Theory of Topographic Vorticity Production by Tidal Currents

摘要

Starting from the shallow water equations of a homogeneous rotating fluid, the equation describing the evolution of vorticity induced by a fluctuating bottom topography is derived. It is shown that nonlinear vorticity advection can be reduced to a quasilinear form in the limit of small amplitude topography and for the topographic horizontal length scale being much smaller than the length of a tidal gravity wave. Topographic vorticity is produced by planetary vortex stretching and differential bottom friction. For both mechanisms, the vorticity response functions at the basic forcing frequency and all its higher harmonics, as well as the residual components, are shown to be given by sums of products of Bessel functions. The exact shape of the residual velocity field for a one-dimensional step-like bottom topography is calculated. In the absence of vorticity diffusion the residual velocity vanishes exactly outside a region of twice the tidal excursion length scale, in contrast of the result derived by the method of harmonic truncation.