Symmetry transformations for magnetohydrodynamics and Chew–Goldberger–Low equilibria revisited

Abstract

Being motivated by the paper [O. I. Bogoyavlenskij, Phys. Rev. E 66, 056410 (2002)] we generalise the symmetry transformations for MHD equilibria with isotropic pressure and incompressible flow parallel to the magnetic field introduced therein in the case of respective CGL equilibria with anisotropic pressure. We find that the geometrical symmetry of the field-aligned equilibria can break by those transformations only when the magnetic field is purely poloidal. In this situation we derive three-dimensional CGL equilibria from given axisymmetric ones. Also, we examine the generic symmetry transformations for MHD and CGL equilibria with incompressible flow of arbitrary direction, introduced in a number of papers, and find that they cannot break the geometrical symmetries of the original equilibria, unless the velocity and magnetic field are collinear and purely

On the linear stability of anisotropic pressure equilibria with field-aligned incompressible flow

“…In addition, we have shown that, if a given equilibrium with field-aligned incompressible flows of constant mass density and constant pressure anisotropy function fulfils the aforementioned condition, and therefore is linearly stable, then all the families of equilibria obtained by the application of the symmetry transformations presented in Evangelias & Throumoulopoulos (2019) to the original equilibrium, are also linearly stable, provided that a parameter, C, appearing in these transformations, is positive-definite.…”

“…In a recent work (Evangelias & Throumoulopoulos 2019) a set of symmetry transformations that map anisotropic plasma equilibria with field-aligned incompressible flows was introduced; specifically, these transformations, when applied to a given equilibrium with parallel incompressible flow and anisotropy function, σ d , uniform on the magnetic surfaces labelled by the function ψ, {B, (ψ), V = λB/( √ µ 0 ), P, σ d (ψ)}, produce an infinite family of respective equilibria with field-aligned incompressible flows, but a density and anisotropy function that may vary on the magnetic surfaces, {B 1 , 1 , V 1 , P 1 , σ d 1 }. These transformations are defined by…”

“…To derive that condition it is also important that the perturbation of the effective pressure, P = (P + P ⊥ )/2, remains arbitrary. Furthermore, in § 5 we show that, whenever a given equilibrium is linearly stable, then all families of equilibria derived by the symmetry transformations introduced in Bogoyavlenskij (2001) and Evangelias & Throumoulopoulos (2019), are also linearly stable provided that a parameter involved in those transformations is positive. In addition, in § 6 we apply the derived sufficient condition to a class of analytic helically symmetric equilibria in order to examine the impact of pressure anisotropy, flow, and certain geometrical parameters characterising helical configurations, such as the torsion and pitch length, on the stability of those equilibria.…”

Μελέτη Ισορροπίας Και Ευστάθειας Ελικοειδώς Συμμετρικού Μαγνητισμένου Πλάσματος

“…Then we construct 3D equilibria by applying the introduced transformations to a known class of axisymmetric equilibria. The results presented in Chapter 4 were published in [168]. In Chapter 5 we derive a sufficient condition for the linear stability of plasma equilibria with incompressible flow parallel to the magnetic field, constant mass density and pressure anisotropy, such that the pertinent anisotropy function remains constant.…”