Modeling ocean surface waves under complex ocean current conditions is of crucial importance to many naval applications. For example, traveling ships and underwater vehicles generate spatially heterogeneous currents behind them through their drag and propeller motions. The strong currents can influence the surface wave pattern in the ship wake. In this study, the nonlinear interactions between waves and complex wake currents are investigated using numerical simulations. An in-house code is developed for high-fidelity simulations of a nonlinear phase-resolved ocean wavefield interacting with subsurface currents. Several typical wake patterns are simulated using the present numerical method, and the influence of complex currents on the waves is analyzed quantitatively using theoretical solutions of wave-current interactions. We also present a method for solving the inverse problem of deducing the current field based on surface-wave data using machine-learning techniques. A deep neural network is designed for processing spatial-temporal surface wave data. Detailed analyses on the distributions of regression errors and the training dataset-dependency show that the proposed neural network can effectively deduce the current field.
The instability and nonlinear evolution of directional ocean waves is investigated numerically by means of simulations of the governing kinetic equation for narrow-band surface waves. Our simulation results reveal the onset of the modulational instability for long-crested wave-trains, which agrees well with recent large-scale experiments in wave-basins, where it was found that narrower directional spectra leads to self-focusing of ocean waves and an enhanced probability of extreme events. We find that the modulational instability is nonlinearly saturated by a broadening of the wave-spectrum, w…
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.
Within a simple model context, the sensitivity and stability of the thermohaline circulation to finite amplitude perturbations is studied. A new approach is used to tackle this nonlinear problem. The method is based on the computation of the so-called Conditional Nonlinear Optimal Perturbation (CNOP) which is a nonlinear generalization of the linear singular vector approach (LSV). It is shown that linearly stable thermohaline circulation states can become nonlinearly unstable and the properties of the perturbations with optimal nonlinear growth are determined. An asymmetric nonlinear response …
Buoyant, finite-size or inertial particle motion is fundamentally unlike neutrally buoyant, infinitesimally small or Lagrangian particle motion. The de-jure fluid mechanics framework for the description of inertial particle dynamics is provided by the Maxey-Riley equation. Derived from first principles - a result of over a century of research since the pioneering work by Sir George Stokes - the Maxey-Riley equation is a Newton-type-law with several forces including (mainly) flow, added mass, shear-induced lift, and drag forces. In this paper we present an overview of recent efforts to port the…
We investigate a new two-dimensional compressible Navier-Stokes hydrodynamic model design to explain and study large scale ice swirls formation at the surface of the ocean. The linearized model generates a basis of Bessel solutions from where various types of spiral patterns can be generated and their evolution and stability in time analyzed. By restricting the nonlinear system of equations to its quadratic terms we obtain swirl solutions emphasizing logarithmic spiral geometry. The resulting solutions are analyzed and validated using three mathematical approaches: one predicting the formation…
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…
We derive scaling laws for the steady spectrum of wind excited waves, assuming two inviscid fluids (air and water) and no surface tension, an approximation valid at large speeds. In this limit there exists an unique (small) dimensionless parameter $ε$, the ratio of the mass densities of the two fluids, air and water, independently of the wind speed. The smallness of $ε$ allows to derive some important average properties of the wave system. The average square slope of the waves is of order $|ln (ε^2)|^{-1}$, a small but not very small quantity. This supports the often used assumption of small n…
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…
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…
Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the Rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension(1D) with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose a two-dimensional(2D), exactly solvable Nonlinear Schrödinger equation(NLS), derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed 2D equation exhibits…
Physics-informed neural networks for phase-resolved data assimilation and prediction of nonlinear ocean waves
