Free-boundary MRxMHD equilibrium calculations using the stepped-pressure equilibrium code

Abstract

The stepped-pressure equilibrium code (SPEC) [Hudson et al., Phys. Plasmas 19, 112502 (2012)] is extended to enable free-boundary, multi-region relaxed magnetohydrodynamic (MRxMHD) equilibrium calculations. The vacuum field surrounding the plasma inside an arbitrary 'computational boundary', D, is computed, and the virtual-casing principle is used iteratively to compute the normal field on D so that the equilibrium is consistent with an externally produced magnetic field. Recent modifications to SPEC are described, such as the use of Chebyshev polynomials to describe the radial dependence of the magnetic vector potential, and a variety of free-boundary verification calculations are presented.

Computation of linear MHD instabilities with the multi-region relaxed MHD energy principle

“…where the magnetic field, B = ∇×A and the C < p,l and C > t,l are circuits about the inner (<) and outer(>) boundaries of Ω l in the poloidal and toroidal directions, respectively. The enclosed poloidal flux ∆ψ p , toroidal flux ∆ψ t and the ideal interface boundary condition √ gB•∇s = 0 are enforced by a set of Lagrange multipliers (e i for the i th Fourier harmonic of interface boundary condition, and c 1 , d 1 for the fluxes) [10]. Additionally, in annular volumes, the gauge dependency of the vector potential defined as A ϑ (−1, ϑ, ζ) = 0 and A ζ (−1, ϑ, ζ) = 0, are enforced by the Lagrange multipliers a i and b i respectively for the i th Fourier harmonics.…”

“…The matrices A l , B l and D l are constructed within each Ω l by inserting the representation for the vector potential given in Eqns. (9) and (10) into Eqn. (11) and computing the volume-dependent integrals.…”

“…The matrices B l and D l do not depend upon interface geometry. The detailed construction of elements in matrices A l , D l and B l can be found in Hudson et al [7,10,21] and Qu et al [22]. The magnetic field that satisfies the Beltrami equation, ∇ × B = µ 0 µ l B in each volume Ω l can be computed by an aforementioned Lagrange multipliers strategy.…”

“…(3) are generated when the first variation of F vanishes. To compute such MRxMHD equilibrium solutions, the Stepped Pressure Equilibrium Code (SPEC) [7] has been developed, and systems with axisymmetric and non-axisymmetric geometry have been explored [8,9,10,11,12,13]. Due to the consideration of partial Taylor's relaxation phenomenon [7] in MRxMHD, now in any sub domain Ω l (where ∇p = 0), magnetic reconnection is permitted, enabling the formation of magnetic islands and stochastic field regions.…”

Combined plasma–coil optimization algorithms

“…In reality, however, the geometry of the plasma boundary is not known a priori. Free-boundary equilibrium calculations require as input the external magnetic field, and the self-consistent plasma boundary is determined as part of the equilibrium calculation (Hirshman, van Rij & Merkel 1986;Hudson et al 2020). Existing free-boundary equilibrium codes invariably perform additional iterations compared with their fixed-boundary analogues.…”