Xiaoping Xu
In this article, we introduce a moving-frame approach to the geophysical equation of two-dimensional uniformly stratified rotational fluid in oceans and find a family of exact solutions containing ten arbitrary parameter functions.
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In this paper we consider fundamental processes of the disturbance and propagation of internal gravity waves in the ocean modeled as a vertically stratified, horizontally non-uniform, and non-stationary medium. We develop asymptotic methods for describing the wave dynamics by generalizing the spatiotemporal ray-tracing method (a geometrical optics method). We present analytical and numerical algorithms for calculating the internal gravity wave fields using actual ocean parameters such as physical characteristics of the sea water, topography of its floor, etc. We demonstrate that our mathematic…
In this paper, we consider a fundamental problem of describing the dynamics of internal gravity waves in the stratified ocean with shear flows. We develop an asymptotic representation of the wave fields in terms of the Green's functions. We explore the far field of the internal gravity waves generated by disturbing sources, and propose asymptotic algorithms for calculating its dynamics.
The high-frequency limit of the Garrett and Munk spectrum of internal waves in the ocean and the observed deviations from it are shown to form a pattern consistent with the predictions of wave turbulence theory. In particular, the high frequency limit of the Garrett and Munk spectrum constitutes an {\it exact} steady state solution of the corresponding kinetic equation.
Hamiltonian description of internal ocean waves with Coriolis force
本文研究了受科里奥利力影响的海洋内部波传播，通过将无旋情形的哈密顿表述扩展为近哈密顿表述推导模型方程，在长波和中等长波机制下获得类似Ostrovsky方程的模型，描述二维水平波传播。
In this article, we propose an inverse technique that accurately reconstructs the ocean's density stratification profile simply from free surface elevation data. Satellite observations suggest that ocean surface contains the signature of internal tides, which are internal gravity waves generated by the barotropic tides. Since internal tides contain the information of ocean's density stratification, the latter can in principle be reconstructed from the free surface signature. We consider a simple theoretical model that approximates a continuously stratified ocean as discrete layers of constant …
The paper is devoted to the presentation of the non-spectral methods of analysis of the natural measurements of the internal gravity waves in the ocean with the purpose to determine characteristics of the wave-trains composing the measured field, the forms and parameters of the ocean along the pass of these wave-trains propagation. The problem of the analysis of the data of the natural measurements of the internal waves with the purpose to separate the single wave-trains from the measured field, and on this basis to determine the characteristics of these wave-trains, to receive the information…
We analyze analytically and numerically the scale invariant stationary solution to the internal wave kinetic equation. Our analysis of the resonant energy transfers shows that the leading order contributions are given (i) by triads with extreme scale separation and (ii) by triads of waves that are quasi-colinear in the horizontal plane. The contributions from other types of triads is found to be subleading. We use the modified scale invariant limit of the Garrett and Munk spectrum of internal waves to calculate the magnitude of the energy flux towards high wave numbers in both the vertical and…
Interacting internal waves explain global patterns of interior ocean mixing
本文利用波-波相互作用理论并结合局地与全球观测，首次从第一性原理解释了全球海洋内部内波破碎驱动的湍流混合模式，为海洋环流模式提供了完全基于物理的波驱动混合参数化基础。
A Versatile Laboratory Approach to Reproduce and Analyze Internal Ocean Wave Dynamics
本文介绍一种适用于本科生实验的装置，展示海洋中常见波生成与破碎机制，并通过改变浮力雷诺数探讨粘性与惯性/浮力力的作用。
A One-Dimensional Model for Investigating Scale-separated Approaches to the Interaction of Oceanic Internal Waves
本文提出一维模型研究海洋内波尺度分离相互作用，通过对比射线追踪数值与波湍流理论，发现海洋振幅下能量传输减弱但仍大于观测，支持用局部相互作用范式取代极端尺度分离范式。
Generalized Transport Characterizations for Short Oceanic Internal Waves in a Sea of Long Waves
针对海洋内波三波相互作用范式争议，利用波湍流动能方程与WKB技术推导两种Fokker-Planck方程，发现谱输运估计因欧拉与射线路径描述差异而相差一个数量级。
Starting from the hydrostatic Boussinesq equations, we derive a time-averaged `hydrostatic wave equation' that describes the propagation of inertia-gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-time-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes that the wave field has a well-defined and non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, neglects nonlinear wave-wave interactions and makes no restriction on either the background density stratification or the rela…
