Narrow Results

The regularized long-wave ($ℝ𝕃𝕎$) equation is a fundamental model in shallow water wave theory, with extended relevance to other physical systems, such as plasma physics. The $ℝ𝕃𝕎$ equation governs key nonlinear wave phenomena, including the propagation and interaction of solitons, or localized solitary waves. In plasma physics, it describes ion-acoustic waves, which are low-frequency compressional waves arising from the interaction between ions and electrons. While the ion-acoustic wave exhibits Korteweg–de Vries soliton behavior in the cold plasma limit, the $ℝ𝕃𝕎$ equation offers a more accurate representation of ion-acoustic solitons and peaked structures in warm plasmas by accounting for thermal effects.

This research presents novel analytical approximations for solitary wave solutions within the $ℝ𝕃𝕎$ equation, which is critical for modeling nonlinear shallow water and plasma waves. The equation’s ability to describe peaked solitary waves, known as “peakons”, represents wave-breaking phenomena. By employing the improved Riccati expansion and modified Fan expansion techniques, this study derives periodic peakon solutions, offering new insights into the equation’s behavior.

A thorough analysis of the $ℝ𝕃𝕎$ equation is provided, emphasizing its importance for nonlinear wave modeling in both fluid and plasma systems. This paper also includes numerical validation using He’s variational iteration method, which enhances the transparency of the findings by addressing the underlying assumptions and limitations. The principal findings contribute to the broader understanding of nonlinear solitary wave theory, with practical implications for nonlinear wave dynamics across multiple physical domains. Suggested extensions are outlined for further investigation within this established theoretical framework. This study maintains consistent notation and terminology to facilitate clear communication of ideas in adherence to academic standards.

The seminal work of J. M. Dawson on sheet crossing and wave breaking is extended to laser wakefield scenarios. We derive a quantitative estimate for the sheet crossing time. We find that the time for the onset of sheet crossing is reduced for lower densities, higher laser intensities and smaller spot sizes. Furthermore, we show that a preformed parabolic plasma channel can decrease the time required for sheet crossing.

The present work is dedicated to the modeling of viscous flow past a NACA0012 foil fixed in a current below a free surface. To this end, the ${\delta }^{+}$-smoothed-particle hydrodynamics (SPH) model has been adopted. This SPH model prevents the inception of the numerical tensile instability in the flow region characterized by negative pressure since a tensile instability control (TIC) has been included. In the TIC, a pressure differencing formulation (PDF) has been adopted for the momentum equation in the flow region characterized by negative pressure. In order to completely remove the numerical noise in the vorticity field, in this work, the PDF is also applied for the region with positive pressure, but except for the free-surface region in order to ensure the free surface stability when wave breaking occurs. The mechanism of PDF being able to eliminate the numerical noise in the vorticity field is also briefly analyzed. In order to reduce the nonconservation of total momentum induced by the PDF, a particle-shifting technique (PST) is implemented in each time step for regularizing the particle position. In the numerical results, ${\delta }^{+}$-SPH results are validated by the experimental data and other verified numerical results. Improvements of the results of ${\delta }^{+}$-SPH with PDF with respect to the ones without using PDF are demonstrated. Parametrical studies based on the ${\delta }^{+}$-SPH model regarding the breaking and non-breaking waves generated by the flow past a submerged foil are also carried out.

We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to transport equations for the Cartesian components of the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the “wave part” of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify the strong null condition. The latter is a nonlinear condition signifying the complete absence of nonlinear interactions involving more than one differentiation in a direction transversal to the acoustic characteristics. Moreover, the same good structures are found in the equations verified by the Euclidean divergence and curl of the specific vorticity. This is important because one needs to combine estimates for the divergence and curl with elliptic estimates to obtain sufficient regularity for the specific vorticity, whose derivatives appear as inhomogeneous terms in the wave equations. The structures described above collectively open the door for our companion results, in which we exhibit a stable regime of initially smooth solutions that develop a shock singularity. In particular, the first Cartesian coordinate partial derivatives of the velocity and density blow up, while relative to a system of geometric coordinates adapted to the acoustic characteristics, the solution (including the vorticity) remains many times differentiable, all the way up to the shock. The good null structures, which are often associated with global solutions, are in fact key to proving that the shock singularity forms. Our secondary goal in this paper is to provide an overview of the central role that the structures play in the proof.

