatment, including the time

species. In particular it is found that the electrons are more rapidly

ion contributions are given separately to show the effects of each

distribution, and a high-energy region with a rapidly decreasing

heated by the alpha particles than are the background ions.

, a

o e

2 ° o

at the

Princeton, NJ 08544

. Thus, the wave can reach

Gerald B. Elder and Francis W. Perkins

Plasma Physics Laboratory, Princeton University

Parametric Decay Heating with an Electron Cyclotron Wave

quencies for high-density plasmas has been slow. As long as

densities such that a) > a) . If the wave is focused to a sufficient inten-

region to be heated; however, development of gyrotrons at the required fre

encounters its first cut off at 0J =t) +t)

normally incident extra-ordinary wave (ti ) propagating from outside the torus

its energy to the plasma. The dispersion relationship for such a decay into

sity in a region near this cut off, parametric decay of the wave can transfer

The standard ECRH schemes for heating tokamaks require 0) > t) pe

A threshold value of the wave intensity for decay to occur is found. Estimates

made assuming a diffraction limited focus. Frequencies as low as 20 GHz could

an ion-acoustic wave and a Langmuir wave is derived. It is found that plasmas

of the power required for an useful coupling of energy to the decay waves are

with a wide range of densities can be heated with a fixed frequency source.

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

*^*0. Eldridge, W. Namkung, A. England, ORNL-TM-6052 (1977).

be used to heat a typical PLT discharge.

PARTICLE SIMULATION OF X-POINT DYNAMICS*

J. N. Leboeuf, J. M. Dawson, T. Tajima and A. T. Lin

Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024

A simulation of the magnetic x-point dynamics has been performed on a 2-1/2 dimensional magnetostatic particle code. The full dynamics of elec trons and ions in a doubly periodic system is represented save the displace ment current. The x-point is created by two temporally rising parallel rod currents in the z-direction with the cross section of the rod oblate in the x-directi on. Proper care is taken to accomodate the k = 0 component of the current which should show up in the Ampere-Maxwell equation in spite of the neglected displacement current. When the current rise time is approximately equal to the wave traveling time from the rod to the x-point, we observe shock fronts converging to the x-point. Jetting from the x-point and into the o-points is apparent from flow vector plots. The plasma dynamics is dominated by the induced Ey x in the x-y plane show a number of vortices which tend to disappear with a slight tilt of the rods in the x -z plane. In this collisionless regime, the x-point seems stable so far. Attempts at x-point heating by ringing the plasma at the appropriate wave traveling time will be reported.

drift current near the rods. The currents

*Work supported by DOE and NSF.

Kamimura

Nagoya 464, Japan

Nagoya University

Robert W. Huff and John M. Dawson

Center for Plasma Physics and Fusion Engineering

SIMULATION STUDY OF THERMAL VERSUS PARTICLE DIFFUSION*

University of California, Los Angeles, California 90024

This is attributed to partial cancellation of ExB drift velocity when

on the CHI computer at UCLA. Ions of 25 to 100 electron masses were

averaged over the large ion Larmor orbits, whereas the smaller orbit

electrostatic particle code with fixed magnetic field as implemented

found to have a diffusion rate as low as 25% of the electron rate.

Multi-species simulations were run using the 2^-dimensional

electrons can move with the full local ExB velocity.

*Work supported by USDOE.

NONLINEAR BEHAVIOR OF BALLOONING MODES IN TOKAMAKS*

For a particular class of equillbra,^ which are diamagnetic, with a

C. C. Mu, P. L. Pritchett and J. M. Dawson Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024

broad current profile and an elongated cross section, our preliminary results indicate that the ballooning instabilities do not saturate at small amplitudes. The results also show filamentation of the toroidal current and its self-reversal at some parts of the cross section.

Ballooning instabilities are believed to be a major limitation on the value of the plasma beta that can be achieved in a tokamak. The non-linear evolution of these instabilities is being investigated by using a 3-D MHD code in toroidal geometry, which integrates MHD equations in time by an explicit, leap-frog finite-difference scheme.

C. H. An and G. Bateman, ORNL/TM-6419 (1978).

*Mork supported by USDOE.

COALESCENCE OF MAGNETIC ISLANDS*

. P. L. Pritchett and C. C. Wu Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024

Tearing instabilities, and the accompanying processes of magnetic field reconnection and island formation, are believed to play an essential role in areas such as magnetic oscillations in tokamak discharges and stability of reversed-field theta pinches. Recently, Finn and Kaw^ studied the tendency of magnetic islands to coalesce into larger units by investi gating the stability of an exact hydromagnetic equilibrium consisting of an infinite chain of magnetic islands in slab geometry. We present the results of an extensive numerical analysis of this configuration which includes nonlinear and finite-resistivity effects.

We treat the coalescence process as an initial-value problem and solve the incompressible MHD equations using a semi-implicit method.^ Our results confirm the existence of the coalescence instability in the ideal MHD limit, but we find no evidence for a threshold in island width. The linear growth rates are found to be large compared to those for purely resistive processes such as the tearing mode. The linear mode structure has only a weak depen dence on resistivity. The resistive contribution to the growth rate has an S dependence similar to the inverse fractional power dependence of the tear In the nonlinear regime, saturation of the mode in the ideal case ing mode. is observed due to flux piling up at the X point, while in the nonideal case the merging process is observed to proceed to completion.

ij. M. Finn and P. K. Kaw, Phys. Fluids 2]3, 72 (1977). ^B. V. Waddell, M. N. Rosenbluth, D. A. Monticello, and R. B. White, Nucl. Fusion 16, 528 (1976).

*Work supported by USDOE and NSF.

A. S. S h a r m aandR. N. Sudan

Stability of Drift Waves in a Field Reversed Configuration*

stabilizing influence. We have modeled such a field reversed

In some field reversed plasma configurations, e.g., 6-pinches,

the short connection length of the poloidal field is an important

geometry are therefore not stabilized by magnetic shear. However,

ion rings, etc. toroidal field is absent. The drift waves in such

Laboratory of Plasmy,‘Studies Cornell University Ithaca, New York 14853

“This work supported under Office of Naval Research Contract N00173-79-C-0096.

Again using the quadratic form method this mode”is shown to be stable

and the modes are now described by two coupled equation in f and Aj j.

aspect ratio. We take account of both radial density and magnetic

forms*** we show that in the low 8 limit the electrostatic universal

configuration by a cylindrical Bennett pinch in the limit of large

lead to ion Landau damping, which accounts for the stability. In

mode is stable. The short connection lengths of the field lines

perturbation from kinetic theory. Using the method of quadratic

field gradients and derive the radial eigenmode equation for the

the finite-8 case the drift shear Alfven mode becomes important

^1. M. Antonsen, Jr., Phys. Rev. Lett. 41, 53 (1978).

under quite general conditions.

ABSTRACT

NEOCLASSICAL TRANSPORT IN EBT*

Science Applications, Inc., La Jolla, California 92037

H. H. Klein, R. D. Hazeltine, **, N. A. Krall, *** and P. J. Catto

We employ a suitably bounce averaged drift kinetic equation to identify three specific neoclassical transport regimes in EBT: a Collisional or Kovrizhnikh^ regime in which the collision frequency, y, is much greater than the poloidal drift frequency, 0; an intermediate or “plateau” regime in which y is less than 0 but still large enough to smooth the particle distribution; a collisionless or “banana” regime in which the guiding centers of slowly processing particles trace out banana shaped orbits before the particles suffer a collision. We employ realistic bounce averaged drift velocities and calculate transport coefficients, correct to lowest order in y/O, for the plateau regime, the operating regime of the present and planned devices. We find that this transport is independent of collision frequency. Results of calculations from a one -dimensional transport code incorporating the plateau transport coefficients are presented and compared to experimental data.

**Present address University of Texas at Austin, Austin, Texas 78712

Work supported by the U. S. Dept, of Energy.

Present address JAYCOR, Del Mar, California 92014

*L. Kovrizhnikh, Sov. Phys. JETP29, 475 (1969).

ABSTRACT

Jack A. Byers

LINEARIZED SIMULATION OF AN AXIS ENCIRCLING ION GYRO INSTABILITY

Lawrence Livermore Laboratory, University of California Livermore, California 94550

We describe particle simulation results for the axis encircling orbit model recently discussed by Aamodt, Catto and Rosenbluth.^* The model employed is identical to the analytic one: perfectly concentric ion orbits, cold electrons, electrostatic model, electron polarization drift neglected. Results from the code have confirmed the analytic results and have helped to elucidate the behavior of the unstable mode when various analytic approximations are of uncertain validity. The code produces details of the Eigenmodes <&(r) along with accurate real frequencies and growth rates as a function of the system parameters and azimuthal mode number a. All specific cases observed agree with the general predictions of the analytic model and in addition there are the following points of detailed agreement: a = 1 stable; a purely growing mode exists at high density and high a; nearly identical predictions for ^(r), is decreased below a critical value. The good agreement between code and analysis illustrates the utility of particle simulation techniques for determining the linear stability of spatially inhomogeneous equilibria. The present equilibria model is of course almost the simplest example; we are now extending the code to allow more general ion orbits.

R. E. Aamodt, P. J. Catto, M. N. Rosenbluth, Bull. Am. Phys. Soc. 23 755

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48.

; for a specific case marginal stability results when R^-j-]

^ w a l l ^ P ’

E. Ott

Tokamaks at U’ = 2 u

H. Hui and K. R. Chu

Cornell University, Ithaca, N.Y.

electron Cyclotron Resonance Heating of

Naval Research Laboratory, Washington, D.C.

Electron cyclotron resonance heating of tokamaks at the fundamental h ar

monic was shown to have great potential.^ If the electron density of future

T. M. Antonsen Massachusetts Institute of Technology, Cambridge, Mass.

at oblique incidence. Results of the numerical and the analytic calculations

tokamaks.is so high that the fundamental harmonic is not accessible, we may

the ordinary mode and the extraordinary mode could be absorbed efficiently

have to use the second harmonic of the electron cyclotron resonance. Under

“Electron Cyclotron Resonance Heating of Tokamaks at uu =

T ^ a few KeV), the second harmonic of

B. Hui and K. R. Chu, to be published.

reactor conditions (n > 1 0 * ^ cm

Work supported by DOE.

will be presented.

E. Ott,

L p ; U r p

<(p/pr<;;p\

pf r/p /<; fAsto;: Coo!f6/pp

Depar/emenV We P/ns/que du

M. TAGGER and R. PELLAT*’

AND LARGE RADIAL WAVENUMBER

.45S0CL4770Y EMMrO.W-CPd SLR LI FIS/OV

large frequencies and large radial wavenumbers.

Rone Posto/e n° 6 . 92260 LOATL_\141 -qLY-ROSLS fFRIVCL)

A NEW TRAPPED-ION INSTABILITY WITH LARGE FREQUENCY

The need for theoretical previsions concerning anomalous transport in

in turn necessitates the knowledge of the linear behaviour of these waves at

large Tokomaks, as well as the recent results of PLT, ask the question of the

process responsible for non-linear saturation of trapped-ion instabilities. This

that the effect of finite banana-width on the usual trapped-ion mode is complex

bining finite banana-width and bounce resonances. Limiting ourselves presently

ves large frequencies (t) ^ (n^) and is destabilized by large radial wavelengths

to the first harmonic expansion of the bounce motion of trapped ions, we show

a new branch of this instability. Essentially due to this new effect, it invol

ly local approximation, but including a term due to a new physical effect, com

1 , where A is the typical banana-width). We discuss the nature of this

In addition we show, analytically and numerically, the appearance of

We study the linear dispersion relation of these modes, in the radial

new mode and its potential relevance of the experiments.

and quite different from what is generally expected.

Ecole Polytechnique de Palaiseau.- France.

(k^ A

and

(3 (a.) and

Small Aspect Ratio*

University of Michigan

are free parameters of the

T. Mizoguchi and T. Kammash

conventional high pressure ordering,

and Stability with High Pressure and

Axisymmetric Sharp-Boundary Toroidal Equilibria

Because of increasing interest in Tokamak plasma with

expansion on the toroidal equilibrium and stability. The

paper the higher order effects of the inverse aspect ratio

small aspect ratio and high pressure we investigate in this

flux function and pressure respectively, is reasonable when

the aspect ratio is large. We introduce in this calculation

Cordey, J. G., Haas, F. A., Proc. of Sixth Int. Conf. on Plasma Physics and Controlled Nuclear Fusion Research, IAEA, Vienna 2^ 423 (1977).

ent of the magnetic axis as well as in the critical equilibrium

cannot be ignored in such toroidal devices as the Two Component

Tokamak. We find that substantial corrections in the displace-

Green, J. M., Johnson, J. L., Weimer, K. E., Physics Fluids

sure. Preliminary results on the stability of high ( H ) mode

ratio are included in a toroidal plasma with isotropic pres

a different ordering which is more suitable for small aspect

anisotropic pressure that includes plasma mass flow^ which

number of such equilibria will be presented and discussed.

-value occur when second order effects of small aspect

and use it to analyze the equilibrium of isotropic

Haas, F. A., Phys. Fluids 15., 151 (1972).

ratio tokamaks, namely cxf

*work supported by DOE

L4, 671 (1971).

-^-<0(3-^ and

and

1 7

ABSTRACT

Seung Kai Wong

ANALYTIC THEORY OF THE TRAPPED ELECTRON MODE

General Atomic Company San Diego, California 92138

Sanae Inoue and Kimitaka Itoh University of Tokyo, Japan

The 2-D problem of the electrostatic trapped electron mode in the limit

included in the analysis. It is first shown that the Pearlstein-Berk type

the trapped electrons and the Landau resonance of the transit electrons are

k^p < 1 is analytically investigated. Both the curvature-drift resonance of

The dispersion relations are solved numerically to determine the growth rates

the solution in a complete set of parabolic cylinder functions. The relevant

near the value given by the usual local approximation. The resultant system

mode rational surfaces and x^ the width of the parabolic cylinder functions.

matrix elements are evaluated with the exact orbit of the^ trapped electrons

and the unstable region for values of parameters representative of tokamak

poloidal angle can be reduced to a single differential-difference equation

rather than the harmonic oscillator approximation.^ The matrix eigenvalut

limits A/x^, << 1 and A/x^, >> 1 where A is the distance between neighboring

cast into the form of a standard matrix eigenvalue problem after expanding

problem can be solved^ and analytic dispersion relations obtained in the

because of a certain symmetry possessed by the system.- This latter is re

of radial differential equations coupling the Fourier harmonics in the

we also adopt here, is justified when L /L is large, in which case 0) is

approach of retaining only the ion sound term in the ion response, which

operation. The 2-D mode structure will also be discussed.

^K. T. Tsang and P. J. Catto, Phvs. Rev. Lett. <3.9 (19

^S. Inoue, K. Itoh, and S. Yoshikawc, Nucl. Fusion If

Work supported by Department of Energy, Contract

Project Agreement No. 38.

No. EY-76-C-03-016

(1978) 755.

s n

GATO

ABSTRACT

F. J. Helton, L. C. Bernard, and R.- W. Moore

General Atomic Company San Diego, California 92138

GATO evaluates stability of a tokamak equilibrium with respect to a

magnetic axes in a general axisymmetric toroidal configuration. A varia

linearized ideal.MHD displacement and can treat equilibria with one o f *two

tional approach to the problem is used; the displacement vector is expanded

^F. J. Helton and R. W. Moore, 8th Conf. on Numerical Simulation of Plasmas, Paper OD-1.

of the sparseness of the matrices. Matrix reordering is being investigated

in terms of a set of basic functions and substituted into the Lagrangian of

Choleski decomposition^. The improvement was obtained by taking advantage

elements^. Since the MHD spectrum is ill conditioned and the matrices are

large, attention has been given to the eigensolver. The problem is solved

the system. GATO uses an orthogonal coordinate system^ and finite hybrid

using an improved v e r s i o n ^ o f the direct method (inverse iteration plus

and may further improve the method. Initial results obtained using GATO

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

Bernard and F. J. Helton, General Atomic Company Report GA-A15257

^R. Gruber, Journal of Computational Physics Pd (1978) 379.

