Stability of a nonlinear magnetic field diffusion wave

Abstract

On the mode stability of a self-similar wave mapThe thermal instabilities that develop in a conductor during nonlinear diffusion of a magnetic field were treated in a linear approximation by solving an eigenvalue/eigenfunction problem and an initial value problem. The limiting increment of thermal instabilities has been determined for the principal mode (for the wave number tending to infinity) as c m $ @d @T ðj max Þ 2 , where @d @T is the temperature derivative of resistivity and j max is the maximum current density. It has been shown that as a nonlinear diffusion wave propagates through a conductor, the long-wave modes whose wavelengths are of the order of the conductor thickness are stable and the short-wave modes are localized near the diffusion wave front. As the diffusion wave arrives at the inner surface of the conductor, the instability increments of all modes with any wave number reach maxima.

Simulations of electrothermal instability growth in solid aluminum rods

“…It is also interesting to note a peak temperature growth rate occurs at the front of the magnetic diffusion wave and peak of the current density which is consistent with theoretical studies of electrothermal instability growth in the presence of a nonlinear magnetic diffusion wave. 34 Although the electrothermal instability growth rate presented in Eq. (1) is based solely on temperature perturbations, it is interesting to compare the observed dominant wavelength of instabilities in Fig.…”

Fast penetration of megagauss fields into metallic conductors

“…Note that the heat-diffusion term is absent from the leadingorder energy balance (32). In fact, it is OðEh Àc Þ relative to the remaining terms and therefore negligible even for E ¼ Oð1Þ.…”

“…In this quasi-one-dimensional problem, the magnetic field propagates inwards into the metal while inducing a transverse electric current as depicted in Figure 1. With pulsed-power applications in mind, the surface field is usually assumed to rise monotonically with time, most often according to the power law 1, 24,26,32 H à e t à t à e r ; r > 0: (2) Disregarding thermal effects, the magnetic field in the above scenario is governed by a simple linear diffusion equation 1,39 in which the apparent magnetic diffusivity is 1=l à 0 r à (l à 0 being the free-space permeability). Considering for specificity a surface field which magnitude rises from zero to H à e during a time period t à e [cf.…”

“…9,12,13,22 Complementary to such heavy-duty computations, tractable model problems which aim to capture the essential peculiarities of nonlinear magnetic diffusion, and to approximately predict field penetration rates, temperatures, and pressures, have been formulated and studied. 1, [23][24][25][26][27][28][29][30][31][32][33][34] The majority of these models hinge upon the understanding that, excluding ultra-high field magnitudes (տ10 MG), field-penetration rate is dominantly determined by a thermo-magnetic coupling via the strong dependence of electrical conductivity on temperature. Prior to evaporation, and ignoring compression effects [see Sec.…”

Numerical simulation of electrical explosions in megagauss magnetic fields

“…For this mode, the time of energy delivery to the conductor is shorter than, or comparable to the time of magnetic diffusion into it. The basic processes inherent in the current skinning mode are the propagation of a nonlinear magnetic diffusion wave [5], the formation of a low-temperature plasma at the conductor surface, and the development of thermal instabilities [6,7].Nonlinear magnetic diffusion features, a speed of field penetration into a conductor are anomalously high compared to a conventional magnetic diffusion. The high diffusion speed is related to a decrease in conductivity of the metal due to its heating by an electric current.…”