This paper proposes a one-dimensional and a horizontally two-dimensional random wave transformation models based on a spectral wave equation with a probabilistic bore-type energy dissipation term, and examines the validity of the wave models by comparing the predictions with the observations. The present spectrum-based prediction models with a probabilistic dissipation term are rather heuristic but robust to use. Since the present random wave transformation models consist of spectral and probabilistic models, the wave model can be called as a hybrid random wave transformation model. The model solves the spatial evolution of complex Fourier amplitudes from which energy spectra are calculated. In addition, water surface elevations are obtained from the calculated complex amplitudes by using the inverse Fast Fourier Transform. Furthermore, representative wave heights and periods are obtained from the water surface elevations.

We apply the one-dimensional wave model to predict transformations of double peak spectral waves, small and prototype scales' experimental waves and field waves over uniform slope and arbitrary bathymetry. In addition, the horizontally two-dimensional wave model is applied to predict transformations over an elliptic shoal, a conic shoal and field bathymetry, and also employed to investigate the mach-reflection. The comparison between the various observations by laboratory and field experiments and the predictions by the wave models shows good agreements.

The large-scale structure of intermittent turbulence below trough level produced by depth-limited wave breaking was investigated in a laboratory wave flume using point velocity measurements and particle image velocimetry. A conditional sampling technique was applied to the instantaneous turbulent kinetic energy per unit mass, k/ρ, estimated from the point velocity measurements where the large scale turbulent fluctuations were separated from the orbital wave motion by phase-averaging. The analysis shows that large, intermittent events exceeding a threshold equal to the mean and standard deviation of k/ρ occur for only 7% of the record but account for approximately 40% of the total turbulent kinetic energy below trough level. The PIV technique was used to quantify the size and vorticity of the large eddies associated with wave breaking. The PIV technique was applied such that the light sheet was in the horizontal plane to quantify the eddy size and vorticity. Observations show that the nominal diameter of the eddy was 0.05 m with a maximum vorticity of 30 1/s for laboratory scale wave breaking.

Organized eddies generated by wave breaking have an important role on sediment transport in surf zones. In order to clarify the characteristics of intermittent sediment clouds generated by vertically-oriented eddies, the Computed Tomography (CT) technique was applied for a laboratory experiment to capture instantaneous two-dimensional distributions of suspended sediment concentration in a near-bottom horizontal plane. The Extended Bayesian Method with a smoothing filter as the prior information model was applied to obtain the optimum solution of the distribution. A specification test showed that concentration distribution was reproduced fairly well. Successive images of concentration distribution due to wave breaking were obtained and movements of sediment clouds were investigated. Concentration at the center of the measuring area was simultaneously measured for verification by a conventional optical forward-transmission-type sedimentmeter. It was found that horizontal distribution of suspended sediment concentration in a surf zone is quite inhomogeneous and the high concentration regions are formed as they indicate separated sediment clouds. Distinctive patterns of sediment clouds were observed in a relatively small number of trials. The length scale of the clouds was found to be in the order of the water depth. The primary cause of sediment cloud formation can be considered sediment pickup by intermittent eddies generated by wave breaking.

Wave overtopping on gentle slope seawalls for both smooth and stepped front faces was investigated in a laboratory wave flume. Overtopping rates were measured by a catchment basin placed behind the seawall. Overtopping velocity and water depth at the top of the seawall were simultaneously measured by using a Laser Doppler Velocimeter and a wave gage. It was found that the velocity of overtopping water has a sharp peak just after initiation of each overtopping event. The water depth on the seawall crown shows relatively gradual decrease.

