Nonlinear and Non-ideal Effects on FRC Stability Page: 4 of 10
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n = 1 tilt motion that does not result in a total loss of confinement . However, the reported
FRC stability for larger values of S* has not been explained so far.
Here we present new results of hybrid and two-fluid (Hall-MHD) simulations of the n = 1
tilt mode in prolate FRCs. The linear instability mechanism and the nonlinear evolution of
unstable modes, as well as the effects of the particle loss along the open field lines, and Hall
stabilization have been studied using a 3D nonlinear hybrid and MHD simulation code (HYM)
2. Linear stability
In order to assess the importance of different stabilizing factors and driving forces on the
n = 1 tilt mode, we have focussed on the three kinetic effects: finite Larmor radius (FLR),
Hall effects, and resonant ion effects. The first two are stabilizing, whereas the third one can be
destabilizing, and tends to obscure the FLR and Hall stabilization in fully kinetic calculations.
Two classes of equilibria have been considered: equilibria with an arbitrary chosen pressure
profile, which, for large elongations, usually corresponds to a configuration with a racetrack-
like separatrix shape; equilibria with large E and an elliptic separatrix shape for special pressure
profile as proposed by Barnes .
0.0 4 8 12 16 H
0.00 0.05 0.10 0.15 0.20 0 I I
0 0.02 0.04 0.06 0.08 0.1 0.12
Figure 1: Growth rate and real frequency of Figure 2: Growth rate of the tilt mode from
the tilt mode from Hall-MHD simulations with hybrid simulations with Hall term (solid line)
E =6. and without Hall term (dashed line) for E=4.
A. Hall effects
To isolate Hall effects from other kinetic effects, we have performed two-fluid (Hall-MHD)
simulations of the n = 1 tilt mode. The MHD version of HYM code has been modified to
include the Hall term in the Ohm's law, and sub-cycling in the induction equation has been used
to insure the numerical stability. Calculations with E=6 and elliptical separatrix shape show a
reduction of the tilt mode growth rate for small S* (Fig. 1) and a significant change in the mode
structure. As S* decreases, the tilt mode becomes more localized, both radially and in the axial
directions, compared to that of the MHD. The unstable mode has a negative real frequency, and
rotates in the direction opposite to that of the equilibrium current. Since the reduction in the
linear growth rate in the strongly kinetic regime, S*/E - 1, is rather modest (50% at most),
Hall stabilization alone cannot account for the experimentally observed stability. Similar results
for different equilibria were obtained in .
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Belova, E.V.; Davidson, R.C.; Ji, H. & Yamada, M. Nonlinear and Non-ideal Effects on FRC Stability, report, October 21, 2002; Princeton, New Jersey. (digital.library.unt.edu/ark:/67531/metadc734450/m1/4/: accessed May 26, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.