We present experimental and computational studies of the propagation of internal waves in a stratified fluid with an exponential density profile that models the deep ocean. The buoyancy frequency profile $N(z)$ (proportional to the square root of the density gradient) varies smoothly by more than an order of magnitude over the fluid depth, as is common in the deep ocean. The nonuniform stratification is characterized by a turning depth $z_c$, where $N(z_c)$ is equal to the wave frequency $ω$ and $N(z < z_c) < ω$. Internal waves reflect from the turning depth and become evanescent below the tur…
The condition of internal gravity waves (IGW) parametric excitation in the rotating fluid layer heated from above, with the layer vibration along the vertical axis or with periodic modulation in time of the vertical temperature distribution, is obtained. We show the dual role of the molecular dissipative effects that may lead not only to the wave oscillations damping, but also to emergence of hydrodynamic dissipative instability (DI) in some frequency band of IGW. This DI also may take place for the localized in horizontal plane tornado-like disturbances, horizontal scale of which does not exc…
Modifications of SPH towards three-dimensional simulations of an icy moon with internal ocean
本文改进SPH方法，通过修正黏性、采用DISPH、实现表面辐射冷却并引入相变状态方程，以支持含内部海洋冰卫星的三维潮汐加热与演化模拟。
An unstructured-grid, nonhydrostatic, GVC ocean model Part I: Model description and application of vertical hybrid coordinates to internal solitary waves
本文介绍了一款整合广义垂直坐标（GVC）的非结构化网格非静水海洋模型，开发了适用于非结构化网格的保守动量平流与保正层高度格式，通过模拟验证了其有效性并展现了混合垂直坐标的优势。
Oceanic internal solitary wave interactions via the KP equation in a three-layer fluid with shear flow
本研究利用KP方程解析三层含剪切流流体中的二维内孤立波，分类五种相互作用类型，发现亮暗模式由剪切流主导，并基于卫星图像验证其存在。
Induced Diffusion of Internal Gravity Waves: Directionality and Role in Ocean Mixing
本研究重新审视了内重力波诱导扩散（ID）机制，发现其包含扩散和尺度分离传递，且尽管作用量扩散方向可变，合并的ID传递始终导致正向能量级联，支持ID作为海洋湍流耗散机制的观点。
Oceanic internal tides: do they get phased at the Equator?
本文研究了赤道喷流与内部潮汐波包的相互作用，探讨了其对相位一致性的影响。
The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg--de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the non…
Testing the consistency of the resonant wave interaction approximation with simulated dynamics of idealized 2D internal wave fields
本文通过理想化二维数值模拟检验共振波相互作用近似的有效性，发现其假设与模拟动力学不一致，部分共振条件下的三元组不显著贡献非线性波能传输。
A fully nonlinear, three-dimensional numerical model is developed for the simulation of tidal flow over arbitrary bottom topography in an ocean with realistic stratification. The model is capable of simulating accurately the generation of fine-scale internal wave tidal beams, their interaction with an ocean thermocline and the subsequent generation of solitary internal waves that propagate on this thermocline. Several preliminary simulation results are shown for uniform and non-uniform flow over an idealized two-dimensional ridge, which are compared with linear theory, and for flow over an ide…
To date, the influence of non-linear stratifications and two layer stratifications on internal wave propagation has been studied for two-dimensional wave fields in a cartesian geometry. Here, we use a novel wave generator configuration to investigate transmission in non-linear stratifications of axisymmetric internal wave. Two configurations are studied, both theoretically and experimentally. In the case of a free incident wave, a transmission maximum is found in the vicinity of evanescent frequencies. In the case of a confined incident wave, resonant effects lead to enhanced transmission rate…
The simulation of large open water surface is challenging using a uniform volumetric discretization of the Navier-Stokes equations. Simulating water splashes near moving objects, which height field methods for water waves cannot capture, necessitates high resolutions. Such simulations can be carried out using the Fluid-Implicit-Particle (FLIP) method. However, the FLIP method is not efficient for the long-lasting water waves that propagate to long distances, which require sufficient depth for a correct dispersion relationship. This paper presents a new method to tackle this dilemma through an …
A new formulation of metriplectic dynamics with an application to quasigeostrophic ocean modeling with advected quantities
提出一种通过两个斜对称括号乘积构造度量-辛四括号的新表述，并将其用于含平流量的广义二维准地转上层海洋模型引入不可逆性，确保能量守恒和熵增。
To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we prov…
Higher-order integrable models for oceanic internal wave-current interactions
本研究针对海洋内波-流相互作用问题，推导了含剪切流与赤道科里奥利力的高阶KdV方程，证明其与多种经典可积方程存在显式变换关系。
This study starts with balances deduced by Baumert and Peters (2004, 2005) from results of stratified-shear experiments made in channels and wind tunnels by Itsweire (1984) and Rohr and Van Atta (1987), and of free-decay experiments in a resting stratified tank by Dickey and Mellor (1980). Using a modification of Canuto's (2002) ideas on turbulence and waves, these balances are merged with an (internal) gravity-wave energy balance presented for the open ocean by Gregg (1989), without mean-flow shear. The latter was augmented by a linear (viscous) friction term. Gregg's wave-energy source is in…
