Natural curvilinear coordinates for ideal magnetohydrodynamics equations. Solutions with constant total pressure

TL;DR

This paper develops a natural curvilinear coordinate system for ideal magnetohydrodynamics equations, deriving new exact solutions for non-stationary flows with constant total pressure, revealing complex magnetic surface geometries.

Contribution

It introduces a novel curvilinear coordinate framework and provides explicit solutions with arbitrary functions for incompressible MHD flows at constant total pressure.

Findings

Derived a nonlinear vector wave equation in natural coordinates.

Calculated symmetry group and group classification.

Presented explicit solutions with complex magnetic surface shapes.

Abstract

Equations of magneto-gasdynamics in the natural curvilinear system of coordinates where trajectories and magnetic lines play a role of coordinate curves are reduced to the nonlinear vector wave equation coupled with the incompressibility condition in the form of the generalized Cauchy integral. The symmetry group of obtained equation, equivalence transformation, and group classification with respect to the constitutive equation are calculated. New exact solutions with functional arbitrariness describing non-stationary incompressible flows with constant total pressure are given by explicit formulae. The corresponding magnetic surfaces have the shape of deformed nested cylinders, tori, or knotted tubes.