Kevin G. Lamb

Numerical experiments of internal wave generation by strong tidal flow across a finite amplitude bank edge

Results of some idealized numerical experiments of strong tidal flow of a stratified fluid across a finite amplitude bank edge are presented. These experiments were motivated by a need to develop an understanding of some of the complex internal wave phenomena observed on Georges Bank (Loder et al., 1992) and at other locations where tidal forcing is strong. The numerical model solves the fully nonlinear, nonhydrostatic Boussinesq equations on an ƒ plane. The model is two-dimensional, with spatial variation in the vertical and cross-bank directions only. Model forcings are based on the Georges Bank observations. A horizontally uniform stratification is used. The model successfully reproduces some observed features including the formation of a large depression and a hydraulic jump over the bank edge during off-bank flow and two on-bank propagating depressions every tidal period. An undular bore propagating away from the bank is in agreement with other observations (La Violette et al., 1990). Rotational effects are shown to be responsible for the formation of the second of the on-bank propagating depressions. Sensitivity of the results to the topographic slope, tidal current strength, stratification, and model initialization is explored.

A numerical investigation of solitary internal waves with trapped cores formed via shoaling

The formation of solitary internal waves with trapped cores via shoaling is investigated numerically. For density fields for which the buoyancy frequency increases monotonically towards the surface, sufficiently large solitary waves break as they shoal and form solitary-like waves with trapped fluid cores. Properties of large-amplitude waves are shown to be sensitive to the near-surface stratification. For the monotonic stratifications considered, waves with open streamlines are limited in amplitude by the breaking limit (maximum horizontal velocity equals wave propagation speed). When an exponential density stratification is modified to include a thin surface mixed layer, wave amplitudes are limited by the conjugate flow limit, in which case waves become long and horizontally uniform in the centre. The maximum horizontal velocity in the limiting wave is much less than the waves propagation speed and as a consequence, waves with trapped cores are not formed in the presence of the surface mixed layer.

The evolution of internal wave undular bores : Comparisons of a fully nonlinear numerical model with weakly nonlinear theory

Abstract The validity of shallow-water, weakly nonlinear theory for describing the evolution of a single large internal wave depression into an undular bore is explored by comparing theoretical results with results obtained from a fully nonlinear numerical model. Inclusion of second-order nonlinear and dispersive terms significantly improves the agreement. Solutions of the KdV and extended KdV equations, which includes second-order nonlinearity, overpredict the wave amplitudes in the undular bore. Inclusion of all second-order nonlinear and dispersive terms significantly improves the predicted amplitudes; however, the resulting evolution equation breaks down for sufficiently large waves. This can be corrected by modifying the linear terms in the equation to give a modified equation. Solutions of this modified second-order equation are in much better agreement with the model results than are the solutions of the KdV equation and the extended KdV equations.

Large fully nonlinear internal solitary waves: The effect of background current

In this paper we consider what effect the presence of a nonconstant background current has on the properties of large, fully nonlinear solitary internal waves in a shallow, stratified ocean. In particular, we discuss how the amplitude of the largest nonbreaking wave that it is possible to calculate depends on the background current as well as the nature of the upper bound. We find that the maximum wave amplitude is given by one of three possibilities: The onset of wave breaking, the conjugate flow amplitude or a failure of the wave calculating algorithm to converge (associated with shear instability). We also discuss how wave properties such as propagation speed, half-width, etc. vary with background current amplitude.

Particle transport by nonbreaking, solitary internal waves

The horizontal transport of particles by solitary internal waves is investigated using a fully nonlinear numerical model. The effect of an additional constant velocity Ud due to another physical mechanism such as surface wind drift or swimming motion in the case of live organisms is also considered. Weakly nonlinear theory is used to derive approximate analytic expressions for the transport distance of surface particles. The theoretical results are compared with transport distances obtained using the numerical model. The numerical model is also used to determine the vertical dependence of the horizontal transport. Two stratifications are considered. For one, with the strength of the stratification increasing monotonically toward the surface, the agreement between the theoretical predictions and the model results is excellent right up to the breaking amplitude. For the second, with a pycnocline between weakly stratified upper and lower layers, the nonlinear waves in the fully nonlinear numerical simulations are much wider than those predicted by weakly nonlinear theory. As a consequence, weakly nonlinear theory significantly underestimates the transport distance. It is found that significant particle transport occurs only when the waves are near the breaking amplitude, when they are very long, or when Ud plus the waves surface current is comparable to the waves propagation speed.

Shoaling solitary internal waves: on a criterion for the formation of waves with trapped cores

Shoaling solitary internal waves are ubiquitous features in the coastal regions of the worlds oceans where waves with a core of recirculating fluid (trapped cores) can provide an effective transport mechanism. Here, numerical evidence is presented which suggests that there is a close connection between the limiting behaviour of large-amplitude solitary waves and the formation of such waves via shoaling. For some background states, large-amplitude waves are broad, having a nearly horizontal flow in their centre. The flow in the centre of such waves is called a conjugate flow. For other background states, large-amplitude waves can reach the breaking limit, at which the maximum current in the wave is equal to the waves propagation speed. The presence of a background current with near-surface vorticity of the same sign as that induced by the wave can change the limiting behaviour from the conjugate-flow limit to the breaking limit. Numerical evidence is presented here which suggests that if large solitary waves cannot reach the breaking limit in the shallow water, that is if the background flow has a conjugate flow, then waves with trapped cores will not be formed via shoaling. It is also shown that, due to a change in the limiting behaviour of large waves, an appropriate background current can enable the formation of waves with trapped cores in stratifications for which such waves are not formed in the absence of a background current.

Conjugate flows and flat solitary waves for a continuously stratified fluid

The concept of conjugate flows is used to determine the vertical structure of solitary internal waves which are horizontally uniform in their center. Continuously stratified fluids are considered and solutions obtained with and without the Boussinesq approximation are compared. Only mode-1 waves are considered. For stratifications with a single pycnocline, conjugate flow solutions are obtained provided the pycnocline is not too close to the upper or lower boundaries. The parallel shear flow in the center of a flat solitary wave is potentially unstable (minimum Richardson number less than 1/4) if the upstream pycnocline is sufficiently narrow. For stratifications with two pycnoclines, cases with three mode-1 conjugate flow solutions have been found. Some conjugate flow solutions for the two-pycnocline case do not seem to correspond to a flat solitary wave. Non-Boussinesq effects were found to be small if the surface to bottom density difference is about 4% for stratifications with one or two pycnoclines.

Nonlinear effects in reflecting and colliding internal wave beams

Using small-amplitude expansions, we discuss nonlinear effects in the reflection from a sloping wall of a time-harmonic (frequency

Breaking of shoaling internal solitary waves

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Instabilities in an Internal Solitary-like Wave on the Oregon Shelf

) plane-wave beam of finite cross-section in a uniformly stratified Boussinesq fluid with constant buoyancy frequency