Theory and discretization of ideal magnetohydrodynamic equilibria with fractal pressure profiles

Self Cite

A brief critique is presented of some different classes of magnetohydrodynamic equilibrium solutions based on their continuity properties and whether the magnetic field is integrable or not. A generalized energy functional is introduced that is comprised of alternating ideal regions, with nested flux surfaces with an irrational rotational transform, and Taylor-relaxed regions, possibly with magnetic islands and chaos. The equilibrium states have globally continuous magnetic fields, and may be constructed for arbitrary three-dimensional plasma boundaries and appropriately prescribed pressure and rotational-transform profiles.

“…x, where x i = i/N and x = 1/N, fails spectacularly, as do higher-order quadratures that are based on regular grids (Kraus & Hudson 2017).…”

Section: Smooth Pressure Smooth Non-integrable Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Three-dimensional magnetohydrodynamic equilibria with continuous magnetic fields

Hudson

Kraus

2017

Self Cite

“…The highly irrational surfaces that persist for large enough perturbations typically show self-similar fractal behaviour (Hudson & Kraus 2017; Morrison 2000). Magnetohydrdynamic equilibrium with such fractal solutions has been recently studied in Kraus & Hudson (2017). Hamiltonians that differ non-perturbatively from integrable ones are of course much more difficult to understand.…”

Section: Introductionmentioning

confidence: 99%

Low-shear three-dimensional equilibria and vacuum magnetic fields with flux surfaces

Sengupta

Weitzner

2019

Stellarators are generically small current and low plasma beta (β = p/B 2 ≪ 1) devices. Often the construction of vacuum magnetic fields with good magnetic surfaces is the starting point for an equilibrium calculation. Although in cases with some continuous spatial symmetry flux functions can always be found for vacuum magnetic fields, an analogous function does not, in general, exist in three dimensions. This work examines several simple equilibria and vacuum magnetic field problems with the intent of demonstrating the possibilities and limitations in the construction of such states. Starting with a simple vacuum magnetic field with closed field lines in a topological torus (toroidal shell with a flat metric), we obtain a self-consistent formal perturbation series using the amplitude of the nonsymmetric vacuum fields as a small parameter. We show that systems possessing stellarator symmetry allow the construction order by order. We further indicate the significance of stellarator symmetry in the amplitude expansion of the full ideal magnetohydrodynamics (MHD) problem as well. We then investigate the conditions that guarantee neighboring flux surfaces given the data on one surface, by expanding in the distance from that surface. We show that it is much more difficult to find low shear vacuum fields with surfaces than force-free fields or ideal MHD equilibrium. Finally, we demonstrate the existence of a class of vacuum magnetic fields, analogous to "snakes" observed in tokamaks, which can be expanded to all orders.

“…Only highly irrational surfaces persist, and they typically show self-similar fractal behaviour (Morrison 2000; Hudson & Kraus 2017). MHD equilibrium with such fractal solutions has been recently studied in Kraus & Hudson (2017). To describe systems where the magnetic shear is weak or zero in some region, mathematical models like non-twist maps (del Castillo-Negrete, Greene & Morrison 1996; Morrison 2000) have been developed.…”

Section: Introductionmentioning

confidence: 99%

Low-shear three-dimensional equilibria in a periodic cylinder

Jaquiery

Sengupta

2019

We carry out expansions of non-symmetric toroidal ideal magnetohydrodynamic (MHD) equilibria with nested flux surfaces about a periodic cylinder, in which physical quantities are periodic of period 2π in the cylindrical angle θ and z. The cross-section of a flux surface at a constant toroidal angle is assumed to be approximately circular, and data is given on the cylindrical flux surface r = 1. Furthermore, we assume that the magnetic field lines are closed on the lowest order flux surface, and the magnetic shear is relatively small. We extend earlier work in a flat-torus by Weitzner [Physics of Plasmas 23, 062512 (2016)]and demonstrate that a power series expansion can be carried out to all orders using magnetic flux as an expansion parameter. The cylindrical metric introduces certain new features to the expansions compared to the flat-torus. However, the basic methodology of dealing with resonance singularities remains the same. The results, even though lacking convergence proofs, once again support the possibility of smooth, lowshear non-symmetric toroidal MHD equilibria.