Instabilities and vortex dynamics in shear flow of magnetized plasmas T. Tajima, W. Horton, P. J. Morrison, J. Schutkeker, T. Kamimura, K. Mima, and Y. Abe Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712 (Received 19 March 1990;a ccepted 1 November 1990) Gradient-driven instabilities and the subsequentn onlinear evolution of generatedv ortices in shearedE X B flows are investigated for magnetized plasmasw ith and without gravity (magnetic curvature) and magnetic shear by using theory and implicit particle simulations. In the linear eigenmodea nalysis, the instabilities considered are the Kelvin-Helmholtz (K-H) instability and the resistive interchange instability. The presenceo f the shear flow can stabilize thesei nstabilities. The dynamics of the K-H instability and the vortex dynamics can be uniformly described by the initial flow pattern with a vorticity localization parameter E. The observedg rowth of the K-H modes is exponential in time for linearly wIsEable modes,s ecular for the marginal mode, and absent until driven nonlinearly for linearly stable modes.T he distance between two vortex centers experiencesr apid merging while the angle 6 between the axis of the vortices and the external shear flow increases.T hese vortices proceed toward their overall coalescence,w hile shedding small-scalev ortices and waves. The main features of vortex dynamics, the nonlinear coalescencea nd the tilt or the rotational instabilities of vortices, are shown to be given by using a low-dimension Hamiltonian representation for interacting vortex cores in the shear flow. 1. INTRODUCTION In this work, we extend the previous work ’ by investi- gating the shear flow effects on the gravitational instability The presenceo f shear in the flow of neutral fluids and and the magnetic shear effectso n the K-H and R-T instabi- plasmasg ives rise not only to instability of the shearedl ayer, lities. Also, the detailed analysiso f the nonlinear evolution of i.e., the Kelvin-Helmholtz (K-H) instability, but also to large size vortices is presentedh ere. stabilization of other instabilities, the interchange mode [ Rayleigh-Taylor ( R-T) instability], for instance. Resis- In magnetic confinement devices the shear flow occurs tive-interchange-driven turbulence has been proposed as a at the boundary between the rotating core plasma and the mechanismf or the anomalous thermal transport in stellara- wall or limiter. The magnitude and direction of the core rota- tors and in edge plasmas of tokamaks. Recent calculations tion is determined by the strength of the nonambipolar loss indicate that a strong nonuniform radial electric field can rates leading to the charge-up of the plasma. The mirror or suppresst he interchange ’ and resistive pressure-gradient- open field line confinement systemh as an intrinsically faster driven instabilities. ’ The fluid dynamics of shear flows under electron loss rate leading to the net positive potential of sev- the influence of gravity is also important for the problem of eral times the electron temperature. In the stellarator with an imploding inertially confined plasma. In the initial phase strong electron cyclotron heating there is also a dominant of implosion, short-wavelength modes are stabilized by the electron Ioss and positive charge to the plasma. In contrast, ablative flow and relatively long-wavelength modes can for stellarators with neutral beam injection or ion cyclotron grow on an ablation surface.3 P4L arge-scalev ortices excited heating and, in general,f or tokamaks, there is a net radial ion by the R-T instability are adiabatically compressed, and loss rate from finite ion orbits size effects and the plasmas thus increase in strength during the implosion. It appears build up a substantial, of order the ion temperature, negative that the shear flows associatedw ith large-scale-lengthv orti- potential. The positive potential plasmas rotates in the ion cess uppresst he short-wavelength R-T mode in the stagna- diamagnetic direction and the negative potential plasmasi n tion phaset hat occurs during the final phase of the implo- the electron diamagnetic direction. In typical stability analy- sion. The presenceo f vortices can also influence the nature of sis the assumption is made that the rotation is sufficiently turbulence and associated transport. In the isotropic two- close to a solid body rotation and sufficiently slow that the dimensional (2-D) Navier-Stokes turbulence the well- only effect is to Doppler shift the wave frequenciesf rom the known Kolmogorov power spectrum of k - 3 developed values calculated in the absenceo f rotation. The conditions for the limit of this approximation are given in Ref. 1 for the from spacef illing small-scale eddies. However, we find that rotating cylindrical plasma wi#h o*, and wlc, drift modes.I n the turbulence power spectrum changest o a steeper power law in kin the presenceo f vertical structure in the fluid in the the presenceo f shearf low we can estimate the condition for a wave number regime on the scale of the vortices. Thus the strong effect of the shear flow on a mode of growth rate yk Y , , presencea nd dynamics of the vortices may strongly affect wave number k,,, and the mode width Ax by the condition the macroscopic behavior of turbulence. k,, Ax u’> yk,. 938 Phys. Fluids B 3 (4), April 1991 0899-8221/91/040938-l 7$02.00 @ 1991 American Institute of Physics 938 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp Here, we consider a configuration of plasma density, shear Shainge ta l. ’note that without considering the stability flow, and magnetic sheara s shown in Fig. 1, where U ’= u/a. problem there may arise improved confinement resulting Applying this condition to the valueso f k, Ax, and yk for the from the shearf low layer. Biglari et al.” also discusst hat the interchange, resistiveg , and the drift wave, gives a first esti- shear flow in itself may reduce the transport. A simple sin- mate for the shear flow required to reduce the growth rate. gle-moded escription of the shearf low reduction in transport Table I shows the condition on U ’obtained from this crite- is given by the convective cell island width formula ’ rion for several forms of plasma turbulence. Sincet he shearedv elocity flow contains a sourceo f free energy, one expectsi nstability to arise from the shear flow, which traps plasmat o form an insulating layer. Here R is the which it does above a critical strength. However, the forms poloidal rotation rate R = (c/rB) (da/&) and @i s the am- of the eigenmodeso f the K-H are sufficiently different from plitude of the vortex wave. those of the interchange-drift-wave-type of instability that there is generally a substantial window betweent he stabiliz- In the presentw ork we consider how the shearf low may ing effect of the shearf low on the interchange modesa nd the strongly modify the strength of the growth rates of the un- onset of the Kelvin-Helmholtz instability, as shown in some derlying turbulence generation from the interchange- and detail for the m = 1 and 2 modes of the rotating cylinder in drift-wave-types of instabilities in the limit of ion gyroradius Ref. 1. small compared with all scale lengths, both in theory and Recent experiment8 in the DIII-D tokamak show that simulation. Theilhaber and Birdsall” studied the K-H in- associated with L (low) to H (high) confinement mode stability with finite Larmor radius effects fully taken into transition, there is a substantial increasei n the perpendicular account but without magnetic shear effect. component of the plasma flow velocity, as measuredb y the A similar charge separationi nduced shearf low appears spectroscopic shifts of helium line radiation. ’ No such ap- in the barium ion injection in the ionosphere.