James T. Kirby

A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves

Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

Observation of undertow and turbulence in a laboratory surf zone

Abstract Undertow and turbulence in the surf zone have been studied in a wave flume for a spilling breaker and a plunging breaker. Fluid velocities across a 1 on 35 sloped false bottom were measured using a fiber-optic laser-Doppler anemometer, and wave decay and set-up were measured using a capacitance wave gage. The characteristics of mean flow and turbulence in spilling versus plunging breakers were studied. The mean flow is the organized wave-induced flow defined as the phase average of the instantaneous velocity, while the turbulence is taken as the deviations from the phase average. It was found that under the plunging breaker turbulence levels are much higher and vertical variations of undertow and turbulence intensity are much smaller in comparison with the spilling breaker. It was also found that turbulent kinetic energy is transported seaward under the spilling breaker and landward under the plunging breaker by the mean flow. The study indicates that there are fundamental differences in the dynamics of turbulence between spilling and plunging breakers, which can be related to the processes of wave breaking and turbulence production. It is suggested that the types of beach profile produced by storm and swell waves may be the results of different relationships between mean flow and turbulence in these waves.

Generation of waves in Boussinesq models using a source function method

Abstract A method for generating waves in Boussinesq-type wave models is described. The method employs a source term added to the governing equations, either in the form of a mass source in the continuity equation or an applied pressure forcing in the momentum equations. Assuming linearity, we derive a transfer function which relates source amplitude to surface wave characteristics. We then test the model for generation of desired incident waves, including regular and random waves, for both one and two dimensions. We also compare some model results with analytical solution and available experiment data.

A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O ( kh ) 4

A Boussinesq-type model is derived which is accurate to O ( kh ) 4 and which retains the full representation of the fluid kinematics in nonlinear surface boundary condition terms, by not assuming weak nonlinearity. The model is derived for a horizontal bottom, and is based explicitly on a fourth-order polynomial representation of the vertical dependence of the velocity potential. In order to achieve a (4,4) Pade representation of the dispersion relationship, a new dependent variable is defined as a weighted average of the velocity potential at two distinct water depths. The representation of internal kinematics is greatly improved over existing O ( kh ) 2 approximations, especially in the intermediate to deep water range. The model equations are first examined for their ability to represent weakly nonlinear wave evolution in intermediate depth. Using a Stokes-like expansion in powers of wave amplitude over water depth, we examine the bound second harmonics in a random sea as well as nonlinear dispersion and stability effects in the nonlinear Schrodinger equation for a narrow-banded sea state. We then examine numerical properties of solitary wave solutions in shallow water, and compare model performance to the full solution of Tanaka (1986) as well as the level 1, 2 and 3 solutions of Shields & Webster (1988).

Dynamics of surf-zone turbulence in a spilling breaker

The structure of turbulence in a spilling breaker has been studied experimentally based on the transport equation for turbulent kinetic energy (the k-equation). We study turbulence transport in the evolving flow from the breaking point to the inner surf zone in the region below trough level and above the bottom boundary layer. The study shows that turbulence transport processes are similar in the outer and inner surf zones. It is found that diffusive transport plays the most important role in the distribution of turbulence, while advection is important mainly near the surface. It is also found that although turbulence production below trough level amounts to only a small portion of the wave energy loss, the production term is not small compared to the dissipation term and the major terms in the k-equation and thus it cannot be neglected. The mixing length is estimated based on the measured rates of vertical advance of the turbulent front, and comparisons of turbulence production and energy dissipation. The results are similar to those found in previous studies. It is shown that the length scale and velocity scale of the large eddies are subject to turbulence transport processes, therefore their distributions cannot be prescribed in an easy way. The relative values of the components of the Reynolds stress tensor are examined. The results of analysis supports the notion that surf-zone turbulence created by spilling and plunging breakers differ primarily in the method of energy transfer from organized wave-induced motion to turbulent motion, and the constraints imposed by the mean flow and the solid bottom on the large-scale turbulence.

