Modeling of propagation and transformation of transient nonlinear waves on a current

A novel theoretical approach is applied to predict the propagation and transformation of transient nonlinear waves on a current. The problem was solved by applying an eigenfunction expansion method and the derived semi-analytical solution was employed to study the transformation of wave profile and the evolution of wave spectrum arising from the nonlinear interactions of wave components in a wave train which may lead to the formation of very large waves. The results show that the propagation of wave trains is significantly affected by a current. A relatively small current may substantially affect wave train components and the wave train shape. This is observed for both opposing and following current. The results demonstrate that the application of the nonlinear model has a substantial effect on the shape of a wave spectrum. A train of originally linear and very narrow-banded waves changes its one-peak spectrum to a multi-peak one in a fairly short distance from an initial position. The discrepancies between the wave trains predicted by applying the linear and nonlinear models increase with the increasing wavelength and become significant in shallow water even for waves with low steepness. Laboratory experiments were conducted in a wave flume to verify theoretical results. The free-surface elevations recorded by a system of wave gauges are compared with the results provided by the nonlinear model. Additional verification was achieved by applying a Fourier analysis and comparing wave amplitude spectra obtained from theoretical results with experimental data. A reasonable agreement between theoretical results and experimental data is observed for both amplitudes and phases. The model predicts fairly well multi-peak spectra, including wave spectra with significant nonlinear wave components.     

The steepness and shape of wind waves

Variations are found in the shape and the steepness of wind-generated surface gravity waves between very young waves, such as seen in a laboratory tank, and larger waves of various wave ages encountered at sea as the result of wind stress over larger fetches. These differences in the characteristic shape of wind waves are presented as a function of the wave age. The wave steepness is also expressed as a function of wave age, the measurement of which is consistent with the 3/2-power law connecting wave height and characteristic period, normalized by the air friction velocity.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

波群内单个波的波陡分布与波破碎

对波群内单个波的波陡分布和波破碎进行了实验研究。研究结果是,波群中波动的最大振幅出现在波群前部而不是出现在波群中央,这种不对称性导致波群前部单个波出现大波陡的概率大于后部单个波出现大波陡的概率;进一步的波破碎统计发现波群前部单个波破碎的频率是后部单个波破碎频率的4倍。因此认为,波群结构的不对称性能够导致单个波发生破碎的...     

Parametric modelling of joint probability density distributions for steepness and asymmetry in deep water waves

Traditional wave steepness s = H/L does not define steep asymmetric waves uniquely. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Parametric models for joint probability density distributions for deep water waves are presented. The joint distributions are for crest front steepness-wave height, vertical asymmetry factor-wave height, total wave steepness-wave height and wave height-wave period. The parametric models are estimated from zero-downcross analysis of wave data obtained from measurements at sea on the Norwegian continental shelf. The results of the analysis presented here can be used in the estimation of the probabilities of occurrence of steep asymmetric waves and breaking waves in deep water. Thus the results are useful for the practical naval architect and ocean engineer who are considering unusual events in the sea, the associated accidents or responses and the probability of occurrence of such events.     

Nonlinear effects occurring during the propagation of trapped topographic waves

Baroclinic topographic waves trapped by a sloping bottom in the case of a real stratification are considered. The dispersion properties of these waves are studied. The characteristic scales and amplitudes of trapped topographic waves observed in the Norwegian Sea are determined. The asymptotic method of multi-scale expansions is used to study nonlinear effects occurring during the propagation of these waves. The wave-induced mean flow is determined in the second order of smallness in the wave amplitude. The evolution equation for the envelope—the nonlinear Schrödinger equation—is derived. Modulation instability of these waves is examined. It is shown that trapped topographic waves are modulationally unstable.     

