Formulation of a variable-function three-dimensional model, with applications to the M2 and M4 tide on the North-West European Continental Shelf

The major steps in the mathematical formulation of a three-dimensional (3D) tidal model using a variable modal approach to represent current profiles in the vertical are presented in the initial section of this paper. Using this method the number of functions through the vertical can be increased locally where enhanced resolution is required, while ensuring mass and momentum conservation, and reducing computational time using a time-splitting technique. Calculations of the M2 and M4 tidal elevations and currents on the North-West European Continental Shelf are performed, initially with bed stress determined from the depth mean current, a hybrid 2D–3D model, the advantages of which are discussed in the paper, and subsequently computing the bed stress from the bottom current (a full 3D model). Computations are made using a range of viscosity formulations and bed-friction coefficients. A rigorous comparison of computed currents against the harmonic analysis of observations from 278 measured time-series is performed in order to determine the accuracy of the computed currents, which were found to agree to within ± 5 cm s−1 with observations at over 200 locations.