**R. Cruber, Computer Physics Communications Id (1975) 30.

Project Agreement No. 38.

will be presented.

C. (1979).

Livermore, California

separate temperature profile.

D. E. Shumaker, M. G. McCoy, J. Killeen and A. A. Mirin

TWO TRANSPORT MODELS FOR NON-CIRCULAR AXISYMMETRIC DEVICES*

The first program, TOAD, writes the transport equations in terms

Two programs are described which solve the differential equations

of Maxwellian ion species which have a common temperature profile. The

of 1-D plasma transport in an axisymmetric toroidal plasma of arbitrary

electrons, whose density is determined through quasi-neutrality, have a

advances these quantities implicitly, which is to say that the equations

cross section. Both programs assume the existence of an arbitrary number

of adiabatic invariants— mass, entropy *” P(V’)5/3^ and magnetic flux, and

National Magnetic Fusion Energy Computer Center Lawrence Livermore Laboratory

non-Maxwellian ion species arising from beam injection. The full nonlinear

optimized Fokker-Planck package. In addition, the equilibrium calculation

implicitly, but splits them into two parts (e.g. P and (V’)5/3). The time

surfaces. The 2-D equilibrium equation is solved by a variational method

*Work performed under the auspices of the U.S. Department of Energy by the

in which the time advanced adiabatic variables are the input parameters.

this value is used during the repetition of the transport cycle. This

are independent of the time rate of change of the volume.between flux

2-D Fckker-Planck operator is solved using, a recently developed CRAY

The second program, FPTE, does not advance the adiabatic variables

Lawrence Livermore Laboratory under contract’ number ¥-7^05-ENG-t8 .

The code FPTE also takes into account the presence of energetic

is determined through the equilibrium calculation, and

Subsequently, one solves for density and temperature.

is designed to accommodate anisotropic pressures.

process is repeated to convergence.

derivative

Shayne Johnston

Columbia University

FINITE-LENGTH THEORY OF COLLECTIVE FREE-ELECTRON LASERS*

Free-electron lasers represent promising sources of tunable

here without limitation on the density of the relativistic electron

beam employed. Expansion of the exact result in powers of the linear

coherent radiation; potential applications of fusion interest include

susceptibility x reproduces the vacuum gain formula”**, and shows that the

gain of such a device operated in the stimulated Compton mode is derived

plasma heating (ECRH) and driving systems for pellets. The small-signal

It is shown that stimulated Compton scattering persists in a finite-length

center approach^, and generalizes past work further by including a static

wavelength laser (visible, ultraviolet, even x-ray) operated in this new

guide magnetic field and an arbitrary distribution of beam momenta; the

Raman gain which is derived for comparison. The possibility of a short-

of the optimum vacuum gain. The theory is founded on the oscillation-

leading plasma correction causes a slight enhancement (not reduction!)

system, and that the Compton gain can easily rival the finite-length

For denser beams (]x[ > I)) a- new regime of operation is discovered.

F.A. Hopf, R. Meystre, M.O. Scully and W.H. Louisell, Opt. Commun.

principal assumption is small gain in the available length (i.e., a

S. Johnston, Phys. Fluids 19, 93 (1976); S. Johnston and A.N.

regime (plasma-modified Compton effect) is discussed.

Work supported by AFOSR contract F44620-75-C-0055.

Kaufman, Sherwood Theory Meeting 1978, paper D23.

recycled system).

18_, 413 (1976).

to:

High-P Tokamak Transport Modelling Studies*

using a module developed by D. Ehst (ANL) [1].

The attainment and maintenance of a high-P (P > 4%) plasma by

As part of this study we have modified our 1 1/2 D transport code

calculate lower hybrid power dissipation and current generation
calculate magnetic reconnection of non-monotone current profiles

J. T. Hogan Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

and magnetic island effects due to saturated tearing instabilities, using criteria developed by Waddell, Carreras and Hicks.

calculate high-P limits self-consistently by using the Bateman-Nelson Ballooning Code ^2] within the transport calculation.

neutral beam injection is an important goal of present tokamak experiments. We discuss work in progress on a number of modelling problems suggested by such experiments: optimization of the configuration by beam deposition and current programming, the role of neutral beam induced currents in determining the shear profile, possible use of Fisch-Bers lower hybrid current drive for tailoring profiles [1], and properties of the final stage of FCT evolution.

We find:

self-consistent transport-generated ‘stable’ D-shaped plasmas with P > 10% (‘stable’ = linearly unstable over less than 20% of the plasma volume), and that the T03CA results do not violate the ballooning criterion.

shear profile at low density for co-injected plasmas, and thus remove some of the profile tailoring capability. Perpendicular injection,, restores it, but a study of anisotropic-pressure equilibrium and stability is needed for this case (cf. A. Cooper, this meeting].

Research sponsored by the Office of Fusion Energy, U. S. Department of Energy under contract #W-7405-sng-26 with the Union Carbide Corporation.

on the post-FCT timescale) should not be accessible in the near future.

that neutral beam induced currents completely determine the

(from external voltage programming or lower hybrid current drive) may be counter balanced by tearing mode-induced enhanced transport.

G. R e v . L. IAE 307 5 (1978). Tneo ry

noncircular version of the Freya module developed by D. Post (PPPL).

that one Zakharov-Snafranov catastrophe ]3l (loss of equilibrium

that benefits for ballooning stability from force-free currents

As before, beam-deposition is calculated with the Fowler-Rome

Ehs t , Argonne Nat ional Laboratory Report, ANL/FFF/T

Bat eman, private -communication; G. Bateman, D.

D. Snafranov,, Kurcnatov Institut

e to. , 9J_ 1809 (1979).

availaDle as ORNL 1’

(En giisn transl.

Z a k n a r o v , V.

smo 79/03.

Abstract

Shear Damping of Drift Waves in Toroidal Geometry

J.W. Connor, R.J. Hastie, K.W. Hesketh and J.B, Taylor

Culham Laboratory, Abingdon, Oxon., 0X14 3DB, UK (Euratom/UKAEA Fusion Association)

In toroidal geometry the magnetic field strength, shear and curvature vary over a magnetic surface. This non-uniformity induces a coupling-!* between drift-modes centred on different rational surfaces, which can inhibit convection of energy away from the mode centre and so reduce shear damping.

For short wavelength modes (kyai > 1 ) the only important coupling effect is due’to trapped electrons, and earlier work^ has shown that this is ineffective in reducing shear damping. For long wavelength modes ion magnetic drift terms provide an important coupling term, and this is shown to be effective in removing shear damping when r^/R > ky? Cg^ (rq’/q) .

drift modes in a large aspect ratio, circular cross section Tokamak (with (3 - 0(e^)) is reduced, using the techniques developed for mhd ballooning modes, to a one dimensional eigenvalue equation from which the shear damping can be calculated. Numerical solution of this equation shows that the analytic ‘strong coupling’ theory developed in reference (1) and employed extensively since, is accurate over much of the range of possible parameters, but that it fails to pre dict an important class of undamped modes.

Application to a typical Tokamak suggests that, for modes of a given wavelength, shear damping may be absent for modes in an inner core region of the cross section, while in the outer regions the normal, slab value of shear damping may prevail, though the break between these two types of behaviour is not necessarily at rq’/q = 1/2 , predicted by strong coupling theory.

dimensional equation, to the global eigenvalue, and to the deter mination of the radial structure of the eigenmode is investigated in higher order of an expansion in (aiLg/r^).

Physics and Controlled Thermonuclear Research, Berchtesgaden (1976).

!* J.B. Taylor, Proceedings 6th International Conference on Plasma

The relationship of the eigenvalue obtained from the reduced one

A two dimensional eigenvalue equation modelling long wavelength

^ D.W. Ross and W.H. Miner, Phys. Fluids 20, 1957 (1977).

References

A.L. Herts

calculations.

Ionic Cross Section Relevant to Plasmas

The diagnostic and the radiative power loss from hot thin plasmas

require cross sections and rate coefficients of the ions present in the

plasma. The electron impact Collisional excitation cross sections appear

cross sections and the analytic forms available for accurate and practical

to be very important. This paper will report on the present status of the

Los Alamos Scientific Laboratory, University of California Los Alamos, NM 87545

1-3

3/2 c

A TRANSPORT ESTIMATE FOR EBT IN THE BANANA REGIME^

In a bumpy torus, the destabilizing vertical drift caused by the toroidal

This cancellation has been identified as the dominant transport mechanism

P. J. Catto and M. N. Rosenbluth Science Applications, Inc., Boulder, Colorado 80302 K. T. Tsang Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

component of the magnetic field is suppressed by the introduction of a poloidal drift due to magnetic field inhomogeneities. As a result of the poloidal drift, the electrostatic potential $ is able to play a sensitive role in the transport since the poloidal component of the E x B drift will always tend to cancel the poloidal magnetic drift of one species for some velocity range.

in the ion plateau regime, are the typical ion-ion c o l l i s i o n and ion Doloidal d r i f t f re q u e n c i e s , and e is the 3/2 inverse aspect ratio. In the low collisionality banana regime v^/n^ < e the ions having canceling E x B and magnetic drifts not only continue to dominate the transport, but are also able to trace out banana-shaped drift surfaces (when their bounce average motion is projected onto a cross section of the torus). Because of the resulting boundary layer structure in velocity 2 3 space previous approaches become inadequate or inappropriate. ’

coefficients are evaluated in the banana regime from the full Fokker-Planck collision operator by assuming that the gradient B drift dominates over the curvature drift, and that the bumpiness of the magnetic field may be treated as small in the collision operator. The model estimates the diffusion coeffi cients from the entropy production, and predicts an ambipolar containment time proportional to 1/v. and independent of aspect ratio, rather than the more

normally assumed. favorable scaling of 1/c

D. A. Spong, E. C. Harris, and C. L. Hedrick, 0RNL/TM-6215, April (1978).
E. F. Jaeger, C. L. Hedrick, and J. S. Tolliver, ORNL/TM-6313, May (1978).
R. D. Hazeltine, N. A. Krall, H. H. Klein, and P. J. Catto, Science

fWork supported by U. S. Department of Energy under contract EY-76-03-1018 at Science Applications, Inc. and under contract with Union Carbide Corporation at Oak Ridge National Laboratory.

In the simplified model presented here the ion and electron diffusion

Applications, Inc., Report LAPS 44, September (1978).

< 1 , where

and

’

D ep a r t m e n t of Physics University of California, Los Angeles, California 30024

It is common practice in RF heating calculations to evaluate the antenna loading solely due to the interaction of the plasma with the far field of the antenna. However, in actual experiments a substantial amount of plasma (poorly confined) always surrounds the antenna structures. The present study considers various kinetic interactions (transit-time damping, boundary absorption, nonlinear generation of ion acoustic waves) that arise in this environment and which can lead to the anomalous heating of the plasma surface. The prototype problem in this class of phenomena (the Landau half- space) is examined in detail and a numerical calculation of a RF heated plasma confined by two reflecting walls (of relevance to multipacting break down) is found to attain a universal equilibrium in which the asymptotic kinetic energy KE satisfies <KE> = aNe§, where N is the number of particles, e the charge, 3 the peak RF potential, and a is a constant which is inde pendent of initial conditions and RF parameters.

*Hork supported by USDOE and ONR.

(2 )

ing of this k-B^ = kj,

We show an analogy between this

= 0 resonance due to stochastic magnetic

particles by electrostatic waves. The implication that a

It is well known that deformation in tokamak flux surfaces

p e r t u r b a t i o n s . A diffusion equation for the field lines

result from magnetic perturbations that are helically resonant

with the equilibrium magnetic field B^. We discuss the broaden

results whose diffusion coefficient has a broadened k^ resonance.

TURBULENT MODEL OF MAGNETIC BRAIDING PART 1: RESONANCE BROADENING E F F E C T S ON STOCHASTIC MAGNETIC FIELDS, D. Tetreault, P. Diamond, T. Dupree, M.I.T.

The diffusion coefficient is similar to that obtained recently by Hirshman and Molvig. stochastic magnetic field model and the velocity scattering of

The model gives the usual expression for island width, as well as

lap. Finally, we suggest the application of this k,, broadening

magnetic island is analogous to a BGK equilibria is discussed.

the criterion for onset of stochasticity, i.e. island over

Hirshman, S. and Molvig, K., Phys. Rev. Letters, 42,

^Tetreault, D., Bult. Amer. Phys. Soc. Oct. ‘77.

model to nonlinear MUD problems.

648 (1979).

for

“clump” of

is constant along field lines, the

TURBULENT MODEL OF MAGNETIC BRAIDING II: PRESSURE CORRELATION FUNCTION AND SELF-CONSISTENCY, P.H. Diamond, D.J. Tetreault and T.H. Dupree, M.I.T., — We have studied the theory of pres sure fluctuations in a cylindrical tokamak in the presence . of stochastic magnetic fields. Using methods from the clump theory of Dupree, we have calculated the pressure fluctuation correlation function, <3p(l)$p(2)>, starting from the equation B-Vp = 0. As pressure structure of the correlation function gives information about field line correlation. It is shown that two lines are correlated when they are within an island (or resonance) width of each other radially and are separated in arc by less than a poloidal wave length. It is shown that large poloidal mode number is necessary to destroy field line correlation. The field line exponentiation length is calculated. The pressure fluctuation correlation function is given by <5p(l)6p(2)>=2Z^D(3<p>/3r)^, where Z^ is the correlation lengt field lines and D is the magnetic diffusion coefficient. The conversion of average pressure (or poloidal field energy) gradients into fine scale fluctuations is seen to be caused by stochastic magnetic diffusion. Similarities to ID Vlasov turbulence are indicated.

of <5pcp>, the possibility of a self-consistent, “stochastic equilibrium” is investigated. The role of marginally stable ideal M.H.D. modes in such a steady state and the effect of stochasticity near the K*B = 0 resonance layer are discussed.

Using momentum balance and Ampere’s Law to obtain <—

in terms

6 B

A.!. Mense, and J.N. Davidson Oak Ridge National Laboratory

THE ELECTRIC SHEATH AND PRE-SHEATH IN A COLLISIONLESS FINITE ION TEMPERATURE PLASMA ,G.A. Emmert*, R.M. Wieland,

The sheath at the interface between a plasma and a wall is a common feature of Langmuir probes, plasma interaction with the limiter or divertor collector plate in tokamaks, and axial electron heat flow in mirror machines. This problem was originally studied by Tonks and Langmuir^ for the cold ion case; a lengthy series of papers has followed on their classic work. Me consider here the formulation of the plasma-sheath equation for a collisionless plasma with arbitrary ion temperature in plane geometry. Outside the sheath this equation is replaced by the plasma equation (quasi-neutral approximation, zero Debye length limit), for which an analytic solution for the electrostatic potential is In addition, the ion distribution function, the wall potential, and obtained. the energy and particle fluxes into the sheath are explicitly calculated. The plasma-sheath equation is also solved numerically with no approximation of the Debye length. The numerical results compare well with the analytical results when the Debye length is small.

