Nested toroidal flux surfaces in magnetohydrostatics. Generation via soliton theory

Abstract

It is shown that the classical magnetohydrostatic equations of an infinitely conducting fluid reduce to the integrable potential Heisenberg spin equation subject to a Jacobian condition if the magnitude of the magnetic field is constant along individual magnetic field lines. Any solution of the constrained potential Heisenberg spin equation gives rise to a multiplicity of magnetohydrostatic equilibria which share the magnetic field line geometry. The multiplicity of equilibria is reflected by the local arbitrariness of the total pressure profile. A connection with the classical Da Rios equations is exploited to establish the existence of associated helically and rotationally symmetric equilibria. As an illustration, Palumbo's 'unique' toroidal isodynamic equilibrium is retrieved.

The Euler Equations of Spatial Gasdynamics and the Integrable Heisenberg Spin Equation

“…However, in the case of constant pressure surfaces of revolution, the solution of the Gilbarg problem has been found. The case of closed constant pressure surfaces of revolution has been dealt with by Palumbo in the above-mentioned context of magnetohydrostatics [10,11]. The case of open constant pressure surfaces of revolution has subsequently been treated by Grice and Schief [21,22].…”

“…In [7,8], it was established that, remarkably, constant speed flows are governed by the integrable Heisenberg spin equation [9] subject to a geometric constraint. The significance of the mathematically equivalent analog of constant speed flows in magnetohydrostatics was discussed in [10,11]. A canonical extension to magnetohydrodynamics was presented in [12].…”

The pulsrodon in 2+1-dimensional magneto-gasdynamics: Hamiltonian structure and integrability

“…30, and references therein). Here, in view of the Steen-Ermakov-Pinney equation, it is convenient to express the solution in terms of the Weierstrass ℘ function.…”

Novel integrable reductions in nonlinear continuum mechanics via geometric constraints

“…27 carry over mutatis mutandis to provide particular spatial geometries that adhere to the kinematic conditions ͑6.1͒ and ͑6.2͒. It is recalled that the conditions ͑6.21b͒ and ͑6.21c͒ encapsulate the integrable Heisenberg spin equation while ͑6.21a͒ is identical to the constraint encountered in the Gilbarg problem.…”