‘* Other mag- preciable change is observed in the toroidal component of netospheric appearancesI and astrophysical ones such as the plasma flow velocity. Taylor et al. ’also report no appre- jets14o f the shear flow instability are noted. When the shear ciable change in the toroidal velocity and a substantial in- flow is sufficiently strong to dominate the stabilizing effects creasei n the poloidal velocity with the onset of H-mode-like of magnetic shear, the growth rate reachesa maximum for wave number k, u 1/2a, where the maximum growth rate is plasma conditions. The abrupt change in the flow speedi s interpreted to be due to a strengthening of the radial electric Ym axC O.2 max~dv,,/dx~~O.224/~. Since the short-wave- field strength. Shaing and Crume ’ have interpreted this length modesw ith k,,a> 1 are stable to exponential growth, change in.the radial field strength with increasedn onambi- vortices excited by the K-H instability extend over all the polar radial ion currents and a bifurcation to a new rota- shear flow region with A, -;1, > a. When the shear flow tional equilibrium. dominates, the density and temperature fields are passively convected with the fluctuations characterized by ep /T, %&z/n, 6T/T. /Vno The fastest growing normal mode forms a perturbed vertical flow pattern with the axis of the vortex tilted with (0) respectt o the flow direction, as shown by theory ’ and simu- lation.5 The tilting of the vertical flow produces a momen- tum flux r = (0, v,,) acrosst he shearl ayer. The momentum flux takes energyo ut of the sheara nd puts it into the vertical flows. Subsequentlyt he vortices coalesce,w ith the dominant wavelength shifting to a multiple of the original wavelength. This shifting to longer wavelengthsi s a configuration space representation of the inverse cascade.O ften the coalescing vortices or islands persist for long times. The effect of the electron parallel motion on stabiliza- tion of the K-H mode is shown to reduce the maximum (b) growth rate. The electron density fluctuations induced by n,(x) l the electron parallel motion (V,, j,, ) balancew ith ion density 1 t fluctuations generated by the ion perpendicular motion (V,*j, ). Namely, for charge neutral currents we have .,,,r V,*j, -I- V,,* j,, = 0. Since j,, -7pE,,= - ne2VIIrj/mveji,, - - (d/d) X (neV,#/Bw, ) from the ion inertia current. The effect of the electron parallel motion is significant when FIG. 1. (a) Slabg eometrys howingc oordinatesu sedt o describet he sheared kiv& 2k:upf. magnetic field B(x) and shearedf low velocity v,,(x) along with the direc- Here Y, is the electron thermal velocity, yei is the electron- tions of Vn,,a nd g. (b) The piecewisec ontinuous profiles of the shearedf low velocity o,(x) and the density n,,(x). ion collision frequency, 4 is the fluctuation potential, wci is 939 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta/ 939 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp TABLE I. Effect of shear flow on other instabilities. Mode Characteristics Shearf low condition k,Ax Yk k,Axu’> yr, Rayleigh-Taylor 1 Resistiveg 7%” = Eta i dlnT 7, z2-. r/, (threshold ) dlnn ’ Collisional drive wave UC k;i.$-cv*ys I, =r. Dissipativet rapped electron mode EZI R the ion cyclotron frequency,p s is the hybrid ion gyroradius the hybrid ion gyroradiusp,r ~0.02 cm for the abovep aram- given by ps = c(m, T, 1“ 2/e3, and 2u is the velocity shear eters The K-H modes in this casea re stabilized when the limits in the B Xl? drift velocity. For a K-H mode with shear length L, Su(2.7v,I,a/up~)“ ‘550 m. As for the in- k, 5 l/a the critical tilt angle 8,, as measuredb etweent he k terchange instability, the flute mode is stabilized by the vector and the ambient magnetic field, is given by strong shear flow, since u/a = 1O ”/sec 2 m 0, z @,/a) [ (u/v, )/( a/l, ) ] I’*, where I, is the mean-free =: 5 X lO’/sec. path of electrons. For 0 2 Q,, the K-H instability will be The second example is the Rayleigh-Taylor instability stabilized. In the case of a sheared magnetic field, the tilt of the imploded laser plasma. A typical acceleration rate angle 19-a/L, is produced by the shearing of the magnetic givesg--z/AR for the target shell thickness AR. The veloc- field, will L, the shear length. Therefore the K-H mode is ity shear will be given by ac,/AR. Since the Rayleigh-Tay- significantly stabilized when L, ~5( a’/~, ) (Z,v,/ua) I’*. The lor mode growth rate is &, the stability criteria is roughly ratio of the parallel diffusion k f I.$/v,~ to the ion inertial given by acceleration k fpfk,,u is sometimes called R, as is given by R = k,,v:a4/v,,up:Ls . Both the resistive g and the K-H c&AR Z 2$+@-. growth rates decreasew ith increasing R. Therefore the unstable modes are limited to short wave- When there exist a density gradient and a gravity force lengths where k 5a2/4 AR, and a k 1 will strongly stabilize as shown in Fig. 1, the interchange modes can be unstable. the Rayleigh-Taylor instability. Here we useg ravity to represente ither the effectivea ccelera- The characteristic time scale of the Kelvin-Helmholtz tion from the VB curvature drift of the ions or the accelera- or interchange processesd o not involve a characteristic os- tion during implosion. The maximum growth rate for the cillation frequency, such as the plasma,c yclotron, or the ion density gradient d In n,/dxE - l/L, and the gravity acoustic frequencies,i n the center of mass frame of the plas- g- vf/R, where vi is the ion thermal velocity and R the ma- ma. The plasma how is due to the E x B drift of the guiding jor radius of tokamak, is K. centersa nd the characteristic time scalesa re those of hydro- When there is a shear flow with [dv,,/dxl= ~/a, the in- dynamic flows, although the elementary processi s that of a terchangem ode can be stabilized. Stabilization by the veloc- magnetized plasma with long range Coulomb interactions. ity shear occurs when u/a > m+ This is related to the The effects of finite pressured ensity gradient and gravity, critical Richardson number. across the magnetic field and the shear flow layer, bring in Let us give two exampleso f the abovei nstabilities. The the drift wave frequencieso *, and @*pi.T hus to study the first examplei s an edgep lasma of the TEXT tokamak. Is The nonlinear evolution of shear flows and vortices associated shear flow layer width 2a-0.6 cm, in which the velocity II with the magnetized plasma through numerical simulation, changesf rom - 3 X ld to 3 X lo5 cm/set, the electron tem- time scalesm uch longer than the plasma oscillation periods perature T, =20 eV, the density n,- 1-2x 1012/cm3,t he are required. We employ the implicit particle simulation density scalel ength L, = 1 cm, and the magnetic field curva- technique with the decenteringa lgorithm,‘6 which system- ture R k 1 m. The electron mean-freep ath Z,- 200 cm and atically removes the characteristic time scales and spatial 940 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima efaf. 940 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp length scalest hat are smaller than the time step At and space u, x>a, scalesA x chosenf or the space-timeg rid. The filtering meth- v” = u(x/a), Ixlca, od has been shown to preserve the accuracy for the low- - u, x-c -a. frequency (w At < 1) dynamics.” The configuration is schematically shown in Figs. 1( a) and In the presentp aper we investigate the nonlinear evolu- 1( b) . Except for especiallyi ndicated cases,w e consider the tions of the Kelvin-Helmholtz and interchange instabilities previous plasma configuration. Also, gravity is applied in as an initial value problem through particle simulation, in the x direction, which destabilizes (stabilizes) the inter- contrast to the previous work,5 where the shear flow was externally fixed with an imposed driver, as would arise from change mode for r ’ = g/L,, ? 0. nonambipolar lossesi n the background plasma. Namely, in In the caseo f low-frequency modesw ith relatively long the present simulation, we assumet hat a space-charges epa- wavelengths/ 2>&, and pi, where &,, and pi is the elec- ration exists initially, which produces an initial E X B shear tron Debye length and the ion Larmor radius, the wave dy- namics can be analyzed by fluid equation for electron and flow. Note that any processesth at induce charge separation ion, and the condition of charge neutrality can be assumed. have not been included in the simulation. The secular Namely, growth and decay of the marginally stable normal modesa re also studied. After the linear stageo f exponential growth of n, = n, = n. (1) the primary normal modes, the growth of secondarym odes From the electron equation of motion along B, we obtain the can be nonlinearly triggered. equation for the parallel electron currentj,, , In order to systematically explore the parametric de- pendenceo f the development in the nonlinear stage,w e iso- me(& -I- V.P)io = - ne2VI14+ eV,,p, - mev,j,, --O, (2) late the evolution of vortex coalescencea nd associatedp ro- cessesc ausedb y vortex formation, which in turn is due to the where vE is the EXB drift, 4 is the electrostatic potential K-H instability and its nonlinear evolution. To investigate perturbation, p, is the electron pressure,a nd vei is the effec- the seconds tage,t he system is initiated from the secondary tive electron collision frequency. Using the electron equation equilibrium of a chain of finite-amplitude vortices. The of continuity and Eq. (2), we obtain chain of vortices is unstable against the coalescencem ode ($+ v .d,)r=r~ , ll~.ill and against the tilt or rotational mode. In this nonlinear regime the growth of coalescencea nd tilt