Verification of a Hall MHD Algorithm in Nimrod for an FRC

The accurate and efficient computation of low-frequency two-fluid phenomena in a confined plasma remains a challenge. As an initial project of the Plasma Science and Innovation Center (PSI-Center), a time-implicit two-fluid version of NIMROD$^{1}$ is being verified and applied to the macroscopic stability of a FRC. FRC macro-stability is a problem of intrinsic interest, both from a fundamental and from a practical standpoint, and also presents a unique verification opportunity. Implicit algorithm issues are briefly discussed. Initial $n$ = 1 linear results have reproduced earlier observed$^{2,3}$ transition from the ``fundamental'' MHD internal tilt mode to modes with higher structure along $B$, with growth rates smaller but comparable to MHD growth rates. Previous long-thin analysis$^{4}$ has been extended to capture the features of these modes and results are compared with NIMROD. Modifications to the analysis suggest effects beyond HMHD, which are important for the stability of these modes. Such effects will be incorporated into future NIMROD versions and verified by comparison with the analysis. $^{1 }$C.R. Sovinec, et al., J. Comp. Phys. \textbf{195}, 355 (2004) $^{2 }$R.D. Milroy, et al., Phys. Fluids \textbf{B1}, 1225 (1989) $^{3 }$Elena V. Belova, et al., Phys. Plasmas \textbf{10}, 2361 (2003) $^{4 }$D. C. Barnes, Phys. Plasmas 10, 1636 (2003)