Breaking and cascade of internal gravity waves in a continuously stratified fluid

The evolution of a few large scale high frequency standing internal waves confined to a vertical plane is studied numerically. The growth of nonlinear interactions leads to a transfer of energy toward small vertical scales and lower frequencies: the result is a steep energy decrease due to wave breaking. Induced mixing is evaluated. A parametric forcing is also introduced in order to compare with laboratory experiments. Wave breaking also occurs but as opposed to the unforced case different phases are next observed: internal wave growth due to constructive forcing alternate with energy decrease.     

Unsteady waves in a viscous fluid of finite depth

Here we study the plane and three-dimensional problems of unsteady waves which arise on the surface of a viscous fluid of finite depth under the influence of a velocity pulse applied on the bottom of the basin.The problem is considered as the simplest scheme for studying, with account for the effect of viscosity, the propagation of waves of the tsunami type which result from an underwater shock.Similar problems on the propagation of waves which arise from initial surface disturbances are considered in [1–9].     

The development of three-dimensional waves generated by perturbations of varying intensity in a flow of continuously stratified fluid

An analysis is made in the linear formulation of the three-dimensional structure of unsteady waves created in a flow of continuously stratified fluid by a region of pressures which are harmonic with respect to time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 71–77, November–December, 1984.     

On the development of plane waves generated by a periodic pressure applied to the flow of a continuously stratified fluid

In the linear formulation, an investigation is made into the development of undamped (in time) plane waves generated by a. harmonically varying pressure applied to the free surface of an initially undisturbed flow of a continuously stratified fluid of finite depth. The cases of a homogeneous fluid and two-layer fluid are considered in [1–3]. Nonstationary waves in a continuously stratified flow generated by a time-independent pressure were investigated in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 99–104, July–August, 1980.     

Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid

The problem of the stabilization of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid is solved using both asymptotic and numerical methods. The analytical solution describes the structure of the main convective cells, including thin meridional jets flowing along the surface and plumes spreading from the flow convergence regions above the upper and lower poles of the sphere which gradually return the fluid particles to the neutral buoyancy horizon. The total width of the flows adjacent to the surface exceeds the thickness of the salinity deficit layer or the density boundary layer. The numerical solution of the complete problem in the nonlinear formulation describes the main convective cells and two systems of unsteady integral waves formed in the vicinity of the sphere poles. At large times, out of the entire system of internal waves only those nearest to the neighborhood of their horizon of formation remain clearly defined. The calculated flow patterns are in agreement with each other and the data of shadow visualization of the stratified fluid structure near a submerged obstacle at rest.     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.     

Micro-scale instability in a continuously stratified fluid

Experimental observations of small-scale structures caused by flow instabilities at layers of high density gradient in the wake behind a cylinder in a fluid with a continuous salt concentration stratification are reported. In the density wake it is possible to discern a number of structures such as wedge-like structures or cusps; small-scale instabilities (breakers) in the zones of interaction of attached internal waves and the high-gradient wake envelope; small-scale instability of the density boundary layer with a complicated density gradient pattern superimposed on a smooth velocity profile, and small-scale wake structures behind attached vortices in the case of a closed (central) wake envelope.Translated from Izvestiya RossiiskoiAkademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–10, May–June, 1995.     

Shear instability of flows of a stratified ideal fluid of finite depth

A linear analysis is made of the stability of flows, stratified with respect to depth, of an ideal liquid of finite depth with a Helmholtz velocity profile. Apart from a Kelvin-Helmholtz wave, additional unstable modes are also discovered. Analytical expressions are obtained for the neutral curve of these modes. Their nature is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 176–179, November–December, 1988.The author is grateful to V. I. Klyatskin and L. Ya. Lyubavin for their interest in the study and for fruitful discussions.     

Internal waves arising in the presence of the unsteady motion of a source in a continuously stratified fluid

A general approach to the investigation of linear internal and surface waves in a stably stratified fluid, arising from different types of perturbations, is presented in [1]. The methods of calculation of the internal waves generated by the motion of a mass source are developed for particular cases of steady horizontal motion in a continuously stratified fluid in [2] and arbitrary motion in an unbounded exponentially stratified fluid in [3]. Internal waves generated by other types of perturbations have also been investigated but only for particular cases of motion (see, for example, [4]). This paper presents a method for the calculation of unsteady, linear, internal gravity waves arising in an inviscid incompressible fluid with continuous stable stratification.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–130, July–August, 1985.     

