Computation of multi-region relaxed magnetohydrodynamic equilibria

Abstract

We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed using the stepped-pressure equilibrium code, SPEC. Convergence studies with respect to radial and Fourier resolution are presented

Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria

“…A numerical implementation of the MRxMHD theory, the Stepped-Pressure Equilibrium Code (SPEC), 21 was recently developed. SPEC is capable of calculating three-dimensional MRxMHD equilibria in slab, cylindrical, and toroidal geometries.…”

“…SPEC is capable of calculating three-dimensional MRxMHD equilibria in slab, cylindrical, and toroidal geometries. SPEC has been benchmarked against VMEC in the axisymmetric case 20,21 and has been used to reproduce selforganized helical states in reversed field pinches. 26 In this paper, we use SPEC in slab geometry in order to benchmark the obtained nonlinear results against those of the semi-analytical, linear model derived in Sec.…”

Experimental and numerical study of error fields in the CNT stellarator

“…Future work will focus on improving the optimization algorithm to achieve even better fits to the observed data. These improvements will include (1) simultaneously optimizing to data from multiple currentratios using F vectors that include geometric parameters for multiple Poincaré cross-sections, (2) operating the field line tracer at greater numerical precision to reduce the uncertainty in numerically generated Fourier coefficients and allow for smaller finite difference intervals for the computation of the Jacobian, (3) expanding the p vector to include other sources of error including coil deformations, uncompensated coil leads, and displacements of the PF coils, and (4) investigating ways of generalizing the X vector to include information about Poincaré data inside islands, possibly analogous to generalizations of 3D toroidal equilibria found in codes like PIES [27], SIESTA [28], and SPEC [29]. Many of these improvements will be more demanding computationally, although much of the algorithm can be parallelized (in particular, the calculation of the covariance and Jacobian matrices).…”

Ideal magnetohydrodynamic stability of configurations without nested flux surfaces

“…In the case of stellarators, for example, the equilibrium is usually calculated using the VMEC [1] code, and its stability is assessed by codes such as CAS3D [2] or TERPSICHORE [3]. There are numerical tools capable of solving the force balance equation J × B = ∇p in three dimensions without requiring nested flux surfaces (the PIES [4], HINT [5], SIESTA [6] and SPEC [7] codes), but there is at the moment no way of directly calculating the stability of the resulting equilibria. In tokamaks, axisymmetric equilibria are obtained from the Grad-Shafranov equation, and their stability is routinely evaluated by a large number of codes, but none of these is applicable if the magnetic topology is broken by error fields or intentionally produced resonant magnetic perturbations creating magnetic islands or regions with an ergodic magnetic field.…”