Nonlinear Fourier Analysis Algorithm and Models for Water Waves in Terms of Surface Elevation, Amplitude Modulations

This paper aims to investigate the accuracy and computational efficiency of three CFD-based numerical codes to accurately simulate extremely large regular and irregular waves of different steepness in deep and shallow water conditions. The work assesses the performance of three numerical techniques with different formulations of the fluid dynamic equations. Firstly, an open-sourced smoothed particle hydrodynamics (SPH) code; secondly, a finite difference method (FDM) based 3D numerical model with the assumption of inviscid and incompressible fluid flow; and thirdly, a commercial CFD code that uses a finite volume method (FVM) to solve the Reynolds-averaged Navier-Stokes (RANS) equations. A suite of metrics and methodologies, considering three key performance parameters: accuracy, computational requirements and available features for providing a consistent framework for the quantitative assessment of different techniques, has been presented. Numerically simulated free surface elevations, wave periods, and spectrum (for irregular waves only) are compared with experimental data previously acquired at an Offshore Engineering Basin (OEB) facility. Extensive convergence studies were carried out for each numerical tool for a selected large wave before predictions were model for all waves. All three models reproduced waves with an accuracy comparable to physical wave makers in the wave basin experiments for the deep-water regular and irregular waves; however, the SPH model performed better than the other two models for the shallow water waves. The challenge remains for wave basins to reduce unwanted basin effects and numerical facilities to accurately model waves with proper account for boundary effects and numerical diffusions. In addition, only flat-bottom domains were considered in the investigation, leaving the wave modelling for uneven bottom for future studies.

Extensive field measurements of wind waves in deep and shallow waters during Hurricane Gustav (2008) in the Gulf of Mexico have been simulated by the spectral wave prediction model, SWAN. First, a parametric asymmetric hurricane wind model with three major improvements is used to generate hurricane wind fields for the wave model. The changes of water level near the coast are taken into account by using a storm surge model. Forced by the verified hurricane winds and hindcasted water levels, the wave model performs fairly well in comparison to the observed wave heights and periods in both deep and shallow waters except a few locations with complex bathymetry and landscape. In addition to the hurricane wind field that controls the accuracy of wave modeling in deep water, wave-surge interaction plays an important role in the wave growth and transformation in shallow water. Wave spectral comparisons show that the white-capping formulation of Westhuysen et al. (2007) generally outperforms the default formulation of Komen et al. (1984) in SWAN under hurricane conditions. The model result indicates that the asymmetry of hurricane winds and the hurricane translation result in the maximum wind waves occurring on the right side of the hurricane track and propagating in the direction parallel to the hurricane translation direction, consistent with field observations.

Recent calculations by Lin and Perrie (1997) on the surface wave spectral energy fluxes due to the wave nonlinearity in deep and shallow water appeared after previously published works by Krasitski (1993), Shrira et al. (1996), and Kalmykov (1993, 1995, 1997), while presenting results that are qualitatively different from those obtained previously. This comment is on these obvious differences and why it appears that the conclusions of Lin and Perrie cannot be justified. Surface waves in deep water as well as in shallow water are very well described by the four-wave kinetic equation as shown by Hasselmann and Hasselmann (1985) and Herterich and Hasselmann (1980). Analogous computation of the five-wave kinetic equation for deep and shallow water show that a five-wave contribution is very small: only 3%–5% of the fourwave one (Kalmykov 1998). Therefore we can conclude that the four-wave kinetic equation still remains dominant for the shallow and deep water wave modeling and that all this discussion is of only academical interest. During the past 15 years many studies have been made in this area, some of which are not cited by Lin and Perrie (1997), including Parts I and II of the series leading to the present article under discussion (Part III). They present results of their own calculations of the spectral transfer rates in a JONSWAP spectrum, which are qualitatively different from various previously published results, while offering no explanation for the differences found. The subject of five wave–wave interactions among surface gravity waves is not new. First discussions concerning the energy transfer by five-wave interactions in wave spectra took place at Zakharov’s seminar in 1993 in Moscow, where the present author made a report (Kalmykov 1993). In experiments, it was first

