Early-time temperature growth due to a three-dimensional, nonlinear perturbation in current-driven metal

The electrothermal instability (ETI) is a Joule heating-driven instability that is important in applications of current-driven metal. Existing ETI theory employs linear stability analysis to predict the growth rate of infinitesimally small temperature perturbations. However, common seeds for ETI are isolated defects in the metal (e.g., resistive inclusions and voids), which constitute three-dimensional (3D), nonlinear perturbations. Hence, studying defect-driven ETI growth necessitates extending ETI theory to the nonlinear regime, which requires self-consistently solving for the evolving electrical conductivity and current density profiles. In this work, we use the analogy between hydrodynamic and electrical current flow, as well as a fortuitous property of the Joule heating profile, to obtain an expression for the temperature growth due to a 3D defect, valid in the early-time, nonlinear regime. Finally, we compare results of the theory with 3D magnetohydrodynamic simulations.