Experimental study on internal waves in a stratified shear flow

This paper describes experiments on interfacial phenomena in a stratified shear flow having a sharp velocity shear at a density interface. The interface was visualized in vertical cross-section using dye, and the flow pattern was traced using aluminum powder. Two kinds of internal waves with different phase velocities and wave profiles were observed. They are here named p(positive)-waves and n(negative)-waves, respectively. By means of a two-dimensional visualization technique, the following facts have been confirmed regarding these waves. (1) The two kinds of waves propagate in the opposite direction relative to a system moving with the mean velocity at the interface, and their dispersion relations approximately agree with the two solutions of interfacial waves in a two-layer system of a linear basic shear flow. (2) The p-wave has sharp crests and flat troughs, and the n-wave has the reverse of this. This difference in wave profile is due to the finite amplitude effect. (3) Phase velocity of each wave lies within the range of the mean velocity profile, so that a critical layer exists and each wave has a “cat's eye” flow pattern in the vicinity of the critical layer, when observed in a system moving with the phase velocity. Consequently, these two waves are symmetrical with respect to the interface. The mechanisms of generation of these waves, and the entrainment process are discussed. It is inferred that when the “cat's eye” flow pattern is distorted and a stagnation point approaches the interface, entrainment in the form of a stretched wisp from the lower to the upper layer occurs for the p-wave, and from the upper to the lower layer for the n-wave.     

On the decay of layered structures in a stratified shear flow

The effect of a vertical velocity shear on the decay of small-scale perturbations in an unbounded thermally-stratified fluid is considered. The behaviour of perturbations at the final stage is shown to be controlled by the combined effects of viscosity, thermal conductivity, collapse, and material line stretching due to velocity shear. The behaviour of density and velocity perturbations has been analysed for the case of dominating viscosity forces and a velocity shear.UDK 532.529+532.517.4Translated by Vladimir A. Puchkin.     

On the onset of the instability of a three-dimensional jet in a stratified fluid

The onset of a three-dimensional jet flow in a stratified fluid is studied with the aid of a direct numerical simulation. An initially cylindrical jet with a Gaussian velocity profile is considered in a fluid with stable linear density stratification. The results indicate that, if an initial small perturbation of the velocity field has a wide spectrum, an exponential growth of the isolated quasi-two-dimensional mode occurs and its spectral maximum is shifted toward smaller wave numbers in comparison with the maximum of the helical mode of the instability of a nonstratified jet. The growth rate is proportional to Ri^0.5, where Ri is the global Richardson number. The onset of the instability leads to the formation of the flow’s vortex structure, which consists of a collection of different-polarity quasi-two-dimensional vortices located in a horizontal plane near the longitudinal axis of the jet. At sufficiently long times (Nt > 100, where N is the buoyancy frequency and t is time), the growth of instability reaches the saturation stage and further fluctuations in velocity and density decay under the effect of viscous diffusion. At this stage, the flow becomes self-similar and the time dependences of the transverse and vertical widths of the jet are consistent with the asymptotic behaviors of integral parameters of the flow that are observed experimentally in the far stratified wake. The results suggest that the onset of the instability of a quasitwo-dimensional mode can play the determining role in the dynamics of flow in the far stratified wake.     

Modeling a neutrally stratified turbulent air flow over a horizontal rough surface

The neutrally stratified boundary layer over a smooth rough surface is consider. The turbulent flow is simulated using a finite-difference eddy-resolving model of the atmospheric boundary layer (ABL). The model includes different turbulence closure schemes and numerical approximations for advection components of the momentum balance equation. We investigate the quality of reproduction of spectral characteristics of the turbulent flow and the model’s capabilities to reproduce the observed profile of mean wind velocity near the rough surface. It is shown that the best result is obtained by coupling a numerical scheme of higher order of accuracy with a mixed closure scheme based on an adaptive estimation of the mixing length for subgrid-scale fluctuations. Here, we are able to reproduce the asymptotics of the fluctuation spectrum of the longitudinal component of wind velocity near the surface and within the boundary layer as well as the logarithmic profile of mean velocity near the surface.     