§1.1 I n t r o d u c t i o n Two equations t ha t have been studied intensively in recent years are the Korteweg-de Vries (KdV; 1805) equation, uT + Buu x + Uxx x = 0 , (1) and a generalization of it due to Kadomtsev and Petviashvili (KP; 1970}, (u~ + 6u~ x + ~xxx)x + 3 ~ = 0. (3) Most of this interest is due to the remarkable fact t ha t each equation can be solved exactly as an initialvalue problem by a method now known as the Inverse Scattering Transform. This method was first discovered for the KdV equation on c ~ < X < vo, in the famous papers of Gardner, Greene, Kruskal and Miura (1967, lg74). The corresponding work for (1) on a periodic interval was published by several people during lg74-1978 [Novikov (lg74), Dubrovin and Novikov (1974), Dubrovin (1975), Lax (1975), Its and Matveev (1975), McKean and van Moerbeke (lg75), McKean and Trubowitz (1976), Dubrovin, Matveev and Novikov (1976)]. For the KP equation on ~ < X, rl < c~, a method of solution was given very recently by Ablowitz, Bar Yaaeov, and Fokas (1982); e/. Ablowitz and Fokas, in these Proceedings). As we shall see below, it happens tha t both (1) and (2) also model the evolution of water waves of moderate amplitude as they propagate in one direction in relatively shallow water. In physical By Harvey Segur, Allan Finkel (Institute for Advanced Study, Princeton), and Hilda Philander (A.R.A.P., Princeton). 1.2. Derivat ion of the equations 213 z :t] (x, y , t ) g ~ z =0 ~ : \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ z : h Figure 1.1. Physical configuration showing notat ion for Equations (3)-(6). terms, the KdV equation arises if the waves are strictly one-dimensional (i.e., one spatial dimension plus time), while the KP equation arises if they are only nearly one-dimensional. Because both equations are completely integrable, they provide very precise predictions about the evolution of water waves under appropiate conditions. This paper has two objetives. The first is to examine the validity of (1) and (2) as models of water waves, by comparing their solutions with some of the available experimental data. The comparison given here will be brief, but much more detailed verifications of (1) have been made elsewhere (e.g., Hammack and Segur, 1974, 1978). The second objetive is to describe in detail a special family of solutions of (2), which are periodic in each of two independent variables. It appears tha t these doubly-periodic solutions may have great practical importance as models of long water waves. Among other things, they describe: L the nonlinear interaction of two trains of finite amplitude waves in shallow water; iL the reflection of a t rain of finite-amplitude shallow-water waves from a vertical wall (by replacing an appropriate line of symmetry with the wall); IlL the reflection of such a wave t rain by a change in bot tom topography; or iv. typical finite-amplitude, short-crested waves in shallow water. In this last sense, they are the natural generalizations to two dimensions of cnoidal waves in one dimension. §1.2 D e r i v a t i o n of t h e e q u a t i o n s The classical problem of water waves is to find the irrotational motion of an inviseid, incompressible, homogeneous fluid, subject to a constant gravitational force (g). The fluid rests on a horizontal and impermeable bed of infinite extent at z --~ h and has a free surface at z = f(x, y, t); see Figure 1.1. In this derivation we neglect the effects of surface tension at the free surface, al though it can be included without difficulty (e.g., see Ch. 4 of Ablowitz and Segur, 1981). The fluid has a velocity potencial, ~b, which satisfies VZ¢---0 , h < z < f ( z , y , t ) ; (3) (irrotational motion of an incompressible fluid). It is subject to boundary conditions on the bottom, Z ~ h : Cz = o, (4) (impermeable bed); and along the free surface, z ~f: D~ Dt s't + ¢~s'~ + ~b~'y = ~ (5)