Modeling of the eddy viscosity by breaking waves

Breaking wave induced nearsurface turbulence has important consequences for many physical and biochemical processes including water column and nutrients mixing,heat and gases exchange across air-sea interface.The energy loss from wave breaking and the bubble plume penetration depth are estimated.As a consequence,the vertical distribution of the turbulent kinetic energy(TKE),the TKE dissipation rate and the eddy viscosity induced by wave breaking are also provided.It is indicated that model results are found to be consistent with the observational evidence that most TKE generated by wave breaking is lost within a depth of a few meters near the sea surface.High turbulence level with intensities of eddy viscosity induced by breaking is nearly four orders larger than υwl(=κuwz),the value predicted for the wall layer scaling close to the surface,where uw is the friction velocity in water,κ with 0.4 is the von Kármán constant,and z is the water depth,and the strength of the eddy viscosity depends both on wind speed and sea state,and decays rapidly through the depth.This leads to the conclusion that the breaking wave induced vertical mixing is mainly limited to the near surface layer,well above the classical values expected from the similarity theory.Deeper down,however,the effects of wave breaking on the vertical mixing become less important.     

The effect of the Airy phase during the propagation of edge waves

Peculiarities of the edge wave evolution a great distance away from the source are considered. It is shown that waveguide dispersion may result in discrimination of the region in a wave train corresponding to the Airy phase. The effect of the discrimination of the Airy wave in the remote zone for the model of a step-like shelf is compared with real edge waves (using the marigrams of tsunami observed in the Sea of Japan in 1983).Translated by Mikhail M. Trufanov.     

The propagation of high frequency internal waves in the Celtic Sea

The continental slope to the south of the Celtic Sea is an area of extremely rough topography and tidal currents of the order of 50cm/s (with components both along and across the slope). This is a region of intense and complicated internal wave and internal tide activity. Historical current meter data from moorings close to the shelf-break show bursts of high frequency, large amplitude internal waves occurring, on average, at either once or twice per M2 tidal cycle. Wave packets at 9 moorings along the shelf-break and further on-shelf are identified using conditional sampling. The paths travelled by these wave packets are calculated using their fluctuation orientation, linear wave theory and the low frequency current. The records are up to 60 days long, allowing the ensemble statistics of propagation direction and wave characteristics to be calculated for a large number of wave packets. This analysis shows that only a fraction of the observed wave packets have orientations consistent with generation by the across-slope barotropic tide. This mechanism accounts for 20% of the wave packets in the north-west Celtic Sea and 29% in the southeast Celtic sea. A similar fraction of the wave packets (23% in the north west and 27% in the south east) have orientations clearly consistent with generation by an along-slope flow over the rough topography on the slope. The remaining wave packets are attributed to generation by tidal flow over topography close to the moorings and possibly internal wave resonance within canyons.     

Extended KdV equation for transient axisymmetric water waves

The equation for non-linear water waves in shallow water with cylindrical symmetry is established in the form of the Extended KdV (EKdV) equation. The method of solution is based on the Split Step Fourier Algorithm. It is applicable in both directions; that is, given a wave record near the origin, the theory predicts the wave evolution. Similarly, given a wave record at a distance from the origin, the method is able to predict the original wave near the origin. Theory is applied to and verified by transient wave records obtained from underwater explosions and by dropping a large cylindrical plate in very shallow water. The difference between the KdV and the Extended KdV equation is emphasized for small valuev of Ursell parameter.     

海啸俘获波在弯曲海脊上的传播规律

越洋海啸会受大洋海脊的引导以俘获波的形式沿其传播上万千米,其携带的巨大能量会严重影响远场地区、威胁海岸地区的安全。本文基于MIKE21-BW模型,分别模拟0°(直海脊)至90°(直角弯曲海脊)不同弯曲角度海脊上俘获波的传播变形过程,并定量比较其能量分配。结果表明,海脊俘获波传至海脊转弯处,少部分能量会泄露出海脊重新以自由波的形式扩散至整个海域;部分能量会被反射回来形成与初始海啸波相反方向的俘获波沿海脊传播,反射的能量会随着海脊弯曲角度的增加而增加;还有一部分能量继续沿着弯曲的海脊向前传播,其随着海脊弯曲的角度增加而减小。     

