Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell II: Effects of temperature boundary conditions

Abstract

Abstract We examine the influence on convection in a rotating spherical shell of various boundary conditions on temperature. In particular, we look at the response to constant heat flux, or fixed temperature gradient, boundaries as opposed to constant temperature boundaries assumed in Part I. We also examine the influence of an upper boundary with linear combinations of temperature and temperature gradient held fixed. To compare with earlier studies, calculations are done at Prandtl number of 1, Taylor number 105, Rayleigh numbers between 104 and 105, and convection zone depth 20% of the outer radius. We find that when a constant heat flux boundary condition is imposed at the bottom of the convective layer, equatorial acceleration is produced by Reynolds stresses in the convection for Rayleigh numbers up to 5 × 104, analogously to the constant temperature bottom case, but over the same range the heat flux differentials in latitude at the top of the layer are greatly reduced. Above this Rayleigh number, equatorial acceleration is converted to deceleration, and a sharp peak in heat flux out the top appears near the equator. Both effects are shown to be produced primarily by an axisymmetric meridional circulation, which grows rapidly in amplitude compared to differential rotation as R is increased. As in the previous convection calculations of the author, the meridional circulation is outward near the equator, and from equator toward the poles near the outer surface. At small R, this circulation is driven against buoyancy by an outward pointing radial Coriolis force associated with the equatorial acceleration. At larger R, the circulation becomes driven by buoyancy forces and can modify the differential rotation from equatorial acceleration to deceleration. Experiments with constant heat flux top as well as mixed top boundary conditions show that certain combinations form larger meridional circulation (particularly constant temperature bottom and constant heat flux top) which, in turn. result in more equatorial deceleration. We find that even when the heat flux is fixed at the bottom, its average value throughout the layer varies with time up to 10%, due to storage and release of energy internal to the convecting layer. The time scale for those variations appears to be the turn over lime for the convection. Each release of energy is quickly followed by a build up of kinetic energy of convection, but the differential rotation changes more slowly, on a “spin up” time scale. Various model assumptions and results are discussed in the context of the problem of global convection and differential rotation on the sun.