S. R. Hudson

Computation of multi-region relaxed magnetohydrodynamic equilibria

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.

Eigenvalue problems for Beltrami fields arising in a three-dimensional toroidal magnetohydrodynamic equilibrium problem

This work was supported in part by the U.S. Department of Energy Contract No. DE-AC02-76CH03073 and Grant No. DE-FG02-99ER54546 and the Australian Research Council.

Equilibria and stability in partially relaxed plasma–vacuum systems

We develop a multiple interface variational model, comprising multiple Taylor-relaxed plasma regions separated by ideal MHD barriers. The magnetic field in each region is Beltrami, ∇× B = µB, and the pressure constant. Between regions the pressure, field strength, and rotational transform may have step changes at the ideal barrier. A principle motivation is the development of a mathematically rigorous ideal MHD model to describe intrinsically 3D equilibria, with nonzero internal pressure, using robust KAM surfaces as the barriers. This article chiefly addresses whether the stability of two interface configurations with continuous rotational transform, but vanishing interface separation, is different from the stability of a single interface configuration with jump in the rotational transform. To make the problem analytically tractable, we derive the equilibria and stability of a multi-interface plasma in a periodic cylinder, generalizing the cylindrical treatment of Kaiser and Uecker (2004 Q. J. Mech. Appl. Math. 57 1–17). For two interfaces with no jump in rotational transform, we show that one eigenmode has in-phase interface displacements, and an eigenvalue that converges to the single barrier case in the limit of vanishing interface width. The complementary eigenmode is out-of-phase, and highly unstable. Physically, the unstable eigenmode is driven by the parallel current, and caused by the high shear required to match the different rotational transform on each interface. In the limit that the interface separation vanishes, the shear and parallel current density become infinite, and the parallel current between the interfaces nonzero. Surfaces with out-of-phase displacements will then collide, unless the amplitude goes to zero as the interface separation goes to zero. These results suggest the hypothesis that KAM barriers with different irrational rotational transform on either side may be allowable without violating nonlinear stability.

Relaxed plasma equilibria and entropy-related plasma self-organization principles

The concept of plasma relaxation as a constrained energy minimization is reviewed. Recent work by the authors on generalizing this approach to partially relaxed threedimensional plasma systems in a way consistent with chaos theory is discussed, with a view to clarifying the thermodynamic aspects of the variational approach used. Other entropy-related approaches to finding long-time steady states of turbulent or chaotic plasma systems are also briefly reviewed.

Stepped Pressure Profile Equilibria in Cylindrical Plasmas via Partial Taylor Relaxation

We develop a multiple interface variational model, comprising multiple Taylor-relaxed plasma regions separated by ideal magnetohydrodynamic (MHD) barriers. A principal motivation is the development of a mathematically rigorous ideal MHD model to describe intrinsically three-dimensional equilibria, with non-zero internal pressure. A second application is the description of transport harriers as constrained minimum energy states. As a first example, we calculate the plasma solution in a periodic cylinder, generalizing the analysis of the treatment of Kaiser and Uecker (2004 Q. J. Mech, Appl. Math. 57. 1-17). who treated the single interface in cylindrical geometry, Expressions for the equilibrium field are generated, and equilibrium states computed. Unlike other Taylor relaxed equilibria, for the equilibria investigated here, only the plasma core necessarily has reverse magnetic shear. We show the existence of tokamak-like equilibria, with increasing safety factor and stepped-pressure profiles.

Boundary modulation effects on MHD instabilities in heliotrons

In three-dimensional configurations, the confinement region is surrounded by the stochastic magnetic field lines related to magnetic islands or separatrix, leading to the fact that the plasma?vacuum boundary is not so definite compared with tokamaks that the various modulations of the plasma?vacuum boundary will be induced around the stochastic region by synergetic effects between a transport around the stochastic region and a large Shafranov shift of the whole plasma, in especially high-? operations. To examine such modulation effects of the plasma boundary on MHD instabilities, high-? plasmas allowing a large Shafranov shift or a large Pfirsch?Schl?ter current are considered in the inward-shifted LHD configurations with the vacuum magnetic axis Rax of 3.6?m, for which previous theoretical analyses based on fixed MHD equilibria indicate that pressure-driven modes are significantly more unstable compared with experimental observations. The concept of the averaged flux surfaces allowing a movement of the equilibrium plasma into the stochastic region is introduced, which induces a boundary modulation and, at the same time, reduces the discrepancy on MHD equilibria between the experimentally obtained and theoretically considered. As a result, it is shown that the boundary modulation, namely, the whole plasma outward-shift due to a large Pfirsch?Schl?ter current has significant stabilizing effects on ideal MHD instabilities, leading to partially resolving the discrepancy on MHD stability between experimental results and theoretical analyses.

Manipulation of islands in a heliac vacuum field

Abstract To confine plasma adequately in toroidal magnetic fields for nuclear fusion experiments, it is essential to control the magnetic islands caused by perturbations and non-axisymmetry inherent to stellarators. A method is introduced which enables the perturbation harmonics of the field line Hamiltonian (which determines the size of islands and the degree of chaos) to be approximated quickly. The method is based upon quadratic-flux minimizing surfaces which pass directly through both the X and O points of the island chains. Using a suitable measure of island width, it is possible to utilize standard numerical optimization methods to achieve desirable configurations. A major island chain in the H-1NF heliac vacuum field has been made to both disappear and to reappear with opposite phase. These results have significance for self-healing phenomena and the method is generally applicable to all stellarators and 1 1 2 - dimensional Hamiltonian dynamical systems.

Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets

We consider the linear and nonlinear ideal plasma response to a boundary perturbation in a screw pinch. We demonstrate that three-dimensional, ideal-MHD equilibria with continuously nested flux-surfaces and with discontinuous rotational-transform across the resonant rational-surfaces are well defined and can be computed both perturbatively and using fully nonlinear equilibrium calculations. This rescues the possibility of constructing MHD equilibria with current sheets and continuous, smooth pressure profiles. The results predict that, even if the plasma acts as a perfectly conducting fluid, a resonant magnetic perturbation can penetrate all the way into the center of a tokamak without being shielded at the resonant surface.

Non-axisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions

We describe a magnetohydrodynamic constrained energy functional for equilibrium calculations that combines the topological constraints of ideal MHD with elements of Taylor relaxation. Extremizing states allow for partially chaotic magnetic fields and non-trivial pressure profiles supported by a discrete set of ideal interfaces with irrational rotational transforms. Numerical solutions are computed using the Stepped Pressure Equilibrium Code, and benchmarks and convergence calculations are presented.

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

Using the recently developed multiregion, relaxed MHD (MRxMHD) theory, which bridges the gap between Taylors relaxation theory and ideal MHD, we provide a thorough analytical and numerical proof of the formation of singular currents at rational surfaces in non-axisymmetric ideal MHD equilibria. These include the force-free singular current density represented by a Dirac δ-function, which presumably prevents the formation of islands, and the Pfirsch-Schluter 1/x singular current, which arises as a result of finite pressure gradient. An analytical model based on linearized MRxMHD is derived that can accurately (1) describe the formation of magnetic islands at resonant rational surfaces, (2) retrieve the ideal MHD limit where magnetic islands are shielded, and (3) compute the subsequent formation of singular currents. The analytical results are benchmarked against numerical simulations carried out with a fully nonlinear implementation of MRxMHD.