Coupled acoustic mode propagation through continental-shelfinternal solitary waves

Three techniques are used to investigate mode coupling as acoustic energy passes through continental-shelf internal solitary waves (ISW's). Results from all techniques agree. The waves considered here are single downward undulations of a thermocline layer separating upper and lower well-mixed layers. Two techniques are numerical: parabolic equation (PE) solution and a sudden approximation joining range-invariant regions at sharp vertical interfaces. The third technique is an analytic derivation of ISW scale lengths separating adiabatic (at large scale) and coupled-mode propagation. Results show that energy is exchanged between modes as ISW's are traversed. The sharp interface solutions help explain this in terms of spatially confined coupling and modal phase interference. Three regimes are observed: 1) for short ISW's, coupling upon wave entrance is reversed upon exit, with no net coupling; 2) for ISW scales of 75-200 m, modal phase alteration averts the exit reversal, giving net coupling; transparent resonances yielding no net coupling are also observed in this regime; and 3) for long ISW's, adiabaticity is probable but not universal. Mode refraction analysis for nonparallel acoustic-ISW alignment suggests that these two-dimensional techniques remain valid for 0° (parallel) to 65° (oblique) incidence, with an accordant ISW stretching     

Finite element analysis of two-dimensional non-linear transient water waves

The two-dimensional nonlinear time domain free surface flow problem is analyzed by the finite element method. Two approaches are used. One is based on the velocity potential which is approximated by means of shape functions. The solution is obtained through use of a variational statement, and the velocity is obtained subsequently by the Galerkin method. The other approach is to write both potential and velocity in terms of the shape functions at the same time. Their solutions are derived from the same equation by using another variational statement. Numerical results are given for the vertical wave maker problem and for a transient wave in a rectangular container. They are compared with analytical solutions, and very good agreement is found.     

Simulation of small-amplitude frequency-dispersive transient waves by means of the mild-slope equation

A numerical model based on the mild-slope equation is applied to reproduce the propagation of small-amplitude transient waves. The model makes use of the Fourier Transform to convert the time-dependent hyperbolic equation into a set of elliptic equations in the frequency domain. The results of two available experimental studies on tsunamis generated by landslides are used to validate the model, which appears to be able of carefully reproducing the effects of the frequency dispersion. An example application of tsunamis propagating around the Stromboli island is also presented to show the applicability of the present approach to real life scenarios. It is finally discussed how this model could be applied as support to a tsunami early warning system.     

Laboratory modeling of the propagation of periodic internal waves over bottom slopes

The experimental investigation of the run-up of periodic internal waves in a two-layer fluid on the coastal slope is performed in an open hydrochannel at the Physical Department of the Lomonosov Moscow State University. The waves are produced by a wave generator. We study the transformation of waves, the vertical structure of the field of velocities of mass transfer, and the behavior of the parameters of internal waves propagating over the sloping bottom. It is shown that the run-up and breaking of internal waves are accompanied by periodic emissions of portions of the heavier fluid from the bottom layer upward along the slope. The Stokes drift velocity changes its sign as a function of depth. Moreover, both the wave length (the horizontal distance between the neighboring crests) and the height of waves over the sloping bottom (the elevation of the crest over the slope along the vertical) decrease as the wave approaches the coast.     

A model for the propagation of nonlinear surface waves over viscous muds

The effect of a thin viscous fluid–mud layer on nearshore nonlinear wave–wave interactions is studied using a parabolic frequency-domain nonlinear wave model, modified to incorporate a bottom dissipation mechanism based on a viscous boundary layer approach. The boundary-layer formulation allows for explicit calculation of the mud-induced wave damping rate. The model performed well in tests based on laboratory data. Numerical tests show that damping of high frequency waves occurs, mediated by “difference” nonlinear interactions. Simulations of 2-dimensional wave propagation over a mud “patch” of finite extent show that the wave dissipation causes significant downwave diffraction effects.     

