Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation

Generation of mode 2 internal waves by the interaction of mode 1 waves with topography

Oceanic internal waves can be decomposed into an infinite set of modes, and the dominant internal mode 1 waves have been extensively investigated. Although mode 2 waves have been observed, they have not received comparable attention, especially the generation mechanisms. In this work, we examine the generation of mode 2 internal waves by the interaction of mode 1 waves with topography. We use a coupled linear long-wave theory with mode coupling through topography, combined with evolution using a Korteweg–de Vries model, to predict the mode 2 wave amplitude, in an ideal three-layer fluid model, in a smooth density stratification and in two realistic oceanic settings. We find that the mode 2 wave amplitude is usually much smaller than the incident mode 1 wave amplitude and is quite sensitive to the pycnocline thickness, topographic slope and background stratification

Generation of nonlinear internal waves by flow over topography: Rotational effects

We use the forced Ostrovsky equation to investigate the generation of internal waves excited by a constant background current flowing over localized topography in the presence of background rotation. As is now well known in the absence of background rotation, the evolution scenarios fall into three cases, namely subcritical, transcritical, and supercritical. Here an analysis of the linearized response divides the waves into steady and unsteady waves. In all three cases, steady waves occur downstream but no steady waves can occur upstream, while unsteady waves can arise upstream only when there is a negative minimum of the group velocity. The regions occupied by the steady and unsteady waves are determined by their respective group velocities. When the background current is increased, the wave number of the steady waves decreases. In addition, the concavity (canyon or sill), the topographic width, and the relative strength of the rotation play an important role in the generation mechanism. Nonlinear effects modulate the wave amplitude and lead to the emergence of coherent wave packets. All these findings are confirmed by numerical simulations

Stable embedded solitons

Stable embedded solitons are discovered in the generalized third-order nonlinear Schroedinger equation. When this equation can be reduced to a perturbed complex modified KdV equation, we developed a soliton perturbation theory which shows that a continuous family of sech-shaped embedded solitons exist and are nonlinearly stable. These analytical results are confirmed by our numerical simulations. These results establish that, contrary to previous beliefs, embedded solitons can be robust despite being in resonance with the linear spectrum.Comment: 2 figures. To appear in Phys. Rev. Let

Accommodating quality and service improvement research within existing ethical principles

Funds were provided by a Canadian Institute of Health Research grant (Nominated PI: Monica Taljaard, PJT – 153045). Funds were also generously provided by Charles Weijer, who is funded by a Tier 1 Canadian Research Chair.Peer reviewedPublisher PD

Using the theory of planned behaviour as a process evaluation tool in randomised trials of knowledge translation strategies : A case study from UK primary care

Peer reviewedPublisher PD

Transcritical flow of a stratified fluid: The forced extended Korteweg-de Vries model

Transcritical, or resonant, flow of a stratified fluid over an obstacle is studied using a forced extended Korteweg-de Vries model. This model is particularly relevant for a two-layer fluid when the layer depths are near critical, but can also be useful in other similar circumstances. Both quadratic and cubic nonlinearities are present and they are balanced by third-order dispersion. We consider both possible signs for the cubic nonlinear term but emphasize the less-studied case when the cubic nonlinear term and the dispersion term have the same-signed coefficients. In this case, our numerical computations show that two kinds of solitary waves are found in certain parameter regimes. One kind is similar to those of the well-known forced Korteweg-de Vries model and occurs when the cubic nonlinear term is rather small, while the other kind is irregularly generated waves of variable amplitude, which may continually interact. To explain this phenomenon, we develop a hydraulic theory in which the dispersion term in the model is omitted. This theory can predict the occurence of upstream and downstream undular bores, and these predictions are found to agree quite well with the numerical computations. © 2002 American Institute of Physics.published_or_final_versio

The effect of a variable background density stratification and current on oceanic internal solitary waves

Large amplitude, horizontally propagating internal waves are commonly observed in the coastal ocean. They are often modelled by a variable-coefficient Korteweg-de Vries equation to take account of a horizontally varying background state. Although this equation is now well-known, a term representing non-conservative effects, arising from horizontal variation in the underlying basic state density stratification and current, has often been omitted. In this paper, we examine the possible significance of this term using climatological data for several typical oceanic sites where internal waves have been observed

The modified Korteweg - de Vries equation in the theory of large - amplitude internal waves

International audienceThe propagation of large- amplitude internal waves in the ocean is studied here for the case when the nonlinear effects are of cubic order, leading to the modified Korteweg - de Vries equation. The coefficients of this equation are calculated analytically for several models of the density stratification. It is shown that the coefficient of the cubic nonlinear term may have either sign (previously only cases of a negative cubic nonlinearity were known). Cubic nonlinear effects are more important for the high modes of the internal waves. The nonlinear evolution of long periodic (sine) waves is simulated for a three-layer model of the density stratification. The sign of the cubic nonlinear term influences the character of the solitary wave generation. It is shown that the solitary waves of both polarities can appear for either sign of the cubic nonlinear term; if it is positive the solitary waves have a zero pedestal, and if it is negative the solitary waves are generated on the crest and the trough of the long wave. The case of a localised impulsive initial disturbance is also simulated. Here, if the cubic nonlinear term is negative, there is no solitary wave generation at large times, but if it is positive solitary waves appear as the asymptotic solution of the nonlinear wave evolution