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.

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Shoaling and breaking of solitary waves is computed on slopes 1:100 to 1:8 using an experimentally validated fully nonlinear wave model based on potential flow equations. Characteristics of waves are computed at and beyond the breaking point, and geometric self-similarities of breakers are discussed as a function of wave height and bottom slope. No wave breaks for slopes steeper than 12. A breaking criterion is derived for milder slopes, based on values of a nondimensional slope parameter S o. This criterion predicts both whether waves will break or not and which type of breaking will occur (spilling, plunging, or surging). Empirical expressions for the breaking index and for the depth and celerity at breaking are derived based on computations. All results agree well with laboratory experiments. The NSW equations fail to predict these results with sufficient accuracy at the breaking point. Pre-breaking shoaling rates follow a more complex path than previously realized. Post-breaking behaviors exhibit a rapid (non-dissipative) decay, also observed in experiments, associated with a transfer of potential energy into kinetic energy. Wave celerity decreases in this zone of rapid decay.

Numerical simulations are performed with a two-dimensional ͑2D͒ fully nonlinear potential flow ͑FNPF͒ model for tsunami generation by two idealized types of submarine mass failure ͑SMF͒: underwater slides and slumps. These simulations feature rigid or deforming SMFs with a Gaussian cross section, translating down a plane slope. In each case, the SMF center of mass motion is expressed as a function of geometric, hydrodynamic, and material parameters, following a simple wavemaker formalism, and prescribed as a boundary condition in the FNPF model. Tsunami amplitudes and runup are obtained from computed free surface elevations. Model results are experimentally validated for a rigid 2D slide. Sensitivity studies are performed to estimate the effects of SMF-shape, type, and initial submergence depth-on the generated tsunamis. A strong SMF deformation during motion is shown to significantly enhance tsunami generation, particularly in the far-field. Typical slumps are shown to generate smaller tsunamis than corresponding slides. Both tsunami amplitude and runup are shown to depend strongly on initial SMF submergence depth. For the selected SMF idealized geometry, this dependence is simply expressed by power laws. Other sensitivity analyses are presented in a companion paper, and results from numerical simulations are converted into empirical curve fits predicting characteristic tsunami amplitudes as functions of nondimensional governing parameters. It should be stressed that these empirical formulas are only valid in the vicinity of the tsunami sources and, because of the complexity of the problem, many simplifications were necessary. It is further shown in the companion paper how 2D results can be modified to account for three-dimensional tsunami generation and used for quickly estimating tsunami hazard or for performing simple case studies.

Abstract. Case studies of landslide tsunamis require integration of marine geology data and interpretations into numerical simulations of tsunami attack. Many landslide tsunami generation and propagation models have been proposed in recent time, further motivated by the 1998 Papua New Guinea event. However, few of these models have proven capable of integrating the best available marine geology data and interpretations into successful case studies that reproduce all available tsunami observations and records. We show that nonlinear and dispersive tsunami propagation models may be necessary for many landslide tsunami case studies. GEOWAVE is a comprehensive tsunami simulation model formed in part by combining the Tsunami Open and Progressive Initial Conditions System (TOPICS) with the fully nonlinear Boussinesq water wave model FUNWAVE. TOPICS uses curve fits of numerical results from a fully nonlinear potential flow model to provide approximate landslide tsunami sources for tsunami propagation models, based on marine geology data and interpretations. In this work, we validate GE-OWAVE with successful case studies of the 1946 Unimak, Alaska, the 1994 Skagway, Alaska, and the 1998 Papua New Guinea events. GEOWAVE simulates accurate runup and inundation at the same time, with no additional user interference or effort, using a slot technique. Wave breaking, if it occurs during shoaling or runup, is also accounted for with a dissipative breaking model acting on the wave front. The success of our case studies depends on the combination of accurate tsunami sources and an advanced tsunami propagation and inundation model.

[1] In this work, we study waves generated by the potential collapse of the west flank of the Cumbre Vieja Volcano (CVV; La Palma, Canary Island, Spain) through numerical simulations performed in two stages: (i) the initial slide motion and resulting free surface elevation are first calculated using a 3D Navier-Stokes model; (ii) generated waves are then input into a 2D (horizontal) Boussinesq model to further simulate propagation to the nearby islands. Unlike in earlier work on CVV, besides a similar extreme slide volume scenario of 450 km 3 , in our simulations: (i) we consider several slide scenarios featuring different volumes (i.e., 20, 40, 80 km 3 ), which partly result from a geotechnical slope stability analysis; (ii) we use a more accurate bathymetry; and (iii) an incompressible version of a multiple-fluid/material Navier-Stokes model. We find wave trains for each scenario share common features in terms of wave directivity, frequency, and time evolution, but maximum elevations near CVV significantly differ, ranging from 600 to 1200 m (for increasing slide volume). Additionally, our computations show that significant energy transfer from slide to waves only lasts for a short duration (order 200 s), which justifies concentrating our best modeling efforts on the early slide motion phase. The anticipated consequences of such wave trains on La Palma and other Canary Islands are assessed in detail in the paper.

Large scale, three-dimensional, laboratory experiments are performed to study tsunami generation by rigid underwater landslides. The main purpose of these experiments is to both gain insight into landslide tsunami generation processes and provide data for subsequent validation of a three-dimensional numerical model. In each experiment a smooth and streamlined rigid body slides down a plane slope, starting from different initial submergence depths, and generates surface waves. Different conditions of wave nonlinearity and dispersion are generated by varying the model slide initial submergence depth. Surface elevations are measured with capacitance gauges. Runup is measured at the tank axis using a video camera. Landslide acceleration is measured with a microaccelerometer embedded within the model slide, and its time of passage is further recorded at three locations down the slope. The repeatability of experiments is very good. Landslide kinematics is inferred from these measurements and an analytic law of motion is derived, based on which the slide added mass and drag coefficients are computed. Characteristic distance and time of slide motion, as well as a characteristic tsunami wavelength, are parameters derived from these analyses. Measured wave elevations yield characteristic tsunami amplitudes, which are found to be well predicted by empirical equations derived in earlier work, based on two-dimensional numerical computations. The strongly dispersive nature and directionality of tsunamis generated by underwater landslides is confirmed by wave measurements at gauges. Measured coastal runup is analyzed and found to correlate well with initial slide submergence depth or characteristic tsunami amplitude.