University of Wisconsin

Georgia Institute of Technology 1 L. Tonks and 1. Langmuir, Phys. Rev. 34, 876 (1929).

Research sponsored by the Office of Fusion Energy, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corp.

-T-

-1*-

with

=v^/(eB/mc)

re . eB 2-\3B

^.nree*ujzcc^ns—ona^.

Ororts and iranspor<-

Princeton, NJ 03544

drift equations become with

Alien E. Boozer and Giuietta Kuo-Eenra”

the coordinates. Transport can be simply evaluated by a Monte

In nonsymmetric systems the evaluation of drift orbits and transport is

difficult by traditional methods. We find the particle drift orbits can be

rapidly evaluated using a magnetic coordinate system B = V a x V p = Vx + gy^ + yPa,

Carlo technique using Lorentz scattering. If V x B = 0 ^ then 6 = y = 0 and the

da 3<P , fc , eB 2^3B dt ” ^ 3 ^ * * - e ^ ^ m c ^1^3^ ’ dt ^ ^ 3 a ^ ^ I ^ * ^ c ^ H 3 a

X = v.,/v is changed after each time step of length i from A to A N

with $(a,^,x) the electric potential and B(a,i<,x) the magnetic field strength.

with the bar a time average of the orbit, then due to the averaging effect .of

with f^(E,p) a local Maxwellian and s(P)d^ the volume element. Moments of

the Lorentz operator and the drifts,the kinetic equation can be written

with ± a random sign and v the collision frequency. Let D(E,i) = (-f

do 22L = _ ii. _fC. + ^3x

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

. . 3f __m _ ^ l _ 3 _ r m^ J

To evaluate transport across the pressure or

A ^ = X ^ ( l - 2 v t ) + [(1-A^)(2VT)]^^

this equation give the transport.

BoozsA rGid kuc-P^ttcLuAc.

dx = eir dt

surfaces, the particle pitch

3B me ^Ir3x

me ^tl ’ dt

s 3 ^ ^

2 —2

)/t

”

—

Princeton, NJ 08544

Three-Dimensional Geometries

Evaluation of Orbits and Transport in

Gioietta Kuo-Petravic and Allen H. Boozer

Plasma Physics Laboratory, Princeton University

The evaluation of a and beam ion drift orbits as well as the heat

transport is essential for the assessment of any magnetic fusion concept.

evaluations of orbits and transport obtained by integrating the drift equa

In nonsymmetric geometries like stellarators, EBT, and the tandem mirror,

symmetric devices like the tokamak and toroidal Z pinch have significant

symmetry breaking ripple effects. In this paper, we report computational

these calculations are subtle and difficult by usual methods. Even ideally

tions in a,^,x coordinates as described in a companion poster by Boozer and

electric field with a potential change across the plasma equal to the particle’s

In the absence of a radial electric field, the trapped particle region of pitch

is in the ripple trapped and untrapped regions. In this way, we were able to

Kuo-Petravic. An eighth-order multi-step Runge-Kutta method was used with a

considerably reduced from naive estimates based on the large radial excursions.

follow single particles for 5 x 10^ ion cyclotron periods with energy conserva

variable timestep to allow for very different time scales when the particle

energy confines all the drift orbits but excursions remain large (A^/^ ** 50%

angle scattering. Consequently, the radial ion transport, though large, is

with ^ the magnetic flux). The drift orbits are quite sensitive to pitch

angle space is filled with unbounded collisionless drift orbits. A radial

The geometry studied was a R = 2 , M = 5 Stellarator with e = l/7 and q = 2 .

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

tion good to 1 in 10^ .

sequence.

for {3 beyond a second

FLUX-CONSERVING TOKAMAK EQUILIBRIA

^he existence of a new stability region^

A SECOND STABILITY REGION FOR A SEQUENCE OF FINITE- 6

A sequence of flux-conserving finite-P tokama): equilibria is

1.^1 and 2.82 over the tlasma cross-section. The members of the

The equilibria are calculated using a procram due to R. Englade”*,

the vacuum toroidal field is 17.3 T; and the toroidal current is 3-6

for.boundary conditions consisting of a conducting surface of circular

MA. The rationalized inverse rotational transform q(t) varies between

tested for stability against ideal M.H.D. modes (so called ballooning).

crcss-section. The major and minor radii of the torus are 50 and 20 cm;

critical value, is confirmed for this numerically generated equilibrium

L. Suciyama, B. Coopi, A. Ferreira and J. W-K. ^‘ark “msachusetts Institute of Technology, Cambridge, Ua.

L. Coppi, J. Filreis, and J. W-K. Mark, 7th International Conference on Plasma Physics and Controlled Nuclear ^usion Research, Innsbrucb” “ustria (1978), Parer IAEA-CN-37-W-4.
J..’. Ramos, B. Ccm-i, A. Ferreira and J.

Meeting of the Division of Plasma Physics (*.n.s.), Colorado (1978’ Bull. Am. Phys. Soc. 2_3, p. 785 (1978). C. Coppi, A. Ferreira, J. W-K. Mark, and d.G.^amos, to be rublisisf in Nuclear Fusion (1979).

dlnq/Jlnr, and G is the pressure gradient parameter, -8 rR^r dp/d^ dr/d.

ballooning modes. It is observed that when p^m 13-15% (pm 12-14%) the

and r(^) is a characteristic scale of a flux-surface. For each curve,

cur ‘aces is determined from the general eigenvalue eouation governing

equilibrium sequence are characterized by different values of the ratio

venerated equilibria. Agreement was found with the results obtained

the numerical solution of the eigenvalue problem. The values of the

iecounle the stability analysis from the equilibrium, and to simtlifv

’. B. Coppi, A. Ferreira, J. W-K. Mark, and B. Sucivama, M.I.T. RLE

equilibrium parameters are obtained by fittinr to the numericallv

directly from the equilibria using the general eigenvalue e o u a t i o n .

r curve in the s,G plane, where s is the magnetic shear parameter,

B.Englade, M.I.T. RLE Retort PRR 77/33 (Cambridge, Ma. 1977).

In addition, we develop a model equation which oermits us to

instability effectively disappears for all values of s.

^erort PRR 78/43 (Cambridge, Ma. 1978).

bcrameterized by the flux coordinate

= 8 m p /B^^. ^ach is represented bv

cf ^lasma to toroidal pressure,

the ran^e of unstable flux-

“ark, 20th Annual

. .

IN THE VICINITY OF THE MAGNETIC AXIS

ANALYTIC TREATMENT OF BALLOONING MODE ^ODEL EQUATIONS

Normal mode model equations for hioh toroidal number

in general, shows the two points of marginal stability.

close to the magnetic axis. After taking this limit, the

the relevant parameters corresponding to magnetic surfaces

For the model configuration with shifted circular magnetic

ballooning modes are studied analytically, in that limit of

of the pressure gradient parameter G = -8mR^r”‘(dp/d^)(dr/d^)

the main features of the full ballooning mode equation, and,

surfaces, the growth rate is convientl’? obtained as a function

eigenvalue equation becomes much simpler, while still retaining

J.J. Ramos, T. Antonsen, B. Coppi and A. Ferreira Massachusetts Institute of Technology

unstable configuration, the squared growth rate is proportional

is shown to have an exact analytical solution. For a given

In this limit, the eigenvalue problem at marginal stability

axis, G tends to zero while the ratio s/G* remains constant.

and the shear parameter s = dlnq/dlnr. Given a specified

ecriuilibrium configuration, as we approach the magnetic

to the fourth power of G.

R. F . l illistic Dampinc - Some Physics Considerations.* Post, H. L. Berk, Lawrence Livermore Laboratory— A technique called “Ballistic Damping” (BD) has been proposed for the control or ion cyclotron instabilities of the highly coherent type characteristically In BD, ion observed in mirror systems at high plasma density . beams, launched parallel to the field lines, transit the plasma, resonantly gaining perpendicular energy and exiting through the far mirror, thereby extracting energy from the wave. Above a critical current the 3D effect can exceed the rate of growth of the unstable wave, leading to suppression of the mode. Critical BD currents may be estimated from simple power balance considerations or by incorporation of BD into the quasi-linear formalism of Berk et al. ; values found lie well within existing technology, both for present experiments and for scaled-up systems. Critical physics issues include the growth rates of the modes involved. Estimates will therefore be made of limits on the growth rate for the fundamental mode (stabilizable by BD) and of effects that weaken or stabilize higher order modes (only weakly influenceable by B D ) .

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-ENG,-48.

R.F. Post, Lawrence Livermore Laboratory UCID 17876, “‘Ballistic

Damping’ - A Proposed Method of Stabilizing Resonant Ion Cyclotron Modes” (July 1978).

H.L. Berk, T.D. Rognlien, J.J. Stewart, Comments on Plasma Physics

^ W.C. Turner, et al., Phys. Rev. Letts. 39, 1087 (1977).

and Controlled Fusion III, 95 (1977).

ABSTRACT

LINEAR THEORY OF HIGH-M TEARING MODES

M. Rosenberg, R. R. Dominguez, W. Pfeiffer, and R. E. Waltz

General Atomic Company San Diego, California 92138

A linear kinetic theory of high-m (m = poloidal mode number) tearing

modes which is valid for arbitrary (M/v ) (v = electron collision fre-

quency) is presented. We investigate the “semi-coilisional”^ regime in

which only the magnetic perturbation enters. Using a pitch-angle scattering

ential equation for the drift-tearing mode result in stable roots for realis

Fokker-Planck electron collision operator, we find that previous results^

are qualitatively incorrect. Numerical solutions of the eigenmode differ

^D. A. D’ippolito, J. F. Drake, and Y. C. Lee, Bull. Am. Phys. Soc. 23

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

^J. F. Drake and Y. C. Lee, Phys. Fluids 22 (1977) 1341.

Project Agreement No. 38.

tic tokamak parameters.

(1978) 867.

B. Lane and T.M. Antonsen Jr.

KINETIC EQUATIONS FOR LOW FREQUENCY

INSTABILITIES IN AXISYMMETRIC PLASMAS*

Kinetic equations for low frequency, high mode number,

intro-differential equations. Included in these eauations

Massachusetts Institute of Technology, Cambridge, Ma. 02139

confined plasma are developed. The analysis makes use of the

electromagnetic perturbations in an axisymmetric magnetically

component of the perturbed magnetic field are retained and the

high toroidal mode number expansion to reduce the lowest order

system of equations to a set of ordinary (along the field line)

fields are represented by a scalar potential and two components

oarticles, and nonuniform magnetic curvature drifts. Perturbed

of the vector potential. Thus, the effects of the compressional

are the effects of finite Larmor radius, magnetic shear, trapped

equations are valid for arbitrary values of nlasma pressure. The

formalism used here is the generalization of that used by Rutherford and Frieman for electrostatic modes. In appropriate

limiting regimes all known ballooning and drift modes are found

P.H. Rutherford and E.A. Frieman, Phys. Fluids 11, 569 (1968).

Work supported by the U.S. Department of Energy

as soecial cases.

The kinetic equations for low frequency instabilities in

magnetically confined plasmas^ are examined for modes with

The familiar electrostatic trapped electron mode is modified

ion inertia can be ignored and the field components are weakly

field components are strongly coupled, and an outer region where

by electromagnetic effects when g (the ratio of nlasma pressure to

treating two asymptotic regions: an inner region where the effects

of ion inertia are important and electrostatic and electromagnetic

asoect ratio. Solution of the mode equations in this case requires

magnetic field energy density) approaches e where s is the inverse

couoled. If 3 is then increased further the drift alfven is excited.

T.M. Antonsen Jr. Massachusetts Institute of Technology, Cambridge, Ma. 02139

B. Lane and T.M. Antonsen Jr. this meeting.

Work supported of USDOE

ABSTRACT

D. K. Bhadra and R. W. Harvey

CURRENT DRIVE WITH ENERGETIC ELECTRONS

General Atomic Company San Diego, California 92138

Toroidal plasma current generation by means other than magnetic

minimize the power dissipation necessary to drive the current. A

interested in current drive (rather than heating), it is desirable to

a tokamak plasma in the steady-state mode. To the extent that one is

possibly efficient way to drive a current is by deposition of momentum

induction is of great significance because of the possible operation of

primarily on high velocity electrons, for which the Collisional drag is

small. In this paper, we consider several alternatives for achieving such

nominal design parameters of RST,. a conceptual steady-state tokamak under study

the electron distribution is used to generate most of the current, we consider

carried by the beam. Finally, we obtain the current to power ratio for each

maintenance by “runaway” electrons for which Collisional drag is negligible

consider the quasi-linear and Collisional absorption of the mode and obtain

self-consistent current drive in the presence of plasma transport, using a

current drive by energetic electrons. These schemes are: lower hybrid rf

but anomolous effects become important, and current drive by relativistic

one-dimensional numerical code. For the case where the “runaway” tail of

Doppler-shifted cyclotron resonance. For REB current drive, we consider

the possibility that a two-stream instability may be excited which would

of these different techniques and arrive at numerical estimates using the

major limiting factors. For the case of lower hybrid current drive, we

affect both the resistivity of the plasma and the directional momentum

The basic mechanisms involved are examined, with attention paid to the

a quasi-linear equilibrium in the presence of anomalous effects due to

Supported by the Electric Power Research Institute, EPRI Contract No.

drive on the tail of the electron Maxwellian distribution, current

at General Atomic Company.

electron beams (REB).

RP 323-3.

-1

-1/3

expanding in n

WKB Theory of the Ballooning Mode Spectrum

between turning points, in n ’ near turning

R. L. Dewar, M. S. Chance, and A. H. Glasser

but also the most localized. Employing the ordering N = 0(n),

There are in general 0(n) unstable eigenmodes with azimuthal

The modes with small radial wavenumber N are the most unstable,

mode number n(>> 1 ) in a ballooning-unstable, axisymmetric torus.

Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544

points and matching the expansions we have derived the quantiza tion conditions^*k dq = EN + (1 /2 )E/n previously postulated.^

This formula applies for phase space trajectories topologically equivalent to trapped particle orbits. There also sometimes

exists a class of modes analogous to passing particles, for which the periodic nature of the ballooning mode dispersion relation makes the WKB analysis highly nonstandard. These modes are

where R is the region of the k - q plane bounded by the lines q = 0, k = ±1/2, and m (k, q) = ^ . The relation to EBK quantization will be discussed, as will the comparison of the

A. N. Kaufman, S. W. McDonald, N. R. Pereira, and N. Pomphrey Paper OB9, Sherwood Meeting 1978.

J. M. Greene, Y-Y. Hsieh, J. L. Johnson, J. Manickam, and A. M. M. Todd, Paper OBI, Sherwood Meeting 1978.

*Work supported by U. S. DoE Contract No. EY-76-C-02-3073.

^M. S. Chance, R. L. Dewar, E. A. Frieman, A. H. Glasser,

quantized according to the formula

above results with PEST.

k dq = N/n ,

and dominant for 6 > s

Princeton, New Jersey 08544

S. Chance and A. H. Glasser

Numerical Studies of Resistive Ballooning Modes*

Plasma Physics Laboratory, Princeton University

Numerical solution of the high-n resistive ballooning

with complex frequency m scaling as fractional prowers of

Matched asymptotic expansions for large and small 6 lead to a

equations shows the existence of pressure-driven instabilities

e = n T /i , with n the toroidal mode number, T, = qR/c the Alfven

6 < e mapped onto an infinite domain by the ballooning representation.

