A Vertical Diffusion Model for Lakes

Abstract

The motion of a fluid in a lake with small depth compared to width is investigated. We prove that when the depth goes to 0, the solution of the stationary Navier--Stokes equations with adherence at the bottom and traction by wind at the surface, once conveniently normalized, goes to a three-dimensional limit which is the solution of an incompressible model with vertical diffusion. The limit velocity is given in terms of the vertical coordinate and of the limit pressure. This pressure, which depends only on the horizontal coordinates, is driven by a two-dimensional equation on the surface degenerating on the shore, which is solved in a weighted space. Thus, a three-dimensional approximation is obtained by a simple two-dimensional computation.