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Relaxed magnetohydrodynamics with cross-field flow

Arash Tavassoli, Stuart R. Hudson, Zhisong Qu, Matthew Hole

Abstract

The phase-space Lagrangian model of Dewar et al. (Phys. Plasmas 27, 062507, 2020) provides a framework for incorporating cross-field flow into relaxed equilibria while retaining ideal magnetohydrodynamics force balance. Here, we characterize the steady-state solution space and identify a solvability condition that couples the prescribed constrained flow to the geometry through the metric tensor. Using this condition, we construct equilibria in slab, cylindrical, and toroidal geometries. In toroidal geometry, the cross-field flow strongly correlates with magnetic-island structure: varying the rotation frequency modifies the dominant Fourier harmonic of the radial component of the magnetic field and can drive a transition from a primary (m = 1) island to secondary (m = 2) islands. In slab and cylindrical geometries, flow parameters weakly affect island width but strongly modify equilibrium profiles.

Relaxed magnetohydrodynamics with cross-field flow

Abstract

The phase-space Lagrangian model of Dewar et al. (Phys. Plasmas 27, 062507, 2020) provides a framework for incorporating cross-field flow into relaxed equilibria while retaining ideal magnetohydrodynamics force balance. Here, we characterize the steady-state solution space and identify a solvability condition that couples the prescribed constrained flow to the geometry through the metric tensor. Using this condition, we construct equilibria in slab, cylindrical, and toroidal geometries. In toroidal geometry, the cross-field flow strongly correlates with magnetic-island structure: varying the rotation frequency modifies the dominant Fourier harmonic of the radial component of the magnetic field and can drive a transition from a primary (m = 1) island to secondary (m = 2) islands. In slab and cylindrical geometries, flow parameters weakly affect island width but strongly modify equilibrium profiles.

Paper Structure

This paper contains 14 sections, 58 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Action principle and Euler--Lagrange equations
  3. 2D equilibrium with purely toroidal constrained flow $\vb{v}$
  4. The coordinate system
  5. Generalized Beltrami equation
  6. The equation of motion
  7. The numerical methods and convergence plots
  8. Example solutions in the Hahm--Kulsrud--Taylor slab geometry and the cylindrical geometry
  9. Hahm--Kulsrud--Taylor slab geometry
  10. The cylindrical geometry
  11. Effect of the $\alpha$ on the flow anisotropy, equilibrium profiles and magnetic islands
  12. The toroidal geometry and Finn -- Antonsen equilibrium
  13. Effect of the angular frequency $\omega$ on the flow anisotropy and the geometry of magnetic island
  14. Conclusion and discussion

Figures (9)

  • Figure 1: Numerical convergence of the Dedalus solver, in the toroidal geometry.
  • Figure 2: (a) HKT slab geometry. (b) Cylindrical geometry. The perturbed boundaries are shown in blue.
  • Figure 3: Example solutions in an HKT slab geometry. (a) Poincaré plot of the magnetic field, (b) magnetic field norm, (c) flow norm, and (d) flow anisotropy around the magnetic field. $\delta=0.02$ and $\alpha=10$ used along with the parameters of \ref{['tab:parameters']}.
  • Figure 4: Example solutions in cylindrical geometry. (a) Poincaré plot of the magnetic field, (b) magnetic field norm, (c) flow norm, and (d) flow anisotropy around the magnetic field. $\delta=0.02$ and $\alpha=10$ used along with the parameters of \ref{['tab:parameters']}.
  • Figure 5: Effect of $\alpha$ on the (a) flow anisotropy, (b) flow norm, (c) magnetic field, and (d) pressure, in cylindrical geometry.
  • ...and 4 more figures