Reply on CC1

The effect of wave breaking on vertical diffusion coefficient was described by using a IDV numerical model with one-equation turbulence closure. The turbulent kinetic energy generated from the wave breaking was supplied at the water surface, and was diffused down into the whole water column. While the breaking-induced turbulence are diffused in the water column, the vertical diffusion coefficient was modified by the additional turbulent kinetic energy. The model was applied to five wave flow cases and three wave and current flow cases to assess the effect of the wave breaking and the current. The eddy viscosity or the vertical diffusion coefficient was obtained from a simple relationship between the eddy viscosity and the mixing length and the turbulent kinetic energy. The model results on the horizontal velocity profile, the turbulent kinetic energy profile, and the diffusion coefficient profile were compared with laboratory measurements, and showed reasonable agreement. The diffusion coefficient was significantly enhanced by about 40000% at the middle of wave boundary layer due to wave-breaking for CASE 1, while it was enhanced by about 200% due to a current for CASE 1-2.

Assessing the performance of wave breaking parameterizations in shallow waters in spectral wave models

In this research study, we focus on the generalized regularized long wave equation and the modified regularized long wave equation, which play pivotal roles in characterizing plasma waves in oceans and ion acoustic waves in shallow water, a domain deeply rooted in physical phenomena. Employing two computational techniques, namely, the optimal auxiliary function method and the Laplace iterative transform method, we approximate these equations. These formulas are used to characterize plasma waves in oceans and ion acoustic waves in shallow water. The results discovered have important ramifications for our comprehension of many physical events. Our results show that both methods are robust, easy to use, and successful. Both methods yield results that are satisfactory to each other. With the use of tables and graphs, we compared the two suggested approaches. The findings suggest that the suggested methods can be widely applied to explore other real-world problems.

The anomaly of radar sea spikes, defined here as the non-Bragg scattering events with backscattering cross section of horizontal polarization exceeding that of vertical polarization, has been associated with steep wave features possibly going through wave breaking process. This property is employed for remote detection of breaking waves. The results are used to quantify the effect of wave breaking on radar returns. Large increase due to breaking is found in the Doppler velocities of both polarizations (about 50% faster with breaking). The effect of breaking on the backscattering cross section is much stronger for the horizontal polarization (with 15 to 20 dB enhancement) and relatively small in the vertical polarization (on the order of 0.5 dB fluctuations). The presence of swell reduces the impact of breaking waves on radar return in comparison to the scattering from wind seas.

In this study, the potential Kadomtsev-Petviashvili (pKP) equation, which describes the oblique interaction of surface waves in shallow waters, is solved by the new extended direct algebraic method. The results of the study show that by applying the new direct algebraic method to the pKP equation, the behavior of the obliquely interacting surface waves in two dimensions can be analyzed. This article fairly clarifies the behaviors of surface waves in shallow waters. In the literature, several mathematical models have been developed in attempt to study these behaviors, with nonlinear mathematics being one of the most important steps; however, the investigations are still at a level that can be called ‘baby steps’. Therefore, every study to be carried out in this context is of great importance. Thus, this study will serve as a reference to guide scientists working in this field.

Numerous practical and mathematical techniques have been piloted to study ships’ behavior in deep water conditions with and without waves, and shallow water conditions without waves, while only limited investigations have been carried out to assess ships’ behavior in shallow waters with wave conditions as the flow around the stern region and appendages and the interaction effects are intricate. Therefore, this study attempts to understand the infrequently explored subset of a vessel’s behavior in regular waves in shallow water conditions (channel depth to ship draft ratio taken as 1.5). A container ship (S175) model scaled at 1:36 was the subject of a numerical study in which it was subjected to static and dynamic maneuver simulations in head sea conditions. The waves were induced using the dispersion relationship of waves in a given depth. The trends of forces and moments acting on the hull while undergoing maneuvering motions were obtained using a smooth particle hydrodynamics-based computational fluid dynamics solver. The resulting periodic trends of forces and moments were analyzed using the Fourier series method to extract the Fourier coefficients and, in turn, calculate the hydrodynamic derivatives. The trajectories in turning circle and zigzag maneuvers were also simulated using a MATLAB code. The results demonstrate an increase in trajectory parameters and improvement in counter maneuverability owing to the complex flow physics around the hull when encountering regular waves in shallow water conditions compared to waves in deep waters and a lack of waves in shallow waters.

