Numerical modeling of waves for a tsunami early warning system

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Numerical modeling of waves for a tsunami early warning system becomes less concentrated in space and time. This also makes the height of the waves to become smaller. Here however, the waves also undergo the so called circumferential dispersion, i.e. the height of the waves decreases in view of the fact that the wave front becomes longer as they propagate. The numerical model is able of carefully reproducing the experimental results. This is especially true for the first gauges (1, 2, 3 and 4). The results are not good for the gauge 5, where the height of the predicted waves is smaller than that measured. This is probably due to the effect of the reflected waves, which were not modeled numerically. Results of the same order of accuracy would be obtained using the BTE of Nwogu (1993), that are not presented here for sake of clarity of the figure. The LSWE appear to provide a wave field different from that observed in the experiment. The wave packet predicted by these equations remains very similar during the propagation, although the circumferential dispersion (properly taken into account by the model) induces a decrease of the wave height, as expected. However the arrival time of the highest waves appears to be predicted with large errors. 4.3 Inclusion of waves generation term - Numerical experiments This section describes some numerical experiments which reproduce the water waves generated by the deformation of the bottom due to the movement of a submerged landslide. The idea is to validate how the effects of the bottom movement are reproduced in the depth integrated model using equation (3.48). Here the model is applied with a ‘direct’ procedure, meaning that the deformation of the sea floor is known, and its effects on the free water surface are simulated. These experiments compare the results of the depth integrated model with those of a three dimensional one, which can be regarded as the reference (i.e. the true) solutions. The three dimensional model solves the Laplace equation within the linearized boundary conditions. It uses the same numerical scheme of the depth integrated model, thus the equations are solved in the frequency domain and are formulated as follows ∇ 2 hΦ+Φzz =0 (4.2) Università degli Studi di Roma Tre - DSIC 51
Numerical modeling of waves for a tsunami early warning system Φz − ω2 Φ = 0 (4.3) g Φn = fft(ht) (4.4) Φn + ikcos (θn)Φ=0 (4.5) where Φ (x, y, z, ω) is the Fourier transform of φ (x, y, z, t). Equation (4.2) is the Laplace equation, equation (4.3) is the free surface dynamic and kinematic boundary condition, equation (4.4) represents the moving bottom boundary condition and equation (4.5) is the radiation condition. In order to further investigate on the source term included in the mild slope equation, some consideration can be made. When the extension of the seismic source is very large in comparison with the water depth a well accepted method for incorporating the effect of the moving seafloor into the depth integrated equations is to assume that the bottom movements instantaneously transfer to the surface elevation. A source term calculated as the time derivative of the water depth is added to the continuity equation (right hand side) ηt + ▽·(vh) =−ht (4.6) where as usual η, v and h represent respectively the instantaneous elevation of the free surface, the depth integrated horizontal velocity of the fluid and the water depth. Since the movements induced by earthquakes are very fast in comparison with the waves celerity, these are alternatively represented as initial conditions for the wave model. The problem becomes much more complicated for tsunamis generated by landslides. These can still be regarded as seafloor deformations, but the spatial extent of the source is usually not much larger than the water depth. Furthermore the time scale of the landslide movements is comparable to that of the waves propagation. Three dimensional numerical models are therefore applied to get detailed descriptions of the flow field, see for example Grilli et al. (2002). Depth integrated models are able also in this case to properly simulate the wave propagation in the far field, but cannot give a detailed description of the flow in the generation area. In these cases it is well accepted that the movements of the bottom do not transfer as they are Università degli Studi di Roma Tre - DSIC 52
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Università degli studi ROMA TRE Ph
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To my father Michele, my mother Ant
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Abstract A numerical model based on
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Sommario Un modello numerico basato
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Contents 1 Introduction 3 1.1 Tsuna
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List of Figures 1.1 Principal phase
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xiii 4.13 Panels a and b: absolute
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4.32 Comparison of the free surface
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xvii 5.8 Numerical results of the S
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List of Tables 4.1 Radial and Angul
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η (m) η (m) η (m) η (m) 20 0
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and Numerical modeling of waves for
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Numerical modeling of waves for a tsunami early warning system

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