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Issues Related to Finite-β Gyrokinetics: 1) MHD and Equilibrium, and 2) Singularly Perturbed Equation

Author: W. W. Lee
Requested Type: Poster Only
Submitted: 2015-01-19 13:13:56

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Princeton Plasma Physics Laboratroy
1 Stellarator Road
Princeton, NJ   08540
USA

Abstract Text:
1) The gyrokinetic vorticity equation, as originally given by Eq. (32) in Ref. [1], can be re-written as

[∂/∂t - (c/B) ∇φ×b·∇](∇_⊥^2)φ − 4π(v_A/c)^2 ∇·(J∥+J⊥)=0

where ∇^2 (A∥ + A⊥)=−(4π/c)(J∥ + J⊥), and J⊥, the pressure driven current, consisted of magnetic drift current and dia-magnetic current, satisfies the pressure balance equation of

J⊥ = −(c/B) b x ∇p.

Along with the collisionless Ohm’s law in the parallel direction,

E∥ ≡−(1/c)∂A∥/∂t −b·∇φ= 0,

and a equation of state of dp/dt = 0, we obtain a simple set of gyrokinetic MHD equations with normal modes of ω = ±k∥ v_A, where v_A is the Alfven speed. To calculate the gyrokinetic MHD equilibrium, we first obtain the pressure driven current for a given pressure profile and then solve the coupled equations for φ and A∥. The solution of φ → 0 and ∂A∥/∂t → 0 gives us the equilibrium state with

∇ · (J∥ + J⊥) = 0.

2) The gyrokinetic Ampere’s law for A∥ based on the canonical momentum p∥, along the direction of the magnetic field, can be written as [2]

(∇_⊥^2 - ω_{pe}^2/c^2)A∥ = −4πe(J∥i − J∥e),

where δ_e ≡ c/ω_pe is the electron skin depth. This is the well-known singularly perturbed equation in mathematics when δe^2∇_⊥^2 ≪ 1. Similarly, this type of equations appears for the split-weight scheme based on v∥[3]. For accuracy purposes, one is required to use a grid of the size of δ_e in solving such a equation [3], which is feasible for low-β plasmas. Otherwise, one has to worry about the cancellation of the large term on the left hand side of the equation by those on the right hand side, as being done in our community.
∗Present work is supported by US DoE Contract No. DE-AC02-09CH11466.

[1] W. W. Lee and H. Qin, Phys. Plasmas 10, 3196 (2003);
[2] T. S. Hahm, W. W. Lee and A. Brizard, Phys. Fluids 32, 1941 (1988);
[3] E. Startsev and W. W. Lee, Phys. Plasmas 21, 022505 (2014).
POSTER PRESENTATION

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