Vertical structure of the quasi-two-dimensional velocity field of a viscous incompressible flow and the problem of nonlinear friction

An approximate theory is constructed to describe quasi-two-dimensional viscous incompressible flows. This theory takes into account a weak circulation in the vertical plane and the related divergence of the two-dimensional velocity field. The role of the nonlinear terms that are due to the interaction between the vortex and potential components of velocity and the possibility of taking into account the corresponding effects in the context of the concept of bottom friction are analyzed. It is shown that the nonlinear character of friction is a consequence of the three-dimensional character of flow, which results in the effective interaction of vortices with vertical and horizontal axes. An approximation of the effect of this interaction in quasi-two-dimensional equations is obtained with the use of the coefficient of nonlinear friction. The results based on this approximation are compared to the data of laboratory experiments on the excitation of a spatially periodic fluid flow.