Yu. V. Kistovich

Linear theory of the propagation of internal wave beams in an arbitrarily stratified liquid

Beams of harmonic internal waves in a liquid with smoothly changing stratification are calculated in the Boussinesq approximation taking into account the effects of diffusion and viscosity. A procedure of local reduction of the beam in a medium with an arbitrary smooth stratification to the case of an exponentially stratified liquid is constructed. The coefficient of energy losses in the case of beam reflection on the critical level is calculated. Parameters of internal boundary flows with split scales of velocity and density that are formed by a wave beam on discontinuities of the buoyancy frequency and its higher derivatives are determined.

An exact solution of a linearized problem of the radiation of monochromatic internal waves in a viscous fluid

Abstract The eigenvalue method is used to construct an exact solution of the linearized boundary-value problem of the generation of internal waves in an exponentially stratified fluid, when the source is part of a plan which vibrates along its surface. The spatial structure of the solution obtained describes two well-known types of wave beams-unimodal and bimodal. In the limiting cases the phase pattern of the waves is identical with well-known asymptotic forms and laboratory experiments. The exact solution is compared with the solution of the model problem of the generation of waves by force sources, constructed using homogeneous fluid theory. The phase patterns of the waves in both cases agree everywhere with the exception of critical angles, when the wave propagates along the radiating surface. The amplitudes of the radiated waves are the same only for certain ratios of the angles of inclination of the plane and the direction of propagation of the beams.

The reflection of beams of internal gravity waves at a flat rigid surface

Abstract The linear problem of the reflection of a beam of monochromatic internal waves at a rigid inclined wall in an exponentially stratified liquid with viscosity and diffusion is considered. In such a medium there is, as well as the reflected wave, a boundary-layer flow at the plane with split spatial length-scales for the velocity and density variation. Viscosity and diffusion restrict the limiting value of the geometrical compression coefficient of the beam. The solution has no singularities at critical angles of reflection. Calculations for the reflected beam and boundary flow are performed for an incident beam radiated by a point mass source.

Mass transport and the force of a beam of two-dimensional periodic internal waves☆

Abstract The steady flow generated by an arbitrary field of monochromatic internal waves in a viscous continuously stratified liquid is calculated in the first order of perturbation theory. The streamline pattern is calculated for a beam of harmonic waves excited by a point dipole. The steady force, which a beam incident on rigid plane surface exerts, is also calculated.

Linear generation theory of 2D and 3D periodic internal waves in a viscous stratified fluid

We investigated analytically and experimentally 2D and 3D periodic internal waves generated by small harmonic oscillations of a plate and of an impermeable vertical cylindrical tube in an exponentially stratified viscous fluid. The linearized governing equations were solved by an integral transform method. The exact boundary conditions on the surface of a body, as well as the governing equations are satisfied if, in addition to propagating internal waves, internal boundary currents on the emitting surface are taken into account. On the basis of these two forms of fluid motion, we constructed a complete linear theory of the wave generation, without any external parameters. We calculated wave amplitudes and evolution along the beam of the so-called ‘St. Andrews Cross’ wave shape, namely the number of maxima in the wave amplitude cross-section. The spatial decay of the wave was different in 2D and 3D problems due to geometry. The distance from the source, where transition from a bi-modal beam to the uni-modal beam takes place, is defined. Small viscosity smoothes out the singularity that arises in the wave field along the inviscid characteristics and in the critical angles. Experimental observations and probe measurements of a periodic wave pattern confirmed the theoretical results for the far field wave structure. The absolute values of calculated wave amplitudes differed from the experimental values by a factor less than 1.5. Indirect evidence of the internal boundary currents in Schlieren photographs of the flow pattern were presented. Copyright

A new mechanism of the nonlinear generation of internal waves

The exact solution to the linearized problem of generating disturbances by a two-dimensional obstacle performing a periodic motion in a viscous continuously stratified liquid describes thin boundary layers [1] as well as progressive waves and makes it possible to estimate the errors of the extensively used method of sources [2]. The three-dimensional boundary-layer flow is split into layers of two types [3]. One of them (viscous periodic flow) is analogous to the Stokes layer in a homogeneous fluid and has a thickness depending on the kinematic viscosity and frequency. The spatial scale of the second (internal boundary) layer depends on the geometric parameters of the problem [3]. Since the governing equations are nonlinear, the moving boundary layers are direct sources of waves. The parameters of waves generated by a boundary-layer flow on a horizontal disc performing torsional vibrations are in satisfactory agreement with measurements [4]. For the first time, we consider here the generation of internal waves through new mechanisms attributed to the nonlinear interaction of boundary-layer flows with each other and with progressive internal waves or residual motions into which the internal waves transform when the disturbance frequency ω exceeds the buoyancy frequency N of a media. As a source of waves, we consider an infinite immovable vertical plane whose part performs complex two-dimensional motion, which is the superposition of two vertical oscillations with frequencies ω1 and ω2 . In this case, only the vertical component of the surface velocity is nonzero:

