Numerical modeling of water wave interaction with a soft mud bed

A vertical 2-D numerical model is presented for simulating the interaction between water waves and a soft mud bed. Taking into account nonlinear rheology, a semi-empirical rheological model is applied to this water-mud model, reflecting the combined visco-elasto-plastic properties of soft mud under such oscillatory external forces as water waves. In order to increase the resolution of the flow in the neighborhood of both sides of the inter-surface, a logarithmic grid in the vertical direction is employed for numerical treatment. Model verifications are given through comparisons between the calculated and the measured mud mass transport velocities as well as wave height changes.

Influence of mean water depth and a subsurface sandbar on the onset and strength of wave breaking

Wave breaking in the open ocean and coastal zones remains an intriguing yet incompletely understood process, with a strong observed association with wave groups. Recent numerical study of the evolution of fully nonlinear, two-dimensional deep water wave groups identified a robust threshold of a diagnostic growth-rate parameter that separated nonlinear wave groups that evolved to breaking from those that evolved with recurrence. This paper investigates whether these deep water wave-breaking results apply more generally, particularly in finite-water-depth conditions. For unforced nonlinear wave groups in intermediate water depths over a flat bottom, it was found that the upper bound of the diagnostic growth-rate threshold parameter established for deep water wave groups is also applicable in intermediate water depths, given by k(0) h greater than or equal to 2, where k(0) is the mean carrier wavenumber and h is the mean depth. For breaking onset over an idealized circular arc sandbar located on an otherwise flat, intermediate-depth (k(0) h greater than or equal to 2) environment, the deep water breaking diagnostic growth rate was found to be applicable provided that the height of the sandbar is less than one-quarter of the ambient mean water depth. Thus, for this range of intermediate-depth conditions, these two classes of bottom topography modify only marginally the diagnostic growth rate found for deep water waves. However, when intermediate-depth wave groups ( k(0) h greater than or equal to 2) shoal over a sandbar whose height exceeds one-half of the ambient water depth, the waves can steepen significantly without breaking. In such cases, the breaking threshold level and the maximum of the diagnostic growth rate increase systematically with the height of the sandbar. Also, the dimensions and position of the sandbar influenced the evolution and breaking threshold of wave groups. For sufficiently high sandbars, the effects of bottom topography can induce additional nonlinearity into the wave field geometry and associated dynamics that modifies the otherwise robust deep water breaking-threshold results.

On the nonlinear integral transform of an ocean wave spectrum into an along-track interferometric synthetic aperture radar image spectrum

We present a new nonlinear integral transform relating the ocean wave spectrum to the along-track interferometric synthetic aperture radar (AT-INSAR) image spectrum. The AT-INSAR, which is a synthetic aperture radar (SAR) employing two antennas displaced along the platform's flight direction, is considered to be a better instrument for imaging ocean waves than the SAR. This is because the AT-INSAR yields the phase spectrum and not only the amplitude spectrum as with the conventional SAR. While the SAR and AT-INSAR amplitude spectra depend strongly on the modulation of the normalized radar cross section (NRCS) by the long ocean waves, which is poorly known, the phase spectrum depends only weakly on this modulation. By measuring the phase difference between the signals received by both antennas, AT-INSAR measures the radial component of the orbital velocity associated with the ocean waves, which is related to the ocean wave height field by a well-known transfer function. The nonlinear integral transform derived in this paper differs from the one previously derived by Bao et al. [1999] by an additional term containing the derivative of the radial component of the orbital velocity associated with the long ocean waves. By carrying out numerical simulations, we show that, in general, this additional term cannot be neglected. Furthermore, we present two new quasi-linear approximations to the nonlinear integral transform relating the ocean wave spectrum to the AT-INSAR phase spectrum.