Wave transport in the South Australian Basin

A numerical model of wave overtopping on seadikes

Under investigation in this paper is the Whitham–Broer–Kaup (WBK) system, which describes the dispersive long wave in shallow water. Through a variable transformation, the WBK system is casted into a general Broer–Kaup system whose Lax pair can be derived by the Ablowitz–Kaup–Newell–Segur technology. With symbolic computation, based on the aforementioned Lax pair, the N-fold Darboux transformation is constructed with a gauge transformation and the multi-soliton solutions are obtained. Finally, the elastic interactions of the two-soliton solutions (including the head-on and overtaking collisions) for the WBK system are graphically studied. Those multi-soliton collisions can be used to illustrate the bidirectional propagation of the waves in shallow water.

The scientific developments carried out in the last years with spaceborne synthetic aperture radar (SAR), as well as the comparisons with in-situ sensors, have demonstrated that SAR is a reliable remote sensing tool to study wave fields on the open ocean. In their so-called SAR image mode, SAR systems on board satellites are able to scan ocean areas about 100 × 100 km2 with a spatial resolution about 20 × 20 m2. These SAR images are able to provide information about the spatial variability of wave fields, as well as other phenomena that occur on the ocean surface, such as the local wind field. This work investigates the capabilities of spaceborne SAR to extract sea state information for those areas close to coastal locations, where the incoming wave fields present high spatial inhomogeneities. For this purpose, ERS-2 SAR and ENVISAT Advanced SAR (ASAR) images acquired over the North Sea and the Bay of Biscay are used to study ocean waves in shallow waters under different climate and oceanographic conditions. In addition, the potential of ENVISAT ASAR capability to scan the ocean surface with dual polarizations (horizontal, HH, and vertical, VV) is analyzed in this work. The relevance of these investigations for different practical applications concerning the analysis of ocean waves is discussed.

Displacements of mechanical waves superposed onto wind waves were measured with a laser displacement gauge in a wind-wave tank. The effects of wave breaking, especially the spilling breaking type, on the wave-variance spectra are investigated. In the absence of wave breaking, the quasi-equilibrium spectrum consists of an f−7/3 subrange in the capillary regime, and its spectral density increases with increasing wind speed. When intense spilling breaking occurs, the water surface is saturated with small-scale features that cause not only an increase in the spectral density but also a reduction in the slope of the spectrum at high frequencies. Velocity components under the water surface were measured with a laser Doppler velocimeter. The energy spectra of the vertical and longitudinal velocity components in breaking waves are practically identical in the frequency range near the dominant wave frequency. At higher frequencies, the spectra generally follow Kolmogorov's −5/3 law. In the intermediate frequency range, we observed a higher spectral density for the vertical velocity component than for its longitudinal counterpart. These results suggest that turbulence energy is transferred from the vertical component to the longitudinal component in breaking waves. The acceleration of the water motion becomes as large as gravitational acceleration when intense wave breaking takes place. The flow field in breaking waves is highly dissipative.

The effects of wave breaking on sediment transport are studied through a new series of mobile-bed experiments in a large-scale wave flume. During the campaign, one experiment involving detailed sand transport process measurements was repeated at 12 different cross-shore location. This procedure allows studying of the cross-shore variation of sand transport processes along the breaking zone. Starting from an initially 1:10 slope followed by a horizontal test section, a breaker bar developed in the breaking region as a result of onshore transport pre-breaking and offshore transport post-breaking. Near-bed suspended sediment fluxes were directed offshore along the complete test section, suggesting that the onshore transport pre-breaking is mainly attributed to bedload. The offshore suspended flux was the sum of an onshore wave-driven component and an offshore current-driven component. The wave-driven contribution to total suspended transport rates seems significant mainly before the breaking point where they account for ~30% of total suspended transport fluxes.

An asymptotic theory of combined wave refraction and diffraction

Bedload and suspended load contributions to breaker bar morphodynamics

Hydraulic experiments using a wave flume with a wind tunnel at a scale of 1/40 were carried out to investigate the effect of wind on wave overtopping of a vertical seawall. To ascertain the possible effects of wave breaking, incident waves with various wave steepness (0.0102 to 0.0371) and three kinds of wind forcing (0 m/s, 6.84 m/s, 10.04 m/s) were used. The authors found that winds significantly increased the overtopping rate in general, and that there is a clear relationship between the overtopping rate and the wave steepness. Velocity fields obtained from PIV analysis revealed that the onshore winds directly enhanced the transport of overtopping water mass. The influence of the wind on the wave hydrodynamic processes that indirectly have an effect on the overtopping rate (such as the formation of a partially standing wave, breaker location, and breaker type) is also discussed. The perspective of treating wave overtopping as a transport of water mass could provide a novel understanding of complex fluid behavior, which would help to formulate countermeasures against future disasters influenced by intensive wind forcing.