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.     

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.     

Asymptotic analysis of transition to turbulence and chaotic advection in shear zonal flows on a beta-plane

The generation of narrow-band Rossby wave packets and the modulated vortex chains induced by them in a weakly-dissipative zonal flow on the beta-plane with a velocity profile in the form of a shear layer is studied. The analysis is performed within the framework of the asymptotic approach based on the distinguishing a thin critical layer inside of which the vortex chains are formed. The evolution equations, describing the simultaneous development of a wave packet envelope and vorticity perturbations in a nonlinear critical layer, are derived for a weakly supercritical flow. A transition to the complex dynamics of a wave packet (low-mode turbulence) is studied within the framework of a numerical solution of the derived equations and its mechanism is revealed. The onset of chaotic advection and anomalous diffusion of passive scalar in the critical layer is considered, and the exponent of the diffusion law is calculated.     

On the quasi-geostrophic dynamics of a stratified two-component medium: The ocean

The quasi-geostrophic dynamics of a stratified two-component medium (salty sea water) are described by formulating a closed system of equations containing the temperature and salinity among the sought field variables. The corresponding system consists of the conservation laws of two Lagrange invariants, namely, the quasi-geostrophic potential vorticity and some “thermodynamic” invariant playing the role of a passive tracer. The temperature and salinity fields determined from the values of these invariants are separated into the density and density-compensated parts. In this case, the density part participates immediately in the dynamics, while the density-compensated part is simply transferred by the geostrophic velocity filed with no contribution to the density field. The system thus formulated is used to describe a number of specific features in the dynamics of thermohaline disturbances in zonal geostrophic flows. These features include the sharp-ening of the spatial gradients of the density-compensated distributions in shear currents and breaking of the initial temperature disturbance into two (density and density-compensated) wave packets propagating with different velocities.     

Trapped quasi-inertial waves in shear oceanic currents

We studied trapped long quasi-inertial waves in horizontally inhomogeneous flows with low Rossby numbers. A simple heuristic derivation of two equations for the wave amplitude is presented. These equations are true for strong and weak density stratifications. A spectral problem is formulated to find the frequencies of trapped waves based on the amplitude equations. Exact solutions of the hyperbolic problem for a free hyperbolic shear layer are found. It is shown that the location of the trapping area principally depends on the stratification. If the buoyancy frequency is greater than the inertial frequency, trapping occurs in the region of anticyclonic velocity shear; if the buoyancy frequency is smaller than the inertial frequency, trapping occurs in the region of cyclonic velocity shear. Thus, in the first case, the frequencies of the trapped waves are smaller than the inertial frequency, while, in the second case, they are greater. The intense wave activity observed in the regions of oceanic fronts and jet currents can be related to the existence of trapped waves.     

Linear dynamics of Eady waves in the presence of horizontal shear

The problem of the instability of the flow of a stratified rotating fluid with constant vertical and horizontal shears is investigated within the framework of a quasi-geostrophic approximation. It is shown that the horizontal shear, when taken into account, leads to a qualitative change in the dynamics of Eady waves, i.e., wave solutions with zero potential vorticity. The main salient feature is related to the effect of the temporary exponential growth of the unstable waves, i.e., the growth effect in a finite time interval. This effect is manifested by an alternation of the stages of a smooth oscillating behavior (in time) with an exponential (explosive) growth of finite duration. A kinematic interpretation of the effect of the temporary exponential growth is suggested which is associated with the passage of a time-dependent wave disturbance vector across the domain of the exponential instability existing in the absence of horizontal shear. Along with the dynamics of individual Eady waves, the generation process of these waves—caused by an initial disturbance defined by one spatial Fourier harmonic—is also investigated. It is shown that this process is accompanied by the formation of nonmodal waves, with time-varying horizontal and vertical wavenumbers and nonzero potential vorticity. The interaction of the nonmodal wave with the background flow leads to an algebraic growth of the Eady wave at the initial cyclogenesis stage.     