Internal gravity wave energy contributes significantly to the energy budget of the oceans, affecting mixing and the thermohaline circulation. Hence it is important to determine the internal wave energy flux $\mathbf{J} = p \, \mathbf{v}$, where $p$ is the pressure perturbation field and $\mathbf{v}$ is the velocity perturbation field. However, the pressure perturbation field is not directly accessible in laboratory or field observations. Previously, a Green's function based method was developed to calculate the instantaneous energy flux field from a measured density perturbation field $ρ(x,z,t…
This paper deals with the dead-water phenomenon, which occurs when a ship sails in a stratified fluid, and experiences an important drag due to waves below the surface. More generally, we study the generation of internal waves by a disturbance moving at constant speed on top of two layers of fluids of different densities. Starting from the full Euler equations, we present several nonlinear asymptotic models, in the long wave regime. These models are rigorously justified by consistency or convergence results. A careful theoretical and numerical analysis is then provided, in order to predict the…
We provide a first-principles analysis of the energy fluxes in the oceanic internal wavefield. The resulting formula is remarkably similar to the renowned phenomenological formula for the turbulent dissipation rate in the ocean which is known as the Finescale Parameterization. The prediction is based on the wave turbulence theory of internal gravity waves and on a new methodology devised for the computation of the associated energy fluxes. In the standard spectral representation of the wave energy density, in the two-dimensional vertical wavenumber - frequency domain, the energy fluxes associa…
Parametric Subharmonic Instability in the Ocean Bottom Boundary Layer
本文通过线性稳定性分析和非线性模拟，研究了海洋底边界层中参数次谐波不稳定性（PSI）的发生机制、增长率及其对近底混合的贡献。
Experimental Investigation of Tidally-Forced Internal Wave Turbulence at High Reynolds Number
本文在大型实验室波槽中模拟潮汐流经海脊地形，利用背景导向纹影法实验研究高雷诺数下内波湍流的完整能量级联路径，揭示了三波共振不稳定性驱动的次谐波生成、频率反向级联与垂直波数正向级联，并强调了局域三波相互作用的重要性。
The energy pathways from propagating internal waves to the scales of irreversible mixing in the ocean are not fully described. In the ocean interior, the triadic resonant instability is an intrinsic destabilization process that may enhance the energy cascade away from topographies. The present study focuses on the integrated impact of mixing processes induced by a propagative normal mode-1 over long term experiments in an idealised setup. The internal wave dynamics and the evolution of the density profile are followed using the light attenuation technique. Diagnostics of the turbulent diffusiv…
We attempt to gain some insight into the modeling of the generation of internal waves produced by submarines traveling in the littoral regions of the ocean with the use of high fidelity numerical simulations. These numerical simulations are shown to be capable of simulating high Reynolds number flow around bodies, including the effects of stable stratification. In addition, we use the results of these detailed numerical studies to test and revise the source distribution parameterizations of the near-field waves that have been used in analytical studies based on linear theory. Such parameteriza…
In this paper, we present the first laboratory experiments that show the generation of internal solitary waves by the impingement of a quasi-two-dimensional internal wave beam on a pycnocline. These experiments were inspired by observations of internal solitary waves in the deep ocean from synthetic aperture radar (SAR) imagery, where this so-called mechanism of 'local generation' was argued to be at work, here in the form of internal tidal beams hitting the thermocline. Nonlinear processes involved here are found to be of two kinds. First, we observe the generation of a mean flow and higher h…
From internal waves to turbulence in a stably stratified fluid
本研究在大型 Coriolis 设施中实验模拟了海洋亚中尺度潮汐驱动的分层湍流，发现强非线性内波的破碎触发了小尺度强湍流，且观测到的能谱应归因于强非线性内波而非弱波湍流。
In this paper, we consider fundamental problems of the dynamics of internal gravity waves. We present analytical and numerical algorithms for calculating the wave fields for a set of values of the parameters, as observed in the ocean. We show that our mathematical models can describe the wave dynamics of the Arctic Basin, taking into account the actual physical characteristics of sea water, topography of its floor, etc. The numerical and analytical results show that the internal gravity waves have a significant effect on underwater sea objects in the Arctic Basin.
