Nonlinear stability threshold for temperature-dependent viscosity convection: Constraints from three-dimensional numerical simulations

We investigate the nonlinear stability threshold for Rayleigh-Bénard convection in infinite Prandtl number fluids with temperature-dependent viscosity by solving numerically the full system of three-dimensional (3D) convection equations below the linear critical Rayleigh number. First, for viscosity contrasts Δ η from 10 to 10¹⁰, we determine the Rayleigh number at which convection ceases in a unit cell of square and hexagonal planforms. The lowest Rayleigh number at which convection exists gives the planform-specific nonlinear stability threshold. For all Δ η considered, hexagons and squares yield lower thresholds than the three previously investigated cases: a unit cell of an infinite array of two-dimensional (2D) rolls, a localized 2D plume, and a localized 3D plume. The variation among thresholds for all planforms is < 15 %. Second, for Δ η = 10¹⁰ we investigate the stability of square and hexagonal planforms in a large box containing hundreds of unit cells. The square planform is stable. However, its threshold is above that found for its unit cell and is close to the threshold for localized plumes. Below the localized plume threshold, the square planform collapses to a conductive state. The hexagonal planform is unstable in a large box. At Rayleigh numbers above the threshold for localized plumes, it breaks down into several isolated localized 3D plumes, which then collapse below their threshold. An initially disordered planform evolves similarly. Thus, the nonlinear stability threshold for localized 3D plumes is a tight upper bound on the global nonlinear threshold for this system, and values much smaller than this bound are unlikely.