A R = a /n the resistive skin time. These modes transit time, and go unstable for 0 < e << 1 , with resistive effects negligible for

dispersion relation similar to that governing low-n resistive

interchange and tearing modes. a numerical study comparing the predictions of three different

A. Glasser, in Proc. Finite Beta Theory Workshop, Varenna, 1977, ed. B. C o p p i and W. S a d o w s k i

numerical solutions for the inner and outer regions. In the third method, the outer region is solved analytically. The numerical

Work supported by the United States Department of Energy Contract No. EY-76-C-02-3073.

A. H. Glasser, J. M. Greene, and J. L. Johnson, Phys. Fluids, 18, 875 (1975)

Fusion Research (Proc. 7th International Conference, Innnsbruck, 1978) IAEA-CN-37-P-2

equations over the whole domain. In the second method, we match

^M. S. Chance, et al., in Plasma Physics and Controlled Nuclear

analytical results provide greater efficiency and understanding.

studies give confidence in the analytical results, while the

methods. In the first method, we solve the full resistive

We present here the results of

with poloidal coordinate 6

A. H. Glasser

Princeton, New Jersey 08544

Ideal MHD Ballooning Mode Theory*

The Role of the Continuous Spectrum in

Plasma Physics Laboratory, Princeton University

A theory of ideal MHD modes with large toroidal mode number

type of periodic representation, the poloidal coordinate 6 is

Floquet theory to the behavior of the solution as 8 ^ +°° shows

mapped onto an infinite domain, with convergence as 6 ^ ±°° replacing toroidal periodicity as a boundary condition. An application of

n in sheared toroidal magnetic fields has recently been developed,^ motivated by an interest in ballooning modes driven unstable by the interaction of the plasma pressure gradient and magnetic field curvature and stabilized by magnetic tension. By means of a new

^M. S. Chance, et al., in Plasma Physics and Controlled Nuclear Fusion Research (Proc. 7th International Conference, Innsburck, 1978) IAEA-CN-37-P-2

conditions the behavior is oscillatory, and convergence cannot be achieved. This behavior is identified with.the continuous spectrum

This work was supported by the United States Department of Energy Contract No. EY-76-C-02-3073.

growing or decaying, and the boundary conditions are the vanish

ing of the coefficients of the growing solutions. Under other

of ideal MHD, which is important for initial value problems,

that, under some conditions, this behavior is exponentially

dissipation and heat absorption, and resistive effects.

M. S. Chance, et al., Nuclear Fusion 17, 1, 65 (1977)

Abstract

and J.C. Wiley

Fusion Research Center

A.A. Ware, R.D. Hazeltine,

Poloidal Rotation Instability in Tokamaks

For plasmas with significant high-Z impurity (Z^^ > 3),

cause weak poloidal rotation is the electron viscous force

the dominant force near the center of the plasma tending to

The University of Texas at Austin Austin, Texas 78712

oxygen impurity in hydrogen, C — 0.5 and the above relationship can be written Jj, - amperes and since at high densities the last three factors are approximately unity, this becomes identical with the Murikami,

proportional to opposing rotation is weakened by electrostatic trapping which, for the ions, is out of phase with the magnetic trapping. The electrostatic potential is caused by the non-uniformity

z where C is only weakly dependent on the plasma parameters. The critical electron temperatures from this relationship accurately predict the experimental values of T ^ . For

at a critical T given by (T /T.)^c,,E„ - C(m./m )^n. ev^ 1 0 T

particularly enhances as the poloidal velocity (V^) approaches

of the Z-ions on a magnetic surface, the non-uniformity being

onset of the disruption in the sawtooth oscillations, occurs

resonance the plasma goes unstable to poloidal acceleration,

(Bg/B)[^ + ^ KgZ /(n^ + — n )]^(T^/m^)^. Somewhat ahead of

the acceleration being towards a new equilibrium with much

higher V^. This instability, which is interpreted as the

= 1.6 * 10^(n^/10^)(2T^/T^(2n^/n^)(T^[eV]/400)”

Callen, Berry relationship, Jj, = 1.6 x 10^(n^/10”^).

resonance with the slow magnetosonic wave velocity

r/eR . The ion viscous force

i, e

e 1

e ^

!! I)

Frozen pellet fuel injection into the I3X-A plasma demonstrated

Pellet Ablation Rate Modifications ion Large Peilees in Tokamak Plasmas

W. A. HoulDerg Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

that the pellets have a strong cooling effect on the plasma. ** This cooling effect is primarily due to dilution of the plasma energy amongst a iarger-number of particles but also due to energy lost- in the ionization process. If the pellets are large enough (local perturbations An/n >> 1), the cooling can reduce further ablation of the pellet, i.e., the ablation process becomes “self-limiting.” Pellets passing through the magnetic axis or crossing a rational flux surface should see a further reduction in the ablation rate due to geometric effects if the ionization process is truly localized in the vicinity of the pellet. Large perturbations in the background plasma may require a kinetic analysis since the electron distribution function can become non-Maxwellian during the ablation process. The relative timescales for the various processes are discussed and compared qualitatively with pellet injection observations in I3X-B.

W-7405-eng-26 with the Union Carbide Corporation. 1 . S. L. Milora, C. A. Foster, P. H. Edmonds, and G. L.

^Research sponsored by the U. S. Department of Energy under contract

Schmidt, Phys. Rev. Lett., 42 (1979) 97-

ABSTRACT

A CYCLOTRON RESONANT FIELD

ADIABATIC AND STOCHASTIC ION MOTION IN

Ion motion in a cyclotron resonant field is of crucial importance

Gary R. Smith, Herbert L. Berk, Jack A. Byers and Yoshiyuki Matsuda

Lawrence Livermore Laboratory, University of California Livermore, California 94B50

in understanding mirror containment in such experiments as r.f. containment, r.f. heating, velocity diffusion in loss cone unstable plasmas and stream penetration into mirror traps. To attack these problems we have developed several analytic and numerical methods which in appropriate limits reproduce such effects as the reversible ponderomotive force equations and the criteria for superadiabatic or stochastic motion. Me find that there exists a low energy region in phase space where particles can be trapped even in the presence of a repelling ambipolar well. This may have an important effect on establishing saturation amplitude of loss cone mode. Analysis also indicates that a superadiabatic region for particle motion exists above a moderate energy (or order SkeV in a mirror machine with 2XIIB plasma parameters), which appears to be contradicted by experimental data. Numerical studies of the equations in a quadrupole field are being performed to see if the discrepancy can be explained.

*Mork performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number M-7405-Eng-48.

u = e - p ,

Charles F.F. Harney

Princeton, NJ 08544

Plasma Physics Laboratory, Princeton University

Lorentz force law for the ions to a set of difference equations

Diffusion of Ions in Velocity Space by a Coherent Lower Hybrid Wave

It has been previously shown that the motion of an ion in a coherent

lower hybrid wave becomes stochastic if the velocity of the ion satisfies

space diffusion coefficient for the ions. We accomplish this by reducing the

v, >o)/k, and if the electric field of the wave exceeds a threshold. In order

to determine the heating rate of the ions in this case, we compute the velocity-

***C.F.F. Karney, Phys. Fluids _21, 1584 (1978).

the bulk ions. The resulting two-dimensional Fokker-Planck equation is reduced

this way, we include collisions as a mechanism for transferring the energy to

These equations give the Larmor radius (which is related to p ) and the phase

Carlo method. Since normally only tail ions can gain energy from the wave in

A >1/4 . These equations allow a rapid numerical determination of the correla

the electric field strength and the proximity of the wave frequency to a cyclo

cyclotron period earlier (the jlh orbit). The parameters A and 6 describe

to a one-dimensional Fokker-Planck equation in v, using a method similar to

(6) of the ion on the (j+l)th cyclotron orbit in terms of these quantities a

tion function and hence the diffusion coefficient. This is checked against

tron harmonic. The stochasticity condition for the difference equations is

that of Fisch.3 Expressions for the heating rates of the bulk ions and elec

the exact equations of motion by solving the diffusion equation by a Monte

C.F.F. Karney, Princeton Plasma Phys. Lab. Rept. PPPL-1528 (1979).

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

v.,, -v. = 2ir6 + 2’n’Acosu.,-, 1+1

u . ,, - u . = 2ir5 - 2irA cos v .

^N.J. Fisch, Ph.D. thesis, M.I.T. (1978).

trons in the steady state are obtained.

v = e + p ,

1+1 1

1+1

(1)

Maxwellian

Young-ping Pao

New York University

Courant Institute of Mathematical Sciences

described as through three successive stages.

A Kinetic Theory of Evolution of Anisotropic Plasmas

In an anisotropic toroidal plasma, collisions and ani

The evolution of the anisotropic plasma can be roughly

First, the anisotropic electron distribution evolves to

tion and motion of the plasma for small collision frequency.

present work describes a kinetic theory for evaluating the evolu

sotropy can cause the plasma distribution functions to evolve in time and at the same time induce a motion in the plasma. The

involving two Fredholm integral equations and a partial differentio-

and the evolution of the distribution functions. It is shown that

is the electron (ion) collision time, and R is the scale length.

vective motion rather than the relaxation of the distribution

the plasma velocity is determined by a boundary value problem

convective transport of mass and energy. It is this global con

A procedure is given for determining the plasma velocity u

where u is the plasma velocity normal to the flux surface, T

In stages (1) and (2), the normal velocity u gives rise to

Work supported by U.S. DOE Contract No. EY-76-C-02-3077.

Next, the anisotropic ion distribution evolves to

function itself that is of primary interest here.

(3) Finally, neo-classical transport takes over.

3_ u 1 T 3t R

integral equation.

3 _ u 1_ 3t R ^ T

(212) 460-7458

Maxwellian

(2)

(Tf)

New York University

New York, New York 10012

Harold Grad and Eliezer Hameiri

Courant Institute of Mathematical Sciences

. ADIABATIC COMPRESSION OF A ROTATING PLASMA^

Current experiments exhibit plasma mass flow, especially

after neutral beam heating. The adiabatic evolution of a

rotating ideal MHD plasma through a sequence of steady state equilibria, can be determined from the initial state by the use of constants of the motion. A specification of a rotating axi-symmetric equilibrium requires the supply of 5 arbitrary

fluid circulation along field lines, and the angular momentum of the ignorable direction. A similar problem can be solved

can be obtained from 5 conservation laws for each moving flux tube, namely: conservation of mass, entropy, toroidol flux,

for a two-pressure guiding center fluid (double adiabatic model),

algorithm based on the “1-1/2 D” concept”** of iterating between

functions of ^ - the poloidal flux function. These functions

geometry and plasma profiles can be used in the present problem.

Grad, Hu and Stevens, Proc. Nat. Acad. Sci., USA 7_2, p. 3789

eralized Differential Equations”**. A plausible numerical

*** Work supported by U.S. DOE Contract No. EY-76-C-02-3077.

The problem is best described by a formulation using Gen

both with and without flow.

(212) 460-7204

(1975).

R. N. Sudan

with Mass Flow*

Stability of Field Reversed, Force Free Plasma Equilibria

G = / d x v*(B + (m/2q)V x v), etc., constant, B = V x A and

The stability of hydromagnetic equilibria is examined in terms

of a variational principle in which the energy is minimized while

keeping a number of global integrals of motion, viz., K = / d x A*B,

a) = V x v is the vorticity. When only K is taken into account, the

usual force free solutions V x B = Kg are obtained, but if G is also

Laboratory of Plasma Studies, Cornell University Ithaca, New York 14853

cusp boundary but the internal fields are equivalent to two vortex rings

magnetic cusp we obtain the interesting case of a plasma with a spindle

^M. N. Rosenbluth and M. N. Bussac, “Spheromak Stability”, to appear

and the surface perturbations are examined by the method of Rosenbluth

match to vacuum fields by using the method of free boundary. For the

These equilibria are stable to internal incompressible perturbations

taken into account then we have in addition v = a B together with

and Bussac.^ Cusp-shaped equilibria are found to be stable to both

pVv /2 + (dp/dp)Vp = 0, where p is the mass density and p is the

special case v = B//p and p constant, we obtain toroidal, D shaped

equilibria where the vacuum field at infinity is uniform. For a

We have obtained axisymmetric incompressible equilibria which

“Work supported in part by U.S. Department of Energy.

surface and internal incompressible perturbations.

with oppositely directed toroidal fields.

in Nucl. Fusion.

pressure.

^ p)

Abstract for 1979 Sherwood Meeting

New York University New York, N.Y. 10012

Both (axially symmetric) MHD and GCF (p^_ ^

George K. Morikawa Courant Institute of Mathematical Sciences

equilibria with spheroidal plasma-vacuum interfaces have been considered in the past by G.K.M. et al. These equilibria apparently have all the external and internal parameters required for viable plasma confinement configurations; and they cover the full range of physical parameters including singularities which of course must be avoided by a practical device. For example: (1) A small vacuum region surrounding the fat toroidal plasma, where the two (or four) stagnation points (or separatrices) are near the singlet plasma boundary, can lead to a square-root singularity (integrable) of the poloidal current density on the plasma-vacuum interface (related to kinks). This high current behavior can be totally suppressed by a modest amount of external axial current, I ; in a practical device, access to the plasma is essential so that the spherical small-vacuum-plasma configuration is imbedded in a topologically straight vacuum region which in turn is bounded by carefully shaped confining coils on an appropriate magnetic flux surface. (2) For MHD configurations there is a practical restriction on pressure or 8 since^in the high-pressure diamagnetic regime, the required I increases excessive; very rapidly with increasing 8 and soon becomes but this behavior is for 8 poloidal > 0.5 (related to incipient reversed currents in the plasma); and (3) In the GCF plasma equilibria the plasma-vacuum interface can be spheroidal, i.e. prolate, spherical or oblate depending primarily on 8- For pro late (p^. > pi<) or oblate (p_,_< P;,) configurations the outer con fining coils must be modified; otherwise kink behavior is immanent on the plasma-vacuum interface. But the plasma-vacuum interface can be maintained inside of a closed spherical region by appropriate choice of 8-

A. Reiman and R. N. Sudan

in a Field Reversed Ion Ring

Instability Driven by the Electron Return Current

It has been shown that electron return currents can increase

the difficulty of field reversal on a Collisional time scaled

We have been studying field reversal by the pulsed injection of

energetic ions. On the relevant time scales we may neglect collisions

Laboratory of Plasma Studies Cornell University Ithaca, New York 148S0

through field reversal by a cusp magnetic field. We find that a strong

electron return current flows throughout the region of closed field

The unstable mode is an electrostatic surface wave, driven unstable

lines. The return current drives an instability having a growth rate

effects for a large aspect ratio ion ring, accelerated azimuthally

but must include electron inertia. We consider the return current

in the return current region. The subsequent nonlinear evolution

-*-D. Baldwin and M. Rensink, Comments Plasma Phys. Cent. Fusion 4,

*This work supported under U.S. Department of Energy Contract

is the electron cyclotron frequency in the cusp field.

by the presence of a resonance,

a) - k v = + (ii ,

is being studied.

EY-76-S-02-3170.