modesa re nonlin- ear instabilities showing the finite time singularity like =--1_(-noeV~4+Vfpe). (3) (t, -t)-“fortimest<f,. m vei In Sec.I I the equations for K-H and interchange insta- Assuming T, constant and nc = n,(x) ( 1 + s,), Eq. (3) is bilities, both for plasmasw ith and without a magnetic field rewritten as are derived. These equations include the effects of velocity shear, magnetic shear, density gradients, gravity, and elec- (P+vE*V)Re = -~$+$vp, -$), (4) tron-ion collisions. The linear dispersion relations derived from the equationsf or the two instabilities are also discussed where $ = eqS/T,, L,, = (d In n,/dx( - ‘, c, = ,,/m, in two analytic limits. In Sec. III we study the initial value and the ion hybrid gyroradius ps = c,/w,+, where w,, is the simulation of the shear flow K-H instabilities, both in its ion cyclotron frequency. In Eq. (4), nonlinearities other linear and nonlinear stages.I n Sec. IV the nonlinear evolu- than the nonlinear polarization drift are neglected. tion of vortices, starting from the periodic chain of vortices, From the ion equation of motion, we obtain vi,, the ion is examineda nd the coalescencea nd tilt instabilities of vorti- drift velocity perpendicular to the magnetic field cesa re studied via the particle simulation method as well as Vi1 =vE +Vg d-v, fvpt (5) theoretical modeling. In Sec. V we summarize the results and discusst he applications of the results to fusion plasmas. where VE= C[ZXV($” + $5,1/B&, (6) II. VORTEX EQUATION IN A PLASMA WITH VELOCITY vg = - (g/w,, 19, (7) SHEAR, MAGNETIC SHEAR, GRAVITY, AND DENSITY vd = [ CgXVp, )/neB,]c, (8) GRADIENT: K-H AND INTERCHANGE STABILITY ANALYSIS and We carry out linear theoretical analysiso f plasmas tabil- v, (40+ 4) ity associatedw ith shear flows. We consider the effects of VP= - $+ (7, +v,)-v, > C. (9) shear flows and gravity both in magnetized and unmagne- tized plasmas.I n the caseo f a magnetizedp lasma, the static Here &, is a background plasma potential. Note here that sheared magnetic field is given by B = B,(f + 9x/L,), when finite ion Larmor radius is included, the convective which is shown in Fig. 1. The initial ion density has a gradi- derivative of Eq. (9) may be replacedb y ( vE + vd + v, )-V, ent of l/L, = - d In n,/dx between x = b and - 6. The as discussedi n Refs. 18a nd 19.T he ion equation of continu- flow velocity is in they direction, and changesa ccording to ity and Eq. (5) imply 941 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajimae t al. 941 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp chaws: a = b and n,(x) = no exp(x/l,) for <b, and & + (v, + vgw , nj -p;v, 1x1 ( > L s--tea. - [ nt ( & +(vE+vg)% > V ,ta+so+o. (10) A. Case 1. Discontinuous density step Setting u(; ’= 0, a solution of Eq. ( 13) is written as fol- Setting nt = b(x) (1 -k 2, ), USing i, = [ (iXV$)/B,lc, lows: and defining 4 =Ae-““, x>O, &JOc= -4--%--- , g (11) B, ax and @ci which is the ambient ion flow velocity, Eq. ( 10) yields, $=Dekx, x<O. -g+(v ,+vg)‘vI The jump conditions at x = 0 are ( > f ( ii - -$V, ln oV, 4 > A=D (14) n0 = 2 ,,a? - w - a$ pf a$ and --PJQy L, ay tpav' (12) where the prime indicates d /dx. Equations (4), ( 12) , and the charge neutrality condition of Eq. ( 1) are our basic equa- = k(n2 - O&P+ + p.$ “vgcs (n, - n,) - tions. Note that only the dominant nonlinearity is retained in 4. (15) w - kv, w(w - kv, ) Eq. (12),asinEq. (4). WhenwelinearizeEqs. (4) and (12) and set v, = 0 and V,, = 0, Eqs. ( 16) and ( 12) can be re- Using Bqs. ( 14) and ( 15), the dispersion relation is written duced to Eq. (31) of Ref. 11 and Eq. (7.17) in Chap. 7 of as Mikhailovski”’ and to the Rosenbluth-Simon equation.” dti-kvg) -cz(tda)(m-kvg) +akg=O, (16) In the absenceo f the gravitational drift velocity and for which yields a uniform vE the coupled equations (4) and ( 12) reduce to the well-known Hasegawa-Wakatani equations” describing c+!ag+$)+fJ~, (17) the collisional drift wave. In the low collisionality-strong shear limit k iv: > vei[ wk 1t he density is forced to be close to wherea = (n, - n,)/(n, $ n,) istheAtwoodnumber.The the local Boltzmann distribution and the equations reduce to interchange mode is unstable when 2-2 /au/a + kv, 1. the single dissipative equation2 ’ often used to study drift Hence the interchange processi s stabilized by the shear flow wave turbulence. Including the gravitational acceleration when g/L,, gives the resistive g mode for the collision dominated plasmaa nd an additional stabilizing or destabilizing effect to (18) the drift wave in the weak collisionahty regime. Let us look at a linearized wave equation for a mode that B. Case 2, Smooth density change varies as &(x)exp[ - iot + iky + ik,, (x)2], where Here a = b and n,(x) = n,exp( - x/L, ) for <a k,,( x) = kx/L,. Eliminating fi = Fz,= ii, from Eqs. (4) and [XI and k,, = kx/L,, as shown in Fig. 1. The eigenmode is (12), we obtain (Ae xp(dP+R , for x>a, ;&!-(no~) = (p;k2_ p:k-j;Ltt) $= Bexp(~~,dx)+Cenp(~~rZdx), for (~/<a, - (ku, + ik f;Q ) (kp,c,/L, ) (~-kvol(~-kv~~ fik&) DexP(&2dx), for x < 0, iki 4, + A (13) (19) w - kv,, f ik f D,, > where the Wentzel-Kramer-s-Brillouin ( WKB) approxi- mation has been used in writing Eq. ( 19). The validity of the where D,, = v~/v,, is the parallel electron diffusion coeffi- cient and uEo = c d&/ax B. Equation ( 13) includes various approximation is discussed later. In Eq. (19), MHD instabilities driven by gravity, shear flow, and density KI = f l/2& - 4, K2 = + 1/2L,, + q, gradient, which correspond to interchange instability, Kel- kvdL, a vin-Helmholtz instability, resistive g mode, and drift wave w - ku, instabilities. In the following discussion,s hear flow stabiliza- tion of the interchange mode is investigated. - (kv, + iv) kc, We derive dispersion relations for the following two (@ - kvo) (W - kv,, + iv)t,cp, cases.C ase1 is the discontinuous density step: b -0, no= It,, Wpf l/2 for x > 0, no= n2 for x < 0, U/a = const as a-+ 00 and no + (201 magnetic shear (L .~-t 00) . Case 2 is the smooth density w-kvEo +iv 1 ’ 942 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima et al. 942 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp v = k [II,, , and we assume Re K, < 0 and Re ~~ > 0. The jump conditions are 4?(a) + t4t I ’ A = B + CeJ’:, (21) X[ex-p2(j -~aq,+-4a1’l q’ + D = Be- *‘I+ C, and (26) (a)A - K; K,(Q)B - KZ(Q)Ceti2 ku/a A where =-- w+ ku - kv, ’ qr(a)= [(s2+ t*)“* +sl’/*/Jz, (27) (22) -a)D K,( -a)Be-“‘l+~*( -Q)C-K;( C?,(a)= - [(S* + t*)“* - s]“*/Jz, (28) ku/a D = + and w- ku - kv, * v(v + yvp5 d/P? Here $,,Z = .f”- o~,,2( x)dx. Equations (21) and (22) yield s=k*+ =k2+ (29) (v + y)* + k*U* y2+ k*u* ’ the dispersion relation vku/p: vku/p: ku/a t= (30) K; (Q) -K,(Q) - (v+y)*+k*u*=yZ+k*u** w - k(u + up> Sinceq i (a) in Eq. (26) is negative, the K-H mode is unsta- ku/a ble when x K2( -Q) -K;( -Q) + w+k(u-v,) > ku/a exp( -2JIoq,dx)- [2q,(a)a- l]*--4qf(a)a’>O. - K; (a) -K*(Q) - w - ku - ku, > (31) ku/a x K,( -a) -K; ( -a) + Roughly speaking, the maximum growth rate is at ( w + ku - ku, > q, (a)~ c ka=J and the threshold with respectt o the parallel xe-“‘+&=o , (23) wave number is where ak,, (Q) < 0.3OJm. (32) K:(Q) =K,(Q) 1, ,L,,=II and K;( -Q> =K2( -a)l,,L,,=o. When the shear scalel ength L, is shorter than This dispersion relation includes both the K-H instability L, = 1.65( V,i,Q/l&) ‘I*, (33) and the interchangeo r the resistivep ressure-gradient-driven all of the K-H mode will be stabilized. instability. 2. Case 2(b) 1. Case 2(a). Magnetic shear stabilization of K-H instability Without shear flow and magnetic shear, the dispersion relation (24) gives the growth rate of the interchange insta- The density gradient l/L, and us are set to zero in Eq. bility for a finite density gradient. Equation (23 ) reducest o (23) to obtain tanh(2aq) = - 2kq/(k* + q*), (34) (ku/a)* exp( - fy, dx) where we assumek L n E L,/p, >>1 and approximate co*- k *u* +2 q++s >( ku/a = -+ 4, 2q- -- = 0, (24) K1.2 (35) w + ku > q = k [ (g/L,/(w - kv,)w)] I’*. 1 + Setting q = iz, Eq. (34) becomes where qr and qi are a real and imaginary part of q, respective- ly, and q * = q( f a). As for the growing mode, we assume f(z) = tan(2az) = 2kz/@ - k*). (36) w = iy is pure imaginary. This assumption is justified since Solutions of the dispersion relation correspond to the cross the imaginary part of the left-hand side of Eq. (24) is pro- points of Fig. 2. As seeni n Fig. 2, the solutions of /z/k ) < 1 portional to the real part of w, which can be set to zero. are approximately given by Equation (24) is rewritten by keeping in mind ZQ-(r/2)l, I= + 1, + 2 ,..., (37) 4+ = qt =q,(Q) •t iq,(Q) as fOllOWS: which yields the frequency 1 - 4Qcl,(Q) -=2+,/m. (38) -exp( -2JIoq,dx)+4a’(q:+qf)] =Q (25) Therefore the models unstable when which yields a2w$i/gL,> k2a2+ (n-1/2)*. 