On deriving flux freezing in magnetohydrodynamics by direct differentiation

TL;DR

This paper presents a clear, direct derivation of the magnetic flux freezing theorem in ideal MHD using Jacobians, emphasizing its pedagogic value and the role of magnetic monopoles in flux breaking.

Contribution

It introduces a straightforward derivation method for flux freezing in MHD that clarifies the underlying assumptions and the role of magnetic monopoles.

Findings

Derivation using Jacobians enhances understanding of flux conservation.

Highlights the importance of magnetic monopoles in flux breaking.

Provides a pedagogically valuable approach to MHD principles.

Abstract

The magnetic flux freezing theorem or Alfv\'en's theorem, is a basic principle of ideal magnetohydrodynamics (MHD), a commonly used approximation to describe the aspects of astrophysical and laboratory plasmas. The theorem states that the magnetic flux---the integral of magnetic field penetrating a surface---is conserved in time as that surface is distorted in in time by fluid motions. Pedagogues of MHD commonly derive flux freezing without showing how to take the material derivative of a general flux integral and/or assuming a vanishing field divergence from the outset. Here I avoid these shortcomings and derive flux freezing by direct differentiation, explicitly using a Jacobian to transform between the evolving field-penetrating surface at different times. The approach is instructive for its generality and helps elucidate the role of magnetic monopoles in breaking flux freezing. The paucity of appearances of this derivation in standard MHD texts suggests that its pedagogic value is underappreciated.