Impulse waves in reservoirs generated by landslides into shallow water

Temporal coherence of acoustic signals determines the processing of a sonar system for various applications. This paper studies the frequency and range dependence of the temporal coherence time of signal propagation in deep water versus that in shallow water. The signal coherence time has been measured in deep water since the 1970s, in terms of a parameter called the phase rate (the rate of phase change of the signal). Signal coherence time in shallow water has received little attention until only recently. We compare measurement data from more than a dozen of experiments covering a wide range of frequencies at different source‐receiver ranges in deep and shallow water. We find that the signal coherence time in both shallow and deep water can be fitted with a universal equation, which exhibits a 3/2 power frequency dependence and 1/2 power range dependence. The signal coherence time in shallow water is two to five times longer than the signal coherence time in deep water at the same frequency and range. The signal coherence times are comparable between shallow and deep water if measured in terms of the range‐to‐depth ratio. [This work is supported by the U.S. Office of Naval Research.]

Forecasting water waves and currents in near shore and off shore regions of the seas and oceans is essential to maintain and protect our environment and man made structures. In wave hydrodynamics, waves can be classified as shallow and deep water waves based on its water depth. The mathematical models of these waves are shallow water and free surface gravity water wave equations which describe the hydrodynamics of waves and currents near shore and off shore regions of seas and oceans. The complexity in these models exist as moving boundaries whose position depends on the solution of the governing equations. For shallow water waves, it is the shore line boundary where the water depth falls dry and for deep water waves, it is the free surface which separates the sea or ocean from atmospheric air. It is often difficult to solve these wave equations analytically while solving them numerically in an efficient and accurate way is a challenging task because of the moving boundaries. The numerical challenges are two fold: one is to develop a numerical method which is accurate and efficient for deforming grids and the other is to design a numerical algorithm for the grid adaptation following the moving boundaries. In this thesis, we aimed at first developing space-time discontinuous Galerkin finite element schemes for shallow water and free surface gravity water wave equations which are accurate and efficient for deforming grids. The shallow water equations are a leading order hydrodynamic model for coastal waves and currents. This is because they can exhibit the complicated flooding and drying phenomena due to the moving shore line boundary, and the wave breaking phenomena in the form of bores. A new space-time discontinuous Galerkin (DG) discretization is first presented for the (rotating) shallow water equations over varying topography and fixed boundaries. We formulated the discretization in an efficient and conservative way with the numerical HLLC flux on the finite element boundaries. We also designed a novel way to apply numerical dissipation around discontinuities, that are present in the form of bores, with the help of Krivodonova's discontinuity indicator such that the spurious oscillations are suppressed. The non-linear algebraic system resulting from the space-time discretization is solved using a pseudo-time integration method. A thorough verification of the space-time DG finite element method is undertaken by comparing the numerical and exact solutions. We carried out a discrete Fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is verified and validated for a number of problems arising in geophysical flows. To demonstrate that the space-time DG method is particularly suitable for problems with dynamic grid motion, we simulated nonlinear waves generated by a wave maker and verified these for low amplitude waves where linear theory is approximately valid. Free surface gravity water wave equations is widely used in marine and offshore engineering to model waves. The mathematical nature of these equations is complex because it consists of a potential flow equation which is of elliptic nature and nonlinear free surface boundary conditions which are hyperbolic in nature. Hence, a space-time discontinuous Galerkin finite element method is presented for simplified linear free surface gravity water waves. The free surface gravity water wave equations also arise from Luke's variational formulation which is associated with the conservation of energy and phase space, under suitable boundary conditions. This variational formulation also provided a basis to obtain a novel space-time variational (dis)continuous Galerkin finite element method. Both the space-time discontinuous Galerkin and the space-time variational finite element discretizations result in an algebraic linear system of equations with a very compact stencil, i.e., the algebraic equations from each element is coupled to its immediate neighboring elements only. Thus, the linear system of equations are built using an efficient block sparse matrix storage routine and solved by using iterative linear solvers using a well-tested PETSc package. Numerical schemes are verified for harmonic waves in a periodic domain and generated in a wave basin. Extension of the space-time discontinuous Galerkin method for flooding and drying in shallow water waves and nonlinear free surface evolution of deep water waves will be the topic of future research.

