Internal gravity waves in critical generation modes and in the vicinity of trajectories of motion of perturbation sources

The field of internal gravity waves in a layer of an arbitrary stratified fluid is studied for critical generation modes and in the vicinity of trajectories of motion of the perturbation sources. The exact solutions describing the structure of a separate mode of the wave field in the vicinity of the perturbation source in the critical generation modes are investigated, and expressions for the total field representing the sum of all wave modes are obtained. In the vicinity of the trajectories of the perturbation sources, asymptotic representations of the eigenfunctions and eigenvalues of the basic vertical spectral problem of internal waves are constructed in the approximation of large wave numbers and asymptotic expressions for a separate mode of the wave field are obtained that describe the spatial structure and features of the fields of internal gravity waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 70–79, September–October, 2008.     

利用多维模态理论分析圆柱贮箱液体非线性晃动

将多维模态理论应用到求解作横向运动圆柱贮箱中液体的非线性晃动问题. 首先通过压力积分变分原理推导出描述液体作非线性晃动的一般形式无穷维模态系统,然后根据Narimanov-Moiseev三阶渐近假设关系,通过选取二阶主模态和三阶次模态,将无穷维模态系统降为五维渐近模态系统. 通过对这个模态系统的数值积分可以看出一些典型的非线性特征(如波峰大于波谷、节径移动等).      

Steady waves in a stratified flow over a combined obstacle

A problem of steady internal waves in subcritical flows of a stratified liquid over a finite sequence of bottom elevations is considered. An asymptotic solution of the second order of accuracy, which takes into account nonlinear effects, is constructed by the method of perturbations in terms of the small parameter of the obstacle height. Near-field interference wave structures are studied.     

Transition modes in stratified compressible fluids

Normal modes which exist in stratified compressible fluids are investigated. For the analysis, the conservation of the number of zeros in an eigenfunction is used. It is generally shown that the condition for transition modes such as Lamb-wave modes to exist is determined only by boundary conditions. This mathematical result is physically explained by boundary waves, and this explanation crucially depends on which is larger, gravity acceleration g or the product of Brunt–Väisälä frequency and sound speed Nc_s. This theory gives a guide to choose boundary conditions free of spurious boundary waves. It also explains why a distinct Lamb wave is not found in the ocean unlike in the atmosphere: it is simply because the ocean is not deep enough, but if the ocean were stratified a little more strongly than it is, the Lamb wave would not exist in the ocean however deep it might be.     

Short-wave modes for wave ducts in quasi-stratified media

Propagation of harmonic waves in three-dimensional ducts, with slowly varying properties in the horizontal directions, is treated. The wave length is assumed to be small in comparison with the width of the duct. A set of asymptotic solutions to the wave equation (generalisations of normal modes of horizontally homogeneous wave-guides) is constructed. The method extends the applicability region of the method of “vertical modes and horizontal rays” and gives a simple estimation of the latter. Oscillations bounded by the wave-guide's boundaries or by the caustics are treated, and a unified formal procedure for construction of appropriate asymptotic solutions is proposed. The vicinity of ‘horizontal caustics’ is investigated. The analysis is carried out for the examples of acoustic and internal gravitational waves in fluids.     

A long wave low frequency motion model for a finitely sheared elastic layer

We consider two-dimensional long wave low frequency motion in a pre-stressed layer composed of neo-Hookean material. Specifically, the pre-stress is a simple shear deformation. Derivation of the dispersion relation associated with traction-free boundary conditions is briefly reviewed. Appropriate approximations are established for the two associated long wave modes. From these approximations it is clear that there may be either two, one or no real long wave limiting phase speeds. These approximations are also used to establish the relative asymptotic orders of the displacement components and pressure increment. Using these relative orders to motivate the introduction of appropriate a scales, an asymptotically consistent model long wave low frequency motion is established. It is shown that in the presence of shear there is neither bending nor extension, or analogues of their previously established pre-stressed counterparts. In fact, both the in-plane and normal displacement components have the same asymptotic orders and the derived governing equation is of vector form.     

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.     