55 (1978).

where

pe’

ABSTRACT

In this Collisional case the

TRANSITION FROM COLLISIONAL TO PASTUKHOV ION CONFINEMENT FOR TMX

The ion confinement time in the central cell of TMX has in the

For TMX the ion mean-free-path is longer than the system, but the

T. D. Rognlien, R. H. Cohen and T. A. Cutler Lawrence Livermore Laboratory, University of California Livermore, California 94550

past been calculated assuming that the distribution function is zero on the loss-cone boundary. We term this Pastukhov confinement^ which is valid for very long mean-free paths. As the collisionality of the^ plasma increases, the filling time of the loss-cone finally becomes shorter than the axial transit loss time. distribution function is Maxwellian everywhere except near the end of the confinement region where the loss-cone is depleted.

volume of the loss-cone is small because of the large mirror ratio. Consequently, ions need to scattering significantly less than 9(P to fill the loss-cone. A simple calculation shows that for initial TMX parameters the loss-cone filling time is somewhat shorter than the axial transit time indicating that the device is in the transition region between Collisional and Pastukhov confinement. We shall present the results of detailed calculations of the confinement time for TMX obtained from a Monte Carlo code which is valid for any collisionality. The total confinement time, T^, is found to be approximated by the sum of the Collisional and Pastukhov confinement

times, scaling of f and T with density, ion temperature and mirror ratio will be discussed. We find that for initial TMX operation, T^/ip >, 1. A self-consistent rate equation code’ is used to calculate the effect this additional confinement has on the steady-state parameters of the system.

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48.

IV.P. Pastukhov, Nucl. Fusion J4, (1974) 3.

^p* The different

i.e.,

and

a 2 ^ = -Y2 ^ 2 ^ - a^a^ 2 ^

J.-M. Wersinger, J. M. Finn, and E. Ott

^ where Y 2 , Y^y and <5 are real and positive.

Instability Saturation by Mode Coupling”

Chaotic, Strange Attractor-Type Behavior in

and reduction of the latter to one dimensional mappings.

The (normalized) equations studied are: a^ = a., + a^a^e

Laboratory of Plasma Studies, Cornell University, Ithaca, NY 14853

We describe the results of an accurate and comprehensive numerical

Exploration to date of the two dimensional parameter space (setting

study of instability saturation by mode coupling in a three wave system.

Particular emphasis is placed upon distinguishing bifurcations leading

investigation include power spectral analysis, surface of section plots,

to motions characteristic of a strange attractor. The tools used in this

Y 2 = Y$ = T) indicates that a number of bifurcations occur before a strange

becomes broad. These results, mapped in the parameter space (<$,r), and

have some thickness (similar considerations have been applied to the Lorenz

all initial conditions examined lead to unbounded solutions. In the range

orbits occur., These orbits manifest themselves as periodic points in the

However, reduction to a one dimensional mapping shows that this arc must

surface of section. As r is increased past some critical value, r^, the

which manifests itself as a single fixed point in the surface of section.

case, the points in the surface of section appear to lie along an arc.

attractor). Qualitative changes are observed in the structure of the

motion becomes chaotic with the characteristics of a strange attractor.

Rabinovich (for Y 2 * Y 3 ) wh o , in contrast, with the results described

< T < Tp, the solutions converge to a simple stable periodic orbit,

This system of equations has also been examined by Vyshkind and

”Work supported under U.S. DOE Contract No. EY-76-S-02-3170.

attractor appears. For fixed 6 , we observe that for T ^

consist of very sharp discrete peaks, while for T >

above, always obtain stochastic motion if 6 ^ 0 .

bifurcations to more complicated periodic

strange attractor as T is increased. For

depend on the value of 6 .) In the chaotic

results for Y 2 ^ Y 3 ”-‘ill be presented.

As T is increased above

the power spectrum

and F > F^ > F^,

the power spectra

(The values

F^, and

> T >

Nonlocal Investigation of the Lower-Hybrid-Drift

Instability in Reversed Field Configurations, J. D. Huba, Science Application, Inc., and J. Drake and N. T. Gladd, Naval Research Laboratory— The lower-hybrid- drift instability is regarded as an important cross field anomalous transport mechanism in a variety of laboratory confinement systems, including theta pinches, RFP’s and mirror machines. This instability may also act as a source of resistivity to drive tearing modes in reversed field systems. We present a fully electromag netic, nonlocal, kinetic theory of the lower-hybrid- drift instability in a reversed field geometry [Bg(x) =

are excited with comparable growth rates. These modes typically occupy a substantial portion of the plasma sheet. Implications of the radial structure in under standing the nonlinear saturation of the instability and influence on tearing modes will be discussed.

(a ^ 2r^), ^ broad spectra of radial wave vectors (k^)

tanh (x/a)j. For moderate sheet pinch widths

; -n t

t.h?

time

could

below

sc ale

3 B 6

d x i s

uni Ty

i n d u c e

lead ing

on tHP

dr oppinv

r-csis tive

insid e ed ?e

signif icantly

Katioral Laborer:

the plasm o pUa

separa trix forming on

Law Midve. Tennessee

ins tability, and a P limiting

RESISTIVE DIFFUSICt-! CF FCT EQUILIBRIA^

resistive diffusion has been questioned. The potential problems which

(FCT concept), the long term maintainance* of finite 6 equilibria under

while controlling the q profile end with it the stability of the plasma

interaction of’ the vertic al field wi t h the Pte s m p ‘s own peloid al f ie l d .

Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

avoided by using higher order confining fields rather than uniform

real, they can be circumvented by combinations of cross section

effects which favorably modify the overly simple relations derived for

D shaped or other noncircular plasmas are also favorable for avoiding

broad temperature profiles are required to keep q ^ ^ g sbove unity as 6

to the current profile are also observed in the transition from FCT to

shaping, profile tailoring, and coil current adjustments. In general

vertical field; such fields do not appear overly difficult to design.

resistive behavior. Some of these may have implications for stability.

Research sponsored by the Office of Fusion Energy (ETM),U.

separatrix formation. This is due to toroidal geometric

Although the problems mentioned above are

large aspect ratio circular cross section.

resi stiv P d iffusicn of FCT Pqui libris

the resistive time scale with Qg^gg of

using both ana lytic al tec hni ques

fui ly toroida 1 1-1 /2D - free

Significant modifications

boundary transport code.

Separatrix formation is

For peaked profiles bg^is

drop tc 0-5 or less on

approaches 1C%.

inve st. is;pted

low baxis

W e ha ve

tc the

and

the

Ronald W. Landau

Ion Streaming Instabilities

less parameter V^= V^//2 v^ where

tures are equal, T^> 4T^ is needed, where

Instability for this mode requires that the electron

example, if the parallel and perpendicular electron tempera

In examining carefully the linear electrostatic dispersion

temperature be at least 4x one of the ion temperatures. For

growth rate than other modes of this type previously considered.

Queens College of the City University of N.Y.,Flushing,N.Y.11367

found a new mode. This mode has a lower threshold and a higher

parallel or perpendicular ion temperature. Defining a dimension

relation of a plasma of counterstreaming ions with an electron background in an unbounded uniform magnetic field, we have

rate an order of magnitude lower. This new mode has kj_- 6k„ and a perpendicular wave length much smaller than the ion larmor radius.

vH^/3. Another mode (with real part Perkins^has a threshold about the same as our mode but a growth

V.> 2.5 is.needed for instability when T > 4T. . The growth rates are large, up to 20^^. This should be compared with the

ion-cyclotron mode of Weibel^, also purely growing, which has a higher streaming threshold (\L> 3) and a maximum growth rate

the ion beams and v^= /T/m, their thermal velocity, we find that

growth rates m ./100, or roughly similar to the other mode.

propagation and the low threshold V> 1.3 . This mode has

When T„ > 4T„ . another mode exists with parallel

E.S. Weibel, Phys. Fluids 13_ 3003 (1970)
F.W. Perkins,Phys. Fluids 1_7 1012 (1976)

is the drift velocity of

may be either the

H^/2) discussed by

P ’ i

i /

We consider electrostatic ion Bernstein waves driven

NONLINEAR STABILIZATION OF THE ION BEAM-CYCLOTRON INSTABILITY

J. R. Myra and C. S. Liu Department of Physics and Astronomy University of Maryland College Park, Maryland 20742

unstable by a cold ion beam with velocity u directed along the magnetic field. A reactive type instability results from the coupling of the beam mode at m=k,,u to the Bern stein mode at M = nQ, with an approximately cubic dispersion relation at the crossover point. The nonlinear analysis assumes a single unstable mode. When the ratio of beam energy density to thermal energy dehsity is small, satur ation by beam trapping is expected. However, when this ratio is large (as in beam heating experiments) the non linear ion gyrofrequency shift saturates the instability first by detuning the resonance. Saturation can occur at values well below those predicted by beam trapping. To describe the nonlinear behavior in detail coupled equations for the Bernstein wave, (}), and beam wave n are derived:

v is related to the group velocity of the waves. The temporal problem is solved numerically and indicates non linear oscillations about the expected saturation level. An exact one-soliton solution exists for the problem in one space dimension. *Work supported by DOE.

id. + iv<{) + d

(j) (j) = n

<j) = 0

ABSTRACT

J. Y. Hsu, K. Matsuda, M. Chu, and T. Jensen

STOCHASTIC HEATING IN A LARGE-AMPLITUDE STANDING WAVE

General Atomic Company San Diego, California 92138

In heating plasmas with high-power radio-frequency waves, a symmetrical

launching structure is often used. This leads to large-amplitude standing

directed traveling waves. As a result, stochastic heating of low energy par

waves in the plasma. Particles may execute random walks in the two oppositely

mechanism should affect the plasma—wave coupling and accessibility conditions.

the stochasticity boundary, where p = eE k /m(i)^ > 0.456, the Fourier spectrum

expansion and scales linearly with p. Modification of the plasma dielectric

of particle trajectory is characterized by the onset of a “stochastic” mode

Comparisons with some experiments^’^ will be discussed. If a standing wave

explained as the scattering off the dense set of unstable fixed points. At

ticles can readily occur. The physical origin of the stochasticity may be

scopic method.^ The energy gain for large p is found from a multiple-time

function due to stochastic electron motions is also obtained. The present

and broadband noise. The stochastic trajectory is presented by the strobo

plasma edge, it may stochastically heat the plasma without any resonance

is created in the plasma core by launching two traveling waves from the

Work supported by Department of Energy, Contract No. EY-76-C-03-0167,

^J. Wesley, g?. a*.. , General Atomic Company Report GA-A14461 (1977).

^W. Hooke and S. Bernabei, Phvs.Rev. Lett. F8 (1972) 407.

*G. Smith and N. Pereira, Phys. Fluids FI (1978) 2253.

Project Agreement No*. 38.

condition.

Viktor K. Decyk and G. J. Morales

COMPUTER SIMULATION OF CURRENT GENERATION BY LOWER HYBRID WAVES*

At the present time the possibility of generating DC plasma currents

Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024

by external RF fields is of considerable interest. A potentially useful RF current driver is the travelling-wave variation of the lower-hybrid heating scheme. As part of our extensive simulation of this heating scheme, we have investigated the simultaneous process of heating and DC current gener ation by a unidirectional lower-hybrid wave of high phase velocity (u/k = 4.1 v ^ ) , which is launched by an external structure that mocks up the role of an “end-fire” waveguide array. Although the present 2-1/2 D electrostatic simulation is rather idealized, a variety of important collisionless effects are isolated; in particular, the role of density profiles and surface heating for the DC current generation process are studied.

*Work supported by USDOE and ONR.

S. Jorna Physical Dynamics, La Jolla

ION BEAM FUSION: BEAM TRANSPORT, THE PENULTIMATE PROBLEM*

W. B. Thompson University of California at San Diego

Energetic ion beams appear to provide an ideal driver for pellet fusion.

We concentrate here on the problem of quasi-ballistic propagation through

Our main conclusion is that at pressures <1 millitorr, the heavy ion beams

can propagate even in a charged mode, as in the accelerating system, and at higher pressures can propagate in a neutralized mode, in spite of a set of possible plasma instabilities.

Either light ion beams of 10^ amps at 1 MeV or heavy (high z) ions of 1000 amps (equivalent) at 10 GeV supply 10 Tw and have ranges compatible with mod erate size « 1 mm pellets.

a 5 m radius target chamber. We consider first what background pressure com prises a vacuum and then discuss the consequences of gradually increasing the background pressure. Self consistent charges and currents will be discussed as well as the effects of a wide group of instabilities.

face instability usually prevents targetting without the help of a background - plasma.

For the light ion case, however, the beam must be neutralized, and a sur

Supported by Occidental Research Corporation

MAGNETIC FLUCTUATIONS EXCITED BY g-PARTICLES

influence on the collective modes that can be excited by a

F. Pegoraro* and B. Coppi Massachusetts Institute of Technology, Cambridge, Ma.

neutral beams in present day hydrogen or deuterium plasma experiments. The general features of the type of magnetic conf inement configurations that are considered, have an important

, The production of 3.5 Mev a-particles in a (D-T) fusion reactor can lead to the excitation of various plasma modes that can be compared to those expected from the injection of energetic

non thermal particle or energy source. For instance, to determine whether modes that would be identified as shear- Alfven waves in an infinite homogeneous plasma are driven unstable by resonant interaction with the a—particles, a number of questions must be addressed such as: a) what form

between these modes and the a-particles; c) how is the theory of these modes related to that of ballooning modes in toroidal

do modes of this kind take in a toroidal oeometry and in the presence of magnetic shear, as they cannot be found as ordinary

waves; b) which is the appropriate form of resonant interaction

Scuola Normale Superiore 56100 Pisa Italy

systems.

G. Ganguli and P. Bakshi

Chestnut Hill, Massachusetts 02167

Department of Physics, Boston College

SHEAR MODIFICATIONS OF ION CYCLOTRON MODES

In continuation of our work^ we have studied the effect of

Shear modifies the normal mode structure by introducing an intrinsic

(S = inverse shear length). An electron drift relative to the ions is

introduced and the effects of shear on the current driven ion cyclotron

damping which is independent of the wavelength for both large and small

shear on the normal mode structure of a uniform, hot, magnetised plasma.

(Pjk) (p^ = ion larmor radius, k = wave number) and is of the order (p^S)

instability are studied. Our results do not agree with a previous study

sharply, from a damping of the order (p^S) for (p^k) > (P^k)^ to a shift

and for nearly perpendicular propagation, we find that depending on the

in the real frequency of the same order for (pjt) < (p^k)^. A shooting

significantly, and indicates that less shear is required for marginal

temperature ratio r (=T^/T^), the effect of the shear changes, rather

code is now being employed to study this problem numerically, without

making any approximation for the potential. Smoothing of this sharp

obtained through our treatment differs from that of Bhadra’s^ quite

stability. Treating the prdblem first at the Weber equation level

(Shear Kinetic Drift) are also introduced and consequences thereof

by Bhadra^ of the same problem. The marginal stability criterion

”^** G. Ganguli and P. Bakshi, Bull. Amer. Phys. Soc. 2J3, 816 (1978).

transition at (p_^k)^_ is expected. Particle orbit modifications^

W. Bellew and P. Bakshi, Bull. Amer. Phys. Soc. 22^ 1089 (1977).

D. K. Bhadra, Plasma Phys. L5, 1185 (1973).

discussed.

ABSTRACT

CURRENT PENETRATION STAGE IN A TOKAMAK*

To study the current penetration stage in a tokamak we use G2M

P. L. Mascheroni, Laura Matte son, and A. L. Sulton Science Applications, Inc., La Jolla, California 92037

with a picture of transport which in addition to neoclassical transport for the ions takes into account the neutral background, wall-plasma effects, and impurities effects. ^ Without the inclusion of resistive tearing modes, runs done in the regime of operation of the Pre-Tex tokamak^ (a = 19 cm, Bp - 10 kG, I =* 75 KAmp at 10 ^sec) shows skin effect in the current. These profiles are unstable to tearing modes. For m > 20 (early times) we enhance the electron transport in the perpendicular direction q ^ = (5B/B)^q^ exp - a (dq/dr). For the remaining modes assuming that the saturated ^ has the same shape as the linear, we solve 7 ^ =(3 at saturation and enhance q ^ in the island width w, q ^ = (5B/B)^, &B being an estimate of the magnetic field in the islands. We have observed the double tearing for which Kadomtsev reconnection

(3) B. Carreras, H. Hicks, B. Waddell, ORNL/TM-6570.