Composite steady waves in a gravity fluid stream of finite depth

The problem of plane steady gravitational waves of finite amplitude, caused by a periodically distributed pressure over the surface of an ideal incompressible gravity fluid stream of finite depth, is considered. It is assumed that these waves do not vanish as the pressure becomes constant, but become free waves, which exist at constant pressure and special values of the stream velocity. As in [1], where a stream of finite depth is considered, such waves will be designated composite as contrasted with forced waves which vanish together with the variable part of the pressure. A general method is given for computing the composite wave characteristics. The first three approximations are computed to the end. An approximate equation for the wave profile is found.     

Parametric excitation of internal waves in a continuously stratified liquid

A study is made of the parametric excitation of internal waves in a continuously stratified liquid in a vessel executing oscillations in the vertical direction. It is shown that vertical oscillations of the vessel will excite oscillatory modes with eigenfrequencies equal to half of the oscillation frequency of the vessel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 167–169, September–October, 1982.I thank S. V. Nesterov for interest in the work.     

Slow motions of a solid in a continuously stratified fluid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 55–60, March–April, 1991.     

On steady periodic waves on the surface of a fluid of finite depth

A solution of Nekrasov’s integral equation is obtained, and the range of its existence in the theory of steady nonlinear waves on the surface of a finite-depth fluid is determined. Relations are derived for calculating the wave profile and propagation velocity as functions of the ratio of the liquid depth to the wavelength. A comparison is made of the velocities obtained using the linear and nonlinear theories of wave propagation.     

Drag coefficient and upstream influence in three-dimensional stratified flow of finite depth

A numerical study of the three-dimensional stratified flow past a vertical square flat plate in a channel of finite depth is described. Particular attention is paid to the anomalous dependence of the drag coefficient C_D on parameter K( = ND/-πU), where N is the Brunt-Väisälä frequency, D is the half depth of the channel and U is the upstream velocity. It is shown that C_D generally increases with K, while it decreases locally at integral values of K. Time development of the upstream columnar disturbance and the corresponding variation of C_D reveals that the periodic variation of C_D with time for K > 1 comes from the successive upstream radiation of the columnar disturbances of the first internal wave mode. Although the propagation speed of the columnar disturbance is consistent with the prediction of linear theory, its time-dependent structure is different from the weakly nonlinear theory as has been shown by laboratory experiments.     

Plane problem of wave motions occurring in a continuously stratified fluid during the flow around a submerged body

The plane stationary problem of wave motions occuring in the flow of a uniform inviscid incompressible gravity fluid with an arbitrary continuous (stable) change in density with depth around submerged sources and sinks of equal intensity is investigated in a linear formulation. An analysis of the structure of the wave motion in a flow with an arbitrary density change is performed in [1].Paper presented at the First Soviet-American Symposium on Internal Waves in the Ocean, Novosibirsk, December 3–8, 1976.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 148–152, July–August, 1978.     

Effect of finite depth of stratified flow on turbulence formation in the wake of a cylinder

The experimental results of investigating the effect of the finite depth of a linearly stratified flow channel on turbulence formation associated with the horizontal motion of a cylinder are presented. The limits of the interval of internal Froude numbers and dimensionless channel depths (with respect to the cylinder diameter) corresponding to local instabilites in the disturbed flow density distribution, leading to the formation of turbulence, are found. The dynamics of formation of the turbulent zones and their evolution are investigated. Unsteady periodic regimes are found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 160–163, September–October, 1990.The authors are grateful to A. T. Onufriev for his interest in their work.     

Evolution of long nonlinear waves on the interface of a stratified viscous fluid flow in a channel

The dynamics of disturbances of the interface between two layers of incompressible immiscible fluids of different densities in the presence of a steady flow between the horizontal bottom and lid is studied analytically and numerically. A model integrodifferential equation is derived, which takes into account long-wave contributions of inertial layers and surface tension of the fluids, small but finite amplitude of disturbances, and unsteady shear stresses on all boundaries. Numerical solutions of this equation are given for the most typical nonlinear problems of transformation of both plane waves of different lengths and solitary waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 49–61, July–August, 2007.