IUMJ

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

Authors: Olivier Lafitte

Issue: Volume 57 (2008), Issue 2, 945-1018

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 (H.J. Kull, \emph{Theory of the Rayleigh-Taylor instability}, Physical Report \textbf{206} (1991), 197--325). This model is used for the study of the ablation front instability, which appears in the problem of inertial confinement fusion. The heat flux $\vec{Q}$ is given by the Fourier law $T^{-\nu}\vec{Q}$ proportional to $\nabla T$, where $\nu > 1$ is the thermal conduction index, and the external force is a gravity field $\vec{g} = -g\vec{e}_x$. This physical system contains a mixing region, in which the density of the gaz varies quickly, and one denotes by $L_0$ an associated characteristic length. The fluid velocity in the denser region is denoted by $V_a$. The system of equations is linearized around a stationary solution, and each perturbed quantity $\tilde{u}$ is written using the normal modes method \[ \tilde{u}(x,z,t) = \text{Re}(\bar{u}(x,k,\gamma)e^{ikz + \gamma \sqrt{gk}t}) \] in order to take into account an increasing solution in time. The resulting linear system is a non self-adjoint fifth order system. Its coefficients depend on $x$ and on physical parameters $\alpha$, $\beta$, $\alpha$ and $\beta$ being two dimensionless physical constants, given by $\alpha\beta = kL_0$ and $\alpha/\beta = gL_0/V_a^2$ (introduced in C. Cherfils-Cl\'erouin, P. Lafitte, and P.A. Raviart, \emph{Asymptotic Results for the Rayleigh-Taylor Instability}, Birkhauser, Boston, 2001). We study the existence of bounded solutions of this system in the limit $\alpha \to 0$, under the condition $\beta \in [\beta_0, 1/\beta_0]$, and the assumption $\text{Re}\gamma \in [0, 1/\beta_0]$, $|\gamma| \leq 1/\beta_0$ (regime that we studied for a simpler model in ibid.) calculating the Evans function $\text{Ev}(\alpha,\beta,\gamma)$ associated with this linear system. Using rigorous constructions of decreasing at $\pm\infty$ solutions of systems of ODE, we prove that, for any $\beta_0 > 0$, there exists $\alpha_1(\beta_0)$ such that, for all $\beta \in [\beta_0, 1/\beta_0]$, $0 < \alpha \leq \alpha_1$, there is no bounded solution of the linearized system such that $\text{Re}\gamma \in [0, 1/\beta_0]$, $|\gamma| \leq 1/\beta_0$. In other terms, for any $M > 0$ and $\beta_0 > 0$ there exists $\alpha_1 > 0$ such that, for $0 < \alpha \leq \alpha_1$ and $\beta \in [\beta_0, 1/\beta_0]$, an admissible value $\gamma(\alpha,\beta)$ such that there exists a bounded solution of the linearized system satisfying $|\gamma| \leq M$ is such that $\text{Re}\gamma \notin [0,M]$.