A generalized action-angle representation of wave interaction 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>

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

Transient development of perturbations in inviscid stratified shear flow is investigated. Use is made of closed form analytic solutions that allow concise identification of optimally growing plane-wave solutions for the case of an unbounded flow with constant shear and stratification. For the case of channel flow, variational techniques are employed to determine the optimally growing disturbances. The maximum energy growth attained over a specific time interval decreases continuously with increasing stratification, and no special significance attaches to Ri = 0.25. Indeed, transient growth can be substantial even for Ri = O(1). A general lower bound on the energy growth attained by an optimal perturbation in a stratified flow over a given time interval is the square root of the growth attained by the corresponding perturbation in unstratified flow. Enhanced perturbation persistence is found for mean-flow stratification lying in the range Ri between 0.1 and 0.3. Small but finite perturbations in mean flow with Ri less than 0.4 produce regions with locally negative total density gradient, which are expected to overturn. Although the perturbations are of wave form, buoyancy fluxes mediate transfer between perturbation kinetic and potential energy during transient development, thus implying that buoyancy flux is not a determinative diagnostic for distinguishing between waves and turbulence in stratified flows.

Direct and large eddy simulations of stratified homogeneous shear flows

In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) At≪O(1) or (ii) Frc2≪O(1), where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted At≪O(1). To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities and vorticities. Furthermore, three kinds of density interfaces—neutral, stable, and unstable—embedded in a background shear layer are investigated. Instabilities ensuing from these configurations, which include Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor, and triangular-jet, are studied in terms of resonant wave interactions. The effects of density stratification and the shear on the stability of each of these flow configurations are explored. Some of the results, e.g., the destabilizing role of density stratification, stabilizing role of shear, etc., are apparently counter-intuitive, but physical explanations are possible if the instabilities are interpreted from wave interaction perspective.In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) At≪O(1) or (ii) Frc2≪O(1), where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted At≪O(1). To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities ...

A mechanism of atmospheric ozone disturbance by internal gravity wave in a stratified shear flow

: Laboratory experiments and numerical simulations of vortex evolution in realistic geophysical flows are reported. Two laboratory experiments are discussed. The first experiment examined three dimensional vortices produced from a wing in a nonstratified nonsheared flow. This study showed that vortices can migrate farther and last longer than previously reported. The second laboratory experiment investigated the evolution of two dimensional vortices in a stratified shear flow. This experiment showed the evolution of a solitary vortex when the background Richardson number was around four or less. Circulation estimates were also obtained for these vortices and showed a more rapid decay of circulation of the countersign vortex. Two numerical studies are also reported. In the first numerical study, we examined the evolution of a two dimensional vortex pair in a stratified shear flow. This study showed that the vortex pair evolved into a solitary vortex when the ambient Richardson number was on the order of unity or less. A second numerical study used a k-epsilon model to examine the evolution of a two dimensional vortex pair. It was found that the standard k-epsilon model did a poor job of representing the turbulent dissipation in vortex flows. Good agreement with experimental results was obtained with this model with a reduction in the eddy viscosity coefficient.

Observations on the lee of a topographic ridge show that the turbulence kinetic energy (TKE) dissipation rate due to shear instabilities is three orders of magnitude higher than the typical value in the open ocean. Laboratory-scale studies at low Reynolds number suggest that high turbulent dissipation occurs primarily within the core region of shear instabilities. However, field-scale studies indicate that high turbulence is mainly populated along the braids of shear instabilities. In this study, a high-resolution, resolving the Ozmidov-scale, non-hydrostatic model with Large Eddy Simulation (LES) turbulent closure is applied to investigate dominant mechanisms that control the spatial and temporal scales of shear instabilities and resulting mixing in stratified shear flow at high Reynolds number. The simulated density variance dissipation rate is elevated in the cusp-like bands of shear instabilities with a specific period, consistent with the acoustic backscatter taken by shipboard echo sounder. The vertical length scale of each cusp-like band is nearly half of the vertical length scale of the internal lee wave. However, it is consistent with instabilities originating from a shear layer based on linear stability theory. The model results indicate that the length scale and/or the period of shear instabilities are the key parameters to the mixing enhancement that increases with lateral Froude number FrL, i.e. stronger shear and/or steeper ridge.

The aim of this contribution is to present the results of experiments conducted to elucidate the conditions of formation and evolution of planar dipolar eddies in a stratified fluid in the presence of a vertical background shear. The flow regime diagram is derived and conditions under which the formation of dipolar eddies is possible are found and explained. The estimates for the dipole lifetime are given. Possible applications include oceanic mushroom-like currents, late stratified wakes and other cases when dipolar or quasi-dipolar eddies are observed in a stratified shear flow.

Abstract

Abstract. 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.

The resonant interactions among internal wave trains in a flow consisting of a middle layer of homogeneous fluid in linear shear flow and continguous upper and lower stratified layers of constant velocities are investigated. Three wave trains satisfying the dispersion relations and the so-called resonant conditions, k1=k2+k3 and σ1=σ2+σ3, are considered graphically, and the resonant triads of the same mode are found to exist. Some multiple resonances, i.e., the resonant interactions among waves which are the members of different triads, are also found to happen. The explosive and decay instabilities of multiple resonances are considered qualitatively, and energy exchange is expected to occur among the resonant triads.

The wavelengths, directions, and phase speeds of internal waves propagating through the Strait of Gibraltar were measured from shore-based marine radar images of their surface features. Shipborne echo sounder images identified these waves as first mode and enabled their amplitudes to be measured. The moving features in the radar images were closely correlated with the waves in the echo sounder image. A numerical solution of the Taylor–Goldstein equation was used to predict the phase speed of the waves, utilizing simultaneous measurements of the density and velocity structure of the water column. Within the bounds of experimental error, there was good agreement between the theoretical and measured phase speeds of the waves on two different tidal cycles. No correlation appeared to be present between amplitude and estimated wave width, indicating that the waves were not behaving as Korteweg–deVries-type solitons.

An analysis is described of long, finite-amplitude internal waves in a stratified shear flow. Both regular and singular modes are considered with a nonlinear critical layer employed in the latter case. A three-layer model is used to develop the theory and closed-form expressions are obtained relating the phase speed to the Richardson number, the latter quantity being taken as O(1). The amplitude evolution equation is found to be either the Korteweg-de Vries equation or the Benjamin-Davis-Ono equation depending upon the distance of the more remote boundary from the edge of the shear layer.

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable. In the present paper it is shown that a well-posed mathematical model is obtained if the finiteness of the adjustment time of the turbulence field to the variations of temperature gradient is taken into account. An evolution-type equation is obtained for the temperature distribution (a similar equation can be derived for the concentration if the stratification is due to salinity or suspended particles). The characteristic property which is obtained from a rigorous mathematical investigation is the formation of stepwise distributions of temperature and/or concentration from continuous initial distributions.