Linearized Instability in the Magnetic Bénard Problem

Abstract.

In the magnetic Bénard problem, a conducting fluid moves in an infinite horizontal layer \( \Omega = R^{2} \,\times (- {1 \over 2}, {1 \over 2}) \), subject to a temperature gradient \( T_0 - T_1 \) and to a magnetic field B perpendicular to the layer. The relevant equations admit a trivial equilibrium state \( w_0 \). We investigate the loss of stability of \( w_0 \) under perturbations in \( L^2(\Omega)^7 \), when \( T_0 - T_1 \) varies in the range \( (0,+\infty) \), and with B kept constant. Since linearized instability entails Ljapounov instability (Theorem 1 and comments), we study the nonselfadjoint linearization around \( w_0 \) on \( L^2(\Omega)^7 \), and the dependence of its spectrum on \( T_0 - T_1 \). This leads via Orr-Sommerfeld to a holomorphic function whose parameter-dependent zeros completely describe the spectrum (Theorems 4, 4' and lemmas). The existence of points of loss of stability then follow (Theorem 5) together with numerical information about their location.