The diffusion of a mega-gauss strong magnetic field into a metal exhibits a nonlinear magnetic diffusion wave form, which is attributed to the strong coupling of the magnetic diffusion and Ohmic heating equations, i.e., the rate of magnetic diffusion influences the rate of Ohmic heating, and the temperature rise of the material affects the rate of magnetic diffusion in turn. In the literature [1], the authors obtained an exact theoretical solution to the set of coupled equations of magnetic diffusion and Ohmic heating in one dimension under the step resistivity model. However, the solution is in the form of a complex implicit form consisting of multiple layers of integrals, and the exact numerical solution can only be obtained with the help of numerical integration. In order to be able to use the theoretical solution of nonlinear magnetic diffusion waves more conveniently, it is necessary to obtain an approximate solution that approximates the exact solution but has a simpler expression. In this paper, a combination of theoretical derivation and numerical fitting is used to obtain the approximate solution. On the one hand, through theoretical analysis and as well as approximation of the integral function, it is argued that the relationship between the solution of a nonlinear magnetic diffusion wave and the input parameters can be approximately simplified to the relationship with a single k-factor. On the other hand, through sampling of input parameters, it is verified that the threshold magnetic field Bc and the h factor representing the magnetic diffusion rate do have an approximately single dependence on the k factor, and by fitting to the results of these scatter calculations, simple expressions of Bc and h depending on k is obtained.
 

megagauss,nonlinear magnetic diffusion,step resistivity model,Partial differential equation