Waves In Stratified Shear Flows Research Articles

Weakly nonlinear waves in stratified shear flows

<p style='text-indent:20px;'>We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.</p>

A generalized action-angle representation of wave interaction in stratified shear flows

In this paper we express the linearized dynamics of interacting interfacial waves in stratified shear flows in the compact form of action-angle Hamilton’s equations. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the interfacial waves to the wave action and the angles are the phases of the interfacial waves. The term ‘generalized action angle’ aims to emphasize that the action of each wave is generally time dependent and this allows for instability. An attempt is made to relate this formalism to the action at a distance resonance instability mechanism between counter-propagating vorticity waves via the global conservations of pseudo-energy and pseudo-momentum.

MKdV Equation for the Amplitude of Solitary Rossby Waves in Stratified Shear Flows with a Zonal Shear Flow

AbstractIn this paper, modified Korteweg-de Vries (mKdV) equations for the amplitude of solitary Rossby waves in stratified fluids with a zonal shear flow are derived by using a weakly nonlinear method. It is found that the coefficients of mKdV equations depend not only on the β effect and the Vaisala-Brunt frequency, but also on the basic shear flow.

A Buoyancy–Vorticity Wave Interaction Approach to Stratified Shear Flow

Abstract Motivated by the success of potential vorticity (PV) thinking for Rossby waves and related shear flow phenomena, this work develops a buoyancy–vorticity formulation of gravity waves in stratified shear flow, for which the nonlocality enters in the same way as it does for barotropic/baroclinic shear flows. This formulation provides a time integration scheme that is analogous to the time integration of the quasigeostrophic equations with two, rather than one, prognostic equations, and a diagnostic equation for streamfunction through a vorticity inversion. The invertibility of vorticity allows the development of a gravity wave kernel view, which provides a mechanistic rationalization of many aspects of the linear dynamics of stratified shear flow. The resulting kernel formulation is similar to the Rossby-based one obtained for barotropic and baroclinic instability; however, since there are two independent variables—vorticity and buoyancy—there are also two independent kernels at each level. Though having two kernels complicates the picture, the kernels are constructed so that they do not interact with each other at a given level.

Nonlinear surface and internal waves in stratified shear flow

Abstract The propagation of solitary waves in a two-layer fluid with shear is studied in the long-wave shallow-water approximation. We show that, to the first order in the perturbation, the wave motion is described by the Korteweg-de Vries equation, whose coefficients now depend on the shear present. Solitary wave solutions are discussed in the following two cases; constant shear flow and Couette shear flow. These results are used to describe the propagation of solution-like internal waves in the Straits of Gibraltar.

The solitary waves in stratified shear flow

In this paper, the basic equation of internal long waves in stratified shear flow is derived under Boussinesq assumption, the first order approximation solution is given for solitary waves with the effects of slowly varying topograph at the sea bottom, weak stratification and basic shear flow.

Propagation of Rossby waves in stratified shear flows

A study is made of the propagation of Rossby waves in a stably stratified shear flows. The wave equation for the Rossby waves is derived in an isothermal atmosphere on a beta plane in the presence of a latitudinally sheared zonal flow. It is shown that the wave equation is singular at five critical levels, but the wave absorption takes place only at the two levels where the local relative frequency equals in magnitude to the Brunt Vaisala frequency. This analysis also reveals that these two levels exhibit valve effect by allowing the waves to penetrate them from one side only. The absorption coefficient exp(2πμ) is determined at these levels. Both the group velocity approach and single wave treatment are employed for the investigation of the problem.

Long nonlinear waves in stratified shear flows

Abstract The governing equations for long weakly nonlinear waves in stratified shear flows are developed for general flows and density stratifications, with the Boussinesq approximation. Solitary wave solutions are found and their character investigated for several particular flows. Both solitary waves of elevation and depression are shown to exist, even for the same flow, depending on the Richardson number. The results suggest a qualitative difference in behaviour with Richardson number between solitary waves in flows with their maximum velocity within the fluid region and those in flows where the maximum velocity is attained at the boundary.

On the instability of internal Alfv�n-gravity waves in stratified shear flows

Internal Alfven-gravity waves and their stability characteristics for an inviscid, nondissipative, Boussinesq fluid undergoing shear in the presence of a lower rigid boundary is studied. The model consists of a layer of constant shear capped by a layer of constant velocity. The Brunt-Vaisala frequency is assumed to be uniform throughout the fluid. The unstable modes have wavelenghts which are close to those of the internal Alfven-gravity waves which propagate from the troposphere into the ionosphere.

Evolution Equations for Long, Nonlinear Internal Waves in Stratified Shear Flows

Evolution equations for long, nonlinear internal waves are derived when the basic stratified shear flow has a slow temporal and spatial variation as well as the usual dependence on the vertical coordinate. When the horizontal waveguide has a limited vertical extent the evolution equation is a variable coefficient Korteweg‐deVries equation, while in the deep fluid case the evolution equation is a variable coefficient Benjamin‐Davis‐Ono equation. Explicit expressions are obtained for the coefficients of these equations.

Long nonlinear waves in stratified shear flows

The propagation of finite-amplitude internal waves in a shear flow is considered for wavelengths that are long compared to the shear-layer thickness. Both singular and regular modes are investigated, and the equation governing the amplitude evolution is derived. The theory is generalized to allow for a radiation condition when the region outside the stratified shear layer is unbounded and weakly stratified. In this case, the evolution equation contains a damping term describing energy loss by radiation which can be used to estimate the persistence of solitary waves or nonlinear wave packets in realistic environments. A continuous three-layer model is studied in detail and closed-form expressions are obtained for the phase speed and the coefficients of the nonlinear and dispersive terms in the amplitude equation as a function of Richardson number.

Weakly nonlinear stability theory of stratified shear flows

AbstractA derivation is given of the first‐order, nonlinear amplitude equation governing the temporal evolution of finite‐amplitude waves in stratified shear flows. the theory has been developed on an essentially inviscid basis by perturbing away from the linear neutral stability curve in Richardson number‐wavenumber space. However, viscosity and heat conduction are still required in order to eliminate the singularities that occur in the inviscid limit. Holmboe's mixing layer model has been studied by applying the present theory and the results show, surprisingly, that subcritical instability can occur, i.e. modes that would be stable on a linear basis (e.g. when the Richardson number is greater than 1/4) become unstable when the initial perturbation amplitude is greater than some critical value. an instability due to resonance which occurs at a Richardson number of 0·22 is also revealed by the analysis. These results have interesting implications in connection with clear air turbulence which are discussed herein.