Symmetries and exact solutions of the model of dynamic convection of the sea

Equations of the mathematical model of dynamic convection of the sea are observed. The model describes incompressible flows of shallow water with variable density under the action of Coriolis force. This approximation is widely applied for modeling of a mit-latitude oceanic and atmospheric flows.
From the group-theoretical point of view this model is exceptional by its infinite-dimensional group of transformations that involves five arbitrary functions of time. The goal of the paper is to demonstrate the physical meaning of the symmetry transformations, to construct the the optimal system of small dimensional subalgebras, and to represent new exact solutions, constructed on the base of the symmetry analysis.