943 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta/. 943 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp ( (ku, - iv) kc,/L,p, q(x)= k’-- (o - kv,) (a - kvEO+ iv) iv/p: -I- (461 w-kv,, -f-iv There are two resonancesin Eq. (44), which are located at X,1 = [b - kv,)/ku]a and X rt = [(w + iv)/ku]a. When 1~1g kv,, i.e., the magnetic shear is small enough, the distanceb etweent he two resonancep oints is au,/[u 1.E valu- FIG. 2. Graphical location of roots of the eigenvaluep roblem given by in- tersection oflc(z) = tan(2az) and the right side of Eq. (37). ating q(x) at the center of the two resonances,n amely, q(- !$+o -2 k2v;;4 W2)“2, the growth rate obtained from Eq. (46) is roughly evaluated to be Since g,cf/R for the magnetic field curvature R and R 2 L,, the mode is unstable up to k Zzl/ p,. 3. Case 2(c) Therefore if V Finally, we briefly discuss the velocity shear effects on the interchange instability in a finite density gradient. By a (481 processs imilar to the derivation of Eq. (36) from Eq. (23), we rewrite Eq. (23) without any approximation to obtain the interchange mode is stabilized by the shear flow. Note tan( -iJIoq(x)dx)=$, (39) that Z? = fL,/gl ( U/Q)* is the Richardson number. Accord- ing to Chandrasekhar’st extbook,* ’ 5? < 1 is given as the sta- where bilization condition of K-H instability as for the gravity in the stable direction. s=q+q- +(K- -&)(G +$--) In unmagnetized plasmas the linearized equation, in- cluding equilibrium shear flow and gravity, is derived from ku K, + 1/2L, the inviscid and incompressiblef luid equations. The similar -- a w - kv, + ku equation has beend erived by Chandrasekhar,**i n which the effect of gravity on the K-H instability is discussed. The ku K- - l/2&, k %*/a* +- , (40) result is a o- kv, - ku T= -- 2;n +K++ -$~+,*~jq?* ( k =P?PO + W + kW, /po -- ku q+ +& q- 141) (co- kv,) ’ o - kv, w - kv, > vx. Q w- kv, + ku a w - kv, - ku ’ (49) K + = [F ’ + iv/p: (0 F ku - iv) ] I’*. (42) As for the shear flow stabilization of the Rayleigh-Taylor Here we used relations instability, the stabilization condition for the configuration of case 1 is exactly the same as that given by Eq. ( 18). The K1.2 = f I/=, t- qt (43) criterion of Eq. ( 48 ) for case2 is also applicable to Eq. (49 ) , and III. INlTiAL VALUE SIMULATION OF SHEAR FLOW 4i =q( fQ). (4.4) INSTABILITIES Since the dispersion relation of Eq. (39) is similar to Eq. The static uniform magnetic field B, is now in the z ( 36)) the solution of Eq. ( 39) is approximately given by 0 direction only. The initial ion density is uniform, n; = n,, in q(x)dx=i:l+S (Z=O,& l,f2 ,... ), (45) the x-y plane. The plasmai s encasedin a metallic box in the x s -a direction with 4(x = + LJ2) = 0 and periodic in theydi- which correspondst o Eq. (37), where S is a phasef actor of rection for most of the computer experiments we present, the order unity. The WKB approximation that was used to unlesso therwise specified.T he particle velocities are reflect- derive Eq. ( 19) is valid when the integer I is sufficiently ed at the x boundaries.W e load the electrons with a density large. Assuming kL, $1, q(x) is approximated by given by 944 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima etal. 944 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp n, (x,t = 0) = no + An,/cosh2(kg), (50) function of time for each mode (m = l-4). Noticet hat be- causeo f the lack of noisei n the implicit particle code, a large where typically An, = 0. In, and k, = l/a, with a being the number of decadeso f exponential growth of the instability is shear layer width of the E X B flow produced by the charge observable.A fter a short period of time modes with m = 1 separation py = e[ n, ( CO) - n,] . The initial flow of the and 2 grow exponentially in time, while modes m = 3 and 4 plasmap roduced by the chargeo r vorticity layer given in Eq. do not grow until well into the nonlinear stage (50) is t-5x 104wP ’; = 31a/u,. Here recall that the threshold u,(x) = (4an,,ec/k&,) (An,/n,)tanh(k+) mode number m, = L,/a = y and m = I,2 are supposedt o = u. tanh (X/Q). (51) be linearly unstablea nd m>3 are stable.T his is in agreement with simulation in Fig. 3. For these simulation parameters Although we vary parameterso ver a wide range, the typical bru,(x=L,)At/A is equal to 0.75. The m =2 mode set of parameters are as follows: the numbers of the grid shows a slight oscillatory feature, as seen near points in the x and y directions L, = L, = 64, the numbers of particles in the x and y directions N, = N,, = 192, the t = 8x 104wP ’; = SOa/u,. The modes with larger mode numbers are triggered unstable after the amplitude of the electron cyclotron frequency o,, = SOW,w, ith oPeb eing the linearly unstable modesb ecomesh igh enough and the vorti- electron plasma frequency, the ion-to-electron mass ratio ces of these modes begin to interfere or overlap around M/m = 1600,t he shear width a = k 0 ’ = 6A with A being t-5x10%,‘. the unit grid separation, the electron and ion Debye lengths Figure 4 exhibits a typical particle plot and the corre- perpendicular to the external magnetic field direction sponding electrostatic potential contours. For clarity, only /2,, = /zm = 0, the electron and ion Larmor radii pe = p, = 0, and the simulation time step At = 2OOw,‘. (In particles with initial velocity uu> 0 on the left half at t = 0 other words, particles are loaded cold with no thermal veloc- are shown. Figure 4 is at t = 1X 10”~~; ’ = 62.5a/u,. In Fig. 5 we show the measureda nd theoretical growth rates of the ities in the directions perpendicular to the B field. No tem- modes.F igure 5 also showst he measuredg rowth rate for the perature development is seen well beyond the linear stage. casew hen the magnetic field B. is tilted toward they direc- The shear width is input as CI= 5A, which gives rise to the tion from the z direction by angle 0 = 0.010 rad. Note that effective shear width of 6 as a result of the finite size particle this is lesst han the critical angle (m,/M, ) “* = 0.025 rad. In effect.) Note that (W,i/W,i)2 = (m,/me)(u,,/u,,)* = 4, this caset he flow is still unstable although the magnitude of which is orders of magnitude smaller than that of the usual the growth rate is reduced by a factor of one order of magni- fusion plasmas.I nstead of assigninga uniform weight of uni- tude and the unstable wave number increases,a s is charac- ty to an individual particle, the weight of the particle is deter- teristic of drift-wave-like modes. On the other hand, when mined by the fraction n, (x)/n, dependent upon its initial we tilt the magnetic field away from the z axis by location. The weight of the particle in the simulation is not 0 = 3 X 10 - *, the system is stable. The electron thermal changedt hroughout the run. In the references imulations we speed uth is taken to be O.O5w,,A in the tilted B field runs chooseA ndn, = 0.1, the size of particles LI, = a,, = 3A, and where electrons can move along the magnetic field line, the decentering parameter”’ yi = 3/e= 0.1. while the thermal velocity perpendicular to B remains zero. The linear theory’*5,23fo r the hyperbolic tangent profile of Eq. ( 5 1) givest hat the Kelvin-Helmholtz mode is unsta- Thus k,,u ,,,= 9.8X 10-4m~pefo r the 8 = 10 - * case,w here m is the mode number in they direction. Thus y,,,,, < k,,u ,,,, ble for the wave numbers k,,, satisfying where yrnax= 1.25X 10 -4~pe = 0.2Ou,/a for the K-H k,a < 1, (52) mode (in Fig. 5). For 8 = 3 X 10 - *,k,,u ,,,g .F -“. The mea- where k, = 2rrm/L, and m is the mode number in the y sured maximum growth rate for 0 = 10 - * is direction. Figure 3 shows the electric potential ]@I2 as a 0.26 x 10 - 4~pe,a bout one-fifth of the 13= 0 case.W hen the normal mode is marginally stable,w e find that the growth of FIG. 4. (a) Contour plot of the electrostatic potential at the critical time FIG. 3. Evolution of the electric potential Qf,, (t) for modesm = 1,2,3,4f or shown in (b). (b) Position of particles in the (x-y) plane with initial fluid the referencep arametersi n Sec.I II. velocities o, <O at a critical stageo f dynamicsj ust before wave breaking. 945 Phys. Fluids 8, Vol. 3, No. 4, April 1991 Tajima eta/. 