Effects of internal waves on sound pulse propagation in the Straitsof Florida

An unexplained result of broad-band transmission experiments made more than ten years ago by DeFerrari in the Straits of Florida (center frequency ~500 Hz, bandwidth ~100 Hz, water depth ~200-m, range ~20 km) is that the measured pulse response functions failed to show the expected multipath replicas of the transmitted pulse and instead were smeared into a single broad cluster (duration ~50-~350 ms) in which the unresolved multipaths fluctuated rapidly in geophysical time (coherence time ≪12 min) leaving only a relatively stable envelope that is useful for oceanographic inversion. It is demonstrated here that the effects of internal waves on sound pulse propagation in the Straits of Florida can explain these observed results, and it is suggested that similar instabilities of acoustic multipaths due to internal waves are to be expected in other shallow-water propagation conditions. The demonstration is based on numerical simulations with the broad-band UMPE acoustic model that includes multiple forward scattering from volume inhomogeneities induced by internal wave fluctuations that are described by a broad spectrum of excitation. The simulated temporal variability, stability, and coherence of acoustic pulse arrivals are displayed on geophysical time scales from seconds to many hours and are qualitatively in agreement with the measured data in the Straits of Florida     

Verification of a parabolic equation for propagation of weakly-nonlinear waves

Verification tests of linear wave propagation models for cases of waves propagating through a refractive focus typically overpredict peak amplitudes in the vicinity of the focus. Using a parabolic equation model for the combined refraction-diffraction of weakly-nonlinear waves [valid for Ursell number U_r < O(1)], we show that, in a particular experiment, much of the discrepancy between laboratory measurements and linear wave model predictions can be accounted for by including the effect of nonlinearity, which is important in the region of the focus.     

Pseudorandom realization of transient acoustic waves scattered fromrandomly rough patches

A simple numerical technique is developed for generating pseudorandom realizations of three-dimensional (3-D) transient acoustic waves that are scattered from two-dimensional (2-D) patches of randomly rough surfaces. The rough surface height of a patch is represented numerically in the 2-D horizontal wavenumber plane by choosing a scheme for interpolation between pseudorandom complex coefficients. Using this approach, the realizations of the patches can be generated from experimentally measured roughness power spectra, and phase information is generated in the frequency domain that leads to time spreads in the time domain. The acoustic scattering is modeled here with first-order perturbation theory. The boundary conditions considered here are pressure-release, rigid, and fluid-fluid. Three different spatial windows are considered for defining the patches. In the time domain, the time spreads of the scattered waveforms agree with predictions. In the frequency domain, the phase is seen as a random walk. The solutions developed here can be used with normal mode propagation models or ray propagation models     

Modeling of nonlinear wave propagation over fringing reefs

The applicability of existing nonlinear (triad) spectral models for steep slopes (0.1–0.2) characteristic of reef environments was investigated, using both deterministic (phase-resolving) and stochastic (phased-averaged) formulations. Model performance was tested using laboratory observations of unidirectional wave transformation over steep and smooth bathymetry profiles. The models, developed for mild slopes, were implemented with minimal modifications (the inclusion of breaking parametrizations and linear steep-slope corrections) required by laboratory data. The deterministic model produced typically more accurate predictions than the stochastic one, but the phase averaged formulation proved fast enough to allow for an inverse modeling search for the optimal breaking parametrization. The effects of the additional assumptions of the stochastic approach resulted in a slower than observed evolution of the infragravity band. Despite the challenge posed by the fast wave evolution and energetic breaking characteristic to the steep reef slopes, both formulations performed overall well, and should be considered as good provisional candidates for use in numerical investigation of wave–current interaction processes on steep reefs.