Transient wave propagation of isotropic plates using a higher-order plate theory

Transient wave propagation of isotropic thin plates using a higher-order plate theory is presented in this paper. The aim of this investigation is to assess the applicability of the higher-order plate theory in describing wave behavior of isotropic plates at higher frequencies. Both extensional and flexural waves are considered. A complete discussion of dispersion of isotropic plates is first investigated. All the wave modes and wave behavior for each mode in the low and high-frequency ranges are provided in detail. Using the dispersion relation and integral transforms, exact integral solutions for an isotropic plate subjected to pure impulse load and a number of wave excitations based on the higher-order theory are obtained and asymptotic solutions which highlight the physics of waves are also presented. The axisymmetric three-dimensional analytical solutions of linear wave equations are also presented for comparison. Results show that the higher-order theory can predict the wave behavior closely with exact linear wave solutions at higher frequencies.     

A new system of equations which predicts the evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption

The problem of the capillary-gravity waves which may arise at an interface between two stratified fluids of different densities is investigated. Particular attention is paid to the case when two different wave modes move at the same speed and to the wave train produced by the ensuing interaction. In contrast to most previous studies, the wave steepness and the wave bandwidth are not taken to be of the same order of magnitude, but the latter is of one order smaller. This leads to a system of nonlinear evolution equations which can be used to predict the subsequent progression of the wave field. These equations may be compared with the more usual nonlinear Schrödinger set which are valid under the equal bandwidth assumption and also a recently derived set which describe broader bandwidth waves. A large class of solutions to the equations is found and the corresponding wave profiles are presented.     

连续分层流体中垂直薄板的水动力特性

研究了在线性连续分层流体中水波与半潜式刚性垂直薄板相互作用的问题. 在 Boussinesq近似下，基于分离变量法，导出了具有自由面的平面前进波的色散关系，建立 了半潜式刚性垂直薄板的反射与透射能量、水平波浪力的计算方法. 对给定的频率，当它大 于浮力频率时，流场中只有一种模态的平面前进波，当它小于浮力频率时，流场 中有无数多个模态的平面前进波，并证明了对每一种模态的入射波，其它每个模态水波的反射与透射能量是 相等的. 对水面漂浮和座底半潜式薄板的反射与透射能量，以及作用在薄板上的水平波浪力 进行了数值计算分析，表明了流体的线性连续分层效应对这些水动力的影响是不可忽视 的. 特别地，在入射波频率小于浮力频率时，与第1模态入射波的能量转化量及其对薄板产 生的水平波浪力相比，其它模态入射波的能量转化量及其对薄板产生的水平波浪力都要大得 多.     

Analysis of micro-structural effects on phononic waves in layered elastic media with periodic nonsmooth coordinates

Propagation of elastic phononic waves in layered composite materials is analyzed by introducing nonsmooth periodic coordinates associated with structural specifics of the materials. Spatial scales of the original (smooth) coordinates are estimated by the wave lengths. In terms of the new coordinates, the homogenization procedure occurs naturally from the continuity conditions imposed on elastic displacements and forces at layer interfaces. As a result, higher-order asymptotic approximations describing spatiotemporal ‘macro’- and ‘micro’-effects of wave propagation are obtained in closed form. Such solutions provide visualizations for the wave shapes illustrating their structure induced local details. In particular, beat-wise mode shapes and effective anisotropy of acoustic wave propagation are revealed. The subharmonic beating in wave modes occur when wave lengths orthogonal to layers is about to ‘resonate’ with layer’ thickness. If the wave speed has a non-zero projection along the layers, then phase shifts between the beats are observed in different cross sections perpendicular to the layers.     

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.     

Vertical Momentum Fluxes Induced by Weakly Nonlinear Internal Waves on the Shelf

Free inertia-gravity internal waves in a two-dimensional stratified flow of an ideal fluid with a vertical velocity shear are considered in the Boussinesq approximation. The boundary-value problem for the amplitude of the vertical velocity of internal waves has complex coefficients; therefore, the wave frequency has an imaginary correction and the eigenfunction is complex. It is shown that the wave is weakly damped, the vertical wave momentum fluxes being nonzero and can be greater than the turbulent fluxes. The Stokes drift velocity component transverse to the direction of wave propagation is nonzero and less than the longitudinal component by an order of magnitude. The dispersion curves of the first two modes are cut off in the low-frequency domain due to the influence of critical layers in which the wave frequency taken with the Doppler shift is equal to the inertial frequency.     

Wave interaction in a nonequilibrium gas flow

By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of non-resonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(ε^-1), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived.     