Numerical wave flume study on wave motion around submerged plates

Nonlinear interaction between surface waves and a submerged horizontal plate is investigated in the absorbed numerical wave flume developed based on the volume of fluid (VOF) method. The governing equations of the numerical model are the continuity equation and the Reynolds-Averaged Navier-Stokes (RANS) equations with the k-epsilon turbulence equations. Incident waves are generated by an absorbing wave-maker that eliminates the waves reflected from structures. Results are obtained for a range of parameters, with consideration of the condition under which the reflection coefficient becomes maximal and the transmission coefficient minimal. Wave breaking over the plate, vortex shedding downwave, and pulsating flow below the plate are observed. Time-averaged hydrodynamic force reveals a negative drift force. All these characteristics provide a reference for construction of submerged plate breakwaters.

Outer wave number spectrum and statistical distribution of apparent wave number vector

Based on the ray theory and Longuet-Higgins's linear,model of sea waves, the joint distribution of wave envelope and apparent wave number vector is established. From the joint distribution, we define a new concept, namely the outer wave number spectrum, to describe the outer characteristics of ocean waves. The analytical form of the outer wave number spectrum, the probability distributions of the apparent wave number vector and its components are then derived. The outer wave number spectrum is compared with the inner wave number spectrum for the average status of wind-wave development corresponding to a peakness factor P = 3. Discussions on the similarity and difference between the outer wave number spectrum and inner one are also presented in the paper. (C) 2002 Elsevier Science Ltd. All rights reserved.

Statistical distribution of wave-surface elevation for second-order random directional ocean waves in finite water depth

Based on the second-order random wave solutions of water wave equations in finite water depth, a statistical distribution of the wave-surface elevation is derived by using the characteristic function expansion method. It is found that the distribution, after normalization of the wave-surface elevation, depends only on two parameters. One parameter describes the small mean bias of the surface produced by the second-order wave-wave interactions. Another one is approximately proportional to the skewness of the distribution. Both of these two parameters can be determined by the water depth and the wave-number spectrum of ocean waves. As an illustrative example, we consider a fully developed wind-generated sea and the parameters are calculated for various wind speeds and water depths by using Donelan and Pierson spectrum. It is also found that, for deep water, the dimensionless distribution reduces to the third-order Gram-Charlier series obtained by Longuet-Higgins [J. Fluid Mech. 17 (1963) 459]. The newly proposed distribution is compared with the data of Bitner [Appl. Ocean Res. 2 (1980) 63], Gaussian distribution and the fourth-order Gram-Charlier series, and found our distribution gives a more reasonable fit to the data. (C) 2002 Elsevier Science B.V. All rights reserved.

Ocean wave imaging mechanism by imaging radar

Analytical representations of the high frequency spectra of ocean wave and its variation due to the variation of ocean surface current are derived from the wave-number spectrum balance equation. The ocean surface imaging formulation of real aperture radar (RAR) is given using electromagnetic wave backscattering theory of ocean surface and the modulations of ocean surface winds, currents and their variations to RAR are described. A general representation of the phase modulation induced by the ocean surface motion is derived according to standard synthetic aperture radar (SAR) imaging theory. The detectability of ocean current and sea bottom topography by imaging radar is discussed. The results constitute the theoretical basis for detecting ocean wave fields, ocean surface winds, ocean surface current fields, sea bottom topography, internal wave and so on.

A parametric method of retrieving ocean wave spectra from synthetic aperture radar images

A parametric method that extracts the ocean wave directional spectra from synthetic aperture radar (SAR) image is presented. The 180 degrees ambiguity of SAR image and the loss of information beyond the azimuthal cutoff can be overcome with this method. The ocean wave spectra can be obtained from SAR image directly by using iteration inversion mapping method with forward nonlinear mapping. Some numerical experiments have been made by using ERS-1 satellite SAR imagette data. The ocean wave direction retrieved from SAR imagette data is in agreement with the wind direction from the scatterometer data.

The nonlinear evolution equation on the shallow water wave

Based on the variation principle, the nonlinear evolution model for the shallow water waves is established. The research shows the Duffing equation can be introduced to the evolution model of water wave with time.

