LLNL FRC Tilt Mode Simulation 1987

UCRL—96819

Mode in FRC***

DE87 011564

E , J. Horowitz D.E. Shumaker

3-D Hybrid PIC Code to :. -lei the T i lt

This Paper was Prepared for Submittal to 3tli Coranacc Toroid Synnosiun June 4 - 5, 1087 College Park, d r y l a nd

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Livermore, California 94550

3-D Hybrid PIC Code to Model the Till Mode in PRCs*

E.J. Horowitz and D.E. Shumalter National Magnetic Fusion Energy Computer Center Lawrence Livermore National Laboratory

The results from QN3D are presented. QN3D is a 3-dimensional hybrid particle-in-cell code designed to run efficiently on the Cray-2 Multiprocessor. The chief application ha* been to the tilt mode instability in FRCs.

In general we would assume that the if the plasma containment vessel were far from the plasma boundary then the shape of the vessel would be of little consequence [1]. However, to facilitate modelling devices with passive mirrors we actually modelled the cylindrical wall as closely as possible.

QN3D accept* as input, the magnetic field, the ion particle density and the ion tem perature on a two-dimensional r-z grid. These quantities are interpolated to the rest of the cartesian grid under the assumption of azimuthal symmetry. The particles are initialised’ with random number* chosen ar.cording to the particle distributions input from the equi librium code. The runs done here used equilibria computed by EQV, a kinetic equilibrium code developed by Dan Shumaker.

One easy method of viewing plasma behavior is to plot contours of constant particle density. This showed very clearly that one case tilted while the other did not (see figure 1). Note that the tilt mode instability observed here grew out of noise introduced by the random nature of the particle intialization. No initial perturbation was employed to help the plasma develop the tilt.

We expect the tilt mode co be unstable when the ion gyroradius becomes small with respect to the size of the plasma. A convenient measure of this relative size is a, which is a measure of the number of ion gyroradii between the o-point and and the seperatrix. Analytically [2]

For a quantitative diagnostic, we simulated an experimental mechod suggested by Michel Tuszewski of the Los Alamos National Laboratory. He suggested that we measure the Faraday rotation of a light beam shot through the plasma. The Faraday rotation of such a beam is proportional to the integral of the density multiplied by the magnetic field [3], i.e.

where R is the o-point radius, r, is the seperatrix radius and p; is the ion gyroradius. We investigated two cases, one with a — 1.6 and another with a = 12.

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In order to pick the tilt mode signature out of the noise we fitted the simulated Faraday rotation data to a polynomial with the same signature. In particular, we used a least-squares fit to

where the integral is along the beam. From symmetry, it is clear that Of is initially zero in an FRC. However, aa the tilt mode develops, some rotation should be noticable. In fact, we should be able to recognize the tilt mode by the signature in a plot of Op as a function of z (figure 2).

where L is predetermined and A and B are found by the fitting procedure. /„ is the polynomial that fits the data found from the Fara/day rotation dignostic done in the x-z plane. A similar function, /u(z), was found for the y-z plane data. To get a magnitude from these functions we simply integrated the sum of the square of these functions, and then took the root, i.e.

|@jr| gives us a magnitude as function of time from which we can get a growth rate. Again, The results are quite clear. The high-s case shows the tilt growing with a growth rate close to MHD predictions (7 = 1.247M H C). The low-s case shows only slight growth if any at all (7 *= .0~7MHC)- Note that comparing the magnitudes of these numbers is not possible since in the density and magnetic fields are orders of magnitude different.

Quite fortunately, we have been able to give strong credence to our initial hypothesis that the tilt mode will exist in regimes of higher s. This result, by itself, is important for those planning to build larger FRC experiments. But, in addition, QN3D has a major advantage in that it should be able to model the nonlinear regime of the tilt mode which will be even more crucial to the future of FRC experiments.

Our results compare favorably with preliminary results from Barnes, et aL [4] (see figure 3). We would not expect exact agreement since their equilibria are different and they are measuring the displacement of the flux surfaces rather than the Faraday rotation. However, the general agreement is very encouraging.

I © F| =

i l /S

References

[1] D.S. Hamed, J. Comput. Phys. 47 (1982), 452.

Top: Opt Lefka = 12

Bottom: 3/ia Right: a = 1.6

Figure 1: Surfaces of Constant Density

fusion 24 (1984), 1537.

*This worked was perforned under the auspices of the United States Department of Energy by the Lawrence Livermore National Laboratory under contract W-7405-ENG-48.

[2] J.T. Slough, A.L. Hoffman, R..D. Milroy, D.G. Harding and L.C. Steinhauer, Nuclear

[4] D.C. Barnes, J.L. Schwarzmeier, H.R. Lewis and C.E. Seyler, Phys. Fluids 29 (1S86),

[3] D.E. Shumaker, “Plasma Physics”, clast notes for UCD DAS 280C, 1985.

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Figure 2: Expected Tilt Mode Signature from Faraday Rotation Diagnostic Qp it plotted at a function of i. The start indicate the simulation results and the curve is the polynomial fit.

Figure 3: Barnes’ Results Thit figure it from the Physics of Fluids 29, August 1986. The stars indicate our results. Used with permission of the authors and the American Institute of Physics. © Copyright 1986 AIP

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