Wave turbulence of inertia--gravity waves: a theory for the oceanic spectrum
本文提出了一种基于动能波理论的海洋内部波湍流理论，推导出适用于所有频率的Garrett-Munk谱，并揭示了科氏参数对非旋转问题的正则化作用。
A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to th…
The adiabatic decay of different types of internal wave solitons caused by the Earth rotation is studied within the framework of the Gardner-Ostrovsky equation. The governing equation describing such processes includes quadratic and cubic nonlinear terms, as well as the Boussinesq and Coriolis dispersions. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth rotation. The characteristic decay time is derived for different types of solitons for positive and negative coefficient of cubic nonlinearity (both signs of that parameter may occur i…
The computation of long wave propagation through the ocean obviously depends on the initial condition. When the waves are generated by a moving bottom, a traditional approach consists in translating the ``frozen'' sea bed deformation to the free surface and propagating it. The present study shows the differences between the classical approach (passive generation) and the active generation where the bottom motion is included. The analytical solutions presented here exhibit some of the drawbacks of passive generation. The linearized solutions seem to be sufficient to consider the generation of w…
Turbulent spectrum of 2D internal gravity waves
利用完整非静力色散关系求得弱相互作用二维内重力波的湍流能谱，通过剔除零频剪切模得到正则化动力学方程的精确解，该解与海洋Garrett--Munk谱吻合，为各向异性系统的湍流能量级联提供了新见解。
Recent developments of the weak turbulence theory applied to internal waves exhibit a power-law solution of the kinetic energy equation close to the oceanic Garrett \& Munk spectrum, confirming weakly nonlinear wave interactions as a likely explanation of the observed oceanic spectra. However, finite-size effects can hinder wave interactions in bounded domains, and observations often differ from theoretical predictions. This article studies the dynamical regimes experimentally developing in a stratified fluid forced by internal gravity waves in a pentagonal domain. We find that by changing the…
The influence of forward speed on stochastic free-surface crossing, in a Gaussian wave field, is investigated. The case of a material point moving with a constant forward speed is considered; the wave field is assumed stationary in time, and homogeneous in space. The focus is on up-crossing events, which are defined as the material point crossing the free surface, into the water domain. The effect of the Doppler shift (induced by the forward speed) on the up-crossing frequency, and the related conditional joint distribution of wave kinematic variables is analytically investigated. Some general…
Characterizing Ocean Flows with the Scattering Transform
针对功率谱无法区分海洋多尺度运动的问题，提出利用小波卷积的散射变换（ST）方法提取流场几何信息，有效区分平衡动力学、内波与湍流，为卫星测高及雷达空间图分析提供了新框架。
Weakly nonlinear internal wave-wave interaction is a key mechanism that cascades energy from large to small scales, leading to ocean turbulence and mixing. Oceans typically have a non-uniform density stratification profile; moreover, submarine topography leads to a spatially varying ocean depth ($h$). Under these conditions and assuming mild-slope bathymetry, we employ multiple-scale analysis to derive the wave amplitude equations for triadic- and self-interactions. The waves are assumed to have a slowly (rapidly) varying amplitude (phase) in space and time. For uniform stratifications, the ho…
Effects of high-frequency and balanced motions on Lagrangian pair dispersion at the ocean surface
研究全球海洋模拟中拉格朗日粒子对的扩散，发现高频惯性重力波不影响相对扩散，后者由准平衡流主导，并指出SWOT卫星测地速估算表面扩散有局限性。
We have found that in two-dimensional Kolmogorov flow a single spatially-localized turbulence (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmet…
One of the pivotal questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose internal wave attractors in the large amplitude regime as a unique self-consistent experimental and numerical setup that models a cascade of triadic interactions transferring energy from large-scale monochro-matic input to multi-scale internal wave motion. We also provide signatures of a discrete wave turbulence framework for internal waves. Finally, we show how beyond this regime, we have a clear transition to a regime of small…
Parametric Subharmonic Instability (PSI) is one of the most important mechanisms that transfer energy from tidally-generated long internal waves to short steep waves. Breaking of these short waves results in diapycnal mixing through which oceanic abyssal stratification is maintained. It has long been believed that PSI is strongest between a primary internal wave and perturbative waves of half the frequency of the primary wave. Here, we rigorously show that this is not the case. Specifically, we show that neither the initial growth rate nor the maximum long-term amplification occur at the half …
