Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the(2+1)-dimensional coupled nonlinear Schr¨odinger(CNLS)equations that can describe interactions of two Rossby waves propagating in stratified fluids.The(2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately.The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method,and the influence factors of modulational instability are analyzed.The results suggest that,when two Rossby waves with slightly different wave numbers propagate in the stratified fluids,the intensity of bright soliton decreases with the increases of dark soliton coefficients.In addition,the size of modulational instable area is related to the amplitude and wave number in y direction.

PIV experimental study on the interactions between ice ridges and stratified fluids

With the rapid decline of Arctic sea ice, the freshwater produced by melting of summer sea ice makes the depth of the halocline under ice become shallower. This has an impact on the drift of sea ice because internal waves may be generated at the interface of the halocline by disturbance from the draft of an ice floe or ridge keel. A laboratory experimental study was carried out to investigate the interactions between an ice ridge and stratified fluid using the method of Particle Image Velocimetry （PIV）. The drift velocity of an ice ridge （U） and the draft of the ridge keel （D） were altered in different experimental cases, and the velocity field in the stratified fluid was then measured by PIV. The results reveal that an obvious vortex exists in the wake field of the ridge keel, and the center of the vortex moves away from the ice ridge with increasing D. Internal waves at the interface of the stratified fluid were observed during the drift of the ice ridge, and the wave height shows a positive correlation with U and D. This study demonstrates that ice ridges could introduce internal waves at the interface of a stratified fluid and thus affect the oceanic drag coefficient and ice drift. It supports improved parameterization of the ice drag coefficients.

The Evolution Equation for Second-order Internal Solitary Waves in Stratified Fluids of Great Depth

By using perturbation methods, the evolution equation is derived for the second-order internal solitarywaves in stratified fluids of great depth, which is a kind of inhomogeneous linearized Belljamin-Ono equation.

Experiment on surface wake of internal waves generated by underwater vehicle in stratified fluids

The surface flow field of internal waves generated by the underwater vehicle is very weak.In order to study the characteristics of the surface wake of internal waves,a surface particle image velocimetry(PIV)technique which can be used to measure the flow field in the order of mm/s is developed.Breakthrough is made with respect to the key technique measuring the micro-velocity flow field of internal waves on the water surface in stratified fluids.The wake generated by SUBOFF model is measured in stratified fluid tank,and the surface flow field of internal waves is successfully measured for the first time.The experimental results are compared with predicted results by the classical Tuck’s internal wave theory.The results show the characteristics of the surface wake signature of internal waves and the variation of the angle between internal wave beams and the surface flow velocity of internal wave with the towing speed of the model are in good agreement.It provides support for further research on wake signature remote sensing of internal waves in laboratory.

WAVES GENERATED BY A SUBMERGED BODY MOVING IN STRATIFIED FLUIDS AND VERTICAL STRUCTURES OF INTERNAL WAVES

This dissertation deals with the internal waves generated by a submergedmoving body in stratified fluids by combining theoretical and experimental methods. Our purpose isto provide some scientific evidences for non-acoustic detection of underwater moving bodies based onthe principles of dynamics of the internal waves. An approach to velocity potentials obtained bysuperposing Green''s functions of sources and sinks was proposed for Kelvin waves at the free surfaceor interface in a two-layer fluid. The effects of interacting surface- and internal-wave modesinduced by a dipole on the surface divergence field were investigated. A new theoretical modelformulating the interaction of a two-dimensional submerged moving body with the conjugate flow in athree-layer fluid was established. An exact solution satisfying the two-dimensional Benjamin-Onoequation was obtained and the vertically propagating properties of the weakly nonlinear long waveswere studied by means of the ray theory and WKB method. The above theoretical results arequalitatively consistent with those obtained in the experiments conducted by the author.

TWO-DIMENSIONAL ALGEBRAIC SOLITARY WAVE AND ITS VERTICAL STRUCTURE IN STRATIFIED FLUID

The algebraic solitary wave and its associated eigenvalue problem in a deep stratified fluid with a free surface and a shallow upper layer were studied. And its vertical structure was examined. An exact solution for the derived 2D Benjamin-Ono equation was obtained, and physical explanation was given with the corresponding dispersion relation. As a special case, the vertical structure of the weakly nonlinear internal wave for the Holmboe density distribution was numerically investigated, and the propagating mechanism of the internal wave was studied by using the ray theory.

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 constant-density 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 socalled quasi-periodic waves and quasi-solitary waves,whose periods,propagation velocities and wave profiles vary slowly.The relations of the period of quasi-periodic 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.

STRONGLY OBLIQUE INTERACTIONS BETWEEN INTERNAL SOLITARY WAVES WITH THE SAME MODE

In this paper, by using the Lagrangian coordinates, the strongly oblique interactions between solitary waves with the same mode in a stratified fluid ape discussed, which includes the shallow fluid case and deep fluid case. It is found that the interactions are described by the KP equation for the shallow fluid case, the two-dimensional intermediate long wave equation (2D-ILW equation) for the deep fluid case and the two-dimensional BO equation (2D-BO equation) for the infinite deep fluid case.

SURFACE EFFECTS OF INTERNAL WAVE GENERATED BY A MOVING SOURCE IN A TWO-LAYER FLUID OF FINITE DEPTH

Based on the potential flow theory of water waves, the interaction mechanism between the free_surface and internal waves generated by a moving point source in the lower layer of a two_layer fluid was studied. By virtue of the method of Green's function, the properties of the divergence field at the free surface were obtained, which plays an important role in the SAR (Synthetic Aperture Radar) image. It is shown that the coupling interaction between the surface_wave mode and internal_wave mode must be taken into account for the cases of large density difference between two layers, the source approaching to the pynocline and the total Froude number Fr close to the critical number Fr 2. The theoretical analysis is qualitatively consistent with the experimental results presented by Ma Hui_yang.

