Search

The paper describes the attempt to develop the accelerated method of simulation of 2-D surface waves with a use of 2-D model derived by simplifications of 3-D equations for potential periodic deep-water waves. The derivation is based on separation of velocity potential in surface-fitted coordinates into linear and non-linear components and analysis of exact Poisson equation for non-linear component of potential on a free surface. This equation contains both the first and second derivatives of velocity potential. The analysis of very accurate multiple solutions for velocity potential obtained with 3-D model shows that these variables are linearly connected to each other, what allows to obtain the 2-D equation for first derivative of potential (i.e., the vertical velocity on a surface), what gives the closed 2-D formulation for 3-D problem of 2-D waves. The connection between first and second variables is not precise; hence, the method as a whole cannot be exact. However, the 2-D model derived is able to reproduce different statistical characteristics of 2-D wave field with good accuracy. The most evident advantage of new model consists of absence of calculation of 3-D structure of velocity potential what increases a speed of calculations by about two orders. [ABSTRACT FROM AUTHOR]

The probability of extremely high waves is calculated by two methods. The first method is based on the direct numerical simulation of two-dimensional wave field using a three-dimensional nonlinear model. The second method consists in calculation of the probability of wave heights over ensemble of fields representing a superposition of linear waves with random phases and a spectrum similar to that obtained in the nonlinear model. It is shown that the integral probability of extreme waves are very close to each other in both cases. This implies that the role of nonlinearity in the generation of extreme waves is probably not so important as it was assumed in most papers considering this phenomenon. [ABSTRACT FROM AUTHOR]

Two-dimensional direct wave model is used for demonstration of the role of reversible interactions which probably is the main process leading to breaking. One-dimensional model was used for performing of thousands of exact short-term simulations of evolution of two superposed wave trains with different steepness, and wavenumbers were performed to investigate the effect of wave crests merging. Nonlinear sharpening of the merging crests is demonstrated. It is suggested that such effect may be responsible for appearance of the typical sharp crests of surface waves, as well as for wave breaking. [ABSTRACT FROM AUTHOR]

A simple and exact numerical scheme for long-term simulations of 3D potential fully nonlinear periodic gravity waves is suggested. The scheme is based on the surface-following nonorthogonal curvilinear coordinate system. Velocity potential is represented as a sum of analytical and nonlinear components. The Poisson equation for the nonlinear component of velocity potential is solved iteratively. Fourier transform method, the second-order accuracy approximation of vertical derivatives on a stretched vertical grid and the fourth-order Runge-Kutta time stepping are used. The scheme is validated by simulation of steep Stokes waves. A one-processor version of the model for PC allows us to simulate evolution of a wave field with thousands degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of nonlinear 2D surface waves, generation of extreme waves, and direct calculations of nonlinear interactions. [ABSTRACT FROM AUTHOR]

A direct wave model based on the one-dimensional nonlinear equations for potential waves is used for simulation of wave field development under the action of energy input, dissipation, and nonlinear wave-wave interaction. The equations are written in conformal surface-fitted nonstationary coordinate system. New schemes for calculating the input and dissipation of wave energy are implemented. The wind input is calculated on the basis of the parameterization developed through the coupled modeling of waves and turbulent boundary layer. The wave dissipation algorithm, introduced to prevent wave breaking instability, is based on highly selective smoothing of the wave surface and surface potential. The integration is performed in Fourier domain with the number of modes M = 2048, broad enough to reproduce the energy downshifting. As the initial conditions, the wave field is assigned as train of Stokes waves with steepness ak = 0.15 at nondimensional wavenumber k = 512. Under the action of nonlinearity and energy input the spectrum starts to grow. This growth is followed by the downshifting. The total time of integration is equal to 7203 initial wave periods. During this time the energy increased by 1111 times. Peak of the spectrum gradually shifts from wavenumber nondimensional k = 512 down to k = 10. Significant wave height increases 33 times, while the peak period increases 51 times. Rates of the peak downshift and wave energy evolution are in good agreement with the JONSWAP formulation. [ABSTRACT FROM AUTHOR]

