Megagauss magnetic-field penetration into a conducting material is studied via a simplified but representative model, wherein the magnetic-diffusion equation is coupled with a thermal-energy balance. The specific scenario considered is that of a prescribed magnetic field rising (in proportion to an arbitrary power r of time) at the surface of a conducting half-space whose electric conductivity is assumed proportional to an arbitrary inverse power γ of temperature. We employ a systematic asymptotic scheme in which the case of a strong surface field corresponds to a singular asymptotic limit. In this limit, the highly magnetized and hot “skin” terminates at a distinct propagating wave-front. Employing the method of matched asymptotic expansions, we find self-similar solutions of the magnetized region which match a narrow boundary-layer region about the advancing wave front. The rapidly decaying magnetic-field profile in the latter region is also self similar; when scaled by the instantaneous propagation speed, its shape is time-invariant, depending only on the parameter γ. The analysis furnishes a simple asymptotic formula for the skin-depth (i.e., the wave-front position), which substantially generalizes existing approximations. It scales with the power γr + 1∕2 of time and the power γ of field strength, and is much larger than the field-independent skin depth predicted by an athermal model. The formula further involves a dimensionless O(1) pre-factor which depends on r and γ. It is determined by solving a nonlinear eigenvalue problem governing the magnetized region. Another main result of the analysis, apparently unprecedented, is an asymptotic formula for the magnitude of the current-density peak characterizing the wave-front region. Complementary to these systematic results, we provide a closed-form but ad hoc generalization of the theory approximately applicable to arbitrary monotonically rising surface fields. Our results are in excellent agreement with numerical simulations of the model, and compare favourably with detailed magnetohydrodynamic simulations reported in the literature.
Some authors1,24,32 employ the slightly different model , where β is a constant. This model essentially corresponds to the special case γ = 1 of (1). It is simple to check in this case that employing the alternative model instead of (1) would leave our main result (5) for the position of the wave front unchanged; the details of the wave-front region profiles would however differ.
The parameter θ is the same as the parameter c defined by Kidder in Ref. 23, and is closely related to the “critical field” defined by Knoepfel in Ref. 1 via the relation ; since this specific strength does not appear to be “critical” in any way, we are reluctant to employ this terminology herein.
According to the Wiedmann-Franz law, the ratio of thermal to electric conductivity is proportional to temperature. Thus, the assumption of a constant thermal conductivity is plausible for γ-values close to one.
These simulations were performed using Matlab's “pdepe” algorithm. Thus, the partial differential equations are transformed into ordinary ones using the so-called “method of lines” technique. The spatial grid is constant and pre-defined by the user; the discretization is second-order. The extent of the domain and the number of nodes was conveniently predetermined by employing the theoretical scalings developed later in this paper for the penetration depth and the wave-front width, respectively. In that way, we have ensured that at any time the narrow wave-front region is covered by at least 50 nodes. The algorithm employs the standard Matlab solvers to advance the solution in time.
The need to have a zero heat flux at the surface, cf. (24), is resolved by the formation of a thin asymptotic layer near it, wherein the heat-diffusion term is not negligible. It is readily verified that the change in the magnetic field and internal energy across this layer are negligible. Thus, the existence of this asymptotic layer has no effect on the leading-order analysis provided in this paper. These arguments are further confirmed by a comparison with our numerical solution, which does include the small heat-conductivity term.
Physically, however, geometrical compression effects are expected to limit the applicability of the idealized model, perhaps less so in the case of a cylindrical shell whose thickness is small compared with the curvature of the metal surface.