Resonant Excitation of Baroclinic Waves in the Presence of Ekman Friction

The quasi-geostrophic dynamics of disturbances of a flow with a vertical shear is described by a transfer equation for potential vorticity. Wave solutions of this equation are represented by edge baroclinic waves (modes in a discrete spectrum) and singular modes in a continuous spectrum. When frequencies of these modes coincide, the effect of resonant excitation occurs in which the amplitude of baroclinic waves increases linearly. This paper studies this effect in the presence of Ekman bottom friction. It is shown that friction suppresses linear wave growth and gives rise to baroclinic waves of finite amplitude.     

A New Class of Edge Baroclinic Waves and the Mechanism of Their Generation

Edge baroclinic waves are generated in a geostrophic flow with a vertical shear near a solid surface. The study investigates a new class of baroclinic waves in flows with horizontal and vertical shears and a linear distribution of potential vorticity. It is shown that taking account of the horizontal shear leads to the appearance of new features of wave dynamics. These include the nonmodal growth of energy in the initial stage of development, the time dependence of the vertical wave scale, and the possibility of generation of stationary or blocked waves. The horizontal shear makes the mechanism of generation of baroclinic waves by initial vortex perturbations more efficient. One important feature is associated with vortex paths, which are formed by the superposition of a baroclinic wave on the flow with horizontal shear.     

Self-localization of planetary wave structures in the ionosphere upon interaction with nonuniform geomagnetic field and zonal wind

The generation and further linear and nonlinear dynamics of planetary magnetized Rossby waves (MRWs) in the rotating dissipative ionosphere are studied in the presence of a zonal wind (shear flow). MRWs are caused by interaction with the spatially nonuniform geomagnetic field and are ionospheric manifestations of ordinary tropospheric Rossby waves. A simplified self-consistent set of model equations describing MRW-shear flow interaction is derived on the basis of complete equations of ionospheric magnetohydrodynamics. Based on an analysis of an exact analytical solution to the derived dynamic equations, an effective linear mechanism of MRW amplification in the interaction with nonuniform zonal wind is ascertained. It is shown that operators of linear problems are non-self-adjoint in the case of shear flows, and the corresponding eigenfunctions are nonorthogonal; therefore, the canonically modal approach is of little use when studying such flows; a so-called nonmodal mathematical analysis is required. It is ascertained that MRWs effectively get shear flow energy during the linear stage of evolution and significantly increase (by several orders of magnitude) their energy and amplitude. The necessary and sufficient condition of shear flow instability in an ionospheric medium is derived. Nonlinear self-localization begins with the development of shear instability and an increase in the amplitude, and the process ends with the self-organization of strongly localized isolated large-scale nonlinear vortex structures. Thus, a new degree of freedom and a way for perturbation evolution to occur appear in medium with shear flow. The nonlinear systems can be a pure monopole vortex, a vortex streets, or vortex chains depending of the shape of the sheared flow velocity profile. The accumulation of such vortices in the ionospheric medium can produce a strongly turbulent state.     

Mass transport in a two-layer wave boundary layer

The transport of a chemical species under the pure action of surface progressive waves in the benthic boundary layer which is loaded with dense suspended sediments is studied theoretically. The flow structure of the boundary layer is approximated by that of a two-layer Stokes boundary layer with a sharp interface between clear water and a heavy fluid. The simplest model of constant eddy diffusivities is adopted and the exchange of matter with the bed is ignored. For a thin layer of heavy fluid, whose thickness is comparable to the surface wave amplitude and the Stokes boundary layer thickness, effective transport equations are deduced using an averaging technique based on the method of homogenization. The effective advection velocity is found to be equal to the depth-averaged mass transport velocity, while the dispersion coefficient can be shown to be positive definite. Explicit expressions for the transport coefficients are obtained as functions of fluid properties and flow kinematics. Physical discussions on their relations are also presented.     

