Ib A. Svendsen

Mass flux and undertow in a surf zone

Abstract The mass of water carried shoreward by the breaking waves in a surf zone will, in a two-dimensional situation, be compensated by a seaward return flow, the undertow. It is shown that the undertow is driven by the local difference between radiation stress and the set-up pressure gradient which only balance each other in average over the depth. Turbulent shear stresses are required to maintain a steady situation. Comparison with measurements confirms the theoretical results.

Wave heights and set-up in a surf zone

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.

An efficient boundary element method for nonlinear water waves

Abstract The paper presents a computational model for highly nonlinear 2-D water waves in which a high order Boundary Element Method is coupled with a high order explicit time stepping technique for the temporal evolution of the waves. The choice of the numerical procedures is justified from a review of the literature. Problems of the wave generation and absorption are investigated. The present method operates in the physical space and applications to four different wave problems are presented and discussed (space periodic wave propagation and breaking, solitary wave propagation, run-up and radiation, transient wave generation). Emphasis in the paper is given to describing the numerical methods used in the computation.

WAVE CHARACTERISTICS IN THE SURF ZONE

The equations describing conservation of mass, momentum and energy in a turbulent free surface flow are derived for a controle volume extending over the whole depth. The effect of the turbulent surface oscillations are discussed but neglected in the following analysis, where the equations are applied to the energy balance in a surf zone wave motion. This leads to results for the wave height variation and the velocity of propagation. The results cannot be reconciled completely with measurements and the concluding discussion is aimed at revealing how the model can be improved.A three-dimensional morphodynamic model of sequential beach changes Is presented. The model Is based on variations in breaker wave power generating a predictable sequence of beach conditions. The spectrum of beach conditions from fully eroded-dissipatlve to fully accreted reflective is characterised by ten beach-stages. Using the breaker wave power to beach-stage relationship the model Is applied to explain temporal, spatial and global variations In beach morphodynamlcs.The agents of initial damage to the dunes are water, which undermines them, and animals (including man) which damage the protective vegetation by grazing or trampling. Of these, man has recently assumed predominant local importance because of the popularity of sea-side holidays and of the land-falls of certain marine engineering works such as oil and gas pipelines and sewage outfalls. The need is therefore increasing for active dune management programmes to ensure that under these accentuated pressures, the coast retain an equilibrium comparable with that delicately balanced equilibrium which obtains naturally at a particular location.

Nearshore mixing and dispersion

Longshore currents have in the past been analysed assuming that the lateral mixing could be attributed to turbulent processes. It is found, however, that the mixing that can be justified by assuming an eddy viscosity vt = l√k where l is the turbulent length scale, k the turbulent kinetic energy, combined with reasonable estimates for l and k is at least an order of magnitude smaller than required to explain the measured cross-shore variations of longshore currents. In this paper, it is shown that the nonlinear interaction terms between cross-and longshore currents represent a dispersive mechanism that has an effect similar to the required mixing. The mechanism is a generalization of the one-dimensional dispersion effect in a pipe discovered by Taylor (1954) and the three-dimensional dispersion in ocean currents on the continental shelf found by Fischer (1978). Numerical results are given for the dispersion effect, for the ensuing cross-shore variation of the longshore current and for the vertical profiles of the longshore currents inside as well as outside the surf zone. It is found that the dispersion effect is at least an order of magnitude larger than the turbulent mixing and that the characteristics of the results are in agreement with the sparse experimental data material available.

Turbulent bores and hydraulic jumps

A theoretical model for the velocity field and the surface profile of bores and hydraulic jumps is developed. The turbulence is assumed to be concentrated in a wedge that originates at the toe of the front and spreads towards the bottom, and the turbulent closure used is a simplified k -e model allowing for non-equilibrium in the turbulent kinetic energy. The flow equations are satisfied in depth-integrated form (method of weighted residuals), and measured deviations from static pressure are analysed and shown to have a negligible effect on the results. Comparison with measurements shows good agreement, but there is a clear need for further experimental results in the highly turbulent region near the free surface. Some basic mechanisms of the flow are discussed and explained from the theory.

