Numerical solution of the linearized three‐dimensional tidal equations using eddy viscosity eigenfunctions

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Abstract
A modification of the spectral method of solving the linearized three‐dimensional tidal equations is described. The resulting expansions, using a basis of eddy viscosity eigenfunctions, are more rapidly convergent than are those that have been previously developed by this method. For a test problem involving a rectangular sea subject to a steady wind stress, an expansion with four eigenfunctions produces errors of not more than about 1% in the computed velocities, including the surface component parallel to the wind. This rate of convergence is comparable to that obtained previously using a basis of Chebychev polynomials or B‐splines, and has the advantage of uncoupling the differential equations for the modal amplitudes. It is also shown that when the bottom drag is zero, the fluid velocities can be obtained in closed form for arbitrary eddy viscosity, apart from the depth‐averaged components. This analytical solution is used to verify the accuracy of the numerical algorithm.