*Work supported by the U. S. Dept, of Energy.

(2) R. Bengtson, private communication.

(1) N. Byrne, private communication.

determine A’(w) = 0

should be used.

dition

to = V^/qR

Princeton, NJ 08544

value computer code.

J. Delucia, S.C. Jardin, and F.W. Perkins

Plasma Physics Laboratory, Princeton University

Simulation of Axisymmetric Alfven Resonance Heating of Tokamaks

is satisfied at one or more radial points. Here we present the results

It was recently reported”** that by oscillating the vertical field in a

tokamak at a frequency to , resonant heating will occur if the resonant con

obtained by simulating this process using a two-dimensional ideal MHD initial

S.C. Jardin, J.L. Johnson, J.M. Greene, and R.C. Grimm, J. Comput. Phys. 29, 101 (1978).

”*“F.W. Perkins and C.F.F. Karney, Bull. Am. Phys. Soc. 23, 864 (1978).

markedly different when the resonance condition is satisfied compared to when

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

It is shown that the time evolution of the plasma is

it is not.

M, C, Vella

BEAM-TURBULENCE ELECTRON HEATING*

Lawrence Berkeley Laboratory, University of California Berkeley, California 94720

The development of electron beam turbulence heating as technique for in situ microwave generation and electron heating in EBT has recently been proposed.* The theory relevant to the interaction of a moderate energy, parallel electron beam with a mirror plasma is reviewed and compared with experiment. Attention is focused on beam-plasma coupling, rather than electron heating as such. Of particular interest is the case of a mirror ratio of 3-5, with

*This work was supported by the Fusion Energy Division of the U. S. Department of Energy under contract No. M-7405-ENG-48.

B. Kunkel, M. C. Vella and B. Feinberg, LBL.

Ri 4.

and

have I nd icated

1-D Reverse Field Pinch Burn S imulations*

high 8, o h m l c a l l y Ignited, and appea r s c o n d u c i v e to q ua s i - s t e a d y o p e r

1 2 Global Reversed Field Pinch (RFP) burn sim ul a ti o ns ’

that the RFP Is a tt r a c t i v e as a r e a c t o r concept. T he RFP Is stable at

R. M. Moses Los A l a mo s S c i en t if i c L a b or a to r y Los Alamos, New M e x ic o 87544

R. A. Nebel and G. H. Mlley Fusion Studies Lab or a to r y N u c le a r E ng in eering Program U n i ve r si t y of Illinois Urbana, Illinois 61801

profiles also peak off -a x is w h i ch s ta bilizes S u y da m modes In the central

near those o f the global studies. However, the r equired m a g n e t i c field

Results have v e r if i ed the f e a s i b i l i t y of ohmic Ignition at p ar ameters

R. L. Hagenson, R. A. Krakowskl, K. I. Thomassen, “A Toroidal Fusion

H. S. Stl mp s on and G. H. Mlley, Trans. Am. Nucl. Soc., 27_, 92 (1977).

In o r d er to f u r t h e r Inv es t ig a te these p roperties, a o ne -d i mensional

React o r Based on the Reversed Field Pinch,” L A - U R - 77 - 23 23 (1977).

R. A. Nebel, G. H. Mlley, and R. W. Moses, Bult. A P S , 23, p. 811

Work supported by U. S. Department of Energy, Contract

Is h i g he r due to off -a x is peaking of the t em p e r a t u r e profile.

Inclusion of a n o ma l ou s t r a n s p o r t enhances this effect.

model has been devised.

No. EY-76-S-022218.

plasma region.

Pressure

ation.

(1978).

max

m m m m

B . )/B . <<1 .

Princeton, NJ 08544

A. Pytte and A.H. Boozer

Helical or Toroidal Symmetry

Neoclassical Diffusion in Plasmas of

Plasma Physics Laboratory, Princeton University

In this case, the neoclassical transport calculations

The neoclassical particle flux across magnetic surfaces has been

magnetic field is assumed to vary only slightly over a magnetic surface

cients calculated previously for the small aspect ratio, axisynnnetric

and toroidally symmetric plasmas. Two cases are considered: First, the

calculated in such a way that the results apply equally well to helically

collision operator is left completely arbitrary, but the magnitude of the

for the two different symmetries become identical, and the diffusion coeffi

tokamak can be carried over to the helically symmetric plasma without change,

variation of the magnetic field and the shape of the magnetic surface left

except for the substitution of appropriate helical parameters for the corre

flux is carried through with the Lorentz collision operator, but with the

sponding toroidal ones. Secondly, a detailed calculation of the particle

completely general, except for the requirements of symmetry, helical or

^Permanent address Dartmouth College, Hanover, N.H. 03755.

Work supported by U.S. DoE Contract No. EY-76-C-02-3073.

toroidal.

ABSTRACT

The equations for low frequency wave propagation (ICRH) have been

LOW FREQUENCY WAVE PROPAGATION IN A HOT TOROIDAL PLASMA*

M. Cotsaftis^ Science Applications, Inc., La Jolla, California 92037

derived in full toroidal geometry with arbitrary cross section. The dielectric tensor^ used here includes both particle effects (Landau damping) and global properties (inhomogeneities, structure of the magnetic field). To this aim, finite Larmor radius expansion has been performed and homogeneity along the field lines has been supposed. To first order, one gets two second order equations, weakly coupled through the toroidal effect, written in a system of intrinsic coordinates related to the flux surfaces, out of which the leading equation, on B^, is an Helmoltz type equation in curved coordinates. Its coefficients are expressed in terms of the components of the dielectric tensor and include, through Z-functions, dissipative effects. Then the singularity in this equation along the ion-ion conversion layer, M = is smoothed out by these dissipative effects, which compete with higher order dispersive terms in Larmor radius expansion. It is shown, by comparing these terms to the lower order dissipative terms, that along the ion cyclotron resonance layer M = O., these later are always the larger, whereas along M = they only overtake as long as the ratio of the minority ions density to the main ions density is: 77 < 77^

(2kyA/rVl07 Z. B. 10^)[§ + (Ak^/l. 7Z^n.)] 10^/cm^, m’\ keV, kG which can be fairly large, =*30% for large enough k^ are observed. So the picture is that for ^ close enough to for M far enough from 0., they significantly lower the available power at M s where ion-ion conversion effects now compete with stochastic effects coming from the collisions of the small wavelength components of the EM fields with the particles, enhancing the lower order absorption affect. So ion-ion conver sion process is masked, if any, and the structure of the field is given by the leading equation for B^ to lowest order in aspect ratio.

*Work supported by the U. S. Department of Energy. *4-

Sy, W. N. C., Cotsaftis, M., Wave Propagation in Hot Nonuniform

lower order absorption effects dominate, whereas

Magnetized Plasma, to be published.

e q u i l i br i um code.

100-150 keV (D*) neutral beams.

We have shown that, by taking adv an t ag e of central

LOW DENSITY IGNITION SCENARIOS USING LNJECTION HEATING*

In o rder to study plasma heating and ignition by neutral injection,

it is possible to ignite a modeled, prototypical reactor plasma using

a-heating, profile effects, and flux surface shifts in e l o ng a te d plasmas,

fluid, one dimensional tra ns p or t code and a two dimensional flux con se r vi n g

a M o n te Carlo neutral injection c omputer code^ has been c o u pl e d to a single

J. A. Holmes, James A. Rome, Y-K. M. Peng, W. A. Houlberg and S. J. Lynch Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

^G. G. Lister, D. E. Post, and R. Goldston, “Computer S i m ul a ti o n of Neutral Beam Injection into Tok am a ks Using M onte Carlo Tec hn i qu e s, ” Paper p re sented at the Third S y m p o s i u m on Plasma Heating in Toroidal Devices (Varenna, Italy, 1976).

Research s p o ns o re d by the Off ic e of Fusion Energy (ETM), U. 5. D e p a r t m e n t of Energy u nder c o n t r a c t W -7 4 0 5 - e n g - 2 6 w it h the Uni on Car bi d e Corporation.

the higher ene rg i es around 150 keV imply f e w er injectors and perhaps lower

beam line e f f i c i e n c y w i t h increasing energy, it is found that a nearly

so that the central core begins to ignite at the time when the neutral

c onstant e x t r a c t e d power is needed for ignition in the range studied.

There is thus little eco no m ic d i f fe r en c e in this energy range.

The density is then increased by peripheral fueling

i mpurity pro du c ti o n rates during heating to ignition.

over the heating req ui r em e nt s in the core region.

To do this, the plasma is started at full

beams no longer pen et r at e to this region.

The fusion a - p a r t ic l es take

Because of the d ec reasing

bore but low density.

However,

1.5 % A $ 3 in circular, elliptic, and D-shaped cross sections. The

The changes in tokamak plasmas undergoing large adiabatic compression

in major radius are examined numerically over the range of aspect ratios:

The fixed boundary approach allows a precise prescription of the plasma

boundary position and shape. During compression the minor radius (a) is

and one-fluid, flux surface averaged energy and particle balance equations.

numerical approach combines the computation of fixed boundary FCT equilibria

TOKAMAK PLASMA VARIATIONS UNDER ADIABATIC COMPRESSION TO SMALL ASPECT RATIOS*

Y-K. M. Peng, J. A. Holmes, D. J. Strickler and S. J. Lynch Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of hnergy under contract W-7405-eng-26 with the Union Carbide Corporation. ^H. P. Furth and S. Yoshikawa, Phys. Fluids

paramagnetic. Despite the large ^ values (^ 30% with q_^.^ - 1, q ^ ^ - 3),

vertical stability of the D-shaped plasma column is enhanced with decreasing A

averaged vertical field decay index (-3tn B^/SRnR) on the mid-plane changes

from -0.7 to +0.2 as A is reduced from 3 to 1.5. This indicates that the

8p(poloidal beta) on the compression ratio (C) differ significantly from

which requires reduced minor radius to preserve the continuity of F at

The present interpretation is that compression to small A dramatically

those proposed by Furth and Yoshikawa^, while the dependences of a, ^

the plasma edge. For D-shaped plasmas with mild elongation (1.6), the

increases the plasma current which lowers gp and makes the plasma more

this tends to concentrate more toroidal flux toward the magnetic axis

(averaged toroidal beta), and P (pressure) show a milder difference.

adjusted iteratively to ensure the invariance of (^g^gg - ^ x i s ^ ’

found that the dependences of Ip(p1asma current) and

(1970), 2593.

F ^ d g e ) ’

and

Laboratory of Plasma Studies

L. Sparks, J. M. Finn, and R. N. Sudan

Cornell University Ithaca, New York 14853

Axisymmetric field reversed equilibria are obtained by a new

Interchange Stability of Axisymmetric Field Reversed Equilibria

method is pertinent to ion rings, field reversed axisymmetric

a toroidal magnetic field. The method allows us to specify the

mirrors, and field reversed theta pinches, all in the absence of

computational method for solving the Grad-Shafranov equation. The

transforms (r,6,z) coordinates to (ijj,e,(j)) coordinates, where ^ identifies

Results of these stability computations are presented for configurations

change criterion assuming unfavorable V” (i.e., unfavorable curvature).

with various pressure profiles and with varying amounts of pressure on

*Work supported under U.S. Department of Energy Contract EY-76-S-02-3170.

pressure on the separatrix as independent parameters. No bifurcations

Jacobian. In particular we have looked at cases where the Jacobian

are observed when this method is employed. As part of a stability

code under development, a mapping routine has been created which

a flux surface and i is determined by specifying the form of the

pressure profile with the pressure on the magnetic axis and the

The mapping routine is used to compute the inter

depends only on

the separatrix.

v a l u e

D. A. D ’ l p p o l i t o , E. A. A d l e r a n d Y. C. Lee

E Q U I L I B R I U M A N D S T A B I L I T Y O F F I N I T E - 6 M U L T I P O L E S

C e n t e r f o r P l a s m a P h y s i c s a n d F u s i o n E n g i n e e r i n g

U n i v e r s i t y o f C a l i f o r n i a , Los A n g e l e s , C a l i f o r n i a 9 0 0 2 4

the v a l u e o f P ’ ( ^ ) c o r r e s p o n d i n g t o m a r g i n a l s t a b i l i t y o f

s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n g i v e n ? b y J o h n s o n e t a l.* f o r the e i g e n

M e h a v e d e v e l o p e d c o d e s t o s t u d y t h e ideal M H D e q u i l i b r i u m a n d s t a b i l i t y

a n d s p e c i f i e d v a l u e s o f t h e coil c u r r e n t s . T h e s t a b i l i t y c o d e s o l v e s the

o f f i n i t e - 8 p l a s m a s c o n f i n e d b y m u l t i p o l e m a g n e t i c f i e l d s w i t h B ^ = 0. T h e

h i g h - n b a l l o o n i n g m o d e s a t t h e g i v e n ^ = c o n s t s u r f a c e . W e w i ll r e p o r t o u r

e q u i l i b r i u m c o d e s o l v e s t h e 6 r a d - S h a f r a n o v e q u a t i o n f o r a g i v e n P(^) p r o f i l e

initial r e s u l t s f o r t h e U C L A t o r o i d a l q u a d r u p o l e e x p e r i m e n t . T h e n u m e r i c a l l y -

ij. L. J o h n s o n , R. M. K u l s r u d , a n d K. E. W e i m e r , P l a s m a Phys. 11^, 4 6 3 ( 1 9 6 9 ) .

q u a r d r u p o l e , w h i c h c a n be s o l v e d a l m o s t c o m p l e t e l y b y a n a l y t i c m e t h o d s .

w i l l b e c o m p a r e d t o t h a t o b t a i n e d f o r t h e l i n e a r

*Mork supported by U S D O E and’ NSF.

o b t a i n e d v a l u e o f

Princeton, NJ 08544

to Stochastic Magnetic Fields

R.G. Kleva, J.A. Krommes, and C. Oberman

Plasma Physics Laboratory, Princeton University

Turbulent Evolution of the Collisionless Tearing Mode due

Magnetic perturbations due to tearing instabilities can lead to the

destruction of flux surfaces and radial diffusion of magnetic field lines.

the stochastic field lines.^ Thus, the tearing instabilities modify the elec

the magnetic turbulence both broadens the layer of particle acceleration and

modify the evolution of the instabilities. In a slab model, it is shown that

As a result, electrons can diffuse radially due to their rapid transport along

tron motion. At the same time, the altered electron motion will self-consistently

J.A. Krommes and R.G. Kleva, Princeton Plasma Physics Lab. Rept. PPPL-1522 (1979).

Work jointly supported by U.S. DoE Contract No. EY-76-C-02-3073 and U.S. AFSOR Contract.No. F 44620-75-C-0037.

adjacent field lines. We have previously shown that L - i and L, - i (k D i ) B

rate. Saturation can occur by quasilinear relaxation of the background current

modes is estimated. Formally, the v^n*Vf streaming nonlinearity in the drift

\j.A. Krommes, R.G. Kleva, and C. Oberman, Princeton Plasma Physics Lab. Rept.

k is a typical wavenumber.^* For a sufficiently low turbulence level, one has

Interaction Approximation by neglecting a mode coupling term, is used to derive

The influence of these effects on the nonlinear growth and saturation of the

magnetic field. A statistical closure approximation, obtained from the Direct

acteristic lengths: L^ , an autocorrelation length which is inversely propor

tional to the spread of the spectrum in parallel wavenumber, and L^ , a non

kinetic equation is studied, where h is the direction of the fluctuating

also causes a ponderomotive renormalization of the background distribution.

linear mixing length which describes the rate of exponential divergence of

In this regime, stochastic diffusion can initially enhance the growth

or by sufficient turbulent broadening of the perturbed current layer.

a nonlinear dispersion relation.

is the magnetic diffusion coefficient, and

The theory depends crucially on two char

is the shear length,

s y m s

PPPL-1389 (1978).

