Abstract
An analytical solution to the sharp-front magnetic diffusion problem with a step-like resistance model is an interesting mathematical physics problem. It also relates to the plasma production at the surface of a metal under strong magnetic loading measured in experiments. In this paper, we make an advance on an earlier work of Xiao et al. [Phys. Plasmas 23, 082104 (2016)], by deriving formulas for the threshold magnetic field strength and the diffusion velocity of the magnetic diffusion wave under a step-like resistance model to the higher order accuracy.
Highlights
The diffusion of a strong magnetic field into a metal is of the highly non-linear property, due to the combined action of the Ohmic heating on the metal and the strong dependence of the metal’s resistance on temperature.1 So, generally speaking, analytical solutions to the magnetic diffusion problem are usually hard to obtain
In the work of Schnitzer,5 asymptotic solution of the magnetic diffusion problem in the asymptotic limit is derived under the condition of a specific temperature dependent resistance model and a specific time dependent outer magnetic field
In the work of Xiao,7 the analytical solution to the magnetic diffusion problem under a step-like resistance model is studied, and based on the approximate solution, the magnetic diffusion velocity is derived in analytical form, and an analytical formula for the threshold magnetic field is derived
Summary
The diffusion of a strong magnetic field into a metal is of the highly non-linear property, due to the combined action of the Ohmic heating on the metal and the strong dependence of the metal’s resistance on temperature. So, generally speaking, analytical solutions to the magnetic diffusion problem are usually hard to obtain. In the work of Schnitzer, asymptotic solution of the magnetic diffusion problem in the asymptotic limit is derived under the condition of a specific temperature dependent resistance model and a specific time dependent outer magnetic field. In the work of Xiao, the analytical solution to the magnetic diffusion problem under a step-like resistance model is studied, and based on the approximate solution, the magnetic diffusion velocity is derived in analytical form, and an analytical formula for the threshold magnetic field is derived. We will make an advance on an earlier work of Xiao, by deriving formulas for the threshold magnetic field strength Bc and the diffusion velocity Vc of the magnetic diffusion wave under a step-like resistance model to a higher order accuracy
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