Results of numerical investigations, based on full dynamic equations, are presented for wave breaking in a one-dimensional environment with a wave spectrum. The breaking is defined as a process of irreversible collapse of an individual wave in physical space, and the incipient breaker is a wave that reached a dynamic condition of the limiting stability where the collapse has not started yet but is inevitable. The main attention is paid to documenting the evolution of different wave characteristics before the breaking commences. It is shown that the breaking is a localized process that rapidly develops in space and time. No single characteristic, such as wave steepness, wave height, and asymmetry, can serve as a predictor of the incipient breaking. The process of breaking is intermittent; it happens spontaneously and is individually unpredictable. The evolution of geometric, kinematic, and dynamic characteristics of the breaking wave describes the process of breaking itself rather than indicating an imminent breaking. It is shown that the criterion of breaking, valid for the breaking due to modulation instability in one-dimensional waves trains, is not universal if applied to the conditions of spectral environment. In this context, the development of algorithms for parameterization of breaking for wave prediction models and for direct wave simulations is more important. [ABSTRACT FROM AUTHOR]

Full nonlinear equations for one-dimensional potential surface waves were used for investigation of the evolution of an initially homogeneous train of exact Stokes waves with steepness AK=0.01–0.42. The numerical algorithm for the integration of nonstationary equations and the calculation of exact Stokes waves is described. Since the instability of the exact Stokes waves develops slowly, a random small-amplitude noise was introduced in initial conditions. The development of instability occurs in two stages: in the first stage the growth rate of disturbances was close to that established for small steepness by Benjamin and Feir [J. Fluid. Mech. 27, 417 (1967)] and for medium steepness [McLean, J. Fluid Mech. 114, 315 (1982)]. For any steepness, the Stokes waves disintegrate and create random superposition of waves. For AK<0.13, waves do not show a tendency to breaking, which is recognized by approaching a surface to non-single-value shape. Sooner or later, if AK>0.13, one of the waves increases its height, and finally it comes to the breaking point. For large steepness of AK>0.35 the rate of growth is slower than for medium steepness. The data for spectral composition of disturbances and their frequencies are given. [ABSTRACT FROM AUTHOR]

Abstract: A method for numerical investigation of nonlinear wave dynamics based on direct hydrodynamical modeling of 1-D potential periodic surface waves is created. The model is a part of an interactive wind–wave model. Using a non-stationary conformal mapping, the principal equations are rewritten in a surface-following coordinate system and reduced to two simple evolutionary equations for the elevation and the velocity potential of the surface; Fourier expansion is used to approximate these equations. High accuracy was confirmed by validation of the non-stationary model against known solutions, and by comparison between the results obtained with different resolution in the horizontal. The method developed is applied to the simulation of waves evolution with different initial conditions. Numerical experiments with initially monochromatic waves with different steepness show that the model is able to simulate breaking conditions when the surface becomes a multi-valued function of the horizontal coordinate. An estimate of the critical initial wave height that divides between non-breaking and eventually breaking waves is obtained. Simulations of nonlinear evolution of a wave field is represented initially by two modes with close wave numbers (amplitude modulation) and a wave field with a phase modulation. Both runs result in the appearance of large and very steep waves, these also break if the initial amplitudes are sufficiently large. [Copyright &y& Elsevier]

A new generic approach to improve computational efficiency of certain processes in numerical environmental models is formulated. This approach is based on the application of neural network (NN) techniques. It can be used to accelerate the calculations and improve the accuracy of the parameterizations of several types of physical processes which generally require computations involving complex mathematical expressions, including differential and integral equations, rules, restrictions and highly nonlinear empirical relations based on physical or statistical models. It is shown that, from a mathematical point of view, such parameterizations can usually be considered as continuous mappings (continuous dependencies between two vectors). It is also shown that NNs are a generic tool for fast and accurate approximation of continuous mappings and, therefore, can be used to replace primary parameterization algorithms. In addition to fast and accurate approximation of the primary parameterization, NN also provides the entire Jacobian for very little computation cost.Three successful particular applications of the NN approach are presented here: (1) a NN approximation of the UNESCO equation of state of the seawater (density of the seawater); (2) an inversion of this equation (salinity of the seawater); and (3) a NN approximation for the nonlinear wave–wave interaction. The first application has been implemented in the National Centers for Environmental Prediction multi-scale oceanic forecast system, and the second one is being developed for wind wave models.The NN approach introduced in this paper can provide numerically efficient solutions to a wide range of problems in environmental numerical models where lengthy, complicated calculations, which describe physical processes, must be repeated frequently. [Copyright &y& Elsevier]

A Correction to this paper has been published: https://doi.org/https://doi.org/10.1007/s10236-021-01450-3 [ABSTRACT FROM AUTHOR]

