Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

This article proposes spectral numerical methods to solve the time evolution of convection problems with viscosity strongly dependent on temperature at infinite Prandtl number. Although we verify the proposed techniques solely for viscosities that depend exponentially on temperature, the methods are extensible to other dependence laws. The set-up is a 2D domain with periodic boundary conditions along the horizontal coordinate which introduces a symmetry in the problem. This is the O(2) symmetry, which is particularly well described by spectral methods and motivates the use of these methods in this context. We examine the scope of our techniques by exploring transitions from stationary regimes towards time dependent regimes. At a given aspect ratio, stable stationary solutions become unstable through a Hopf bifurcation, after which the time-dependent regime is solved by the spectral techniques proposed in this article.