Time Evolution of Nonlinear Horizontal Convection: Its Flow Regimes and Self-Similar Solutions

Abstract

The basic processes of time-dependent nonlinear horizontal convection caused by differential bottom cooling in a two-dimensional stratified Boussinesq fluid are investigated both theoretically and numerically. The fluid is initially at rest, and horizontal convection is induced by suddenly cooling one half of the bottom boundary. It is shown that three distinct flow regimes exist according to the nondimensional elapsed time and the nondimensional stratification parameter: diffusion regime, gravity current regime, and gravity wave regime. In each regime, existence of a self-similar solution is predicted theoretically, and its realizability is confirmed by a numerical experiment. The vertical length scale of the circulation in each regime is given by the diffusion length scale. The horizontal length scales of the circulation in the diffusion, gravity current, and gravity wave regimes are determined by the diffusive spread, horizontal propagation of the gravity current, and horizontal propagation of the gravity wave, respectively. The presence of a self-similar solution for each flow regime gives a useful perspective about the development process and dynamics of horizontal convections in the atmosphere. For example, the formation process of a steady heat/cool island circulations is understood as a merging of two horizontal convection initiated from both ends of the heat/cool island. In fact, this interpretation is proved to give correct estimates for the vertical scale of the circulation as well as the time required for the formation of the steady heat/cool island circulation.