Three-dimensional instability of shear flows with inflection-free velocity profiles in stratified media with a high Prandtl number

In stably stratified media with a Prandtl number Pr ≫ 1, vertical scales of the density (ℓ) and horizontal velocity variation (L) are quite different, ℓ/L = O(Pr^−1/2) ≪ 1, and this influences the flow stability. In particular, shear flows without inflection points on the velocity profile are unstable even in an ideal incompressible fluid. The maximum instability growth rate for sufficiently small ℓ/L is of the same order as in homogeneous mixing layers, with mainly three-dimensional rather than two-dimensional oscillations increasing in a wide range of parameters. This paper focuses on the three-dimensional instability of such flows. It is shown that the spectrum of unstable oscillations is essentially anisotropic in the case of a relatively weak stratification when the bulk Richardson number J ≤ O[(ℓ/L)^3/2]. The results of the asymptotic analysis are illustrated by calculations for a model flow in a two-layer medium (ℓ = 0) as well as for flows with values of ℓ/L corresponding to a temperature or salinity stratification of the water.     

Long nonlinear topographic planetary waves in a rotating stratified ocean

Long nonlinear topographic waves in a continuously stratified ocean with a linear bottom slope are investigated. It is shown that odd cross-channel modes are governed by the Korteweg-de Vries (K-dV) equation. The solitary waves are those of a low pressure type. The long waves are shown to be modulationally stable because of the nonlinear effect due to irrotational motion. All these results are missed if the conventional quasi-geostrophic approximation is adopted.     

Laboratory, numerical, and theoretical modeling of the flow in a far wake in a stratified fluid

The far-wake flow past a sphere towed in a fluid with high Reynolds and Froude numbers and with a pycnocline-form salt-density stratification is studied in a laboratory experiment based on particle image velocimetry and in numerical and theoretical modeling. In the configuration under consideration, the axis of sphere towing is located under a pycnocline. Flow parameters, the profiles of density and average velocity, and the initial field of velocity fluctuation in numerical modeling are specified from the data of the laboratory experiment. The fields of fluid velocity at different times and the time dependences of integral parameters of wake flow, such as the average velocity at the axis and the transverse width of the flow, are obtained. The results of numerical modeling are in good qualitative and quantitative agreement with the data of the laboratory experiment. The results of the laboratory experiment and numerical modeling are compared to the predictions of a quasi-linear and quasi-two-dimensional theoretical model. The time evolution of both the average velocity at the axis and the transverse width of the wake is obtained with the model and is in good agreement with the experimental data. The results of numerical modeling also show that, under the effect of velocity fluctuation in the wake, internal waves whose spatial period is equal to the characteristic period of the wake’s vortex structure are excited efficiently in the pycnocline.     

Resonant interaction of waves of continuous and discrete spectra in the simplest model of a stratified shear flow

The equations of dynamics of eddy—wave disturbances of two-dimensional stratified flows in an ideal incompressible fluid that are written in a Hamiltonian form are used to study the resonant interaction of waves of discrete and continuous spectra. A gravity—shear wave generated at a jump of the density and vorticity of the undisturbed flow and a wave generated at a weak vorticity jump, which is similar to a wave of a continuous spectrum, participate in the interaction. The equations are written in terms of normal variables to obtain the system of evolution equations for the amplitudes of the interacting waves. The stability condition for eddy—wave disturbances is derived within the framework of the linear theory. It is shown that a cubic nonlinearity may lead to the stabilization of unstable disturbances if the coefficient of the nonlinear term is positive.     

Stationary waves in the flow of a homogeneous fluid with vertical shear of the velocity

A plane problem of free stationary gravitational waves in a horizontal current with vertical shear of the velocity is studied in the linear statement. The determination of the parameters of waves is reduced to the solution of the Sturm–Liouville boundary-value problem. For some vertical distributions of current velocity, we obtain analytic solutions. We propose a numerical algorithm for finding the parameters of waves. On the basis of the performed analysis, we establish the possibility of existence of stationary surface waves in currents for certain ranges of the Froude number. As the Froude number decreases, the waves become shorter, which leads to a faster attenuation of waves disturbances with depth. Under the actual conditions, the waves are short and suffer the influence of shear currents only in the subsurface layer of the ocean.     

