Barotropic Magnetohydrodynamics as a Four Function Field Theory with Non-Trivial Topology and Aharonov-Bohm Effects

TL;DR

This paper develops a simplified variational framework for magnetohydrodynamics using four functions, capturing complex topologies and Aharonov-Bohm effects, and reducing the number of variables compared to standard formulations.

Contribution

It introduces a four-function Eulerian variational principle for magnetohydrodynamics that accounts for non-trivial topologies and Aharonov-Bohm effects, simplifying the traditional variable set.

Findings

Reduced variables from seven to four in the variational formulation.

Inclusion of non-single valued functions representing topological effects.

Connection between MHD functions and Aharonov-Bohm phase phenomena.

Abstract

Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. In previous works [1] Yahalom & Lynden-Bell and later Yahalom [2] introduced a simpler Eulerian variational principle from which all the relevant equations of Magnetohydrodynamics can be derived. The variational principles were given in terms of four independent functions for non-stationary flows and three independent functions for stationary flows. This is less than the seven variables which appear in the standard equations of magnetohydrodynamics which are the magnetic field, the velocity field and the density . In the case that the magnetohydrodynamic flow has a non trivial topology such as when the magnetic lines are knotted or magnetic and stream lines are knotted, some of the functions appearing in the Lagrangian are non-single valued. Those functions play the same rule as the phase in the Aharonov-Bohm celebrated effect [3].