Study on dynamics of shear waves——Ⅰ. Scale analysis and dispersion relation

- Starting from satellite remote sensing data, the dynamical processes of shear waves occurring at the boundary between the western boundary current and the shelf slope water are studied and dynamically analyzed in this study. The average wavelength is 75 km, and the average amplitude (from crest to trough )17 km. the average phase speed 100 cms-1 for the shear waves along the north wall of the Gulf Stream to the east of Cape Hatteras measured from NOAA satellite IR (infrared ) images. The average wavelength of shear waves along the north wall of the Kuroshio Current is 57 km, and the average amplitude 17 km. For the shear waves occurring along the west wall of the Gulf Stream to the south of Cape Hatteras, the average wavelength is 131 km, and the average amplitude 33 km measured from Seasat SAR (synthetic aperture radar )images. The time for one cycle of shear wave event is about one week.In order to explore the dynamical mechanisms of shear waves, we solved the vorticity equation for a stratified flu     

Stabilization of a linearly unstable wave participating in a three-wave resonant interaction

An analytical study of the influence of three-wave resonant interactions on the evolution of unstable wave disturbances is presented in the Kelvin-Helmholtz model. These results may be of interest in analyzing the dynamics of disturbances at the ocean-atmosphere interface and in two-layer flows which arise in the ocean and are characterized by large gradients of flow velocity at the boundary of layers. In the case under consideration, the instability arises when eigenfrequencies coincide in the framework of a single mode and the instability is algebraic. The amplitudes of the two other interacting stable waves are assumed to be small compared to the amplitude of the third, unstable, mode. The system of amplitude equations for this case is investigated using the WKB method. As a result, we obtain the formulas coupling the solutions for the time before and after a transition through a singular point, where the amplitude of the linearly unstable wave has a local minimum. These formulas give the rule of transformation of the parameter that characterizes a phase shift between fast and slow modes and determines the behavior of the system. It is shown that, in a transition through a singular point, this parameter changes randomly. As long as the parameter is positive, the amplitude of the linearly unstable wave remains limited and oscillates stochastically. In a transition of the parameter through zero, we exit the stabilization region and have an infinite growth of amplitude. The transition into the instability region is random. However, if the time interval where the amplitude remains limited is large enough, the scenario of the behavior of the system we have obtained can be treated as the partial stabilization of instability. The results make it possible for us to investigate the stochasticity caused by the nonlinear interaction of gravity-capillary waves in a two-layer model of a shear flow. These results are also of interest in analyzing secondary flows in laboratory facilities modeling the ocean and atmospheric processes.     

Numerical simulation of wave-induced laminar boundary layers

The three-dimensional numerical model with σ-coordinate transformation in the vertical direction is applied to the simulation of surface water waves and wave-induced laminar boundary layers. Unlike most of the previous investigations that solved the simplified one-dimensional boundary layer equation of motion and neglected the interaction between boundary layer and outside flow, the present model solves the full Navier–Stokes equations (NSE) in the entire domain from bottom to free surface. A non-uniform mesh system is used in the vertical direction to resolve the thin boundary layer. Linear wave, Stokes wave, cnoidal wave and solitary wave are considered. The numerical results are compared to analytical solutions and available experimental data. The numerical results agree favorably to all of the experimental data. It is found that the analytical solutions are accurate for both linear wave and Stokes wave but inadequate for cnoidal wave or solitary wave. The possible reason is that the existing analytical solutions for cnoidal and solitary waves adopt the first-order approximation for free stream velocity and thus overestimate the near bottom velocity. Besides velocity, the present model also provides accurate results for wave-induced bed shear stress.     

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.     

Internal flow structure of short wind waves

Characteristic features of the internal flow field of short wind waves are described mainly on the basis of streamline patterns measured for four different cases of individual wave. In some waves a distinct high vorticity region, with flow in excess of the phase speed in the surface thin layer, is formed near the crest as shown in Part I of this study, but the streamlines are found to remain quite regular even very near the water surface. The characteristics of flow in the high vorticity region are investigated, and it is argued that the high vorticity region is not supported steadily in individual waves but that growth and attenuation in individual waves repeats systematically, without no severe wave breaking. Below the surface vorticity layer a quite regular wave motion dominates. However, this wave motion is strongly affected by the presence of the high vorticity region. By comparing the measured streamline profiles with those predicted from wave profiles by the use of a water-wave theory, it is found that the flow of the wind waves studied cannot be predicted, even approximately, from the surface displacements, in contrast to the case of pure irrotational water waves.     