On the stability of vortex streets in a stratified fluid

On the basis of the perturbation theory developed previously by the authors for localized hydrodynamic vortices, the influence of a specified jet flow and of the structure of individual vortices on the stability of the Karman street is investigated. It is shown that, for a street of vortices with a power law of decrease in the azimuthal velocity, the jet flow suppresses instability only with respect to perturbations with wavelengths from a certain range determined by the parameters of the flow. At the same time, for streets formed from vortices with a Gaussian profile of the azimuthal velocity, even in the absence of a specified flow, there is a certain region of the street’s parameters in which the street is stable against perturbations of all scales. Thus, for the purposes of modeling quasi-two-dimensional flows in a stratified fluid by a sequence of localized vortices, which is discussed in this study, vortices with a Gaussian profile of the azimuthal velocity turn out to be preferable. The results of this study are consistent with numerous experiments on the structure of a quasi-two-dimensional wake behind a body in a stratified fluid at large Reynolds and Froude numbers.     

Wave-induced boundary layers in a rotating homogeneous fluid

The structure of the Eulerian streaming induced by inertial standing waves over a flat plate perpendicular to the rotating axis is investigated by using the matched asymptotic technique. It is shown that the Eulerian streaming driven by the divergence of the Reynolds stress in the Stokes layer vanishes outside the Stokes and the Ekman layer due to the role of Coriolis force.     

Selective withdrawal from a rotating stratified current with applications to OTEC

The results of a simple theoretical model of axisymmetric withdrawal from a rotating stratified current are presented. A rotating stratified flume was used to test the validity of the theory. Good agreement was found and when the theory is applied to the fluid motion into an OTEC plant, it is found that when the current going by the plant is small, large vortices develop near the inlets. The torques on the plant due to these vortices and the degradation of temperature due to selective withdrawal processes are estimated to be potentially significant.     

Trapped symmetric disturbances in rotating shear flows

The structure of trapped symmetric disturbances in rotating stratified shear flows is investigated theoretically. It is shown that the arrangement of the trapping region is determined by atmospheric stratification. For example, if the characteristic Brunt-Väisälä frequency is greater (smaller) than the inertial frequency, waves are trapped in the region of anticyclonic (cyclonic) velocity shear. Accordingly, in the first (second) case, the frequencies of trapped waves are smaller (greater) than the inertial frequency. The problem of finding the frequencies of trapped waves is reduced to solving the Schrödinger equation but with a more complex dependence on a spectral parameter. Exact solutions to the problem are obtained for a triangular jet and a hyperbolic shear layer.     

Nonlinear adjustment of a heavy rotating fluid to equilibrium

We consider the problem of geostrophic adjustment of nonlinear axially symmetric formations in a two-layer fluid with infinitely deep lower layer and demonstrate the existence of a new integral of motion, which enables us to determine the final equilibrium state of perturbations without analyzing the transient (wave) stage of the process. It is shown that the final motions caused by initial perturbations of the opposite signs are not equivalent. Translated by Peter V. Malyshev and Dmitry V. Malyshev     

Fine structure of a conical beam of periodical internal waves in a stratified fluid

Stratified fluid flows caused by torsional or linear harmonic oscillations of a ring along the surface of an infinite vertical cylinder have been calculated by methods of the perturbation theory. The complete solutions of the linearized system of equations with sticking boundary conditions for velocity and impermeable boundary conditions for substance have been obtained taking into account viscosity and diffusion. Disturbances forming a conical beam of three-dimensional internal waves and families of small-scale components are identified. Formulas for calculating waves in media with different Schmidt numbers are described.     

四层成层水域内波的研究

运用小振幅波理论研究了四层成层水域的内波运动，给出了四层成层状态下的各界面波波面位移和各层速度势的解析表达式及各层深度平均流速分布和内波波动频散关系，并与三层成层水域内波的解析结果进行了比较。     

On the spatial instability of internal waves in continuously stratified flows

A stationary parallel current in a continuously stratified incompressible fluid of finite depth is considered. The instability of internal waves is studied with the assumption that the current has only a vertical velocity shear.Translated by Vladimir A. Puchkin.