945 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp this amounts to the following initial electron density: An,(l -2) a, (4.Y) = nof (53) [cosh(kfi) + Ec os(k#) I2 ’ where O<E< 1. The ion density is taken to be uniform ni = n,. The electrostatic potential resulting from these charge densitiesi s #(x,Y) = (v&,)ln[cosh(kg) + ~cos(k~)l. (54) Thus the secondarye quilibrium flow is given by VX= eUcsin(kg)/[cosh(kG) + Ecos(kg)], u. sinh ( k,,x) I uv= I cosh(kfi) $ E cos(ky,) * k o- For E = 0, Eq. (53) reducest o Eq. (50). As E is increased, Y the island structure of the equidensity contours becomes FIG. 5. Comparisono f the measuredg rowth rates with theoretical growth wider in the x direction. We load electrons of nonuniform rate for the collisionlessp iecewisel inear slab flow. For the * data the tilt weight to describet he nonuniform density distribution, and angle 6 of the magnetic field is zero and for the A data the tilt angle is 0 = O.OlO/rad. the parametersa re the samea si n Sec.I II. With thesep aram- etersf ixed, we vary Ef rom zero to the following set of values: 0.08,0.3,0.5,0,6,0.7,0.85, and 0.95. The unit of frequency wPeis measuredw here N, = n,, L, /a = 64.5, and the hydro- the mode becomess ecular, as seen in Fig. 6. The electric dynamic unit of time a/u, at n: = 0 or L, is given by potential resulting from the marginally unstable mode in- creasel inearly with time. a/v,- 1.6x IO%+; ‘. As hasb eens hown in theearlier driven simulations, ’the Figure 7 showss napshotso f particles with (x,y) coordi- stage at which the nonlinear triggering of other mode nates and the corresponding electric potentials at various numberss etsi n is coincident with the developmento f a vor- times for the caseo f E = 0.08. The instability is triggered by tex chain. In order to better control the study of the nonlin- ear problem of vortex evolution, our approach here is to separatet he linear and nonlinear stages. We idealize the problem by starting from the secondarye quilibrium of a vor- tex chain. IV. NONLINEAR EVOLUTION OF VORTICES A. Comparison with implicit particle simulation In this section we initialize the simulation near the Stu- art-Kelvin cat’s eye equilibrium.24*25In the plasma context FIG. 7. (a)-(c) Contourosf the electrostaptioc tentiafol r the period-dou- bling coalescencefr om the m = 2, e = 0.08 island chain. (d)-(f) Particles FIG. 6. Secularg rowth of the potentia1w ith a linear increasei n time for a starting with uv <:0 at later stageso f the coalescencein stability evolving marginally stable mode. from the period island chain of strength E = 0.08. 946 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta/. 946 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp noise due to numerical truncation that results upon loading the particles, as there is no noise associatedw ith the particle motion perpendicular to B with the initialization and the subsequent decentering algorithm. I6 At t = 4.2 X 104wP ’; = 26&c the contour lines show a recon- nection of the flow lines, reminiscent of the tearing instabil- ity of magnetic islands, which yields vortices (islands) with smaller wavelengths( m = 4). Note the original vortices had mode number m = 2. At a later time the smaller induced islands (m = 4) are absorbed by the original islands (m = 2). At a later time the smaller induced islands (m = 4) are absorbed by the original islands (m = 2). In Figs. 7(c) and 7(f) at time t = 6.6X 104wP ’; = 41a/v,, we L I I I I 1 I I I 1 I Oo 0.2 0.4 0.6 08 IO observe that the larger original vortices coalescei nto one E-- vortex with m = 1 in the direction of the exterior flow. We FIG. 9. Plot of normalized linear growth rate G vs E: X representsa data also seet he vortex tilts in the clockwise sensea s a result of point extracted from the simulation results, 8 is a data point where it is the ambient external flow, which is downward on the right difficult to ascertaint he growth rate becauseth e growth is faster than expo- side of the vortex and upward on the left side. nential. The dashedc urve is the linear theory of Ref. 25; the solid curve is During the coalescencep rocesst he perturbed electro- that derived from the localized vortex model. static potential energy grows exponentially in time as shown in Fig. 8 (a). Figure 8 summarizest he growth of the electro- static energy for casesw ith various values of E. The satura- physics include (i) the vortex dynamics is describedb y the tion of this energy in Fig. 8(a) occurs shortly after a com- single field 4, while the magnetic island dynamics requires at plete coalescence. least two fields 4 and the z component of vector potential A,; As we raise the value of E, with other parametersb eing (ii) consequently, there exists a magnetic repulsive force fixed as before, the growth rate of the electrostatic energy upon magnetic island coalescence,w hile in the vortex dy- increases,a s shown in Fig. 9. This figure will be further dis- namics there is none; and (iii) the magnetic flux conserva- cussedi n Sec.V . In frames (b)-(d) of Fig. 8 we also observe tion inhibits the reconnection of the magnetic flow lines. On that a slight bump develops in the middle of the otherwise the other hand, the presenceo f the Kelvin-Helmholtz insta- exponential growth phase.I n particular, Fig. 8(d) shows a bility and the coalescencein stability for the vortex dynamics faster than exponential growth in the early stage,w hile set- is similar to the presenceo f the tearing instability and the tling into a nearly exponential growth later. This indicates a magnetic coalescencein stability, except for the frozen flux transient growth that is faster than exponential growth for constraint that d$/dt = vV*$+O. The importance of the E> Lt ~0.5. Another feature to be noticed in Fig. 8 is the flux conservationc onstraint is easily seeni n the formulas for amplitude oscillations after the coalescenceT. hesea re asso- the linear growth rates where yk-“- k, hvv but ciated with the ringing of the vortex shape.T his is reminis- y’a~k:/3~~‘~. cent of the coalescencep rocesso f magnetic islands, although In light of the above the actual dynamics of the tearing any parallelism of the coalescenceo f vortices in the present mode is, in principle, different from that for vortex coales- investigation with that of the magnetic island coalescenceis cence dynamics. Ideas used to study the tearing, however, perhapsf ortuitous since the dynamical equations are rather can be used to measuret he vortex dynamics observedh ere. different. Some conspicuous differences in the governing One measurei s the vorticity difference betweent he original 0 point and the innermost X point (e.g., betweenA and B in Fig. 7 and Fig. 12), and another measure is the vorticity differenceb etweent he original 0 point and the outermost X point (e.g., between A and C in Fig. 7 and Fig. 12). The former measureo f vorticity is indicated by circles and the latter measureb y crossesi n Fig. 10, for various E cases.B y definition of the former, the measure of vorticity vanishes when the two vortices complete their coalescenceC. ompare Fig. 10 with Fig. 8. On the other hand, the latter measureo f ,~~~~1 vorticity may or may not vanish. In small Er uns [Figs. lO( a) and 10(b) 1, we see that it decreasesu ntil a certain point (t- 5 X 104wP ’; = 38.6a/v,) and then begins to increase. This manifests itself in a larger vortex at t = 6.6 x 104wP ’; = 5 la/v, [e.g., Fig. 7(c) ] than the origi- nal vortex. In larger E experiments [Figs. 10(c) and 10(d) ] 0 kdp- twp- 9x104 the two measuresd epart, but both measuresd ecrease,o r at least not increase,e ven well after the coalescence. FIG. 8. Evolution of the electrostatic potential energy from the nonlinear island chain as a function of increasingv ortex strength e. Let us further examine the cases with E = 0.3 and 947 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima et a /. 947 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp I 6 20.60 (ok=008 64 =! 1:1:11T_1.::;:~~:::~:~~ twp=5.6xId two=6r103 64 0 +wp- 9xto’“o twp- 9 FIG. 10.A measureo f the vorticity in the trappedc oherents tructures.T he vorticity differenceb etweent he original 0 point to the innermostX point is given by the circles (0) and vorticity differenceb etweent he original 0 point and the outermostX point is given by the crosses( x ). 