实验室造波条件对内孤立波发展影响的直接数值模拟

内孤立波具有振幅尺度大、能量集中的特点,其引起流场和密度场的迅速变化可能对海洋工程结构物以及水下潜体造成严重威胁.因此研究不同造波条件下生成的内孤立波运动的流场特征具有重要的学术意义和实际应用价值.采用直接数值模拟方法和给定的初始密度场密度跃迁函数,对重力塌陷激发内孤立波的运动过程进行研究,探讨了不同造波条件下,激发产...     

SPH方法的孤立波与部分淹没结构物相互作用数值计算研究

为研究孤立波作用下结构物周围流场特征,基于无网格SPH方法,建立孤立波与海洋结构物相互作用模型,对不同波幅孤立波作用下部分淹没矩形结构物周围波面、流速、涡量及结构受力特征进行计算分析,探索了相对波高对非淹没结构物周围流场的影响规律。结果表明:流场特征与相对波高密切相关,相对波高较小时,波面、流速、涡量及结构荷载均较为光滑,相对波高在0.2以上时,波峰爬升至结构物顶部并在越过结构物后与水槽内水体碰撞造成流场波动,波面、流速、涡量及结构荷载的波动幅度随着相对波高增大而增大,流场更加复杂,结构物水平和垂向负压也越大,且结构物周围涡分布逐渐向深度方向和下游方向发展。     

The mixing mechanism in the formation of ocean shear waves

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Strong near-inertial oscillations in geostrophic shear in the northern South China Sea

With observational data from three Acoustic Doppler Current Profiler (ADCP) moorings, we detected strong near-inertial oscillations (NIO) in the continental shelf region of the northern South China Sea in July 2008. The amplitude of the near-inertial current velocity is much greater than that of diurnal and semi-diurnal tides. The power of the NIOs is strongest in the intermediate layer, relatively weak in the surface layer, and insignificant in the near-bottom layer. The spectral analysis indicates that the NIOs have a peak frequency of 0.0307 cph, which is 2% lower than the local inertial frequency, i.e., a red-shift. The near-inertial wave has an upward vertical phase velocity, which involves a downward group velocity and energy flux. The estimated vertical phase velocity is about 43 m day⁻¹, corresponding to a vertical wave length of about 58 m. The horizontal scale of the NIOs is at least hundreds of kilometers. This NIO event lasted for about 15 days after a typhoon’s passage. Given the northeastward background flow with significant horizontal shear, both Doppler shift and shear flow modulation mechanisms may be responsible for the red-shift of the observed NIOs. For the shear flow mechanism, the observed negative background vorticity and the corresponding effective Coriolis frequency reduce the lower limit of admissible frequency band for the NIOs, causing the red-shift. Meanwhile, the mooring area with the broadened frequency band acts as a wave-guide. The trapping and amplification effects lead to the relatively long sustaining period of the observed NIOs.     

On Instability of Geostrophic Current with Linear Vertical Shear at Length Scales of Interleaving

The instability of long-wave disturbances of a geostrophic current with linear velocity shear is studied with allowance for the diffusion of buoyancy. A detailed derivation of the model problem in dimensionless variables is presented, which is used for analyzing the dynamics of disturbances in a vertically bounded layer and for describing the formation of large-scale intrusions in the Arctic basin. The problem is solved numerically based on a high-precision method developed for solving fourth-order differential equations. It is established that there is an eigenvalue in the spectrum of eigenvalues that corresponds to unstable (growing with time) disturbances, which are characterized by a phase velocity exceeding the maximum velocity of the geostrophic flow. A discussion is presented to explain some features of the instability.