A numerical model of three-dimensional Large Eddy Simulation was developed for evaluating the instantaneous overtopping discharge and compared to the experimental results. The calculated root-mean-square value of water surface elevation at the toe of the seawall agreed very well with the measured value, but large discrepancies were found for individual wave deformation in some waves. The magnitude of the calculated velocity on the top of seawall was generally smaller than that of the measured velocity. The calculated volume of the overtopping water was between the volumes measured for the smooth seawall and the stepped seawall. Since the two-dimensional Large Eddy Simulation model far overestimated the overtopping volume, it was considered that appropriate representation of three-dimensional large eddies generated by wave breaking is important for wave overtopping evaluation.

A particle method, or a gridless Lagrangian method, shows the high performance in describing the complicated behavior of water surface with the fragmentation and coalescence of water. In this paper, a wave overtopping process on a vertical seawall is numerically simulated on the basis of the Navier–Stokes equation with surface-tension term, which is discretized by the MPS (moving particle semi-implicit) method belonging to the category of the particle method. An improvement of the listing process of neighboring particle is introduced to reduce the computational load. Wave overtopping process in the experiments are well reproduced by the MPS method. The predictions of the MPS method of the overtopping volume agree well with the experimental results.

The wave-induced pressure gradient, ∂p/∂x, at the bottom is related to fluid acceleration and sediment movement in the surf zone. Following similar large-scale laboratory work by Suzuki et al. [2008a], this paper deals with the observations and analysis of bottom pressure gradients on a natural sandy beach. The cross-correlation coefficients between ∂p/∂x and the water surface elevation are high even in the surf zone, and the coefficients are higher than the coefficients between ∂p/∂x and the vertical velocity component or ∂p/∂x and du/dt. The observed nonlinear characteristics of ∂p/∂x are weaker than the laboratory experimental data but extreme values of ∂p/∂x are larger than the experiments. The distributions of exceedance probability of ∂p/∂x are evaluated using the two-parameter Weibull distribution. The modulus of the Weibull distribution is evaluated as a function of local significant wave height normalized by the offshore significant wave height. The exceedance probability distributions of ∂p/∂x show a broader distribution for the field data compared to the laboratory, but are, nevertheless, predicted reasonably well with the Weibull distribution.

Statistics of breaking waves across the surf zone are reanalyzed on the basis of various sets of field and laboratory data so as to provide coastal engineers with reliable information on breaking waves. The breaker index or the ratio of wave height to water depth is to be expressed as a function of the two parameters of beach slope and relative depth, and Goda's breaker index formula is revised slightly to reduce the slope effect. The breaker index for regular waves has inherent variability as expressed with the coefficient of variability, which increases from 6% to 14% as the beach slope becomes steep up to 1/10. The incipient breaking height of the significant wave is about 30% lower than that of regular waves, but the ratio of significant wave height to water depth gradually increases within the surf zone toward the shoreline. The wave height distribution is the narrowest in the middle of the surf zone, but it returns to the Rayleigh distribution near the shoreline owing to wave regeneration after breaking. The nonlinearity of random waves is strongest at the outer edge of the surf zone, but it is destroyed by the wave breaking process inside the surf zone. The ratios of statistical wave heights H_1/10, H_1/3 and H_rms to the spectral significant wave height H_m0 are shown to increase as the wave nonlinearity parameter increases up to the outer edge of the surf zone.