64 E = 0.6, which are shown in Figs. 11 and 12, respectively. Figure 11( c) showss kewnesso f eachv ortex, as well as the tilt of the axis of the two vortex centers,m easuredi n the negatived irection. This is similar to the prediction by Liu et al. ’ and one found in the simulation.5 A more pronounced tilt may be seeni n Fig 12.T he rotation of the axis continues evena fter the completion of the coalescenceT. hus we find that the chain of vortices is unstablea gainstt he tilt or rota- FIG. 12.A s in Fig. 11,b ut a caseo f e = 0.6 with strongert ilting, wheret he rotation continuesa fter coalescence. tional instability. In Figs. 12(a)-12(d) we observet herota- tion of the axis that connectst he two 0 points as they ap- proach eacho ther. Even after the coalescenceth e rotation continues. At the same time the overshooting oscillation (squashingo f the droplet) continues.I n this particular case, whereE = 0.6, during the courseo f thesed roplet vibrations, fissiono f the vortex occurs,a ss eeni n Fig. I2 (d) . Observein Fig. 12 that as the m = 2 vortices coalescei nto an m = 1 vortex, much smaller-scalev orticess pring up. As the energy inverselyc ascadefsr om the m = 2 vortices to the m = 1v or- tex, the enstrophyc ascadefsr om m = 2 to higher m’s, since both the overall enstrophya nd energya re conserved.F rom the distribution of particlesi n plots in Figs. 7, 11,a nd 12,w e note that even when the potential contours show fairly co- herentp atterns,t he particlesa re strongly mixing in complex structures. tw, - 3.6 x IO ’ ‘- In Fig. 13 the negativeo f the rotational angle of the FIG. 11.P otentialc ontours (a)-(c) and particle plots (d)-(f) for theperi- vortex-vortex axis asa function of time is measuredA. s csis od-doublingc oalescencoef the m = 2, E = 0.30 islandc hain. The skewness increased,s o is the growth of the angle.T he rate of increase of eachv ortexa st he axis betweent wo vortex centersr otatesd uring coales- cence. of the angleb efore8 = 180” is found to be faster than expo- 948 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima et& 948 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp t=o (a) t=o (c) 64 64 t Y 0 0 2.4 4.8 7.2 0 X- 64 +a.-- 64 FIG. 13. Simulation measurementsf or the rotation angle as a function of time 0(r) for labeled valueso f E. nential. Following the terminology of magnetic coalescence 0 x-+ 64 this growth is called explosive growth. This explosive in- crease of 8 saturates at or near 8 = 180”. In some cases0 FIG. 15. Equipotential contours obtained from the simulation: (a) stays around 180” after it reaches this position. In other E = 0.95, t = 0; (b) E = 0.95 at a later time; (c) E = 0.7, t = 0; (d) 6 = 0.7 cases, after a brief pause at 8 = 180” the angle again in- at a later time. creases.T he higher value of 8, the stronger is this tendency for continued rotation. Figure 14 displays the distance be- tween the two 0 points as a function of time. Once again this distance Sr(t) = J-z also grows faster than expo- nential during the coalescence,i ndicating the explosive na- systems.T his is a plausible goal, as the system is nearly dissi- ture of the transients in the coalescencep rocess. Note that pationless and low wave number modes dominate the dy- different from 0 in Fig. 13,t he dependenceo f Sr asa function namics. The specific application of these techniques to the of E is not monotonically increasing. Also noted is that the near cat’s eye simulations are considered.T he goal here be- increaseo f 6r as a function of time is sometimes not mono- ing to elucidate the dynamical mechanisms that are active tonic. during the coalescenceo r tilt instabilities. Figure 15 shows the potential 4 at t = 0 and a later time The simulation results suggestt hat there is a large tem- at which point a nearly ?r/2 rotation of the axis of a pair of poral regime where the two localized vortices of the initial vortices is realized for the E = 0.3 and E = 0.7 cases.T he condition remain isolated and maintain their integrity while evolution will be later compared with the theoretical model moving. This suggestsa few degree-of-freedomm odels for in Sec. IV B. the flow, composedo f localized interacting vortices, perhaps subject to an external field. B. Localized vortex model Sincet he computational studies of this paper are period- We now present techniques for analytically modeling ic in they direction, but not the x direction, theseb oundary vortex simulation results, such as those presentedi n Sets. III conditions must be incorporated into a model of the dynam- and IV A, by simple few degree-of-freedom Hamiltonian ics. The x boundary condition is straightforward, since to a large degree the motion of the localized vortices is far enough removed from the boundaries for us to assume - 00< x < COM. ore generally, one can satisfy finite metal- lic boundary conditions by appropriate configurations of im- 3(I - agev ortices. This is not pursued here for the x direction, but the periodic y boundary condition does require images. Theseb oundary conditions are perhapsa bit confusing since the simulations have a periodicity length of 4r/k,, while the initial conditions of interest are nearly 2n/k, periodic. This + sr latter condition near periodicity is not a constraint of the dynamics; thus unlike the 4r/k0 periodicity should not be built into the vortex model. Begin by supposing that there are two vortices in the simulation domain: one denoted by “0” and the other by “1.” Periodicity in they direction requires that each of these vortices be tracked by an infinite chain of equal strength O t -J image vorticities. Vortices that track vortex 0 will be denoted 0 6~10~ twp + by an even subscript, while those that track vortex 1 will be denoted by an odd subscript. The periodicity requirement FIG. 14. The distanceb etweent wo 0 points as a function of time E+(t), as measuredi n the simulation for the labeled valueso f E. thus demands the following constraints: 949 Phys. Fluids 5, Vol. 3, No. 4, April 1991 Tajima eta/. 949 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp y,,,(f) = ye(t) + Bn-n/k,, relative motion of the vortices. Sincef or this system physical space (&f;rl) is the phases pace,a nd since trajectories lie on ~rn+i(t) =~,(t) -i-4rn/k,, (55) curves of constant H, onec an attempt to fit H to the simula- x*n (t) = x,(t), tion output. Alternatively the vortex-vortex interaction $ %I+,(0 =x,(t), can be postulated, perhaps, by examination of the vortex shape.K nowing @,t he sum of Eq. (63 ) must be evaluatedi n order to determine the dynamics. where n = + 1, + 2, + 3, ... . Sometimes the sum of Eq. (63) can be evaluated in In general,s upposet hat the velocity at vortex n located at x,- = (x, ,y, ), resulting from vortex m located at x, , is closed form; for exampIe,i n the caseo f point vortices where given by $(@I) = q. InG ’ + $1, (661 V nm= V(x, - x,). (56) this is the case. Defining z = (kd4)(77 + y), it is evident Upon superposing,t he total velocity at the location of vortex that 0 is given by J - ai! jag - ko~o i 1. (67) a7j af 2 mz-oc.z-mmlT v,= -g V(x,-x,1. The sum of Eq. (67) is the Mittag-Leffler expansion for m= --r rn#O cot z, which implies Similarly, for vortex 1, (68) cosh( k,{ /2) - cos(k ,T/2) > ’ v,= 2 V(x,-xx,). (58) m= --a: and thus from the first equality of (67) we obtain, to within mp I an additive constant, Supposet hat V is derivable from a streamfunction, defined H(&q) = z,&l n[cosh(k,CfZ) - cos(k,q/2) 3. (69) by The summation performed abovei s related to the solved V(x,y) = ~XV?NX,V), (59) classical problem of obtaining the velocity field as a result of where $ is even in both of its arguments, an infinite chain of point vortices,26b ut here the context is ?KcY) = $( - KY) = $(x, -VI. (60) different, in that H determines the dynamics subject to the constraints (55). Here H is not the streamfunction. Below Making use of these symmetry conditions and the periodic- we will construct the streamfunction as a function of time. ity constraints of Eq. ( 55) yields the following for Eqs. (57) We proposet he following form for H in the casew here E and (58): can differ from unity: 4v7m v()= ;xv+ x,---xl,yo- y1 -- , m=2 --no ( ko > H(S,q;c) = 11,I n [ $;(E + cash k,c) - $ cos(k ,q/2) ] . (61) (701 v, = i PXV$(X, - XO,Yl -Y. - T). Arguments in favor of this seemingly obscure choice will in= -* 0 shortly be given, but first consider the inverse problem for We ignore the self-interaction of the vortices and assume obtaining $( ~,v;E). Suppose$ has the form of the point vor- they move with the local flow. This yields the following tex interaction, except isotropy is broken by warping the x Hamiltonian equations of motion: dependenceo f the interaction, i.e., dH dH i;=--, * i= O,l, (62) ljrG77;~) = fjo lnlf2(C) + ~~1, (71) ay; yt=z' where the functionf(g) is yet to be determined. Substitution where off for 6 in our treatment of the point vortex casey ields H(x,-x,,yo-y,)= 2 t+o-xlYYo-Yl -g. -a=H koh sin ( k,q/2 1 (72) ,?I= -0e 877 2 cosh(k,f/2) - cos(k,q/2) ’ ( ’(63) aff sinh (k,f/2) The form of the Hamiltonian of Eq. (63) suggests the intro- - a = f ko 2 @ o cosh(k,f/2) - cos(k,v/2) ’ (73) duction of the “center-of-mass” coordinates, defined by Equations (72) and (73) imply $5=x0--XI, 71=Yo--YI, ( \- 6 4 .I 1 H&W) = Jt, In[cosh(k,f/2) - cos(k,q/2)] -I-c onst. 2-=x, +x0, 3=y, +y(y (74) The coordinates (F, 9) remain fixed in time while ({,v) Choosing the functionfas follows: satisfy cosh(k,f) = (l/e)cosh(k$), (751 & 2Q77) ?j=~({@ (65) a7 ’ ’ ac ’ ’ results in, apart from an unimportant additive constant, the In order to effect the modeling, the function H({,r]) H given by Eq. (70). remainst o be determined. This can be achievedi n two ways: There are several favorable attributes that lead one to first, a model H can be obtained directly by tracking the chooset he H of Eq. (70). To begin with, it is a continuous 950 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta /. 