Numerical modeling of waves for a tsunami early warning system

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Numerical modeling of waves for a tsunami early warning system derivative of the Fourier Transform of the free surface elevation η along the normal to the reflective boundary should be zero (see Mei, 1983): Nn =0. (3.32) The radiation boundary condition can be obtained by using a mathematical formulation that allows the waves that propagate toward the open boundaries to freely exit the computational domain. This condition can be easily formulated for progressive outgoing waves (Sommerfeld, 1964; Van Dongeren & Svendsen, 1997) c ηt + cos (θn) ηn = 0 (3.33) where θn is the angle the wave direction forms with the outgoing normal to the considered boundary. The Fourier Transform of equation (3.33) provides the radiation condition in the frequency domain (see Beltrami et al., 2001; Steward & Panchang, 2000) Nn + ikcos (θn)N =0. (3.34) Please note that the equation (3.34) is nonlinear in the sense that θn is not known a priori and depends on the solution itself. Iterative techniques can therefore be applied or a reasonable estimate of this parameter can be used to solve the indeterminacy. The waves generation in the numerical domain can be modeled with two different approaches: by means of a wave-maker boundary condition or by means of a source term included in the field equation. The latter approach is discussed in the next section 3.1.3. A wave-maker boundary condition can be seen as a wave paddle used in physical models (although here evanescent modes are not reproduced) and can be applied when the wave generation mechanism occurs close to a boundary of the numerical domain. The wavemaker boundary can be ‘open’ in the sense that generates the waves inside the domain and even allows the waves to exit the domain through it, i.e. the vertical section of an harbor opening, modeling the wave field inside the harbor. Otherwise the wave-maker boundary can be ‘closed’ thus reflects the incoming waves. The latter is the case of the wave field generated by an aerial landslide entering into the sea water when it is known the point of the impact. Università degli Studi di Roma Tre - DSIC 29
Numerical modeling of waves for a tsunami early warning system The wave-maker boundary condition is conveniently formulated in terms of the velocity potential at z = 0 as follows ϕn = u I , z =0, (3.35) where u I is the velocity at z = 0 of the desired wave field orthogonal to the wave-maker boundary. In order to obtain a mathematical expression involving η and consequently N we make use of the dynamic boundary condition at the free surface (3.2) which if transformed in the frequency domain results as follows providing the usual relationship between Φ and N: iωΦ =−gN, (3.36) Φ=− g N iω which allows rewriting of the wave-maker condition as (3.37) Nn = − iω I U g (3.38) being U I the Fourier Transform of the desired time series of velocity at z = 0. It is worth to remind that the proposed wave-maker condition is used to specify the fluid velocity at z = 0, and that the velocity field for −h
Page 1 and 2: Università degli studi ROMA TRE Ph
Page 3 and 4: To my father Michele, my mother Ant
Page 6 and 7: Abstract A numerical model based on
Page 8 and 9: Sommario Un modello numerico basato
Page 10 and 11: Contents 1 Introduction 3 1.1 Tsuna
Page 12 and 13: List of Figures 1.1 Principal phase
Page 14 and 15: 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
Page 23 and 24: Numerical modeling of waves for a t
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Numerical modeling of waves for a t
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ηS15(m) ηS24(m) ηS16(m) −0.01
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η(m) η(m) η(m) 50 0 Numerical mo
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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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