Predicting the breaking onset of surface water waves

Why do ocean waves break? Understanding this important and obvious property of the ocean surface has been elusive for decades. This paper investigates causes which lead deep-water two-dimensional initially monochromatic waves to break. Individual wave steepness is found to be the single parameter which determines whether the wave will break immediately, never break or take a finite number of wave lengths to break. The breaking will occur once the wave reaches the Stokes limiting steepness. The breaking probability and the location of breaking onset can be predicted, properties of incipient breakers measured. Potential applications to field conditions are discussed

Fine scale inhomogeneity of wind-wave energy input, skewness, and asymmetry

Analysis of measured sea and lake wind wave data reveals large variability of the wind energy input, as well as the waves skewness and asymmetry. The spatial and temporal third moments alternate in sign over a few wave periods and over a few wavelengths, respectively. Simulation through a 2D Wave Boundary Layer model in which the air flow is modeled by 2nd order Reynolds equations (Chalikov, 1998) conforms to these findings and exposes a rich structure. We found clear correlation of the variations of the skewness and the asymmetry with the wind input

Advanced wave modeling, including wave-current interaction

The paper outlines principles of phase-resolving and phase-average wave models, with emphasis on the state of the art of wave-current interaction physics. We argue that these interactions are the least well-developed part of such models. Linear and nonlinear dynamics of waves on currents are discussed; depth-integrated and depth-varying approaches are described. Finally, examples of numerical model performance for waves on currents in realistic oceanic scenarios are presented

Three-dimensional periodic fully-nonlinear potential waves

An exact numerical scheme for a long-term simulation of three-dimensional potential fully-nonlinear periodic gravity waves is suggested. The scheme is based on a surfacefollowing non-orthogonal curvilinear coordinate system and does not use the technique based on expansion of the velocity potential. The Poisson equation for the velocity potential is solved iteratively. The Fourier transform method, the secondorder accuracy approximation of the 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. The model requires considerable computer resources, but the one-processor version of the model for PC allows us to simulate an evolution of a wave field with thousands degrees of freedom for hundreds of wave periods. The scheme is designed for investigation of the nonlinear two-dimensional surface waves, for generation of extreme waves as well as for the direct calculations of a nonlinear interaction rate. After implementation of the wave breaking parameterization and wind input, the model can be used for the direct simulation of a two-dimensional wave field evolution under the action of wind, nonlinear wavewave interactions and dissipation. The model can be used for verification of different types of simplified models

Numerical investigation of turbulence generation in non-breaking potential waves

Theoretically, potential waves cannot generate the vortex motion, but scale considerations indicate that if the steepness of waves is not too small, Reynolds number can exceed critical values. This means that in presence of initial non-potential disturbances the orbital velocities can generate the vortex motion and turbulence. In the paper, this problem was investigated numerically on basis of full two-dimensional (x-z) equations of potential motion with the free surface in cylindrical conformal coordinates. It was assumed that all variables are a sum of the 2D potential orbital velocities and 3D non-potential disturbances. The non-potential motion is described directly with 3D Euler equations, with very high resolution. The interaction between potential orbital velocities and non-potential components is accounted through additional terms which include the components of vorticity. Long-term numerical integration of the system of equations was done for different wave steepness. Vorticity and turbulence usually occur in vicinity of wave crests (where the velocity gradients reach their maximum) and then spread over upwind slope and downward. Specific feature of the wave turbulence at low steepness (steepness was kept low in order to avoid wave breaking) is its strong intermittency: the turbulent patches are mostly isolated and intermittency grows with decrease of wave amplitude. Maximum values of energy of turbulence are in agreement with available experimental data. The results suggest that even non-breaking potential waves can generate turbulence, which thus enhances the turbulence created by the shear current

Simulation of one-dimensional evolution of wind waves in a deep water

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

Improved 3-D model for periodic waves based on initial equation

The improved method of modelling of three-dimensional surface waves is described. Contrary to approximate HOS approach, where the vertical velocity on the surface is calculated with Taylor expansion, the given method allows to calculate the exact surface vertical velocity with estimated accuracy. The advantages of new method are discussed by comparison with HOS method. Method is based on dynamic and kinematic surface boundary conditions and elliptic equation for velocity potential written in surface-fitted coordinate system. A direct iterational solution of equation for potential usually needs a high vertical resolution what significantly slows down the speed of calculations. The separation of velocity potential into linear and nonlinear modes allows to reduce the problem to the same equations but written for nonlinear correction for linear solution. Since the correction is typically smaller by 2 decimal order than a linear constituent, the solution needs less accuracy and smaller number of vertical levels. Such improvement makes the 3-D model well suitable for performing the multi-mode and long-term calculations. Such calculations allows to investigate the nonlinear properties of 3-D wave field, mechanics and statistics of extreme waves, formation of angular spreading of waves. Being supplied by well developed algorithms of wind input and breaking dissipation model allows the simulation of long-term development of waves initially assigned as a monochromatic train of initially small waves. The details of numerical scheme as well as the results of model validation and the examples of long-term simulations of 3-D wave field are given

Numerical modeling of 3D fully nonlinear potential periodic waves

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

A neural network technique to improve computational efficiency of numerical oceanic models

A new generic approach to improve computational efficiency of certain processes in numerical environmental medols 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 emprical relations based on physical or statistical models. It is shown that, form 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 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