Properties of the exact wave-type solutions in the theory of stratified flows☆

The general properties of the wave-type solutions in the theory of internal waves for flows in continuously stratified media are analysed. In addition to the well-known cases of the equivalence of the conditions for the summation of plane non-linear periodic waves and the principle of the superposition of linear waves, the conditions for the existence of wave-type solutions for non-stationary and attached waves in dissipative media are determined. The sets of relations of the physical parameters which can be used as expansion parameters when constructing approximate (asymptotic) solutions of the equations of internal waves in dissipative media are determined.

Generation of Periodic Internal Waves by an Oscillating Strip of Finite Width

The traditional approach to the calculation of inter-nal-wave generation, which is based on the use of forceand mass sources with parameters adopted from thehomogeneous-liquid theory, enables one to determine afar field accurate to empirical constants [1–3]. Amethod for constructing the solutions to a linearizedproblem that exactly satisfy boundary conditions, wasproposed in [4, 5]. As a wave source, part of an infiniteplane positioned at an arbitrary angle ϕ to the horizon-tal and executing periodic oscillations with a frequency ω was considered. A finite-width strip oscillating alongits surface emits unimodular and bimodal beams into aliquid of a constant buoyancy frequency N ; the beamstravel at the angle θ = to the horizontal. In anarbitrary case ( ϕ ≠ θ ), when all beams separate from theemitting surface, the wave pattern and particle-dis-placement amplitudes are consistent with the measure-ments [6, 7]. In the critical case ( ϕ = θ ), when two wavebeams propagate along a plane separating the liquid,the calculations result in overstated values of the sepa-rated-beam amplitudes and give no way of finding adja-cent-beam parameters [5, 7]. The critical-angle case isof particular interest for problems of geophysicalhydrodynamics [8] and calls for special consideration.In the present paper, a solution to the more physi-cally-based problem of internal-wave generation by afinite-width oscillating strip is constructed for the entirerange of variation of the strip slope including the criti-cal one.A system of two-dimensional equations of motionfor an exponentially stratified incompressible liquid inthe Boussinesq approximation [1] is brought to the fol-lowing equation for the stream function Ψ in the emit-ωNarcsin---- ting-surface axes coordinate system ( ξ , ζ ) (see figure): (1) Here, ∆ = + and ν is the kinematic viscosity.The gravity g is opposite to the z -axis; the relationbetween the coordinate systems ( x , z ) and ( ξ , ζ ) isshown in figure.The adhesion conditions at the emitting surface(which is a strip with a width a inclined at an angle ϕ and executing oscillations along its surface) and thedamping of all perturbations at infinity constitute theboundary conditions for the velocity u

Experimental study of the generation of periodic internal waves by the boundary layer on a rotating disk

The exact solution to the linearized problem of the generation of internal waves, which involves internal waves and internal boundary flows [1], allows us to estimate errors intrinsic to the well-known method of force (or mass) sources [2]. In the case of small displacements, calculations of perturbances excited by an oscillating bar satisfactorily agree with the measurement results of [3]. There exist situations when a body moving periodically in continuously stratified viscous fluid does not radiate (in the linear case) and generates only isopycnic boundary flows. This situation takes place, e.g., in the case of a horizontal disk performing torsional vibrations [4]. However, by virtue of the nonlinear nature of the hydrodynamic system of equations, various forms of motion interfere with one another. In particular, thin-layer boundary flows can be a source of periodic waves [4]. Previously, experimental studies of such internal-wave generators were not conducted. Therefore, it is of interest to investigate the practical feasibility of the principles for nonlinear generation. In the present paper, the possibility of generation of threedimensional beams of periodic internal waves by torsional vibrations of a horizontal disk are studied and the principal regularities connecting wave-field characteristics with the properties of a medium and source motion parameters are established.

Localized and volume internal waves in a stratified fluid contiguous to a mixed layer

Abstract Dispersion relations, the rate of energy transfer, orthogonality and completeness relations and, also, the functions describing the vertical structure of two types of internal waves (perturbations localized close to the boundary and volume perturbations), which exist in an exponentially stratified medium contiguous to a homogeneous layer of finite thickness without any discontinuity in the density are calculated without recourse to the Boussinesq approximation.