A method of submersion in the problem on the determination of dispersion curves of internal waves in the presence of a current with vertical velocity shear

The previously proposed method [1] of submerging aimed at determining internal wave dispersion characteristics is generalized to consider the case of a background horizontal flow with a vertical velocity shear. The results of calculating, dispersion curves of discrete spectrum modes for the profiles of the Brunt-Väisälä frequency and the mean current velocity are given; they agree well with the results of studies carried out in ref. [5].Translated by Mikhail M. Trufanov.     

Decay of layered features in temperature- and salinity-stratified fluid with a current velocity shear

The decay of small-scale perturbations in salinity- and temperature-stratified shear flow is considered. The fundamental system of solutions (modes) has been derived asymptotically for the case of predominant viscosity contribution. The main modes of velocity, temperature, and salinity perturbations are identified, and the decay rates of these modes are analysed.Translated by V. Puchkin.     

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.     

Hamiltonian description of shear and gravity shear waves in an ideal incompressible fluid

Methods of studying the dynamics of wave disturbances in st;ratified shear flows of an ideal incompressible fluid are considered. The equations governing the motions of interest represent Hamilton equations and are derived by writing the velocity field in terms of Clebsch potentials. Equations written in terms of semi-Lagrangian variables are integrodifferential equations, which make it possible to consider both continuous and discontinuous solutions, as well as the cases where the parameters of the undisturbed medium are step functions. Two dynamic systems are presented. The first, canonical system of equations is most suitable for describing gravity waves in a shear flow in the case where the undisturbed medium is characterized by sharp gradients of density and flow velocity. The simplest model in which disturbances obey this system of equations is the well-known Kelvin-Helmholtz model. The second dynamic system describes, in particular, gravity-shear waves and, in the case of a homogeneous medium, shear waves in a two-dimensional flow. This system is most suitable for studying the dynamics of disturbances in models with sharp gradients of vorticity. On the basis of the approach developed in this study, the problem of the dynamics of disturbances in a flow with a continuous distribution of vorticity in a finite-thickness layer is solved. If the thickness of this layer is small compared to the characteristic wavelength and the gradient of the undisturbed vorticity in this layer is large, the solution has the form of a mode whose frequency is close to the frequency of the shear wave on a vorticity jump that would be obtained by letting the layer’s thickness approach zero. The results obtained allow, in particular, the estimation of the range of validity of finite-layer approximations for models with smooth profiles of flow and density. In addition, these results can be interpreted as the basis for the development of nonlinear aspects of the theory of hydrodynamic stability.     

Rotational instability of a horizontal shear flow in a stratified rotating fluid

The rotational instability of a thermally stratified, viscous, conducting, rotating fluid is investigated by means of linearized perturbation equations. It is assumed that the basic horizontal flow is vertically uniform and that the horizontal shear is confined in a thin layer. By solving a simplified boundary value problem as a model of rotational instability in the sea, we have shown that the vertical wave length of the neutral disturbance is of order 10 times as large as the laminar Ekman layer thickness, and that this scale is proportional to (L/N)^1/3, whereL is the width of the shear layer andN is the Brunt-Vaisala frequency.     

Transparent lateral boundary conditions for baroclinic waves III. Including vertical shear

The problem of how to adapt the so-called transparent boundary conditions to the physical reality that the horizontal advection velocity varies in the vertical, is addressed. A practical demonstration is presented of the transparency of the resulting 'shear' boundary conditions.     

Generation of vertical fine structure in inhomogeneous flow by inertial-gravity internal waves

Corrections to the vertical distribution of mean density and current velocity non-oscillating on the wave's time scale have been derived in the approximation, quadratic by the wave's steepness. It has been shown that the horizontal volume transport of both the induced current and the Stokesian drift, integrated over depth, equates zero in the Boussinesq approximation.Translated by V. Puchkin.     

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.     

Flow over mountains with the stream velocity shear

The dependence of orographic disturbances of the atmosphere on properties of the incident flow is studied within the semianalytic approach. Reducing the initial system of equations of hydrothermodynamics to a single equation for an associative stream function makes it possible to consider a class of solutions of a sufficiently general type when the background velocity of the wind and the Lyra’s scale vary with height. It is shown that the dependence of the solution on the indicated factors can be not only strong, but also sufficiently unexpected. In particular, with the monotonic growth in the wind velocity in the troposphere, which corresponds to conditions of a jet stream near the tropopause, disturbances at low and medium heights can acquire an almost resonant and waveguide nature.     

Tide-induced vertical circulation in a bay with homogeneous water: Theory and experiment

The structure of the tidal residual current due to vertical viscosity is investigated both theoretically and experimentally. It is found that the interaction between the vertical component of the oscillatory current and vorticity,wT ₁ T ₁, induces a strong residual constituent outside the boundary layer and forms a circulation which is quite similar to gravitational circulation and that the vertical profile of the oscillatory current not only affects the magnitude of the residual constituent but also decides the direction of the circulation. In the hydraulic experiment, the residual constituent is larger than the theoretical prediction and a phase difference in the oscillatory constituent between the upper and lower layer is observed. The amplitude difference is caused by the strong nonlinear effect of the residual constituent and the phase difference is caused by the interaction between the residual current and the basic oscillatory current. The principal generating force of the residual constituent outside the boundary layer,wT ₁ T ₁, is observed in a bay where the tide is nearly a standing wave.     

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.     

