Runup Of Non-breaking Waves Research Articles

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

This is the first of three papers on the modelling of various types of surf zone phenomena. In this first paper, part I, the model is presented and its basic features are studied for the case of regular waves. The model is based on two-dimensional equations of the Boussinesq type and it features improved linear dispersion characteristics, possibility of wave breaking, and a moving boundary at the shoreline. The moving shoreline is treated numerically by replacing the solid beach by a permeable beach characterized by an extremely small porosity. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is based on the surface roller concept for spilling breakers using a geometrical determination of the instantaneous roller thickness at each point and modelling the effect of wave breaking by an additional convective momentum term. This is a function of the local wave celerity, which is determined interactively. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches and over submerged bars. Model results comprise time series of surface elevations and the spatial variation of phase-averaged quantities such as the wave height, the crest and trough elevations, the mean water level, and the depth-averaged undertow. Comparisons with physical experiments are presented. The phaseaveraged balance of the individual terms in the momentum and energy equation is determined by time-integration and quantities such as the cross-sectional roller area, the radiation stress, the energy flux and the energy dissipation are studied and discussed with reference to conventional phase-averaged wave models. The companion papers present cross-shore motions of breaking irregular waves, swash oscillations and surf beats (part II) and nearshore circulations induced by breaking of unidirectional and multidirectional waves (part III).

Modeling of Breaking and Nonbreaking Long-Wave Evolution and Runup Using VTCS-2

We present a variable grid finite-differences approximation of the characteristic form of the shallow-water-wave equations without artificial viscosity or friction factors to model the propagation and runup of one-dimensional long waves, referred to as VTCS-2. We apply our method in the calculation of the evolution of breaking and nonbreaking waves on sloping beaches. We compare the computational results with analytical solutions, other numerical computations and with laboratory data for breaking and nonbreaking solitary waves. We find that the model describes the evolution and runup of nonbreaking waves very well, even when using a very small number of grid points per wavelength. Even though our method does not model the detailed surface profile of wave breaking well, it adequately predicts the runup of plunging solitary waves without ad-hoc assumptions about viscosity and friction. This appears to be a further manifestation of the well-documented but unexplained ability of the shallow water wave equations to provide quantitatively correct runup results even in parameter ranges where the underlying assumptions of the governing equations are violated.

Run-up of non-breaking waves — A finite-element approach

An important aspect of the study of wave-structure interaction is the extent to which the wave-induced motion carries water up the structures. This is commonly referred to as the wave run-up. Several studies in the past have produced numerical solutions to this problem under restrictive assumptions. One of the important restrictions in those studies is that the bottom is gently sloping — such as slopes of 1 to 10 or flatter. The traditionally used shallow-water equations cannot be applied to regions where the vertical accelerations become significant. Such accelerations are significant when water moves over dykes or levees with fairly steep slopes. In the present study an approach has been made to develop the appropriate momentum equation which will account for the vertical acceleration and hence replace the shallow-water equation for the situation mentioned above. The newly developed equation is also non-linear in nature. This, along with the continuity equation, forms a system of coupled non-linear time-dependent partial differential equations for the problem of wave run-up. In order to solve this system the Finite-Element Technique is adopted for the space domain. The Finite Difference Euler Predictor-Corrector is used to step up the variables in the time domain. Such a combination, known as the semi-discrete method, is found to produce a powerful model for the wave run-up. An important aspect of this problem is that the problem domain changes with time. In other words, it is a moving boundary problem and gives rise to peculiarities in handling the boundary conditions. In the present model this aspect is handled using a time-dependent end-element. Numerical results are obtained using sinusoidal waves, solitary waves (tsunamis) and surges as the forcing functions. Single-slope faces as well as composite-slope faces (as is often the case in levees) are included in the study. The results of these cases (i.e. particle velocities and heights of run-up) are given in the form of charts. In the design of coast protective structures with sloping sides, this model will be found particularly useful in predicting the run-up and particle velocities.

A Synthesis on Wave Run-Up

A critical survey of existing theories and experiments on wave run-up is carried out. The theories of run-up of nonbreaking waves are analyzed first. Breaking criteria and the main theories of run-up of breaking waves are presented, including the theories for the run-up of a bore, the nonsaturated breaker theory, and the numerical method for calculating run-up. This set of various theories is compared with experimental investigations. A simple case illustrates how the wave run-up can be enhanced by resonance effects.