Simulations and observation of nonlinear internal waves on the continental shelf: Korteweg–de Vries and extended Korteweg–de Vries solutions

Numerical solutions of the Korteweg–de Vries (KdV) and extended Korteweg–de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation and mean shear. The KdV model is run for a number of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolve. Comparisons between KdV and eKdV solutions are made. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock-like front, while nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.

Simulations and observation of nonlinear waves on the continental shelf: KdV solutions

Numerical solutions of the Korteweg-de Vries (KdV) and extended Korteweg-de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation, and mean shear. The KdV model is run for a variety of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolves. Comparisons between KdV and eKdV solutions is explored. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Mid Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock like front, while nonlinear high frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.

Bispectral Analysis of Energy Transfer within the Two-Dimensional Oceanic Internal Wave Field

Bispectral analysis of the numerically reproduced spectral responses of the two-dimensional oceanic internal wave field to the incidence of the low-mode semidiurnal internal tide is performed. At latitudes just equatorward of 30°, the low-mode semidiurnal internal tide dominantly interacts with two high-vertical-wavenumber diurnal (near inertial) internal waves, forming resonant triads of parametric subharmonic instability (PSI) type. As the high-vertical-wavenumber near-inertial energy level is raised by this interaction, the energy cascade to small horizontal and vertical scales is enhanced. Bispectral analysis thus indicates that energy in the low-mode semidiurnal internal tide is not directly transferred to small scales but via the development of high-vertical-wavenumber near-inertial current shear. In contrast, no noticeable energy cascade to high vertical wavenumbers is recognized in the bispectra poleward of ∼30° as well as equatorward of ∼25°. A new finding is that, although PSI is possible equatorward of ∼30°, the efficiency drops sharply as the latitude falls below ∼25°. At all latitudes, another resonant interaction suggestive of induced diffusion is found to occur between the low-mode semidiurnal internal tide and two high-frequency internal waves, although bispectral analysis shows that this interaction plays only a minor role in cascading the low-mode semidiurnal internal tide energy.

Theory of the Eulerian tail in the spectra of atmospheric and oceanic internal gravity waves

Observed atmospheric and oceanic internal wave spectra, when analysed in an Eulerian frame of reference, exhibit a large-wavenumber ‘tail’. In one-dimensional vertical-wavenumber (k3) spectra, it is typically proportional to |k3|−3.In 1989, K. R. Allen and R. I. Joseph showed that a large-wavenumber tail was to be anticipated as a consequence of Eulerian nonlinearity, and they derived relations for the coefficients of both horizontal and vertical spectra of the form |k|−3. The coefficients were obtained only for the wave-induced vertical-displacement spectra, and only for an input spectrum having a certain ‘canonical’ frequency variation derived on other grounds.The present work builds on that of Allen & Joseph. It is more general in some respects, more limited in others. It provides a more transparent form of analysis, it treats a broad class of wave variables, and it does so for input (Lagrangian) spectra that can be chosen by the user, free from any constraint to canonical or other restricted forms. It provides relations whereby the full Eulerian spectrum may be determined numerically, once the input spectrum has been chosen, and it provides analytic forms applicable at large wavenumbers for horizontally isotropic spectra. The derived one-dimensional vertical-wavenumber spectra are discussed in relation to observations.Certain shortcomings in the development, both as given by Allen & Joseph and as found here, are identified and discussed.