Applications of improved 2-D numerical wave tanks in wave breaking

Song and Banner (2002, henceforth referred to as SB02) used a numerical wave tank (developed by Drimer and Agnon, and further refined by Segre, henceforth referred to as DAS) to study the wave breaking in the deep water, and proposed a dimensionless breaking threshold that based on the behaviour of the wave energy modulation and focusing during the evolution of the wave group. In this paper, two modified DAS models are used to further test the SB02's results, the first one (referred to MDAS1) corrected many integral calculation errors appeared in the DAS code, and the second one (referred to MDAS2) replaced the linear boundary element approximation of DAS into the cubic element on the free surface. Researches show that the results of MDAS1 are the same with those of DAS for the simulations of deep water wave breaking, but, the different values of the wavemaker amplitude, the breaking time and the maximum local average energy growth rate delta(max) for the marginal breaking cases are founded by MDAS2 and MDAS1. However, MDAS2 still satisfies the SB02' s breaking threshold. Furthermore, MDAS1 is utilized to study the marginal breaking case in the intermediate water depth when wave passes over a submerged slope, where the slope is given by 1 : 500, 1 : 300, 1 : 150 or 1 : 100. It is found that the maximum local energy density U increases significantly if the slope becomes steeper, and the delta(max) decreases weakly and increases intensively for the marginal recurrence case and marginal breaking case respectively. SB02's breaking threshold is still valid for the wave passing over a submerged slope gentler than 1 : 100 in the intermediate water depth.

Wave breaking on turbulent energy budget in the ocean surface mixed layer

As an important physical process at the air-sea interface, wave movement and breaking have a significant effect on the ocean surface mixed layer (OSML). When breaking waves occur at the ocean surface, turbulent kinetic energy (TKE) is input downwards, and a sublayer is formed near the surface and turbulence vertical mixing is intensively enhanced. A one-dimensional ocean model including the Mellor-Yamada level 2.5 turbulence closure equations was employed in our research on variations in turbulent energy budget within OSML. The influence of wave breaking could be introduced into the model by modifying an existing surface boundary condition of the TKE equation and specifying its input. The vertical diffusion and dissipation of TKE were effectively enhanced in the sublayer when wave breaking was considered. Turbulent energy dissipated in the sublayer was about 92.0% of the total depth-integrated dissipated TKE, which is twice higher than that of non-wave breaking. The shear production of TKE decreased by 3.5% because the mean flow fields tended to be uniform due to wave-enhanced turbulent mixing. As a result, a new local equilibrium between diffusion and dissipation of TKE was reached in the wave-enhanced layer. Below the sublayer, the local equilibrium between shear production and dissipation of TKE agreed with the conclusion drawn from the classical law-of-the-wall (Craig and Banner, 1994).

Traveling wave front for the Fisher equation on an infinite band region

Instead of discussing the existence of a one-dimensional traveling wave front solution which connects two constant steady states, the present work deals with the case connecting a constant and a nonhomogeneous steady state on an infinite band region. The corresponding model is the well-known Fisher equation with variational coefficient and Dirichlet boundary condition. (c) 2006 Elsevier Ltd. All rights reserved.

Application of effective medium approximation theory to ocean remote sensing under wave breaking

Based on the effective medium approximation theory of composites, the empirical model proposed by Pandey and Kakar is remedied to investigate the microwave emissivity of sea surface under wave breaking driven by strong wind. In the improved model, the effects of seawater bubbles, droplets and difference in temperature of air and sea interface (DTAS) on the emissivity of sea surface covered by whitecaps are discussed. The model results indicate that the effective emissivity of sea surface increases with DTAS increasing, and the impacts of bubble structures and thickness of whitecaps layer on the emissivity are included in the model by introducing the effective dielectric constant of whitecaps layer. Moreover, a good agreement is obtained by comparing the model results with the Rose's experimental data.