Dynamics of surf-zone turbulence in a strong plunging breaker

Abstract The characteristics of turbulence created by a plunging breaker on a 1 on 35 plane slope have been studied experimentally in a two-dimensional wave tank. The experiments involved detailed measurements of fluid velocities below trough level and water surface elevations in the surf zone using a fibre-optic laser-Doppler anemometer and a capacitance wave gage. The dynamical role of turbulence is examined making use of the transport equation for turbulent kinetic energy (the k-equation). The results show that turbulence under a plunging breaker is dominated by large-scale motions and has certain unique features that are associated with its wave condition. It was found that the nature of turbulence transport in the inner surf zone depends on a particular wave condition and it is not similar for different types of breakers. Turbulent kinetic energy is transported landward under a plunging breaker and dissipated within one wave cycle. This is different from spilling breakers where turbulent kinetic energy is transported seaward and the dissipation rate is much slower. The analysis of the k-equation shows that advective and diffusive transport of turbulence play a major role in the distribution of turbulence under a plunging breaker, while production and dissipation are not in local equilibrium but are of the same order of magnitude. Based on certain approximate analytical approaches and experimental measurements it is shown that turbulence production and viscous dissipation below trough level amount to only a small portion of the wave energy loss caused by wave breaking. It is suggested that the onshore sediment transport produced by swell waves may be tied in a direct way to the unique characteristics of turbulent flows in these waves.

Boussinesq modeling of a rip current system

In this study, we use a time domain numerical model based on the fully nonlinear extended Boussinesq equations [Wei et al., 1995] to investigate surface wave transformation and breaking-induced nearshore circulation. The energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves. Wave run-up on the beach is simulated using a moving shoreline technique. We employ quasi fourth-order finite difference schemes to solve the governing equations. Satisfactory agreement is found between the numerical results and the laboratory measurements of Haller et al. [1997], including wave height, mean water level, and longshore and cross-shore velocity components. The model results reveal the temporal and spatial variability of the wave-induced nearshore circulation, and the instability of the rip current in agreement with the physical experiment. Insights into the vorticity associated with the rip current and wave diffraction by underlying vortices are obtained.

A parabolic equation for the combined refraction-diffraction of Stokes waves by mildly varying topography

A parabolic equation governing the leading-order amplitude for a forward-scattered Stokes wave is derived using a multiple-scale perturbation method, and the connection between the linearized version and a previously derived approximation of the linear mild slope equation is investigated. Two examples are studied numerically for the situation where linear refraction theory leads to caustics, and the nonlinear model is shown to predict the development of wave-jump conditions and significant reductions in amplitude in the vicinity of caustics.

Propagation of obliquely incident water waves over a trench

The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries ; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission ; however, reflection is not total owing to the finiteness of the obstacle. Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution. 1. Introduction The problem of the diffraction of incident waves by a finite obstacle in an otherwise infinite and uniform domain remains of general interest in linear wave theory. Several geometries of interest can be schematized by domains divided into separate regions by vertical geometrical boundaries, with the fluid depth being constant in each subdomain. Representative two-dimensional problems, with the boundary geometry uniform in the direction normal to the plane of interest, include those of elevated rectangular sills, fixed or floating rectangular obstacles at the water surface, and submerged trenches. The approach to the solution of problems of this type has typically been to construct solutions for each constant-depth subdomain in terms of eigenfunction expansions of the velocity potential ; the solutions are then matched at the vertical boundaries, resulting in sets of linear integral equations which must be truncated to a finite number of terms and solved numerically. One of the earliest solutions of this type was given by Takano (1960), who studied the cases of normal wave incidence on an elevated sill and fixed obstacle at the surface. In this study, we employ a modification of Takano’s method, discussed in

Nonlinear evolution of shear instabilities of the longshore current: A comparison of observations and computations

3. Newman (19653) also employed an integral-equation approach to study reflection and transmission of waves normally incident on a single step between finite- and infinite-depth regions. A variational approach, developed by Schwinger to study discontinuitiesin waveguides (see Schwinger & Saxon 1968) has been used by Miles (1967), to study Newman’s single-step problem, and by Mei & Black (1969), who studied the symmetric elevated sill and a floating rectangular cylinder. Lassiter (1 972), using the variational approach, studied waves normally incident on a rectangular trench where the water depths before and after the trench are constant but not necessarily equal, referred to here as the asymmetric case. Lee &