This study investigates wave transformations over an elliptic shoal on a sloping bottom using numerical calculations. The theoretical model is based on the fully nonlinear Boussinesq equation that was applied by Chen et al. [2003]. This equation is expressed by the velocity at any level below the surface. The momentum equation is derived using modified vertical vorticity terms from the equation by Wei et al. [1995]. Bottom friction, wave breaking, and subgrid lateral turbulent mixing proposed by Kennedy et al. [2000] are also included in the equations. Several numerical experiments are conducted for waves with incident angles of 0°, 15°, and 30°. The numerical results demonstrate the phenomena of wave focusing in the rear of the elliptic shoal. The wave focusing not only results in a smaller wave-induced current on top of the shoal than that at the rear of the shoal, but also a strong return flow at the back of the shoal. It is found that the non-dimensional wave height in the wave-focusing zone increases as the relative water depth decreases. Moreover, the orientation of wave height contours in the focusing region is almost identical to that of the incident wave angle. This phenomenon stays true for different incident wave periods.

A similarity parameter is derived to describe surf zone dissipation using the classical energy dissipation model for surf zone bores. This parameter can also be interpreted as a relative beach slope parameter, β_γ, and, for shallow water sinusoidal waves, is the ratio of the local beach slope and the local wave steepness (H/L). β_γ = 1 defines the boundary between two different energy dissipation regimes. Conditions with β_γ<1 represent over-dissipative conditions, where the classical bore dissipation model provides more dissipation than that required for depth-limited waves (constant H/h) to be maintained. Conditions with β_γ>1 represent under-dissipative conditions, where the bore model provides insufficient dissipation for depth-limited conditions to occur. Conditions with β_γ = 1 at the breakpoint lead to locally saturated but not depth-limited surf. Hence, the new similarity parameter distinguishes between saturated and unsaturated surf conditions. Based on this bore dissipation model, an analytical model for the wave height transformation of monochromatic waves on planar beaches is derived. The cross-shore variation in wave height obtained from this model show different functional forms; concave upward for over-dissipative conditions and convex upward for under-dissipative conditions. Further, the analytical model shows that depth-limited conditions within the inner surf zone are not possible with this bore dissipation model and the model assumptions. Additional work is required to determine if this parameter is a useful predictor of other surf zone characteristics.

Highlights: We derive a similarity parameter to describe surf zone dissipation. The similarity parameter defines the boundary of two dissipation regimes. The parameter distinguishes between saturated and unsaturated surf conditions. The classical bore dissipation model cannot predict depth-limited wave heights on a plane beach.

The rate of sediment pickup for suspension by breaking waves is shown to be proportional to the dissipation rate of wave energy flux within the surf zone. A database of mean concentration of suspended sediment has been compiled by collecting various data of field measurements and large-scale laboratory tests and by calculating the depth-averaged mean concentration. The wave energy flux dissipation rate is computed with the PEGBIS model for random wave breaking on beaches of arbitrary profile. By comparing the mean sediment concentration and the energy flux dissipation rate, a sediment pickup coefficient is assessed at a value of 0.0045 on the average. Use of this coefficient makes it possible to estimate the cross-shore variation of mean sediment concentration. Peak values of mean sediment concentration within measured cross-shore profiles agree well with the prediction based on the wave energy flux dissipation rate.

This study aims to develop a new numerical model for predictions of the time-varying shear current field under breaking and broken waves. The present model differs from the other similar existing models in that the model explicitly determines the bottom shear stress so that it satisfies both force balance and mass-conservation equations while most of the other existing models leave the bottom shear stress as one of the calibration parameters to be fitted. The present model consists of two sub-models, a wave model and a flow model. The wave model is based on modified Boussinesq equations with evolution and dissipation of surface rollers. The flow model is based on Reynolds' equations with turbulence closure models and computes vertical profiles of wave-induced shear current field. The numerical model is applied to various existing and newly performed experimental cases which cover various breaker types and bed profiles with different bed roughness. The present model showed overall good predictive skills of various surf zone hydrodynamics such as wave heights, mean water levels and undertow profiles. It should also be highlighted that the model has only a few calibration parameters and the predictive skill of the model is relatively less sensitive to these parameters.