Boundary layer turbulence in coastal regions differs from that in deep ocean because of bottom interactions. In this paper, we focus on the merging of surface and bottom boundary layers in a finite-depth coastal ocean by numerically solving the wave-averaged equations using a large eddy simulation method. The ocean fluid is driven by combined effects of wind stress, surface wave, and a steady current in the presence of stable vertical stratification. The resulting flow consists of two overlapping boundary layers, i.e. surface and bottom boundary layers, separated by an interior stratification.…
This paper presents specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients in relation to surface and internal waves in a rotating ocean with a variable bottom topography. For solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analyzed in detail and supported by direct numerical modelling. One of them is a constant-slope bottom and the other is a specific bottom profile providing a constant amplitude solitary w…
Shear-induced breaking of internal gravity waves
本研究通过直接数值模拟揭示正弦剪切流与大振幅内重力波相互作用导致波破碎的机制，发现线性射线理论可定性预测早期演化，且剪切驱动的卷曲结构主导混合，混合效率与开尔文-亥姆霍兹不稳定性相当。
With the aim of assessing internal wave-driven mixing in the ocean, we develop a new technique for direct numerical simulations of stratified turbulence. Since the spatial scale of oceanic internal gravity waves is typically much larger than that of turbulence, fully incorporating both in a model would require a high computational cost, and is therefore out of our scope. Alternatively, we cut out a small domain periodically distorted by an unresolved large-scale internal wave and locally simulate the energy cascade to the smallest scales. In this model, even though the Froude number of the out…
Horizontal currents in the Bay of Bengal were measured on eight cruises covering a total of 8600 $km$ using a 300 $kHz$ Acoustic Doppler Current Profiler (ADCP). The cruises are distributed over multiple seasons and regions of the Bay. Horizontal wavenumber spectra of these currents over depths of 12--54 m and wavelengths from 2--400 km were decomposed into vortical and divergent components assuming isotropy. An average of across and along track spectra over all cruises shows that the spectral slope of horizontal kinetic energy for wavelengths of 10--80 $km$ scales with an exponent of $-1.7 \p…
Wave-kinetic dynamics of forced-dissipated turbulent internal gravity waves
本文通过受迫耗散动理学方程模拟弱非线性内重力波，揭示早期非局域相互作用与后期局域各向异性能量级联，并发现小水平波矢处由波-波相互作用介导的层化凝聚现象。
Laboratory experimental results are presented for nonlinear Internal Solitary Waves (ISW) propagation in deep water configuration with miscible fluids. The results are validated against direct numerical simulations and traveling wave exact solutions where the effect of the diffused interface is taken into account. The waves are generated by means of a dam break and their evolution is recorded with Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV). In particular, data collected in a frame moving with the waves are presented here for the first time. Our results are representa…
Triadic resonance is one mechanism via which internal waves dissipate their energy, often at locations away from their generation sites. In this paper, we perform a combined theoretical and numerical study of triadic resonance in internal wave modes in a finite-depth ocean with background rotation and an arbitrary stratification profile. The spatial evolution of the modal amplitudes within a resonant triad are first derived based on the requirement that the nonlinear solution at leading order cannot diverge. The amplitude evolution equations are then numerically solved for two different cases:…
Here we show that there exist internal gravity waves that are inherently unstable, that is, they cannot exist in nature for a long time. The instability mechanism is a one-way (irreversible) harmonic-generation resonance that permanently transfers the energy of an internal wave to its higher harmonics. We show that, in fact, there are countably infinite number of such unstable waves. For the harmonic-generation resonance to take place, nonlinear terms in the free surface boundary condition play a pivotal role, and the instability does not obtain for a linearly-stratified fluid if a simplified …
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