Numerical modeling of waves for a tsunami early warning system

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Numerical modeling of waves for a tsunami early warning system first crest arrives earlier than that measured in the experiments, and the waves are higher than those expected. Of course the non proper reproduction of the frequency dispersion leaves all the wave components to propagate together. As the distance from the generation area grows these errors become unacceptable. aFourier x 10−3 2.5 2 1.5 1 0.5 0 0 10 20 30 40 50 60 70 80 ω (rad/s) Figure 4.5: Amplitude spectrum of the surface elevation at gauge 1 for the experiment with water depth sets at 0.23 m and the box released from the still water level. The dashed line indicates the highest frequency component reproduced in the model. It is worth to briefly discuss how the energy of these considered transient waves is distributed in the frequency domain. The modulus of the Fourier Transforms coefficients of the measured water level oscillations at gauge 1 is reported in the figure 4.5 against the angular frequency. The figure is cut at the angular frequency of 80 rad/s, because it is considered not advisable to show the wave spectra until the 2π · 10 3 angular frequency, and it can be noted that very high frequency components, as expected, receive minor energy. The accurate reproduction of these waves may therefore be avoided in order to save computational time. For the present experiments it has believed sufficient to solve the equation (3.29) up to frequencies corresponding to kh =92.6 (0.1Hz < f < 10Hz). This limit is shown in the Figure 4.5 by the vertical dashed line. The higher frequency component corresponds to a wave length (for the considered water depth) of 0.0156 m, thus an appropriate finite element grid (the maximum distance from the computational nodes) of 0.0015 m has been used since a resolution of about 10 nodes per wave length is required; the resulting number of Degrees of Freedom (DOF) is 42,600. The solution of the elliptic equation has been carried out, using always the Università degli Studi di Roma Tre - DSIC 45
Numerical modeling of waves for a tsunami early warning system same, very fine grid, for 100 discrete frequencies. The total computational time is of the order of 8 minutes on an AMD Opteron 246 2GHz computer equipped with 4GB of RAM. aFourier x 10 2.5 −3 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 2 1.5 1 0.5 kh 0 0 1 2 3 4 5 6 7 8 9 10 Figure 4.6: Upper panel: amplitude spectra of the surface elevation equal to that described in caption of Figure 4.5 versus the wave length parameter. Lower panel: distribution of the wave celerities over the wave length parameter; the three vertical solid lines represent the frequencies corresponding to h/L =1/20 (blue), h/L =1/4 (green) and h/L =1/2 (red). The figure 4.6 looks in more detail how the wave energy is distributed in frequency. In the upper panel the same amplitude spectra of the water level oscillation at gauge 1 is reported now versus the wave length parameter written as kh. The figure is zoomed only for those frequencies where there is the major wave energy content. In the lower panel are reported the wave celerity functions against the same wave length parameter: the solid line represents the wave celerity in the shallow water approximation (csw = √ gh), while the dash-dot line and the dashed line represent respectively the phase Università degli Studi di Roma Tre - DSIC 46 kh csw cg c
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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
Page 16 and 17: 4.32 Comparison of the free surface
Page 18 and 19: xvii 5.8 Numerical results of the S
Page 20: List of Tables 4.1 Radial and Angul
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Page 65: η1 (m) η2 (m) η3 (m) η4 (m) η5
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Page 95 and 96: half an hour. ηS12(m) ηS7(m) ηS1
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η(m) η(m) η(m) 50 0 Numerical mo
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Numerical modeling of waves for a t
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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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