A numerical study on scale selection of convection in a shallow sea

Abstract

We investigate the fundamental properties of gravitational convection in a shallow sea cooled from above and their dependence on two parameters, the eddy Rayleigh number, Ra = F ρ gH 4/ρ 0κ z 2ν z , and the anisotropic parameter, δ = κ h / κ z , executing two-dimensional numerical experiments. Starting with a homogeneous initial condition, we obtain three stages in evolution of convection; initial, transient and quasi-steady stages. In the initial stage there appear numerous small-scale convections in the whole basin. These are governed by the linear dynamics. In the transient stage, they repeat to merge with each other and increase their horizontal scales. That is, the larger cell survives to grow up and the adjoining smaller cell is absorbed. Horizontal advection of vorticity and difference in scale between adjoining cells are essential to the merging process. When horizontal advection time becomes comparable to vertical diffusion time, convections stop merging (quasi-steady stage). This result gives a possible scale-selecting rule of convection cell in the non-linear regime. In this stage, the aspect ratio of convection cell generally increases with Ra and δ, though some complicated features are found. Different initial conditions lead to different values of the aspect ratio in the quasi-steady stage, i.e. multiple equilibrium state. We also discuss the influence of difference in thermal boundary condition on the aspect ratio.