950 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp deformation away from the casew here E = 1, a casew here Thus the linear growth rate basedo n this model, normalized the model agreesw ith the simulation [cf. Eq. (69) 1. Thus at by k i q&/4,i s given by least near E = 1 we expect reasonablea greement.I nspection p = 23’4E”4/(1 + E) I“+( Jr+E + $G,. (82) of Eq. (75) revealst hat for E# 1 the logarithmic singularity A at CJ= 77= 0 has beene liminated. Any distributed vorticity For E = 1, y = 4, the classical result for the maximum arrangement will have this feature. Also, the anisotropy in- growth rate of a row of point vortices. The localized vortex troduced is in agreementw ith that observedi n the simula- model selectst he maximum becauset his is the only motion tion (cf. Fig. 15). In particular, cY$/&$I I =e dominates allowed by the periodicity constraints of Eqs. (55). Exami- d$/&7] I = o. A convincing argument in favor of the choice of nation of Fig. 16 revealst hat the simulation is in agreement Eq. (70) is that it leadst o a #(x,y,t), which, as we shall see, in this limit. As e-+0, p-23’4&4. This vanishing growth looks like the simulation. We emphasize,h owever, that this rate is in disagreementw ith the simulation; not a surprising choice is not unique and only qualitative agreement is result since the assumption of localization of the vortices sought. breaks down. Becauseo f E1 ’4 behavior this disagreementi s The construction of d(x,y,t) requires that the contribu- confined to O(E 5 0.2. The theory is in reasonablea greement tions from the double infinity of vortices be summed at each for a large range of e away from unity. In Fig. 16w e have also moment of time. We conclude that plotted the results of Ref. 25, where the linear eigenvalue problem for the cat’s eye equilibrium was solved numerical- W,YJ) = 2 [ d(x - XOYY - Yo - 9) ly. Observe that for e-O.3 there appearst o be a transition ,n= -0c 0 from localized vortex behavio: to what we refer to as K-H 45-m behavior, i.e., sustainment of y. This is further evidencedi n +q x-X,,Y-Y, -7 * (76) >I Fig. 10. On the other hand, the theory2 ’ by Pierrehumbert ( 0 Without loss of generality the arbitrary constant can be and Windall is correct at E = 0 and agreesr easonably with dropped. Assuming simulation for E < 0.3, but is unable to convergeb eyond. Now consider the nonlinear behavior. Since H is con- .P=yY=o, served,o ne can obtain the orbits in physical spaceb y simply x0 = g, XI = -g, (77) plotting surfaceso f constant H. In Fig. 17 we have done so for different values of E. This figure shows that the energy Yo= 4% Yl = - pi% surface decreasesin width as E decreasesfr om 1 to 0. In the yields simulation it was observedt hat for small values of E the in- Ql(x,YJ;E=) H[x -&t)/2,y- 7(t)/2;el +H[x+&W2,y+ 7j(t)/2;~1. (78) At time t = 0, Eq. (78) should representt he cat’s eye initial condition. Using g( t = 0) = 0, q( t = 0) = 2r/k,, Eq. (70) implies qVx,y,O;~)= $. In [cash k,,x + Ec osk g], (79) the equilibrium state desired. At later times, ko77 &x,y,t;e) = qboln (2[Jm - J; ..p(y - Y)) x[J~ - J; cos$(y + Y))) . (80) Now consider the comparison of the linear theory of the localized vortex model with the simulation. As noted above, 6 = 0 and 7 = 2n-/k, correspond to the cat’s eye equilibri- um, but, also, thesec orrespond to dynamical equilibrium of the localized vortex model. This is evident upon differentiat- ing Eq. (70). Moreover, expandingH to secondo rder yields the following Hamiltonian for the linearized dynamics: kc& kc& G$o@ h(w1?;E) = -&6#+gy* FIG. 16. Contours of the vortex-vortex interactiopno tentia$l givenb y 8[-+&1 E Eqs. (75) and (79),forthecases (a) ~=0.05, (b) ~=0.30, (c) l =0.70, (81) and (d) e = 1.0. 951 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima eta/. 951 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp betweent he simulation and the localized vortex model with increasinglyb etter agreementf or larger e. V. SUMMARY AND CONCLUSIONS We have derived linear theoretical stability conditions and growth rates for a plasmaw ith shearf lows, taking into accountg ravity and magnetics hear.W e analyzedt he stabil- ity problem with a discontinuousb ackground density and with a smoothly varying density. The dispersion relation shows the presenceo f E XB shear flows can stabilize the interchangea nd other relatedi nstabilities. The linear analy- siso f the gravitational instability showst hat the interchange (R-T) modei s stabilizedb y theshearf low when the velocity shearu/a>fiorm[seeEqs. (18)and (48)J.Inthe case of the interchange mode, much shorter wavelength modes (much higher azimuthal modes) are destabilized. The mode is strongly locahzedn ear the mode rational sur- face. The interchange mode can be stabilized when u/a 2 (g/L, ) ‘/2. Implicit particle simulation results of the shear flow ( K-H) instability resultingf rom E X B drift in a magnetized plasma show good agreementw ith the linear theory. The kO6 kOC maximum growth (0.2&a) and the threshold waven umber FIG. 17. Contours of constant interaction Hamiltonian, H, for (a) ( l/a) agreew ell with the observedg rowth rate and thresh- ~~0.05, (b) e-0.30, (c) E=0.70,and (d) E= 1.0. old. The linear K-H instability with k,, = 0 has a sharp boundary betweent he stable and unstable wave numbers, with the marginally stable modes having a local secular ( - t) growth arising, perhaps,f rom a ballistic resonanceI.n contrast, three-dimensional (3-D) modes with k,!/ k < 8, stability that occurs is of the pairing or coalescenctey pe, as have a reducedg rowth rates. In the simulation, if the tilt shown in Fig. 10. For valueso f E near unity the instability angle 6 2 0.02, the modesb ecomed rift-wave-like and the that occursi s of the tilt or rotational type. Figure 17e xplains growth rate is greatly reduced, while the unstable wave- this tendencyw ith E. For all finite valueso f E the vortices length band expands.I n the K-H instability, vortices that approache acho ther and movet ransverset o eacho ther. For grow to a sufficient sizet rigger a secondaryn onlinear insta- small Et he later motion decreasews ith the width of the ener- bility with smaller, subharmonicw ave numbers. gy surfacea nd we observet he predominant coalescence. The nonlinear instability is analyzed using a dynamic In Fig. 18(b) we show the results of integrating Eqs. periodic chain of localizedv ortex structures with an equilib- (65). Here 8(t) and Sr(t) = r(0) - r(t) for various values rium equivalentt o the Kelvin-Stuart cat’s eyet ype solution, of e areg iven.M any featureso f the correspondingq uantities This equilibrium is observedt o be unstablea gainst the co- for the simulation Figs. 13a nd 14a re reproduced,a ss eeni n alescencae nd tilt modes.T he electrostatice nergyi ncreaseo f Fig. 18(a). The initial conditions herew erec hosenn ear the a lower wave number mode (m = 1) (the growth rate of separatrix,e itherj ust insideo r outside.I n the casew heret he m = 1 mode) is in reasonablea greementw ith the theory by initial condition is just inside, B can increaseb eyond 180”. Pierrehumberta nd Windal12 ’and the analysiso f Sec.I V B. This behavior is seen in Fig. 18(a) for the case where In a small-amplitude regime,t he tilt and coalescencein sta- E = 0.95 and in Fig. 19(b) for the casew ith E = 0.3. When bility and shearf low instability coexist. Even after the com- the initial condition is outsidet he separatrix,i n the localized pletion of coalescencet,h e shearf low instability continues. vortex model, 8 can only approach 180”. This behavior is In the caseo f large amplitude, the tilt and coalescencien sta- indicated in Fig. 18(a) by the flat spot near 8 = 180” and bilities dominate the shearf low instability. Upon overshoot shown in Fig. 18(b) for the casew here E = 0.3. Similarly, of the tilt and coalescencet,h e coalescedv ortices can again Figs. 14a nd 18f or SP(t ) show qualitative comparison. separatein to two. Given the resultso fthe orbit integration we canp lot the The growth rate of the tilt angle is in good agreement streamfunctiona s a function of time by making useo f Eq. with the coalescencien stability ofa point vortex modeli n an ( 78). We haved ones oi n Fig. 19( a) for the point vortex case appropriate range.T he angleB increasesfa ster than the ex- where E = 1 and t = 0, while Fig. 19(b) shows4 for E = 1 ponential function of time. The angle B approachesr r and and t = 1, a later time chosens o that 8=:~/2. Similarly, in staysf or a long time. The point vortex model can accurately Figs. 19(c) and 19(d) we plot 4 for the casew here E = 0.6 predict the time profile of the rotation angle of the vortices. and t = 0 and t = ?, respectively (7 is chosena gain so that The time scalep rediction of the relative rotation of two vor- 8~=/2). Figure 19 should be comparedt o Fig. 15. tices also providesa goode xplanationo f the simulation. In summary,w e seet hat there is qualitative agreement The resultsf or the stability of the transitional layer of a 952 Phys. Fluids B, Vol. 3, No. 4, April 1991 Tajima et al. 952 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp t3 8 0 -50 0 50 t- I I (c) (d) t t 8 FIG. 18.N onlinear resultso fthe localized vortex model. (a) 8(f) for E = 0.05,0.30,0.70, and 1. O.