· · · Repository Hydraulic Engineering Reports

This report documents the computer program FUNWAVE based on the fully nonlinear Boussinesq model of Wei et al. (1995). The documentation provides a description of the governing equations and the numerical scheme used to solve it. A user's manual is provided and gives instructions on how to use various preprocessors and postprocessors to set up and read data files needed for model runs. Fortran code is provided for one and two-dimensional versions of the model, as well as for the additional data processing programs. Finally, several example calculations are documented.

Using a multiple-scale perturbation method we derive a set of governing equations describing the transformation of long wave and short wave components in a wave group. These equations are derived from Boussinesq equations with the assumption that the length scale of wave group modulation is in the same order of magnitude as that of the bottom variation, and is much longer than the length scale for the carrier (short) waves. Therefore the reflection of carrier waves by the topographical variation is small and neglected. Numerical examples are presented to show that long waves associated with wave group can be reflected resonantly by a field of periodic sandbars.

Analytical representation of shallow water wave phenomena is complicated due, in part, to the fact that nonlinear features are important in a predominant number, if not all , problems of shallow water wave motion. It therefore may not even be approximately valid to utilize the Airy wave theory and to assume that various wave components behave independently of one another. At present, even for periodic wave motion, predictions of shallow water wave phenomena based on various available theories differ by disturbing amounts. In this paper, several features of a numerical wave theory (Stream function) are reviewed and explored to demonstrate: (a) the agreement between theory and laboratory measurements, and (b) the differences between the numerical wave theory and the Airy wave theory. In particular, total wave energy, momentum and momentum flux , pressure response factors, maximum drag forces, shoaling coefficients, etc. are examined. The purpose of this paper is to direct attention to the very significant differences that exist between the two theories in shallow water and the need for additional research to resolve the differences noted.

Review paper on the recent progress in the field of wave modelling in the surf zone. The authors have the following concluding remarks (taken directly from the rapport): Wave breaking provides the forcing for larger scale motions in the surf zone. It is therefore probably both somewhat ironic as well as unfortunate that at the present time there exists no satisfactory theory to describe breaking and broken waves in the surf zone. This is currently a topic of intense research interest and we are confident that substantial progress will be made in the near future. While our overall understanding of wave-induced nearshore circulations seems to be fairly sound, there are a number of phenomena that clearly require further study. These include, but are not limited to, quantitatively accurate predictions of rip currents; the predictions of longshore currents on barred beaches; and the importance of alongshore inhomogenieties on nearshore circulations. Once again, these topics are currently being pursued by a number of investigators, and we expect considerable progress in the near future. Recent work has demonstrated that the surf zone is an important region for the generation of infragravity motions. While the present indications are that a substantial fraction of the infragravity energy seems to be generated in and near the surf zone, the existing models of surf zone generation of infragravity motions have not been verified. Shear waves seem to be amenable to an interpretation as a manifestation of an instability of the longshore current. Ongoing work on the nonlinear development of the instability and the importance of wave group forcing on these motions promises to yield interesting results. In conclusion, the subject of surf zone hydrodynamics is at an exciting stage of development right now and we expect that many of the issues will be clarified in the near future. We may also expect that the ongoing and future work will discover phenomena which we are currently unaware of.

A theoretical model is developed for wave heights and set-up in a surf zone. In the time averaged equations of energy and momentum the energy flux, radiation stress and energy dissipation are determined by simple approximations which include the surface roller in the breaker. Comparison with measurements shows good agreement. Also the transitions immediately after breaking are analysed and shown to be in accordance with the above mentioned ideas and results.

Surface gravity waves breaking in the nearshore region force a longshore surf zone current. This current can be unstable to longshore periodic perturbations. The continuity and momentum conservation equations averaged over the short wave time scales and over depth present a suitable basis for the modeling of these motions. The governing equations are in the form of the well-known shallow water equations with additional terms accounting for short wave forcing and dissipation effects. The objective of this study is to analyze the finite amplitude behavior of instabilities of the surf zone longshore current utilizing numerical experiments. For this purpose a solution method for the shallow water equations governing wave motions in the nearshore environment is developed. Spatial derivatives contained in these equations are computed using spectral collocation methods. A high-order time integration scheme is used to compute the time evolution of the velocities and water surface elevation given initial conditions. The model domain extends from the shoreline to a desired distance offshore and is periodic in the longshore direction. Properly posed boundary conditions for the governing equations are discussed. A curvilinear moving boundary condition is incorporated at the shoreline to account for wave run-up. An absorbing-generating boundary is incorporated offshore. The boundary treatments are tested using analytical and numerical results. The model is applied to the prediction of neutral stability boundaries and equilibrium amplitudes of subharmonic edge waves. Numerical results are compared to weakly nonlinear theory and are found to reproduce the theory well. The solution method is utilized to simulate instabilities of an analytic longshore current profile over a plane beach. The instabilities are observed to grow and equilibrate at amplitudes up to 50% of the original peak mean longshore current. For long domains in the longshore direction the long time behavior is observed to be dominated by subharmonic transitions that result in a reduction of the number of waves in the domain. The resulting longshore periodic fiow structures exhibit strong offshore directed velocities and propagate in the longshore direction at a fraction of the peak current speed. Details of the subharmonic transitions as well as the effect of non-linearity on the flow structures are analyzed. Next, the shear instability climate during the SUPERDUCK field experiment is simulated. Observations of undulations in the longshore current were first made during this field experiment by Oltman-Shay et al. (1989), who stated that the frequency range less than 0.01 Hz is dominated by these motions. Due to uncertainties in the friction and lateral mixing coefficients, numerical simulations are carried out for a realistic range of values for these coefficients. The resulting flow structures can be characterized as unsteady vortices propagating in the longshore direction. These vortices interact, occasionally merge and are shed offshore. During the shedding process, locally strong offshore directed currents are generated. Lateral mixing induced by the finite amplitude shear instabilities is analyzed and found to be of comparable magnitude to other mixing processes in the surf zone. Results from simulations of shear instabilities on plane and barred beaches show the existence of localized, migrating, offshore directed currents. Since the short wave field can be affected by these flow features, the modeling effort is extended to include the effects of time-varying short wave forcing and interactions between the short wave and current fields. The extension involves the solution of the time-dependent energy equation for the short wave motions and refraction equation due to variations in the bathymetry as well as current fields. The inclusion of a more realistic bottom friction treatment is also discussed.