Study of the linear ablation growth rate for the quasi isobaric model of Euler equations with thermal conductivity

TL;DR

This paper analyzes the linear growth rate of ablation fronts in a quasi-isobaric Euler model with thermal conduction, relevant for inertial confinement fusion, by computing the Evans function and studying stability conditions.

Contribution

It provides a rigorous analysis of the linearized system's stability, including the calculation of the Evans function and conditions for the absence of bounded solutions.

Findings

For small alpha, no bounded solutions exist, indicating stability.

The Evans function is explicitly computed for the linearized system.

The model advances understanding of ablation front instability in fusion contexts.

Abstract

In this paper, we study a linear system related to the 2d system of Euler equations with thermal conduction in the quasi-isobaric approximation of Kull-Anisimov [14]. This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by L0 an associated characteristic length. The system of equations is linearized around a stationary solution, and each perturbed quantity is written using the normal modes method. The resulting linear system is not self-adjoint, of order 5, with coefficients depending on x and on physical parameters $\alpha, \beta$. We calculate Evans function associated with this linear system, using rigorous constructions of decreasing at $\pm \infty$ solutions of systems of ODE. We prove that for $\alpha$ small, there is no bounded solution of the linearized system.