It is known that drag coefficient varies in broad limits depending on wind velocity and wave age as well as on wave spectrum and some other parameters. All those effects produce large scatter of the drag coefficient, so, the data is plotted as a function of wind velocity forming a cloud of points with no distinct regularities. Such uncertainty can be overcome by the implementation of the WBL model instead of the calculations of drag with different formulas. The paper is devoted to the formulation of the Wave Boundary Layer (WBL) model for the parameterization of the ocean-atmosphere interactions in coupled ocean-atmosphere models and wave prediction models. The equations explicitly take into account the vertical flux of momentum generated by the wave-produced fluctuations of pressure, velocity and stresses (WPMF). Their surface values are calculated with the use of the spectral beta-functions whose expression was obtained by means of the 2-D simulation of the WBL. Hence, the model directly connects the properties of the WBL with an arbitrary wave spectrum. The spectral and direct wave modeling should also take into account the momentum flux to a subgrid part of the spectrum. The parameterization of this effect in the present paper is formulated in terms of wind and cut-off frequency of the spectrum. [ABSTRACT FROM AUTHOR]

In this study, the relation between rogue wave occurrence statistics and properties of the directional wind-wave spectrum is investigated. The study is conducted by means of the fully nonlinear phase-resolving numerical model and the high-order spectral model. Benjamin-Feir Index (BFI ), Π numbers and Modulation Index, which are designed to represent instabilities in nonlinear wave systems, are used as indicators for the occurrence of rogue waves. Rogue wave statistics and indicator parameters' performance are assessed based on numerical simulations and compared against in situ measurements. Directional Π 2 is found to be the best performing indicator. It is also observed that probability of rogue wave occurrence further peaks up at high directionality where modulational instability should be suppressed. It is argued that this condition is associated with the linear superposition of waves coming from different directions. Therefore, both directionality and instability, i.e. both linear and nonlinear effects, appear to be important physical mechanisms contributing to the extreme wave statistics. However, it is also argued that rogue wave occurrence is a transient (fast) event and cannot be accurately represented with conventional probability distribution functions. [ABSTRACT FROM AUTHOR]

The concept of a constant-flux layer is usually employed for vertical profiling of the wind measured at some elevation near the ocean surface. The surface waves, however, modify the balance of turbulent stresses very near the surface, and therefore such extrapolations can introduce significant biases. This is particularly true for buoy measurements in extreme conditions, when the anemometer mast is within the wave boundary layer (WBL) or even below the wave crests. In this paper, field data and aWBL model are used to investigate such biases. It is shown that near the surface the turbulent stresses are less than those obtained by extrapolation using the logarithmic-layer assumption, and the mean wind speeds very near the surface, based on Lake George field observations, are up to 5% larger. The behavior is then simulated by means of a WBL model coupled with nonlinear waves, which confirmed the observations and revealed further details of complex behaviors at the wind-wave boundary layer. [ABSTRACT FROM AUTHOR]

We show that interaction of two monochromatic waves at the water surface enters a different dynamic regime if their wavenumbers become very close. The study is conducted by means of a fully nonlinear wave model. In the course of evolution of the two waves, downshifting of the initial wave energy and growth of the first mode occur depending on wave steepness and dk/k. Behaviour of these features changes if dk/k < 0.0025: both downshifting and growth rate become independent of dk/k, accompanied by rapid transfer of wave energy to large scales. [ABSTRACT FROM AUTHOR]

Evolution of nonlinear wave groups to breaking under wind forcing was studied by means of a fully nonlinear numerical model and in a laboratory experiment. Dependence of distance to breaking and modulation depth (height ratio of the highest and the lowest waves in a group) on wind forcing was described. It was shown that in the presence of a certain wind forcing both distance to breaking and modulation depth decrease; the latter signifies slowing down of the instability growth. It was also shown that wind forcing significantly reduces the energy loss in a single breaking event. [ABSTRACT FROM AUTHOR]

Progression of nonlinear wave groups to breaking was studied numerically and experimentally. Evolution of such wave group parameters as a function of distance to breaking and modulation depth-the height ratio of the highest and the lowest waves in the group-was described. Numerical model results demonstrated good agreement with experimental results in describing the behavior of the distance to breaking and modulation depth as functions of initial wave steepness. It was shown that energy loss appears to be a function of the modulation depth at the breaking onset. Energy loss grows with modulation depth up to a certain threshold of the latter. It was also shown that breaking probability for wave groups with modulation depth below 2.2 is very low. [ABSTRACT FROM AUTHOR]