The vertical mode method in the problems of flexural-gravity waves diffracted by a vertical cylinder

The linear three-dimensional problem of ice loads acting on a vertical circular cylinder frozen in an ice cover of infinite extent is studied. The loads are caused by an uni-directional hydroelastic wave propagating in the ice cover towards the cylinder mounted to the see bottom in water of constant depth. There are no open water surfaces in this problem. The deflection of the ice cover is described by the Bernoulli–Euler equation of a thin elastic plate of constant thickness. At the contact line between the ice cover and the surface of the cylinder, some edge conditions are imposed. In this study, the edge of the ice plate is either clamped to the cylinder or has no contact with the cylinder surface, with the plate edge being free of stresses and shear forces. The water is of finite constant depth, inviscid and incompressible. The problem is solved by both the vertical mode method and using the Weber integral transform in the radial coordinate. Each vertical mode corresponds to a root of the dispersion relation for flexural-gravity waves. It is proved that these two solutions are identical for the clamped edge conditions. This result is non-trivial because the vertical modes are non-orthogonal in a standard sense, they are linearly dependent, the roots of the dispersion relation can be double and even triple, and the set of the modes could be incomplete. A general solution of the wave-cylinder interaction problem is derived by the method of vertical modes and applied to different edge conditions on the contact line. There are three conditions of solvability in this problem. It is shown that these conditions are satisfied for any parameters of the problem.     

A new method to estimate phase speed and vertical velocity of internal solitary waves in the South China Sea

A new method of estimating the phase speed and vertical velocity of internal solitary waves (ISW) based on dynamic governing equations using Acoustic Doppler Current Profiler (ADCP) data was developed. This method was applied to a representative ISW case, which was captured in the South China Sea in the 2007 summer experiment. The result shows that this ISW had a phase speed of approximately 2.6?m?s^?1. This method provides an excellent alternative in estimating the phase speed of ISWs, especially useful in the absence of vertical stratification needed in solving the Korteweg–de Vries (KdV)-type equation, e.g., long-term mooring observation. Analyses show that the new method is fairly self-consistent, and it can be applied when only a part of the ISW observations is available. The computed vertical velocity of the ISW using the new method has a good agreement with the ADCP observations both in magnitude and pattern.     

Wave forces acting on a vertical circular cylinder with a constant forward velocity

Wave forces acting on submerged circular cylinders moving forward with a constant velocity in regular waves are investigated experimentally. Hydrodynamic forces acting on the cylinder forced to surge in a steady are also measured and hydrodynamic coefficients were obtained. Wave force coefficients obtained from wave force measurements are compared with the hydrodynamic coefficients from surging tests, and the similarity and difference between them are discussed. Experiments show that these coefficients are quite different from those of the cylinder without a forward velocity.     

Horizontal and vertical cylinders in waves

The present investigation examines a vertical cylinder and a horizontal cylinder in progressive waves. The physical differences between the flows are explored and experimental results are compared to previous planar harmonic flow measurements. It is found that modifications to the usual Morison approach are required in some cases to adequately account for the orbital motions of the fluid and to account for the orientation of the orbits with respect to the cylinder axis. The axial variations of the wave force on vertical cylinders are considered in order to evaluate the common practice of assuming constant values of $\text{C}{}_{\mathrm{m}}\text{and}\text{C}{}_{\mathrm{D}}$ over the entire span. Lastly, the methods of computing force transfer coefficients from a force record are examined and several sources of error are identified and briefly discussed.     

Stationary Rossby waves and shocks on the Sverdrup coordinate

Gyre-scale frontal structures, e.g., the Subtropical Front, have been well documented in the oceans. Although the generation mechanism of such a front remains unclear, it may be ascribed to the steepening of nonlinear planetary waves as discussed by Dewar (1992). To discuss the stationary wave characteristics and their shock formation in a two-layer wind-driven gyre, the present paper introduces a new coordinate, referred to as theSverdrup coordinate here, in which the Sverdrup function is used instead of the longitudinal coordinate, and especially, investigates the possibility of the frontgenesis caused by the Rossby waves emanating from the western boundary region. On theSverdrup coordinate, since the advection by the Sverdrup flow is in the direction normal to that of the Rossby wave propagation, the system becomes much simpler than that on the (x,y)-coordinate, and the solution to this coordinate does not explicitly depend on the distribution of the Ekman pumping, i.e., we can treat cases, in which the Ekman pumping is a function of bothx andy, in a similar way to the case with zonally uniform Ekman pumping.     

两层流体中内孤立波与潜体相互作用数值模拟

提出基于双推板的内孤立波数值造波方法,对两层流体中内孤立波与潜体的相互作用进行了数值模拟,分析不同潜深下潜体所受的内孤立波载荷特性.结果表明,内孤立波对水下潜体的载荷作用是不可忽视的.特别地,当潜体位于内孤立波中时,其所受的垂向力要远比潜体位于内孤立波下时的大.     

Wave runup on a surging vertical cylinder in regular waves

The wave runup caused by a vertical cylinder surging in regular waves is studied both experimentally and numerically. The so-called DualSPHysics Smoothed Particle Hydrodynamics (SPH) code is used for the 3-D numerical modelling. A wide range of cylinder sizes and wave conditions is investigated with results comparing favourably between the experimental and SPH model under both fixed and forced-surge conditions. The experimental and SPH results are further used to predict the maximum runup amplification, in particular the ratio of the runup caused by the surging cylinder to that of the fixed, over the phase difference between the incident wave and surge motion. This maximum runup ratio has been analysed for its dependence on factors such as wave steepness, wave scattering and surge amplitude. An empirical equation is proposed for predicting the maximum runup ratio from known incident wave and surge conditions. Comparison with results from linear solvers suggests that the linear solvers under-predict the full nonlinear runup by a factor of 1.3–1.5.