Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature
This work provides numerical tools for geophysical and astrophysical convection problems where viscosity strongly depends on temperature, though it is incremental as it applies spectral methods to a specific symmetry (O(2)) in 2D.
The authors propose spectral numerical methods for solving time-dependent convection problems with temperature-dependent viscosity at infinite Prandtl number, demonstrating their effectiveness on transitions from stationary to time-dependent regimes via Hopf bifurcation.
This article proposes spectral numerical methods to solve the time evolution of convection problems with viscosity strongly depending on temperature at infinite Prandtl number. Although we verify the proposed techniques just 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. This introduces a symmetry in the problem, 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.