In this paper a new shallow water wave model is described which uses nonlinear dissipation derived from turbulent diffusion as damping mechanism. The source functions of the model are presented in detail. Analytical results of the dynamical equation for simple cases illustrate basic features of the model. Academic test runs in deep and shallow water are performed. The designed cases are identical to the ones used in previous wave model intercomparison studies and thus allow comparison with other wave models. Results of a hindcast of a North Sea storm event illustrate the model behaviour in nonuniform real shallow water systems. In this case we can compare with field data and with the community wave model WAM cy. 4, which has been run parallel to our model. Our study shows that the concept of wave modelling with nonlinear dissipation is consistent with common knowledge of wave evolution in oceanic and shelf sea applications.

Juan is recorded as one of the most damaging storms in the modern history of Nova Scotia, Canada. In this paper, the spectral evolution characteristics of waves generated by hurricane Juan are studied, based on the observed 1D wave spectra along Juan’s track in deep open ocean waters (buoy 44137) and the 2D wave spectra in shallow coastal waters at the directional waverider (DWR) location. Valuable results are obtained for cyclone-generated wave spectral properties, in both deep and shallow waters. In deep water, as illustrated at buoy 44137, the spectral variation, spectra pattern development, spectral peak frequency, cut-off frequency coefficient and high frequency spectral tail of the wave spectra are analyzed, before, during and after the cyclone’s passing. Thus, the spectral variation characteristics during the entire cyclone processe are obtained. Properties of the high frequency spectral tail are discussed, using average frequency and peak frequency as the cut-off frequency parameters under different cut-off coefficient conditions, respectively. We suggest reasonable values for the cut-off frequency parameter. Cyclone-generated 2D wave spectra in shallow water (at DWR location) are investigated, shoaling effects, 2D spectral pattern variations, swell and wind waves spectral evolution. Our study shows the invalidity of presently accepted spectral formulae, in describing cyclone-generated waves.

Growth responses of Nymphoides indica seedlings and vegetative propagules along a water depth gradient

Abstract. Accurate description of the wind energy input into ocean waves is crucial to ocean wave modeling and a physics-based consideration on the effect of wave breaking is absolutely necessary to obtain such an accurate description. This study evaluates the performance of an improved formula recently proposed by Xu and Yu (2020), who took into account not only the effect of breaking but also the effect of airflow separation on the leeside of steep wave crests in a reasonably consistent way. Numerical results are obtained through coupling an enhanced atmospheric wave boundary layer model with the ocean wave model WAVEWATCH III (v5.16). The coupled model has been extended to be valid in both deep and shallow waters. Duration-limited waves under controlled normal conditions and storm waves under practical hurricane conditions are studied in detail to verify the improved model. Both the representative wave parameters and the parameters characterizing the wave spectrum are discussed. It is shown that the improved source-term package for the wind energy input and the wave energy dissipation leads to more accurate results under all conditions. It performs evidently better than other standard source-term options of ST2, ST4 and ST6 embedded in WAVEWATCH III. It is also demonstrated that the improvement is particularly important for waves at their early development stage and waves in shallow waters.

