Sherwood 2015

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approvedechowell.pdf2015-01-15 11:14:43Eric Howell

Extended MHD Analysis of the Gravitational Interchange.

Author: Eric C. Howell
Requested Type: Poster Only
Submitted: 2015-01-15 11:13:15

Co-authors: C.R. Sovinec

Contact Info:
University of Wisconsin-Madison
1500 Engineering Drive
Madison, Wi   53706

Abstract Text:
In a recent letter, Zhu, et al. derive a dispersion relation for the g-mode in a slab using an extended MHD model that includes gyroviscosity and a two-fluid Ohm's law [P. Zhu et al, PRL 101, 2008]. The authors analyze the dispersion relation in the simplified case that only includes gyroviscosity. They show that complete gyroviscous stabilization can fail with finite-β. Here we analyze the dispersion relation that results from including two-fluid effects but neglecting gyroviscosity, and we analyze the dispersion relation that results from including both effects. We also present calculations of the g-mode comparing results from the NIMROD code [Sovinec and King, JCP 229, 2010] with the analytic model.
The two-fluid dispersion relation reduces to a two-parameter model X3+G(1+H)X2+X+GH=0, where X=ω/γM is the mode frequency normalized to the MHD growth rate, γM=gn'/n, G=ωg/γM is the normalized gravitational drift frequency, ωg=-g ky/Ω, and H=ω*/ωg is a normalized ion diamagnetic drift frequency, ω*=ky(Pi'- γ Pi n'/n )/(Ωmn). The behavior of the solutions to the dispersion relation is determined by H, which only depends on equilibrium quantities. Increasing G corresponds to increasing ky. At small G the solutions are the unstable g-mode, its damped complex conjugate counterpart, and a stable ion drift wave. For H>0 a second instability exists due to the interaction between the g-mode and the ion drift wave. This instability grows at a rate comparable to the MHD growth rate and persists at infinite ky.
A second instability also exists in the full model. Its growth rate greatly exceeds the MHD growth rate and results from the interaction between the two branches of the stabilized g-mode. The instability occurs when kyρi > 1, where ρi is the ion Larmor radius. While the fluid model is not physically valid in this regime, this mode is a concern for codes that use extended MHD.


March 16-18, 2015
The Courant Institute, New York University