THE WAVE ATTENUATION ON THE MUD BED

In the paper the wave attenuation in a two layer fluid system is studied.The fluid in the top layer is ideal and that in the lower layer is the Voigt model of the viscoelastic medium.A dispersion relation is derived and the rate of the wave decay is computed.The approximate explicit expressions of the decay rate for different water depth are given,where the viscoelasticity is either very large or very small.Compared with the numerical results,our results are very accurate,which can be used by an engineer.

Hydrodynamic performances and wakes induced by a generic submarine operating near the free surface in continuously stratified fluid

The wake of a submarine is a crucial element that greatly affects its stealth and hydrodynamic performance.To investigate the propagation and evolution characteristics of submarine wakes in a stratified fluid,particularly under free surface conditions,a numerical method to simulate the linearly stratified environment was developed.A thermocline model based on Boussinesq assumption was introduced to match the continuously stratified fluid.The volume of fluid(VOF)method was utilized to capture free surface waves,while a fully structured grid and the SST k-ωturbulence model were combined to solve the complex flow of submarines.Grid independence study in homogenous flow near the free surface was initially conducted.Furthermore,the hydrodynamic performance,wake evolution characteristics,free surface signatures as well as decay of velocity under different stratification levels were analyzed.Results demonstrated that the numerical method employed was efficient in simulating stratified flows.Moreover,it showed that density stratification had a significant impact on the hydrodynamics and wake characteristics of a submarine,especially under strong density stratification circumstances.

EXPERIMENTAL RESEARCH ON INTERNAL WAVES GENERATED BY A MOVING BODY IN A STRATIFIED FLUID

Two-dimensional internal waves generated by a moving flat body in a continuous- ly stratified fluid are investigated by using three kinds of flow-visualization methods:electrolytic precipitation,hydrogen bubbles,and dye streaks.Attentions are paid mainly to the generation and propagation process in the upstream region under low internal Froude number(1/10π<F_r<1/2π).The features of the upstream disturbances as well as their relationship with stratification number K (K=1/F_r) are illustrated.The corresponding theoretical analysis is briefly presented,and by comparison,the experimental and theoretical results agree well.

Time-frequency analysis of internal waves generated by a towed and self-propelled submerged body model

The spectrogram,based on a short-time Fourier transform,can visualize the time-dependent frequency spectrum of waves and is easy to compute.This time-frequency analysis method provides crucial information about waves generated by moving vessels and has been utilized to analyze Kelvin ship waves and internal waves.To further study the internal waves induced by a submerged body,an experiment is conducted for the towed and self-propelled SUBOFF model in a stratified fluid.The internal wave elevation signals are captured using electronic conductivity probes.Comparing with the calculation results of theoretical model,the high-frequency component of internal waves is identified.The high-frequency component has the exact same characteristics in both the towed and self-propelled model experiment and is consistent with the theoretical results for all Froude numbers.Therefore,this component is composed mainly of lee waves.Through spectral characteristics identification,a low-frequency component is discovered in the spectrogram in addition to the lee wave component.The intensity of the low-frequency component is tightly related to the vortex structure behind the submerged body.The vortex structure depends on the net momentum imparted by the submerged body.Therefore,this component is composed mainly of wake waves induced by the vortex structure.

Modulation instability analysis of Rossby waves based on(2+1)-dimensional high-order Schrodinger equation

Modulational instability is an important area of research with important practical and theoretical significance in fluid mechanics,optics,plasma physics,and military and communication engineering.In this paper,using multiscale analysis and a perturbation expansion method,starting from the quasi-geostrophic potential vortex equation,a new(2+1)-dimensional highorder nonlinear Schrodinger equation describing Rossby waves in stratified fluids is obtained.Based on this equation,conditions for the occurrence of modulational instability of Rossby waves are analyzed.Moreover,the effects of factors such as the dimension and order of the equation and the latitude at which Rossby waves occur on modulational instability are discussed.It is found that the(2+1)-dimensional equation provides a good description of the modulational instability of Rossby waves on a plane.The high-order terms affect the modulational instability,and it is found that instability is more likely to occur at high latitudes.

THE WAVE-MAKING CHARACTERISTICS OF A MOVING BODY IN A TWO-LAYER FLUID

The Wave-making characteristics of a moving body in a two-layer fluid with free surface is investigated numerically and experimentally. The numerical analysis is based on the modified layered boundary integral equation system. The wave characteristics on the free surface and interface generated by a moving sphere and an ellipsoid is numerically simulated in both finite depth and infinite depth of lower layer model. The numerical results of the sphere are compared with the analytical results for a dipole with the same velocity in a two-layer fluid of finite depth. The dependence of the wave systems and structures on the characteristic quantities is discussed. Three kinds of measurement techniques are used in model experiments on the internal waves generated by a sphere advancing in a two-layer fluid. The effects of the varying velocity and stratification on the wavelength, wave amplitudes and the maximum half angles of internal waves are analyzed qualitatively.

Effects of oceanic ice cover and density stratification on the propagation of hydroacoustic waves

The propagation characteristic of hydroacoustic waves is studied for an ideal compressible two-layer fluid with different densities covered by an elastic ice sheet.Boundary conditions are simplified by adopting linear assumption and then the dispersion relation is derived.The analysis and visualization of the dispersion relation present the introduction of compressibility leads to the appearance of hydroacoustic wave modes while the density stratification leads to the appearance of interfacial wave mode.Larger ice thickness and the density ratio of the two fluid layers increase the wave number and group speed of hydroacoustic waves at the same frequency while the phase speed decreases.