Local ITG-like Instability in the Two-fluid and Extended MHD Models in Slab Geometry

We have verified the NIMROD code by direct comparison with analytic solutions for the case of a plasma with uniform equilibrium density and electron temperature, but with a small gradient of the equilibrium ion temperature, in Cartesian slab geometry. Analytic solutions of the two-fluid and Extended MHD equations in the local approximation and ballooning ordering are developed. The requirement that the corresponding ideal MHD solution remain stable places a constraint on the allowable magnitude of the ion temperature gradient for the local approximation to remain valid. Within this constraint, it is found that the system is unstable if k⊥ρi is sufficiently large, but still small enough for the fluid model to be valid. Qualitative agreement with kinetic theory requires that both the ion gyro-viscosity and the ion diamagnetic heat flux be included in the model. NIMROD results are presented for this problem and are compared with the growth rates obtained with the local analytic model. With fixed β = βe + βi = 0.05 reasonable agreement is found for the variation of the growth rate as a function of inverse equilibrium ion temperature scale length 1/LT i0, the electron β fraction 0 < fe = βe/(βi + βe) < 1, and k⊥ρi. There is disagreement between the shape of the global eigenfunction obtained with NIMROD and Department of Physics TriAlpha Corp.