L ^ < L ^ .

where

o s

-1/3

DIFFUSE VLASOV-FLUID SCREW PINCH*

We present a method for the numerical solution of linearized stability problems

economical with computer storage, a main difficulty is solving inhomogeneous

large amounts of storage by computing the dispersion matrix elements as numerical

the manner in which the eigenvalue appears in the equations, requiring that large

collisionless plasma stability problems. The reason for this difficulty is due to

arrays must be stored for later iteration upon the eigenvalue. We manage to save

applied with considerable success to the sharp boundary screw pinch^ and the rotating

for the diffuse Vlasov-fluid screw pinch. The theory behind the technique has been

diffuse theta-pinch.^ Here we present a version of the method which is much more

C. E. Seyler andH. R. Lewis University of California Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87545

inner product of the force operator with a finite element representation of the

functions of the eigenvalue. Since the matrix elements are smooth functions of the

eigenvalue only a few points in the complex plane are required to give an accurate

representation of the dispersion matrix. We form the dispersion matrix by taking the

electron fluid displacement. A complete discussion of the analytical techniques and

H. R. Lewis and J. P. Freidberg, Proc. of the Fifth European Conf. on Controlled

*Work performed under the auspices of the U.S. Department of Energy.

Fusion and Plasma Physics (Grenoble, France, Aug. 21-25, 1972).

C. E. Seyler, submitted to Physics of Fluids.

the numerical algorithms will be presented.

^pcn.v

Oak Ridge , Ten.r

P. uirshner*., ^nu L.

Oak Ridge la tier;2l L<obcr3torv

excursion of thos? croscont

i . 1 s ” r. ? , J . D . C a 1l^p., C . L .

orbits coinoorod vrith ^*h?t nf

CFECCENI SpspF O^-EIT niFP!PPM in ?PT#

collision operators or are valid.in more Collisional regimes.

tekamaks), they make significant contributions to diffusion.

In the ccllisionless regime, particles in hET with magnetic poloidal

the banana orbits in tckamaks). Bee?us? of the relatively large radial

orrC€ssicn drift v ** i c c 11 i p s th?t p ? P r 1 y cr?rc^i tnp P y p ^ ^ i f r ^ < r ^ i ^ f-

in their poloidal motion, forming crescent-shaped drift orbits (analogous tc

simultaneously included the effects of these orbits and differential

ccllisionless regime. Previous neoclassical transport calculations ha^e net

circulating particle orbits (analogous to the untrapped particle orbits in

^Research sponsored by the Office of Fusion Energy (ETM), U.S. of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

entropy production arguments.^ However, the overall scaling depends also on

the radial and poloidal ambipolar potential and hence on the transport of

drift kinetic equation in the small collision frequency limit, using

aspect ratio , which is significantly different from the scaling in most

the radial width of the crescent shaped orbit is essential in the lowest

order drift kinetic equation; and (ii) the longitudinal adiabatic invariant

major differences distinguish this from the tokamak banana calculation: (i)

ccllisionless regime is found to scale as the square root of the inverse

An analytic calculation of the diffusion coefficient due tc the

techniques similar to those employed in calculations for tokamaks.

scaling agrees with the scaling obtained from

^Catto, Rosenbluth, and Tsang, this meeting.

crescent shaped orbits is presented here.

determines the trapping boundary.

We solve the bounce averaged

The diffusion coefficient in

Collisional regimes. This

the off-resonance species.

Department

the

Two

r r

ee 3*830

ABSTRACT

Oak Ridge National

OXE-DIMEXSIOXAL TRANSPORT SOLUTIONS FOR EBT-11

Jaeger and C. L. Hedrick Laboratory, Oak Ridge, Tenne

Recently [1], one-dimensional radial transport solutions for the

sionality (plateau regime [3]). In this paper, we extend these calcula

regime as observed in experiments [2]. In these calculations, resonant

are small. For ion temperatures characteristic of EBT-1, this leads to

ion transport coefficients which are approximately independent of colli-

diffusion of ions is included in regions where poloidal drift frequencies

ELMO Bumpy Torus (EBT-1) have been obtained in the collisionless electron

Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of Energy under contract W-7405-end—26 with the Union Carbide Corporation.

^Hazeltine, R. D. and Krall, N. A., SAI Report SAI-78-855-LJ/LAPS 44 (1978).

tions to the higher temperature and density regime proposed for the EBT-11

experiment [4]. Results show somewhat hollow density profiles in steady

exchange neutrals in EBT-11 is increased by a factor of 20-30 over that

state due to the presence of off-diagonal neoclassical transport coeffi

cients. In addition, the sputtered flux of aluminum due to charge

“EBT Experimental Croup, 0RNL/TM-64S7 (1978).

^Jaeger, E. F. et al., ORNL/TM-6806 (1979).

^Dandl, R. A. et al., 0RNL/TM-5955 (1978).

found in EBT-1 calculations.

ABSTRACT

ENHANCED TAIL FOR IONS IN EBT

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

C. L. Hedrick, R. A. Dory, E. F. Jaeger, and D. A. Spong

Experimental observations of EBT suggest that the distribution

dominates the diffusive loss processes and that the source of particles

and sinks which yields such an “enhanced tail”. The two features which

are critical to this analysis are that a narrow region of velocity space

function for well trapped ions can have an “enhanced tail” [1]. Here we

present analytic kinetic calculations based on simple models of the sources

d by the Office of Fusion Energy (ETM), U.S. Department mtract W-7^05-eng-26 with the Union Carbide Corporation.

slope of the distribution function is approximately constant (as a function

of energy) to either side of this critical energy and is smaller in magni-

(ionization of neutrals) is at lower energies. For well trapped particles

from a Maxwellian could lead to larger electric fields (factor of 2) than

“enhanced tail” has implications for further development of neoclassical

into account this distortion of the lowest order distribution function

(v^. - 0) this zone occurs for energies approximately equal to eb- The

bution is a natural consequence of neoclassical theory for EBT. An

theory for EBTS. For example, simple arguments suggest.that taking

These calculations suggest that an “enhanced tail” on the ion distri

.1.. “Measurements of Plasma Properties in BBT-1”, submitted

presently obtained from 1-D transport calculations.

tude for higher energies (i.e. an enhanced tail).

“Dane!. R. A. et a tc Nuclear Fusion.

Research sponsore

of Energy under co

DRHL.‘TM-ni”.

IN A MAGNETIZED PLASMA WITH A TEMPERATURE GRADIENT

A ONE-FLUID MODEL OF MAGNETIC FIELD FLUCTUATIONS

The time-dependent thermal fluctuations in a magnetized

I . M. Tkachenko * Department of Physics & Astronomy University of Maryland College Park, Maryland 20742

(Bo=J3oz) plasma with a temperature gradient (VT,-,=ay) are considered. A one-fluid MHD model with all damping effects is utilized. The system under consideration is supposed to be stationary and spatially homogeneous. The assumptions of local equilibrium [1] and ^ocal charge neutrality are made. The Fourier-transform duced by the following equation:

fluctuations is obtained explicitly. The influence of the temperature gradient on the propagation and damping of the modes in the plasma is analyzed. The static correlations of the magnetic field are also studied [2].

The dynamic and static correlation functions of the various hydrodynamic quantities may also be investigated using the same method (for comparison see [3]).

'''Permanent address: Odessa University. Odessa, 270000, U.S.S.R [1] A. 1. Akhiezer,et. al., Plasma Electrodynamics, §11.6,

where k = (k^,ky,k^), K = (kg,kg), & is the length of the sys tem in^the gradient direction.

[3] V. P. Leshikov. I. Z. Fisher, Sov. Phys.-JETP.. j40. 667

The results obtained will be useful in explaining trans

The dynamic correlation function of the magnetic field

The author wishes to acknowledge here the hospitality

[2] L. D. Landau, E. M, Lifshitz, Statistical Physics,

of the University of Maryland.

(2t) dtcdm<i)(k,m)exp(ik-r-imt)

of any quantity <)) is intro

Pergamon Press, 1975-.

Pergamon P r e s s , 1958.

port in tokamaks.

‘b *b

4<(t „t) = &

(1975).

*b ‘b

—3

-ll

‘b

New York University

Rotation of a Toroidal Plasma

Shih-liang Wen*and Young-ping Pao

Courant Institute of Mathematical Sciences

It is found that the poloidal velocity component is

classical formulas of stress tensor in the MHD equations.

completely determined in terms of the instantaneous plasma

variables (B,P,T), independent of the initial conditions or viscosity coefficients, while the toroidal velocity

The toroidal and poloidal velocity components of an axisymmetric toroidal plasma are calculated by using the

toroidal velocity components as well as the formula for determining the rate of change of the toroidal velocity component. The special case of a large-aspect ratio plasma

This work was partially supported by DOE Contract No. EY-76- C-02-3077. *

instantaneous plasma variables and is independent of the initial conditions of the toroidal velocity and viscosity.

On leave from the Mathematics Department, Ohio University Athens, Ohio.

mining whether it increases or decays depends only on the

Explicit expressions are obtained for the poloidal and

can increase or decay in time. The criterion for deter

On the other hand, the toroidal velocity component

component depends on the initial conditions.

is also examined.

(212) 460-7127

ABSTRACT

Livermore, CA

D. Schnack and J. Killeen

IN FINITE BETA REVERSED FIELD PINCHES*

THE NONLINEAR EVOLUTION OF RESISTIVE INSTABILITIES

National MFE Computer Center Lawrence Livermore Laboratory

There is currently renewed interest in the Reversed Field Pinch

concept as a means of confining a hot plasma. Such magnetic field configurations are characterized by a reversal of the axial field in the outer regions, and allow for higher values of plasma B than do tokamaks. The use of ohmic heating to achieve thermonuclear temperatures is also a possibility.

We use a non-linear, two-dimensional, MHD computer code4 to study the non-linear behavior of resistive modes in the equilibria described above. For the Pitch and Pressure model we find that the m=l tearing mode can trigger the slow resistive interchange, which leads to localized interchange vortices near the x-point with subsequent convolution of flux surfaces. For the tearing mode stable equilibria, the m=l inter change remains relatively benign, while the m=0 mode can lead to large magnetic islands and highly distorted flux surfaces.

The linear stability of these devices is well known. An analytic equilibrium (the Pitch and Pressure model) which is stable to ideal MHD modes has been found with B - 30%1. This was subsequently found to be unstable to tearing modes^. More recently, equilibria which are stable against both ideal and tearing modes at values of central B up to 18% have been found^. However, these equilibria are unstable to slow resistive interchange modes.

D. C. Robinson, Plasma Phys. 1^, 439 (1971)
J. A. Dibiase, LLL Report UCRL-51591 (1974).

D. C. Robinson, Nucl. Fusion 18^ 939 (1978). 4. D. Schnack and J. Killeen, submitted to J. Comp. Phys.

by the Lawrence Livermore Laboratory under contract number M-7405-LNb-4H

*Work performed under the auspices of the U.S. Department of Energy

Octupole* M.W. PHILLIPS, University of Wisconsin- Madison— A computer code has been developed for looking at ideal magnetohydrodynamic equilibria in the Levitated Octupole. We use this code to study high beta equilibrium in conjunction with experiments now going on in this device. Results for betas up to 8%, as defined locally on the separatrix in the bridge regions, are considered. We also look at ideal MHD stability with respect to ballooning modes and attempt to set an upper limit for beta for this device.

MHD Equilibrium and Stability of the Levitated

*Work supported by USDOE.

Abstract

T. E. Stringer

The effect on

Since the total fluxes most

In the standard neoclassical

Impurity Transport in Tokamaks.

JET Joint Undertaking, Abingdon, Oxon,England, U.K.

Interaction between Anomalous Loss and Neoclassical

Neoclassical transport theory for an impure toroidal plasma predicts that high-Z impurities diffuse towards the centre until their density profile becomes sharply peaked. this transport of anomalous electron loss, such as is observed in all Tokamaks, will be examined. theory the dependence on the radial electric field is eliminated by using the ambipolar condition. be ambipolar, this ambipolar electric field may be changed by the presence of anomalous electron loss. influence the diffusion of impurity ions, because of their large ionic charge. addition of the two independent fluxes.

break-up of magnetic surfaces by MUD,resistive, or kinetic instabilities, leading to ergodisation of the field lines. outward directed ambipolar electric field may be needed to restrain the outward electron flow along field lines, reversing the field found when neoclassical transport is considered alone. The effect of this on impurity transport will be evaluated. This offers a possible explanation of some experimental results.

One possible mechanism for anomalous electron loss is the

The effect may be much larger than a simple

This would strongly

ABSTRACT

POWER REQUIREMENTS OF EBT ELECTRON RINGS*

Two related models are developed to describe the rings of

for Model B electrons can always reach resonant surfaces even in

for Model A heating is limited by the rate of pitch angle scattering;

energetic electrons necessary for MHD stability of the EBT plasma.

Both assume electron cyclotron resonance heating and differ in that:

G. W. Stuart Science Applications, Inc., La Jolla, California 92037

microwave power was available), and 1000-1500 M W for the EBTR-48

estimate the Collisional power loss between the ring and the toroidal

monotonically from approximately the toroidal electron temperature

conceptual reactor (4000 M W thermal output). Most loss occurs at

plasma. These estimates are insensitive to the Model A/Model B

the absence of pitch angle scattering. The resulting ring particle

relatively low energies where Collisional rates are large. These

results suggest that contrary to previous estimates EBT reactor

to some high energy cutoff. The calculated spectra are used to

-difference, and show loss rates of 10-20 kW in EBT-1 (60 kW

distributions resemble cosmic ray type spectra, and decrease

operation may require large recirculating power fractions.

*Work supported by DOE.

ABSTRACT

J. J. Stewart, Y. Matsuda and H. L. Berk

QUASILINEAR RADIAL TRANSPORT SIMULATION OF TMX PLUGS

Lawrence Livermore Laboratory, University of California Livermore, California 94550

The end plugs of the Tandem Mirror Experiment (TMX) are expected to follow 2XIIB scaling based on a quasilinear theory of DCLC turbulence.; It has been realized recently that radial plasma transport due to the DCLC turbulence and charge-exchange on cold background gas near the plasma surface may contribute significantly to particle and energy loss. These effects are important in determining the plug profiles which will affect the DCLC stability as well as confinement. We study this problem using the two-dimensional quasilinear radial transport code. At fixed turbulence level, the code was first verified to produce steady states consistent with 2XIIB experimental observations. We then run the code with the TMX design parameters to obtain steady state density, temperature and confinement times. The results in comparison with the design parameters will be reported. The simulation results with self-consistent calculation of the DCLC turbulence will also be reported.

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48.

