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

TL;DR

This paper develops spectral numerical methods to simulate time-dependent convection with temperature-dependent viscosity, focusing on exponential dependence, in a 2D periodic domain, capturing bifurcations and regime transitions.

Contribution

The paper introduces spectral schemes tailored for convection problems with viscosity depending exponentially on temperature, extending to other laws, and analyzes regime transitions.

Findings

Spectral methods effectively capture bifurcations in convection regimes.

Stable stationary solutions become unstable via Hopf bifurcation.

The methods are applicable to various viscosity dependence laws.

Abstract

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.