On the application of Maxwell's theory to many-body systems, or why the resistive magnetohydrodynamic equations are not closed

TL;DR

This paper examines the application of Maxwell's theory to many-body systems in resistive MHD, revealing that the standard equations are incomplete without Gauss's law, which is essential for closing the system and determining key variables.

Contribution

It clarifies that the resistive MHD equations are not inherently closed and emphasizes the necessity of reinstating Gauss's law for a complete and solvable model.

Findings

Standard resistive MHD equations are incomplete without Gauss's law.

Reinstating Gauss's law closes the system and allows for determining current and velocity.

Analysis of electromagnetic hydrodynamic models including magnetization force is provided.

Abstract

The resistive magnetohydrodynamic (MHD) equations as usually defined in the quasineutral approximation refer to a system of 14 scalar equations in 14 scalar variables, hence are determined to be complete and soluble. These equations are a combination of Navier-Stokes and a subset of Maxwell's. However, one of the vector equations is actually an identity when viewed from the potential formulation of electrodynamics, hence does not determine any degrees of freedom. Only by reinstating Gauss's law does the system of equations become closed, allowing for the determination of both the current and mass flow velocity from the equations of motion. Results of a typical analysis of the proposed electromagnetic hydrodynamic model including the magnetization force are presented.