MOST PROBABLE MED EQUILIBRIA AND THEIR STABILITY

John Ambrosiano and George Vahala (William and Mary)

A systematic (statistical) approach to determining ideal incompres

sible MHD equilibria subject to given information on the values of a few- global constraints is presented. This is not only more realistic experimentally but also in marked contrast to the standard theoretical MHD approach of arbi trarily specifing two profiles and then determining the remaining third profile from the Grad-Shafranov equation. In our approach, utilizing the information theory formulation of statistical mechanics-^-we obtain (for ID systems) a set of coupled ordinary differential equations whose solution yields the “most probable” MHD equilibrium consistent with the given information on the global constraints: total energy E, magnetic helicity H^, toroidal current 1^ and toroidal flux ^ on some subset of these. Typically, if H^ is prescribed, the most probable states are of the screw pinch or reversed-field-pinch type with finite pressure while if H^ is unspecified the profiles are those for a low P Tokamak. Relaxation of constraints E and H^ results in uniform J^ and B^, consistent with the information theoretic approach^.

generated by varying the constraints E, H^ and 1^. These are plotted on the usual F - 9 diagram and the stability of these profiles to the m = 1 kink mode is examined. The relation of most probable (P ^ 0) states to the force- free P = 0 Taylor*^ states is also considered.

Approach (San Francisco; W. H. Freeman, 1 9 6 7 ). 3j. B. Taylor, in Pulsed High Beta Plasmas ed. D. E. Evans (Pergamon, Oxford,

^“Montgomery, Turner, and Vahala, Journal of Plasma Physics, April 1979. ^Ammon Katz, Principles of Statistical Mechanics: The Information Theory

For fixed (f>g, profiles with varying degree of field reversal are

1976), p. 59.

Abstract

and R.D. Hazeltine

Austin, Texas 78712

Anomalous Current Penetration

Swadesh M. Mahajan, Daniel A. Hitchcock,

Recent experiments on PLT indicate that the current

penetration in the start-up phase of a Tokamak is much

more rapid than predicted by magnetic field diffusion and

Fusion Research Center The University of Texas at Austin

by even extremely small fluctuations can provide effective current penetrations. Further application of the newly derived turbulent transport coefficients to Tokamak plasmas’ is also discussed.

neoclassical resistivity. We propose here that this could be due to the turbulent electron viscosity. An expression for the turbulent viecosity is derived, and is used in the electron equation of motion, which is solved alongwith

Maxwell’s equations, using a one-dimensional code. Preliminary results show that anomalous viscosity caused

i ^R.J. Hawryluk, Bull. Am. Phy. Society 6B7, 23(78).

This work is supported by the U.S. Department of Energy

Contract DE-AC05-79ET53036.

PEST II

Princeton, New Jersey 08544

R. C. Grimm and R. L. Dewar

Plasma Physics Laboratory, Princeton University

Theoretical understanding of the nature of ideal MED instabilities^ and considerable experience with existing numeri cal techniques and their limitations has enabled significant improvements in our ability to formulate ideal MHD instability problems computationally. As a first step in extending our computational treatment of ideal linear MHD modes to allow for resistive instabilities in general axisymmetric toroidal configu rations, we have constructed what is essentially a second generation PEST c o d e . 2?3 This uses a scalar version of 6W similar to that found by B i n e a u . 4 Although restricted in its ability to find exact normal modes and growth rates, it is directly appli cable to the determination of marginal stability and thus addresses many problems of practical importance (including that of 8-limits in tokamaks). Generalizations in the formulation enable it to be applied to configurations like spheromaks and reversed-field pinches for which the earlier version had limitations. These improvements have resulted in considerable reductions in computer requirements and thus, while it is now possible to apply these methods in a more routine fashion with other activities (e.g., plasma modeling codes, comparison with experimental data), it should also be possible to study more complex situations, such as behavior very close to marginal stability and larger toroidal mode number instabilities. Here, we describe the basic features of the formulation, present the results of a comparison study with the previous version of PEST and, if time permits, give some examples of its application.

^M. S. Chance, R. L. Dewar, E. A. Frieman, A. H. Glasser, J. M. Greene, Y-Y. Hsieh, J. Manickam, and A. M. M. Todd, Paper OBI, Sherwood Meeting 1978. R. C. Grimm, J. M. Greene, and J. L. Johnson, in Methods in Computational Physics, Vol. 16, J. Killeen, ed. (Academic, NY 1976), p. 253^

Work supported by U. S. DoE Contract No. EY-76-C-02-3073.

Princeton, NJ 08544

in Axisymmetric Geometries

portional to J^(k,p) (where

Barry E. Hynick and John A. Krommes

Plasma Transport by Stochastic Magnetic Fields

Plasma Physics Laboratory, Princeton University

A canonical framework developed by Kaufman for particle diffusion in

square of field-particle coupling coefficients g . The dominant g is pro

a typical perpendicular wavelength of the turbulent spectrum, and p is the

into the formalism. The expression for the diffusion tensor D involves the

In particular, the particle drifts present in any realistic geometry are built

axisymmetric geometries is adapted to enable a more systematic study of plasma

transport due to magnetic perturbations of axially symmetric equilibrium fields.

*^*A.N. Kaufman, Phys. Fluids L5, 1063 (1972).

A.B. Rechester and M.N. Rosenbluth, Phys. Rev. Lett. 4B, 38 (1978).

Work jointly supported by U.S. DoE Contract No. EY-76-C-02-3073 and U.S. AFOSR Contract No. F 44620-75-C-0037.

for runaway electrons in the presence of drift or tearing turbulence, one may

vious estimates. This may provide an explanation, alluded to in Ref. 2, for

gyroradius of any given particle), and thus D is down from the zero gyro-

have (k,p) *“1, so D for these particles will be greatly reduced from pre

radius result of previous t h e o r i e s ^ b y a factor J^(k^p) . For ions or

the anomalously long confinement times of runaway electrons in tokamaks.

J.A. Krommes, Princeton Plasma Phys. Lab. Rept. PPPL-1462 (1978).

is,the Bessel function of index 0, k^ is

and

S. Rehker

J. A. Tataronis

D. T. Anderson and J. L. Shohet

Max Planck Institute fur Plasmaphysik

Garching-bei-Munchen, Federal Republic of Germany

NEUTRAL BEAM HEATING CALCULATIONS FOR TORSATRONS*

The University of Wisconsin, Madison, Wisconsin 53706

Courant Institute, New York University, New York 10012

It has been demonstrated, both theoretically and experimentally,

These results have been obtained with a fully three-dimensional code originally developed by Dei-Cas^ for tokamak geometry and modified by S. Rehker^ to take into account Stellarator fields and geometries. The method of calculation used in the code is based on the single par ticle guiding center equations and employs Monte-Carlo techniques to model the Collisional transfer of the beam energy to the plasma. a 16 field period a=3 torsatron of minor radius 35 cm, major radius 350 cm and toroidal field of 30 kG with plasma density of l O ^ / c n ^ and ion temperatures of 1 keV, 97% heating efficiency was obtained with tangential injection. The efficiency decreased to under 20% when the injection angle with respect to the magnetic axis was increased to 65 degrees. The results show that ionized particles born near the outer edge of the plasma tend to escape, but good confinement occurs near the plasma center. As the angle of injection is increased, one observes confinement only for those particles born well inside the plasma.

that tokamak plasmas can be heated efficiently through neutral beam in jection. The status of this method is less clear for Stellarator plasmas, although recent computational results imply heating efficien cies of more than 97% for tangential injection of the beam in a torsa- tron of moderate aspect ratio. Significant heating has been observed experimentally for tangential injection in the Cleo Stellarator.^

*This work was supported in part by the National Science Foundation under gfant ENG 77-14820 and in part by the U.S. Department of Energy under Contract No. ET-73-S-02-5069.

iD. J. Lees, et al., Proc. IAEA Innsbruck Meeting (1978).

2J. Dei-Cas,Varenna School of Plasma Physics (1976).

3s. Rehker, Max Planck IPP Report 2/237 (1978).

For

ABSTRACT

COMPUTER MODEL OF A SLOW RFP*

An old code (G2M) has been adapted to the calculation of the

ZT-40 size. The basic equations are those of Ref. 1, though the

slow diffusion of profiles to be expected of a reversed field pinch of

boundary conditions have been altered to allow for a resistive shell

R. N. Byrne and C. K. Chu** Science Applications, Inc., La Jolla., California 92037

**Permanent address Columbia University, New York, New York 10027.

^Christiansen, J. P. and Roberts, K. V., Nucl. Fusion N3 (1978) 181.

and the transport coefficients are enhanced, following Christiansen

and Roberts in Suy dam-unstable regions. The roles of impurities

^Byrne, R. N. and Klein, H. H., J. Comp. Phys. 26 (1978) 352.

*Work supported by U. S. Dept, of Energy.

and neutrals are examined.

Windows for MHD Kink Modes

Princeton, New Jersey 08544

Effect of Toroidal Curvature on Stability

Plasma Physics Laboratory, Princeton University

behavior of the stable window for kink modes for different

As part of a parametric survey to investigate the behavior

of ideal MHD instabilities in tokamaks, we are studying the

J. Manickam, J. M. Greene, J. L. Johnson,’ and A. E. Miller

current and pressure distributions and aspect ratios. These

4- ‘On loan from Westinghouse Research and Development Center.

we find agreement with cylindrical calculations.^ The width of

dominant poloidal mode number. In the large aspect ratio limit

plasma-vacuum interface, n the toroidal mode number, and m the

^E. A. Frieman, J. M. Greene, J. L. Johnson, and K. E. Weirner,

the stable window decreases with decreasing aspect ratio, and

should occur near nq ” m - 1 with q the safety factor at the

Work supported by U. S. DoE Contract No. EY-76-C-02-3073.

Phys. Fluids 16^ 1108 (1973).

also with decreasing shear.

E. Ott

The Goodness of Ergodic Adiabatic Invariants

For a “slowly” time dependent Hamiltonian system exhibiting

(This invariant has proven to be useful for discussing particle

ergodic motion, the 2N dimensional phase space volume inside the

to other plasma fusion problems where ergodic particle motion is

motion in field reversed geometries,*** and should have application

hypersurface, Hamiltonian equals constant, is an adiabatic invariant.

Department of Electrical Engineering, Cornell University Ithaca, New York 14853

and scales like (i^/i) ”, where r^. is a certain correlation time of

the ergodic motion, and r is the time scale over which the Hamiltonian

prevalent.) It is shown that the error in the constant is diffusive

lR. V. Lovelace, Phvs. Fluids (1979).

changes.

1/7

= 7

D,.!

3 D i v e r t o r Toka:

Fueling of a long-pulse divertor tokamak is modeled with the 1-D

H. C. Howe Oak Ridge National Laboratory Oak Ridge, Tennessee 37330

transport code PROCTR. ’ Flow to the divertor plate is at the ion sound speed and cross-field diffusion in the scrapeoff is of the order of Bohm (^10^ cm*/sec). We consider fueling oy neutral beams MW, Eggj^ = 50 keV), gas and pellets of a machine with major radius R^ = 150 cm”, wall radius a^ - 55 cm and separatrix radius a^ - 45 cm. Neutral beam fueling alone results in a scrapeoff transparent to impurities. Gas puffing thickens the scrapeoff but does not fuel the plasma when the puffing rate is limited by an assumed divertor pumping rate of 10*^ sec”**. Pellets are shown to fuel the plasma using existing technology (v ^ = 1 km/sec). We conclude that gas puffing is needed to control tne opacity of the scrapeoff to sputtered impurities while pellets are needed to fuel the plasma.

of Energy under contract W-7 ^0 !5-eng-26 with the Union Carbide Corporation.

“Research sponsored by the Office of Fusion Energy, U. S. Department

3 B 4 5

ABSTRACT

Arthur E. Walstead and William A. Newcomb

The one dimensional cubic nonlinear Schrodinger equation ( i E ^ + p E ^ + q]E]^E=o)

MODULATIONAL THEORY OF THE CUBIC NONLINEAR SCHR0DIN6ER E Q U A T I O N

Lawrence Livermore Laboratory, University of California Livermore, California 94550

has two well known solutions in the form of uniformly translating cnoidal waves classified by the sign of the constant pq. These waves are determined, apart from phase constants, by the values of four parameters. We use Whitham’s averaged variational principle^ to derive a determined quasilinear system of evolution equations for the values of these parameters when they are regarded as slowly varying in space and time. The stability of the waves is also determined by examining the quasilinear system for local hyperbolic or elliptic character.

*Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48.

^G. B. Whitham, Linear and Nonlinear Waves, (Wiley, New York, 1974)

TOROIDAL PINCH EQUILIBRIA WITH FLOW*

theta pinch is examined in particular.

Rotation is found to have interesting

consequences on the forms of the equilibrium.

*Wo rk performed under the auspices of the U. S. Department of Energy.

The Morozov-Soloviev^ magnetohydrodynamic equations with ideal compressible

stationary flow are solved for some axisymmetric configurations. The reversed field

R. Y. Dagazian University of California Los Acientific Laboratory, Los Alamos, New Mexico 87545

A. 1. Morozov and L. S. Soloviev, Soviet Phys., Doklady, .8, 243 (1963).

Coupiinp and Penetration of Whistter Waves in

Inhomogeneous Piasma.^ K. S. THEtLHABER, MiT— Linear Excitation of Whistter waves in the tower-hybrid frequency range is described. These waves are excited at the piasma edge by eiectric fietds perpendicuiar to those required for the tower-hybrid excitation. W e consider coupiing from a waveguide array and find power refiection as a function of array design and density gradient at the edge. W e then consider propagation into the ptasma and caicuiate fieid structure as a function of distance of penetration.

^ Work supported by U. S. Department of Energy Contract

(ET78-S-02-4682).

3 B 4 8

A new method for efficient computer simulation of long time-scale

Orbit-Averaged Particle Codes for Long-Time Simulations* T. A. Brengle, B. 1. Cohen, D. B. Conley, and R. P. Freis Lawrence Livermore Laboratory

phenomena has been proposed and has proved successful in one- and two-dimensional magneto-inductive particle codes. The method relies on orbit-averaging charge and current densities in Maxwell’s equations before solving for the self-consistent electric and magnetic fields, in otyder to both filter out high-frequency phenomena and reduce the number of simulation particles necessary to adequately fill phase space. This activity is motivated by the desire to efficiently simulate evolution of plasma over long time intervals compared to particle orbit periods.

We have modified the one- and two-dimensional magneto-inductive particle codes MAGIC and SUPERLAYER . These codes use a Darwin model to calculate self-consistent fields from the plasma current provided by finite-orbit ions, assume charge neutrality, and neglect electron dynamics (appropriate assumptions for open magnetic field lines and T^ >> T^). SUPERLAYER also models neutral beam deposition, r-f, and other mirror physics. the cyclotron orbits of the ions. To approximately time-center the difference equations, we use a predictor-corrector method. We have achieved numerically stable code operation averaging over several cyclotron periods with one or two corrector iterations. Results of the new codes agree with those of their predecessors which do not orbit-average, but some physics improvements are evident. The new codes very effectively filter high-frequency noise associated with discreteness of the injection models used when studying neutral beam build-up to high beta and field-reversal. This results in cleaner and more realistic ion orbits. A factor of two reduction in the number of particles needed has been achieved, but this is problem dependent. Refinements such as temporal interpolation of the fields on corrector iterations and digital smoothing of the currents will be described, along with details of the equations, algorithms, and tests made so far.

T. A. Brengle and B. 1. Cohen, Lawrence Livermore Lab. Report UCID-17795, Rev. 1.(1978) J. A. Byers, Phys. Rev. Lett. 39, 1476 (1977)

“Work performed under the auspices of U.S. Department of Energy by the Lawrt Livermore Laboratory under contract nun W-7405-ENG-48.”

In the orbit-averaged codes, the plasma current is averaged over