( b) &(t) for the sameE v alues. (c) Plots of 6( I) that show trajectories with initial conditions on the two sides of the separatrix. (d) &r(t) for two initial conditions, one just inside and the other just outside the separatrix. resistive plasma with a substantial change in density and perpendicular E X B flow velocity is given in terms of the transcendentald ispersion relation in Sec.I I. The roots of the dispersion relations in the absenceo f shear flow describet he resistiveg instability and the collisional drift wave instability with their different dependenceo n collisionality and mag- netic shear. In the presenceo f a sheared flow, the growth rates are strongly affectedw hen the condition k,,A xu ’> yk is satisfied, where k,,, Ax, and rkY are the parameters of the instability in the absenceo f the shear flow. The critical shear flow u ’ obtained from this condition is shown in Table I. For the shearf lows reported in the TEXT tokamak plas- ma with the new higher resolution probe measurementst he condition given previously is marginally satisfied so that we conclude that shear flow may have an influence on the edge turbulence, even if it is not sufficiently strong to excite the FIG. 19. Thestreamfunction for-thecases: (a) 4(q~,O;l), (b) qS(x,y,^t;l), K-H instability in that experiment. (c) 4(xg,O;O.6), and (d) 4(x9&0.6). Here I and tare chosens o that the rotation is approximately 180”. There are two effects of the shearedE x B flow on the 953 Ws. Fluids B, Vol. 3, No. 4, April 1991 Tajima et al. 953 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp edge turbulence. Here, as in Ref. 1, we consider the direct strong shear flow are given for slab approximation to toroi- effect of the shear flow on the wave dispersion relation, da1c onfinements ystems. showing that the growth rate of the mode present in the absenceo f shear flow can be strongly reduced. The second aspect is that even for a given fixed level of background ACKNOWLEDGMENTS wavest he transport across the magnetic field is reduced by The authors thank Dr. E. Solano and members of the the shearf low. This decorrelation effect of the transport has Edge Physics Study Group of the IFS for useful discussion, been calculated by Shaing and Crume ’ and Biglari et a[.” This researchw as supported by US. Department ofEn- using the ideaso f relative diffusion or clump turbulence the- ergy Contract No. DE-FGO5-80ET-53088.T he work was ory. A simpler estimate of the reduction of the transport also supported by Joint Institute for Fusion Theory and Na- comes from considering the single-particle motion of test tional ScienceF oundation Grant No. ATM88-11128. particles in a dominant fluctuation of mode number k and strength &k; the radial excursion size is reduced from the distanceT ./k, betweenn odeso f the radial modest o the size ‘J. Liu, W. Horton, and J. E. Sedlak, Phys. Fluids 30, 467 ( 1987); W. Horton and J. Liu, Phys. Fluids 27,2067 (1984). Ar = ( c$~/Bu’) “2 given by the strength of the shearf low u ’ *K. C. Shaing,E . C. Crume, and W. A. Houlberg, Phys. Fluids B 2, 1492 and the amplitude of the potential fluctuation. Taylor eta i., (1990). Shainga nd Crume,9B iglari et al.,l o and Burrell et al. ’argue ‘H. Takabe,L . Montierth, and R. L. Morse, Phys.F luids 26.2299 ( 1983). 4H. Takabe, K. Mima, L. Montierth, and R. L. Morse, Phys. Fluids 28, that the reduction in the plasma transport associatedw ith L 3676 (1985). to H mode transition occurs as a result of the increased ‘W. Horton, T. Tajima, and T. Kamimura, Phys. Fluids 30,3485 ( 1987). sheared flow velocity resulting from the deepening of the ‘R. J. Groebner, P. Gobil, K. H. Burrell, T. H. Osborne,R . P. Seraydarian, negative electrostatic potential well of the toroidal system. and H. St. John, in Proceedingso f the 16th European Conferenceo n Con- trolled Fusion and PIasmaP hysicsB. udapest,H ungary f European Phys- We show here how the increaseds trength of the shear flow ical Society,Geneva, 1989), Vol. 1,~. 245. changest he stability conditions of the plasma. ‘K. H. Burrell, T. N. Carlstrom, E. J: Doyle, P. Gohil, R. J. Groebner,T . We consider the unstable shearedf low regime with im- Lehecka,N . C. Luhmann, Jr., H. Matsumoto, T. H. Osborne,W . A. Pee- plicit particle simulations and describet he resulting vortex bles, and R. Philipona, Phys. Fluids B 2, 1405 ( 1990). ‘R. J. Taylor, M. L. Brown, B. D. Fried, H. Grote, J. R. Liberati, G. J. dynamics by the motions of the vortex cores. To make an Morales, P. Pribyl, D. Darrow, and M. Ono, Phys. Rev. Lett. 63, 2365 analytic treatment of the vortex core dynamics, we idealize (1989). to the caseo f point vortex dynamics, including the vortex- 9K . C. Shaing and E. C. Crume, Jr., Phys. Rev. Lett. 63,2369 ( 1989). vortex interactions and the vortex-shear flow dynamics. ‘OH. Biglari, P. H. Diamond, and P. W. Terry, Phys. Fluids B 2, 1 ( 1990). ” K. Theilhaber and C. K. Birdsall, Phys. Rev. Lett. 62, 772 ( 1989); Phys. This appearst o give a good description of the principal pro- Fluids B 1.2244 (1989). cesseso f mutual rotations and coalescenceo f the vortices. “J. S. Wagner, R. b. &iora, T. Tajima, T. Hallinan, L. C. Lee, and S. 1. Comparisonw ith the simulations shows that a lowest-order Akasofu, J. Geophys. Res.8 8.8013 11983). description of the turbulence follows from the vortex core I3J . L. Burch, Rev. SpaceS ci. SpaceP hys. Zi, 463 ( 1983). 14T.T ajima and J. N. Leboeuf, Phys. Fluids 23,884 ( 1980). dynamics with treating the density and pressure fields as “Ch. P. Ritz R. D. Bengtson,S .J . Levinson,a nd E. J. Powers,P hys.F luids passively convected. The possibility of describing the rela- 27,2956 (i984). tionship of the density and potential fluctuations measured lbD C Barnes.T . Kamimura, J. N. Leboeuf, and T. Tajima, J. Comput. P&s: 52.480 ( 19831 . in TEXT with the density being passivelyc onvected in the “T %aji&a, Cokputakonal PlasmaP hysics( Addison-Wesley, Redwood E x B flows given by the potential fluctuations hasb eenp re- City, CA, 1989). viously suggestedb y Bengtsona nd Rhodes.27 ‘*A . B. Mikhailovski, in Theory of Plasma Instabilities (Plenum, New Previous attempts to explain edget urbulence in TEXT York, i974), Vol. 2. 19M. N. Rosenbluth and A. Simon, Phys. Fluids 8, 1300 (1965). by resistive hydrodynamic modes and by collisional modes *“A Hasegawaa nd M. Wakatani, Phys.R ev. Lett. 50,682 ( 1983); M. Wa- have not beenc ompletely successful.T he theoretical formu- katani and A. Hasegawa,P hys. Fluids 27, 611 ( 1984). las used have neglected the effects of shear flow assuming ” P. W. Terry and W. Horton, Phys.F luids25.491 ( 1982); 26,106 ( 1983). that the background velocity simply Doppler shifted the fre- * ’S. Chandrasekhar,H ydrodynamica nd HydromagnericS tability (Dover, New York, 1961), Chap. 11. quencieso f these instabilities. In view of the present theory *‘P. G. Drazin and L. N. Howard, J. Fluid Mech. 14,257 ( 1962). and new measurements of Ritz et a/.‘* (reporting ‘*J. T. Stuart, J. Fluid Mech. 29,417 (1967). du,/dr<106/sec,a reexamination of the comparison be- “R. T. Pierrehumbert and S. E. Windall, J. Fluid Mech. 114,59 ( 1982). tweenf luctuation theory and experiment is necessaryt,a king *“H. Lamb, Hydrodynamics( Dover, New York, 1945), 6th ed. p. 225. *‘R. Bengtsona nd T. Rhodes (private communication). into account the finite value of dv,/dr. Here the first results “Ch. P. Ritz, H. Lin,T. L. Rhodes,a nd A. J. Wootton, Phys.R ev. Lett. 65, of the drift wave resistive g turbulence in the presenceo f 2543 (1990). 954 Phys. Fluids B,Vol.3,No.4,April 1991 Tajimas f al. 954 Downloaded 17 Dec 2009 to 128.83.61.179. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp