A generalized quasi-geostrophic model of thermal convection

Abstract:

It is well known that, under the influence of planetary rotation, the primary force balance for large scale convective planetary flows is between pressure gradients and the Coriolis force; these flows are termed geostrophic. Convective flows are never purely geostrophic because buoyancy (which powers convection) is necessarily present and so is viscous dissipation. Nevertheless, provided rotation is dominant, the first order geostrophic balance is preserved and these flows are often referred to as quasi-geostrophic (QG). When buoyancy is perpendicular to rotation, the non-axial QG flows are rigid, that is, they have small variations along the direction of rotation. QG numerical models of thermal convection, in which only the non-axial flows are evolved, have been developed to take advantage of this 2D structure of QG flows. These models can reproduce faithfully some of the features of fully three-dimensional (3D) numerical models. The chief advantage of such QG models is that, because of their 2D nature, a much higher numerical resolution is achievable than for 3D models for the same computing cost and can thus be used to study aspects of convection under a regime not accessible in 3D models. In existing QG models, buoyancy is restricted to its non-axial component and the modelled region of convection is limited to that outside the tangent cylinder. Here, we present an extension on these models by incorporating the axial component of buoyancy and by modelling convection inside the tangent cylinder. When buoyancy is parallel to rotation, the non-axial QG flows are no longer rigid and include an axial gradient. To capture the first order dynamics inside the tangent cylinder, we must also track the evolution of non-rigid flows. We show that our model can reproduce the salient features of 3D numerical models near onset. Further, our model also captures features of well developed, fully turbulent convection, such as production of zonal jets.