This reply is aimed at clarifying and further discussing the methodological aspects of this neural network application for a better understanding of the technique by the journal readership. The similarities and differences of two approaches and their areas of application are discussed. These two approaches outline a new interdisciplinary field based on application of neural networks (and probably other modern machine or statistical learning techniques) to significantly speed up calculations of time-consuming components of atmospheric and oceanic numerical models. [ABSTRACT FROM AUTHOR]

A new approach based on a synergetic combination of statistical/machine learning and deterministic modeling within atmospheric models is presented. The approach uses neural networks as a statistical or machine learning technique for an accurate and fast emulation or statistical approximation of model physics parameterizations. It is applied to development of an accurate and fast approximation of an atmospheric longwave radiation parameterization for the NCAR Community Atmospheric Model, which is the most time consuming component of model physics. The developed neural network emulation is two orders of magnitude, 50–80 times, faster than the original parameterization. A comparison of the parallel 10-yr climate simulations performed with the original parameterization and its neural network emulations confirmed that these simulations produce almost identical results. The obtained results show the conceptual and practical possibility of an efficient synergetic combination of deterministic and statistical learning components within an atmospheric climate or forecast model. A developmental framework and practical validation criteria for neural network emulations of model physics components are outlined. [ABSTRACT FROM AUTHOR]

The potential of Neural Networks (NN) to provide accurate estimates of nonlinear interactions for wind wave spectra by means of direct mapping is considered. Expanding on a previously reported feasibility study, an Empirical Orthogonal Functions (EOF) based NN for single peaked spectra is shown to be much more accurate than the well known Discrete Interaction Approximation (DIA), at the expense of a moderate increase of computational costs. This Neural Network Interaction Approximation (NNIA) gives reasonable results for modeled wave spectra, but is not yet capable of providing acceptable model integrations. Methods to expand the NNIA to be suitable for model integration are discussed. [Copyright &y& Elsevier]

Probability of rogue wave occurrence due to modulational instability in unidirectional JONSWAP sea states are investigated. This investigation has been conducted based on the quantitative indicators of Benjamin-Feir index (B F I), Π -number and kurtosis, which are used to represent instability in wave trains. Evolution of wavefields is simulated using the fully nonlinear phase-resolving numerical model of Chalikov-Sheinin (CS) and the high order spectral model of HOS-Ocean. Effects of the high-frequency end (tail) of the spectrum on modulational instability and rogue wave evolution are discussed considering four different tail lengths. According to our simulation results, indicators considered for both model results perform as proxy for the occurrence of extreme events. Therefore, it is possible to define a certain interval for the indicators, where the probability of rogue wave occurrence is the highest. Nonlinearity (described in terms of steepness) is the dominant parameter on probability of rogue wave occurrence since it is related to various active physical processes in sea states (both modulational instability and breaking). Rogue wave occurrence increases with increased nonlinearity until it is restricted again by high nonlinearity which causes waves to reach breaking onset earlier. • BFI, Π-number and kurtosis perform as a proxy for the occurrence of extreme events. • Rogue wave occurrence both increases and restricted due to changes in nonlinearity. • Rogue wave modelling are highly dependent on models' ability to simulate breaking. • In highly nonlinear cases, rogue wave probability drops with longer spectral tail. [ABSTRACT FROM AUTHOR]

A brief historical overview of numerical wind wave forecast modeling efforts at the National Centers for Environmental Prediction (NCEP) is presented, followed by an in-depth discussion of the new operational National Oceanic and Atmospheric Administration (NOAA) "WAVEWATCH III" (NWW3) wave forecast system. This discussion mainly focuses on a parallel comparison of the new NWW3 system with the previously operational Wave Model (WAM) system, using extensive buoy and European Remote Sensing Satellite-2 (ERS2) altimeter data. The new system is shown to describe the variability of the wave height more realistically, with similar or smaller random errors and generally better correlation coefficients and regression slopes than WAM. NWW3 outperforms WAM in the Tropics and in the Southern Hemisphere, and they both show fairly similar behavior at northern high latitudes. Dissemination of NWW3 products, and plans for its further development, are briefly discussed. [ABSTRACT FROM AUTHOR]