The principal lunar semidiurnal tide and its harmonics: baseline solutions for M 2 and M 4 constituents on the North-West European Continental Shelf

Abstract

A 2D numerical model of the shelf seas around the U.K. is used to derive the M 2 tide and the M 4 and M 6 tidal harmonics. The accuracy of the tidal harmonics is shown to be related to the form of the dissipation. In particular a quadratic form of the bottom friction results in unrealistically high amplitudes for the M 6 tide. It is shown that this effect can be reduced by use of a combination of a linearised friction tensor and quadratic friction without significantly affecting the M 2 constituent. The role of diffusion in controlling the amplitude of the M 4 tide is demonstrated. The 2D results provide a baseline or reference solution for further development using 3D and/or multiconstituent models. Amplitude and phase diagrams derived from the model show a progression with frequency in the North Sea and the English Channel suggestive of a transition from Kelvin wave to Poincaré wave propagation. It is shown that both the North Sea and the English Channel can be considered from the point of view of waves propagating in a closed channel in the presence of friction. For the inviscid case, the critical frequency above which Poincaré waves are able to propagate is calculated to be between the M 2 and M 4 frequencies for the North Sea and between the M 6 and M 8 frequencies for the English Channel. A dispersion relation is derived for Poincaré and Kelvin waves in the presence of friction. Both types of wave possess decaying and propagating characteristics when friction is present with wavelength and decay scale dependent on frequency and the coefficient of friction. There is no longer a sharp critical frequency, but instead a transition region where wavelength and decay scales vary relatively rapidly with frequency. The response of a semi-infinite rectangular channel to Kelvin and Poincaré wave propagation is calculated providing a partial explanation of the numerical model results. © 1997 Elsevier Science Ltd