Hamburg University of Technology（汉堡工业大学）；Imperial College London（帝国理工学院）；University of Oxford（牛津大学）；Universitat Politécnica de Catalunya（加泰罗尼亚理工大学）；China University of Petroleum（中国石油大学）；College of Oceanography（海洋学院）；Space Informatics（空间信息学院）
Breather solutions of the nonlinear Schrödinger equation (NLSE) are known to be considered as backbone models for extreme events in the ocean as well as in Kerr media. These exact determinisitic rogue wave (RW) prototypes on a regular background describe a wide-range of modulation instability configurations. Alternatively, oceanic or electromagnetic wave fields can be of chaotic nature and it is known that RWs may develop in such conditions as well. We report an experimental study confirming that extreme localizations in an irregular oceanic JONSWAP wave field can be tracked back to originate …
In this study we discuss the shapes and statistics of the rogue (freak) waves emerging due to wave-current interactions. With this purpose, we use a simple governing equation which is a nonlinear Schrodinger equation (NLSE) extended by R. Smith (1976). This extended NLSE accounts for the effects of current gradient on the nonlinear dynamics of the ocean surface near blocking point. Using a split-step scheme we show that the extended NLSE of Smith is unstable against random chaotic perturbation in the current profile. Therefore the monochromatic wave field with unit amplitude turns into a chaot…
We review the study of rogue waves and related instabilities in optical and oceanic environments, with particular focus on recent experimental developments. In optics, we emphasize results arising from the use of real-time measurement techniques, whilst in oceanography we consider insights obtained from analysis of real-world ocean wave data and controlled experiments in wave tanks. Although significant progress in understanding rogue waves has been made based on an analogy between wave dynamics in optics and hydrodynamics, these comparisons have predominantly focused on one-dimensional nonlin…
In this paper, we show that a revised convolutional recurrent neural network (CRNN) can decrease, by orders of magnitude, the time needed for the phase-resolved prediction of waves in a spatiotemporal domain of a nonlinear dispersive wave field. The problem of predicting such waves suffers from two major challenges that have so far hindered analytical or direct computational solutions in real time or faster: (i) the reconstruction problem, that is, how one can calculate from measurable wave amplitude data the state of the wave field (wave components, nonlinear couplings, etc.), and (ii) if suc…
In this Letter we study the form of the energy spectrum of Riemann waves in weakly nonlinear non-dispersive media. For quadratic and cubic nonlinearity we demonstrate that the deformation of an Riemann wave over time yields an exponential energy spectrum which turns into power law asymptotic with the slope being approximately -8/3 at the last stage of evolution before breaking. We argue, that this is the universal asymptotic behaviour of Riemann waves in any nonlinear non-dispersive medium at the point of breaking. The results reported in this Letter can be used in various non-dispersive media…
Wave attenuation by ice floes is an important parameter for modelling the Arctic Oceans. At present, attenuation coefficients are extracted from linear models as a function of the incident wave period and floe thickness. Recent explorations in the Antarctic Mixed Ice Zone (MIZ) revealed a further dependence on wave amplitude, suggesting that nonlinear contributions are non-negligible. An experimental model for wave attenuation by a single ice floe in a wave flume is here presented. Observations are compared with linear predictions based on wave scattering. Results indicate that linear models p…
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…
Nonlinear instability and refraction by ocean currents are both important mechanisms that go beyond the Rayleigh approximation and may be responsible for the formation of freak waves. In this paper, we quantitatively study nonlinear effects on the evolution of surface gravity waves on the ocean, to explore systematically the effects of various input parameters on the probability of freak wave formation. The fourth-order current-modified nonlinear Schrödinger equation (CNLS4) is employed to describe the wave evolution. By solving CNLS4 numerically, we are able to obtain quantitative predictions…
We review the theory of wave interaction in finite and infinite depth. Both of these strands of water-wave research begin with the deterministic governing equations for water waves, from which simplified equations can be derived to model situations of interest, such as the mild slope and modified mild slope equations, the Zakharov equation, or the nonlinear Schrödinger equation. These deterministic equations yield accompanying stochastic equations for averaged quantities of the sea-state, like the spectrum or bispectrum. We discuss several of these in depth, touching on recent results about th…
