The sharp-front magnetic diffusion wave of a strong magnetic field diffusing into a solid metal

When a mega-gauss magnetic field diffuses into a solid metal, the Joule heat would rise rapidly the temperature of the metal, and the rise of temperature leads to an increase of the metal’s resistance, which in turn accelerates the magnetic field diffusion. Those positive feedbacks acting iteratively would lead to an interesting sharp-front magnetic diffusion wave. By assuming that the metal’s resistance has an abrupt change from a small value $\eta_{\rm S}$ to larger value $\eta_{\rm L}$ at some critical temperature $T_{\rm c}$, the sharp-front magnetic diffusion wave can be solved analytically. The conditions for the emerging of the sharp-front magnetic diffusion wave are $B_0>B_{\rm c}$, $\eta_{\rm L}/\eta_{\rm S} \gg 1$, and $\frac{\eta_{\rm L}}{\eta_ {\rm S}}\frac{B_0-B_{\rm c}}{B_{\rm c}} \gg 1$, where $B_{\rm c} = \sqrt{2\mu_0 J_{\rm c}}$, $B_0$ is the vacuum magnetic field strength, and $J_{\rm c}$ is the critical Joule heat density. The wave-front velocity of the diffusion wave is $V_{\rm c} = \frac{\eta_{\rm L}}{\mu_0}\frac{B_0-B_{\rm c}}{B_{\rm c}}\frac{1}{x_{\rm c}}$, where $x_{\rm c}$ is the depth the wave have propagated in the metal. In this presentation we would like to discuss the derivation of the formulas and its impact to magnetically driven experiments.