Abstract
An algorithm is given for an analytical solution to the problem of the eigenvalue and eigenfunction for the Navier-Stokes equation within a rectangle under the adherence condition at its boundaries. The resulting system of orthonormal functions can be used for correctly deriving the divergence-free component of flow at the free surface of laboratory currents. This system makes it possible to apply the spectral approach to the solution of the Navier-Stokes equation, strictly calculate the primary regimes, investigate their stability, and calculate the secondary currents for small above-criticalities for the conditions of a laboratory experiment on simulating geophysical flows.
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