Significant wave height (SWH) in shallow waters is assessed by generating two wave hindcasts; the first uses ERA-Interim wind fields and the second one from ERA5 to quantify the improvement of the ERA5 surface winds on the SWH representativeness, both in deep and shallow waters along the Chilean coastline. Additionally, wind field predictions from the Global Forecast System (GFS) were used to assess the representativeness of shallow waters. Oceanographic buoys were used to validate SWH in deep waters, while Acoustic Doppler Current Profiler (ADCPs) was equipped to measure waves in shallow waters. Energy spectrums coupling Wavewatch III and Simulating Waves Nearshore (SWAN) models were transferred to evaluate the performance of shallow water simulations. In general, the SWH from both wave hindcasts showed good performance. Nonetheless, those forced by ERA5 presented a better qualitative comparison of sea state temporal variability, which increased the correlation coefficients (&gt;0.9), coefficients of determination (&gt;0.8), and minor errors (RMSE, MAE, and BIAS) compared to oceanographic buoys and ADCPs. Additionally, in simulations forced by GFS, the temporal variability of the waves in shallow waters was successfully reproduced. Nevertheless, an increase in the RMSE, MAE, and BIAS error was statistically verified compared to ERA-Interim and ERA 5.

Relationships between narrow band directional energy spectra and probability densities of deep and shallow water waves

Fixed offshore structures are operated in shallow water depths up to 500 feet and are often subjected to huge wave loading. The huge waves destabilize the structures, which then can cause widespread damage to the local ecology, coastal towns, and the environment. Mitigating the impact of wave loading requires the accurate prediction of wave induced forces, which is used for the structural assessment. Determination of wave induced forces requires solution of two problems. The prediction of wave kinematics by using an appropriate wave theory and then the prediction of the pressure and viscous forces due to the wave impact loading. This paper focuses on the hydrodynamic wave loading of fixed offshore structures in shallow water. Accurate modeling of huge waves in shallow water is more challenging compared to deep water due to higher relative wave height (wave height to water depth ratio). In this paper, we propose a Computational Fluid Dynamics (CFD) solution using Volume of Fluid (VOF) method and fifth order solitary wave theory for modeling huge waves in shallow water with very high accuracy. This model is validated with experiment for the physical mechanisms in the wave loading of the structures such as wave propagation, run up and interactions. Finally, this model is used for the wave- in-deck analysis of a fixed offshore oil rig in shallow water. In this study, we use a 20m (65.6 feet) wave in 41m (134.5 feet) water depth (Relative wave height of 0.49), which is not possible to model using Airy and Stokes wave theories. The wave-in-deck analysis is carried out for three different wave heights and the effect of wave height on the wave run-up and loads is analyzed. Introduction Hydrodynamic wave loading on fixed offshore structures has been an issue of concern to the offshore oil and gas industry. A huge wave hitting the offshore platform leads to high wave-in-deck loads that can eventually result in significant platform damage and collapse. Fatalities and damages costing hundreds of millions of dollars can occur. From 2004–2008, five major hurricanes (Ivan, Katrina, Rita, Gustav, Ike) destroyed 180 structures and 1,070 wells in Gulf of Mexico [Kaiser, M.J, 2011]. About 50 Russian crew members were killed after a jack up oil rig capsized and sank in a 6m (19.68feet) wave hitting [News: Reuters, December 2011]. Some of the possible causes of the accidents are:Some areas of the Gulf of Mexico floor have experienced several feet of subsidence, which leads to lower deck height and are more vulnerable to wave loading [Laurendine, T., 2007].The wave crest height according to RP 2A of the American Petroleum Institute (API) is higher than the lower deck elevations of many existing platforms [Bea, R.G., et all,1999]With the occurrence of a tropical storm or hurricane, the wave height exceeds the design height.Old structures are designed to a lower environmental criterion and have lower strength characteristics. So a structural assessment is necessary to determine whether the structure can withstand the peak loads during the huge wave impact. The wave-in-deck loading is very complex and difficult to model with traditional analytical tools. It is also very difficult to assess the loading very accurately in physical model tests carried out in a small scale wave basin, which introduces inaccuracies in the measurements [Grønbech, M. J.et all. 2011]. With the recent advancements in CFD and increased computing power, CFD can be a valuable tool for the assesment of the structures subjected to wave loading.

Hogner model of wave interferences for farfield ship waves in shallow water