The combined effectiveness of model reduction and the quasilinear approximation for the reproduction of the low-order statistics of oceanic surface boundary-layer turbulence is investigated. Idealized horizontally homogeneous problems of surface-forced thermal convection and Langmuir turbulence are studied in detail. Model reduction is achieved with a Galerkin projection of the governing equations onto an subset of modes determined by proper orthogonal decomposition. For less than 0.2% of the modes retained, the reduced quasilinear model is able to reproduce vertical profiles of horizontal mea…
Direct phase-resolved simulations are performed to investigate the propagation and scattering of nonlinear ocean waves in fragmented sea ice. The numerical model solves the full time-dependent equations for nonlinear potential flow coupled with a nonlinear thin-plate representation of the ice cover, and neglects dissipative processes. The two-dimensional setting with incident wave groups on deep water is considered, in view of applications to wave attenuation along transects of the marginal ice zone. A spatially-varying weight is assigned to the surface pressure so that irregular distributions…
Through ensemble-based data assimilation (DA), we address one of the most notorious difficulties in phase-resolved ocean wave forecast, regarding the deviation of numerical solution from the true surface elevation due to the chaotic nature of and underrepresented physics in the nonlinear wave models. In particular, we develop a coupled approach of the high-order spectral (HOS) method with the ensemble Kalman filter (EnKF), through which the measurement data can be incorporated into the simulation to improve the forecast performance. A unique feature in this coupling is the mismatch between the…
A formulation is developed to assimilate ocean-wave data into the Numerical Flow Analysis (NFA) code. NFA is a Cartesian-based implicit Large-Eddy Simulation (LES) code with Volume of Fluid (VOF) interface capturing. The sequential assimilation of data into NFA permits detailed analysis of ocean-wave physics with higher bandwidths than is possible using either other formulations, such as High-Order Spectral (HOS) methods, or field measurements. A framework is provided for assimilating the wavy and vortical portions of the flow. Nudging is used to assimilate wave data at low wavenumbers, and th…
Exact Lagrangian in compact form is derived for planar internal waves in a two-fluid system with a relatively small density jump (the Boussinesq limit taking place in real oceanic conditions), in the presence of a background shear current of constant vorticity, and over arbitrary bottom profile. Long-wave asymptotic approximations of higher orders are derived from the exact Hamiltonian functional in a remarkably simple way, for two different parametrizations of the interface shape.
We analyze with the tools of lobe dynamics the velocity field from a numerical simulation of the surface circulation in the Northwestern Mediterranean Sea. We identify relevant hyperbolic trajectories and their manifolds, and show that the transport mechanism known as the `turnstile', previously identified in abstract dynamical systems and simplified model flows, is also at work in this complex and rather realistic ocean flow. In addition nonlinear dynamics techniques are shown to be powerful enough to identify the key geometric structures in this part of the Mediterranean. In particular the N…
We analyze shallow water wind waves in Currituck Sound, North Carolina and experimentally confirm, for the first time, the presence of $soliton$ $turbulence$ in ocean waves. Soliton turbulence is an exotic form of nonlinear wave motion where low frequency energy may also be viewed as a $dense$ $soliton$ $gas$, described theoretically by the soliton limit of the Korteweg-deVries (KdV) equation, a $completely$ $integrable$ $soliton$ $system$: Hence the phrase "soliton turbulence" is synonymous with "integrable soliton turbulence." For periodic/quasiperiodic boundary conditions the $ergodic$ $sol…
The long-term goal of this work is the development of high-fidelity simulation tools for dispersive tsunami propagation. A dispersive model is especially important for short wavelength phenomena such as an asteroid impact into the ocean, and is also important in modeling other events where the simpler shallow water equations are insufficient. Adaptive simulations are crucial to bridge the scales from deep ocean to inundation, but have difficulties with the implicit system of equations that results from dispersive models. We propose a fractional step scheme that advances the solution on separat…