An ultrasonic velocity profiler (UVP) acquires instantaneous distributions of axial velocity along the ultrasonic beam emitted from the transducer. In this paper, organizations of the transverse flows over a span of a laboratory wave flume in the surf zone were identified experimentally on the basis of UVP measurements. Ensemble mean divergent and convergent flows were observed over the span of the flume in the surf zone regardless of breaker conditions. By analogy with the flow evolution in a previous computation, by Watanabe and Saeki [1999], we identified that the organization of the wave-breaking-induced counter-rotating vortices determined the spatial and temporal variations in the transverse flows at the early stages of the breaking process regardless of the widths of wave flumes and friction on the sidewalls and bottom. The observed transverse flows were driven by pairs of counter-rotating vortices produced under the breaking waves and the change in the orientation during the passage of breaking waves. The fundamental features of the transverse flows evolving in the breaking process, depending on the breaker type, are discussed statistically to parameterize velocity, time, and length scales of the variations of the transverse velocity. The dimensionless length and time scales had negative linear correlations with the Froude number (defined by relative transverse convection with respect to gravity) as well as a surf similarity parameter, which indicates that the gravity effect associated with the projections of the breaking wave crests defines the primary roller vorticity and shear intensity at the plunging location, resulting in destabilization of the flows in the transverse direction to form organized transverse flow structures.

In this paper, a hybrid finite-volume-finite-difference scheme for the solution of the one-dimensional form of the Nwogu’s extended Boussinesq-type equations was developed to investigate the breaking of a solitary wave propagating over a fringing reef system. The adoption of a high-order finite volume WENO schemes with HLL Riemann solver for advection terms as well as the third-order Runge–Kutta time-stepping scheme makes the present scheme shock-capturing. From the validation case, it is shown that the present shock-capturing model could simulate the shoreline movement and wave breaking as well as bore propagation very well. In order to comprehensively understand the hydrodynamic characteristics when a solitary wave travels over a fringing reef system, a parametric study has been carried out to analyze the interaction process under various conditions. Effects of different physical parameters on wave breaking are analyzed and the reflection, transmission and dissipation (RTD) coefficients are calculated based on the integration of kinetic and potential energies in different zones.

Based on a large amount of published laboratory results, reliable models are developed for computing the average rate of energy dissipation in regular and irregular breaking waves. The average energy dissipation rate is assumed to be proportional to the difference between the local mean energy density and stable energy density. Wave height transformation is computed from the energy flux conservation law based on the linear wave theory. The models are examined and verified extensively for a variety of wave and bottom conditions, including small and large scale laboratory and field experiments. Reasonable good agreements are obtained between the measured and computed wave heights and root mean square wave heights.

A numerical wave model for two-dimensional wave field in the vertical plane is proposed in this study. The model combines a VOF method with a non-reflective wave generator in addition to the open boundary treatment based on an added dissipation zone to achieve stable computations. Not only wave breaking due to a submerged breakwater, but also post-breaking wave deformation are numerically investigated using the proposed model. Extensive laboratory experiments were also conducted to verify the validity of the model. The breaking characteristics and the frequency spectra of wave amplitude are examined for various incident wave conditions and breakwater configurations. The computed and measured results have revealed that second harmonic free wave component is generated in the region of non-breaking reformed wave, and that a breaking wave-induced circulating flow is formed at the onshore side of the submerged breakwater. It is demonstrated that the proposed numerical wave model can reproduce well the wave deformation before and after wave breaking through comparison with the experimental results.

A three-dimensional large eddy simulation (LES) of wave breaking was carried out. A numerical method for LES is proposed in this paper. The following characteristics of vorticity and velocity field after wave breaking are discussed on the basis of results of the LES: (1) generation and evolution of the widthwise (shore direction) velocity component; (2) transition from a two-dimensional velocity field to a three-dimensional one after wave breaking; (3) evolution process of large-scale eddies comprised by horizontal, vertical and helical eddies; and (4) a coherent eddy structure involving a turbulent bottom and wall boundary layer.