On the failure of Laplace's tidal equations to model subinertial motions at a discontinuity in depth

Abstract

Abstract Infinitesmal amplitude, inviscid, subinertial oscillations over a discontinuity in depth are considered distinguishing three ocean models: (i) the Laplacian model in which the flow is governed by Laplace's tidal equations (LTE); (ii) the more realistic geophysical model in which the Vaisala frequency is assumed to be much greater than the inertial frequency; and (iii) the laboratory model in which the fluid is homogeneous with the Vaisala frequency equal to zero. The Laplacian model supports free waves perfectly trapped at the step while the laboratory model supports no perfectly trapped waves. The two approximations nevertheless predict similar behavior, because in the presence of forcing, the surface mode of the laboratory model is highly excited at frequencies and wavenumbers close to those of the perfectly trapped solutions predicted by LTE. Only in the limit of very long, low-frequency motions does the Laplacian model describe the barotropic modes of the geophysical model well, qualitatively, and even here the quantitative disagreement in predicted phase speeds and group velocities is substantial. At shorter wavelengths, LTE qualitatively misrepresent the dispersion properties by erroneously predicting topographically trapped motions with vanishing group velocity at a subinertial upper limit to the frequency of free oscillation. In particular, the results indicate that, for the step topography, there is in general no trapped, barotropic mode (double-Kelvin wave) in the geophysical model. Thus, LTE fail in modeling even the barotropic parts of subinertial motions at a depth discontinuity and should not be used in such calculations.