Tide-topography interactions in a stratified shelf sea I. Basic equations for quasi-nonlinear internal tides

By means of a multiple-scale analysis of the shallow water equations for a uniformly rotating, stratified fluid, subject to a time-periodic advection over a small-amplitude topography, it is shown that the inclusion of quasi-nonlinear advection by the barotropic (tidal) current is a necessary ingredient of the dynamics, once the internal wave length, the barotropic tidal excursion amplitude and the topographic wave length are all of the same order of magnitude. The basic set of equations describing the generation of internal tides by the interaction of barotropic tidal currents and topography thus derived, is extended with damping both by bottom- and internal friction. The effect of bottom friction is parametrized in a Rayleigh damping term for each of the separate vertical modes, thereby allowing the vertical structure of the baroclinic tidal currents to remain expressible in terms of vertical modes. The spectral forcing equation for damped internal motions is then derived. Finally the characteristic roots (dispersion relation) of the homogeneous spectral equation are discussed and summarized in a dispersion diagram. It is shown that these consist in general of two damped gravity wave modes and a transient, the asymptotic regimes of which are discussed. The transient gives rise, among other things, to baroclinic residual currents which are the subject of a second paper, whereas the structure of the quasi-nonlinear gravity wave modes is treated in a third part.