The flow in surf-zone waves

An extended set of Boussinesq equations, which are used to model breaking waves, is derived. The wave breaking is described by accounting for the effect of vorticity generated by the breaking process. The vorticity field in the domain is obtained by solving the vorticity transport, which is based on the Reynolds equations. The boundary conditions to solve for the vorticity are obtained from measurements in hydraulic jumps. In addition to predicting the wave height decay and profile deformation, the present model also provides information about the velocity field. Comparisons with laboratory measurements show that the model results give good agreement with experimental data for wave height, set-up and the velocity profiles. The undertow profiles predicted by the model are also in good agreement with the results from experimental data. The cross-shore variation of the radiation stress calculated from the model results gives a good representation of the results from experimental data.

Quasi-three-dimensional modeling of rip current systems

[1] The focus of the paper is the analysis of the flow in rip current systems generated by channels in longshore bars on a beach. The horizontal variations of rip current systems are described through the use of the quasi-three-dimensional nearshore circulation model SHORECIRC. Model predictions are compared to laboratory measurements of waves and current velocities throughout the entire rip current system and show reasonable agreement. The rips in the two channels are found to behave differently because of the depth variation across the basin. It is found that higher bottom stress leads to more stable flow where the rip current meanders less and fewer eddies are generated. The wave current interaction creates forcing which reduces the distance rip currents flow offshore and can lead to a slow pulsation of the rip current. This pulsation is in addition to the instabilities of a jet which can also be present in rip currents. The three dimensionality of the rip current system is found to have a significant effect on the overall circulation patterns. INDEX TERMS: 4255 Oceanography: General: Numerical modeling; 4512 Oceanography: Physical: Currents; 4546 Oceanography: Physical: Nearshore processes; KEYWORDS: rip currents, nearshore circulation, numerical modeling, waves

Corner problems and global accuracy in the boundary element solution of nonlinear wave flows

Abstract The numerical model for nonlinear wave propagation in the physical space, developed by Grilli, et al. 12,13 , uses a higher-order BEM for solving Laplaces equation, and a higher-order Taylor expansion for integrating in time the two nonlinear free surface boundary conditions. The corners of the fluid domain were modelled by double-nodes with imposition of potential continuity. Nonlinear wave generation, propagation and runup on slopes were successfully studied with this model. In some applications, however, the solution was found to be somewhat inaccurate in the corners and this sometimes led to wave instability after propagation in time. In this paper, global and local accuracy of the model are improved by using a more stable free surface representation based on quasi-spline elements and an improved corner solution combining the enforcement of compatibility relationships in the double-nodes with an adaptive integration which provides almost arbitrary accuracy in the BEM numerical integrations. These improvements of the model are systematically checked on simple examples with analytical solutions. Effects of accuracy of the numerical integrations, convergence with refined discretization, domain aspect ratio in relation with horizontal and vertical grid steps, are separately assessed. Global accuracy of the computations with the new corner solution is studied by solving nonlinear water wave flows in a two-dimensional numerical wavetank. The optimum relationship between space and time discretization in the model is derived from these computations and expressed as an optimum Courant number of ∼0.5. Applications with both exact constant shape waves (solitary waves) and overturning waves generated by a piston wavemaker are presented in detail.

A turbulent bore on a beach

A theoretical model is developed giving a moderately detailed description of the flow in a turbulent bore, the velocity profiles, the shear stresses, the energy dissipation, etc. An analysis of the flow conditions at the toe of the turbulent front indicates significant differences from the usual description based on the finite-amplitude shallow-water equations, and it is shown that the present model gives a closer description of the actual physical conditions. Finally, numerical results are presented that illustrate how the model works, and test its validity on an example with known properties.