Free Oscillations of a Thin Luid Layer of Finite Electric Conductivity under the Action of an External Magnetic Field

The problem of free oscillations of a thin layer of a heavy, incompressible, inviscid fluid of finite electrical conductivity in a horizontal magnetic field is reduced to a system of integrodifferential Fredholm equations with variable coefficients. A numerical analysis is performed over a broad range of input parameters, and the results obtained are supplemented with asymptotic formulas with large and small magnetic Reynolds numbers. A classification of the resulting wave modes is proposed. It is shown that certain conditions can lead to the occurrence of unstable oscillations of the fluid layer that grow in time.     

The coupling between turbulent,penetrative convection and internal waves

Experiments aimed at exploring the coupling of penetrative convection with internal waves in the adjoining, stable layer were performed in a long convection cell. The experiments are motivated by preliminary theoretical results suggesting that an intrinsic phase instability may exist in the coupled system in which case long internal waves modulate the height and strength of convective plumes. Using a temperature-controlled, stably stratified experimental apparatus, measured temperature data reveal the presence of long internal wave modes that persist for many convective time scales. The frequencies of these waves increase linearly in time during the energy transfer between the convective and stratified regions as the depth of the stratified region diminishes and the depth of the mixed layer increases. Temporal variations in the heat flux, interface rise characteristics, and frequencies of internal wave motions are reported. A natural temporal modulation of the thickness of the transition layer separating the mixed layer from the stratified layer occurs following commencement of heating, with the amplitude and frequency of the modulation varying with the initial stratification. Temperature variance data suggest that a fairly strong interaction between convection and internal waves occurs, especially when the interface region is midway between the upper and lower boundaries of the cell and the no-slip boundary conditions play a less influential role on the dynamics of the coupling.     

Propagation and localization of Rossby waves over random topography (two-layer model)

We study propagation of Rossby waves over randomly stratified bottom topography in a two-layer system. The problem is reduced to coupled stochastic wave equations in latitudinal variable with suitable boundary conditions. When the two-dimensional system is broken into its basic barotropic and baroclinic modes, each one is scattered by the medium, and generates the like-wise and the cross-wise components. We study statistics of the reflection and transmission coefficients for both types of generated modes, and show localization for the like-wise components, and propagation for the cross-wise components. The localization lengths and the transmission rates are estimated in terms of the basic parameters of the system and the correlation-function of topographic fluctuations.     

Buoyancy-Opposed Darcy’s Flow in a Vertical Circular Duct with Uniform Wall Heat Flux: A Stability Analysis

An analytical and numerical study is presented to show that buoyancy-opposed mixed convection in a vertical porous duct with circular cross-section is unstable. The duct wall is assumed to be impermeable and subject to a uniform heat flux. A stationary and parallel Darcy’s flow with a non-uniform radial velocity profile is taken as a basic state. Stability to small-amplitude perturbations is investigated by adopting the method of normal modes. It is proved that buoyancy-opposed mixed convection is linearly unstable, for every value of the Darcy–Rayleigh number, associated with the wall heat flux, and for every mass flow rate parametrised by the Péclet number. Axially invariant perturbation modes and general three-dimensional modes are investigated. The stability analysis of the former modes is carried out analytically, while general three-dimensional modes are studied numerically. An asymptotic analytical solution is found, suitable for three-dimensional modes with sufficiently small wave number and/or Péclet number. The general conclusion is that the onset of instability selects the axially invariant modes. Among them, the radially invariant and azimuthally invariant mode turns out to be the most unstable for all possible buoyancy-opposed flows.     

Shock waves in reactive hydrodynamics

Using the weakly non-linear geometrical acoustics theory, we obtain the small amplitude high frequency asymptotic solution to the basic equations in Eulerian coordinates governing one dimensional unsteady planar, spherically and cylindrically symmetric flow in a reactive hydrodynamic medium. We derive the transport equations for the amplitudes of resonantly interacting waves. The evolutionary behavior of non-resonant wave modes culminating into shock waves is also studied.      

Natural oscillations of a liquid of finite electrical conduction in the presence of an external magnetic field

The effect of the finite electrical conduction (finiteness of the magnetic Reynolds number), which is considered a dissipative factor, on small natural oscillations of an ideal heavy liquid of finite depth whose free surface borders on vacuum is studied. A constant external horizontal magnetic field is applied to the liquid. The energy-balance equation is derived, and the theorem of wave attenuation with time is proved. Numerical calculations and the resulting asymptotic formulas give a complete pattern of the spectrum, including its continuous part. The amplitude-frequency characteristics of the wave modes are presented. Rostov State University, Rostov-on-Don 344007. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 3–10, March–April, 2000.