Numerical modeling of waves for a tsunami early warning system

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Numerical modeling of waves for a tsunami early warning system The Laplace equation within the free surface boundary conditions and the so modified bottom boundary condition, represent a problem as that formulated in the section 3.1.1. If the same procedure of depth integration is followed, it will appear an extra term. From the Green Theorem it results 0 −h f ∂2 φ ∂z 2 − φ∂2 f ∂z 2 dz = f ∂φ ∂z − φ∂f ∂z − f 0 ∂φ ∂z − φ∂f ∂z which, using the modified boundary conditions, now becomes 0 −h f∇ 2 hφ + k 2 fφ dz = 1 g ϕtt + ϕ ω2 g − [fht] −h − [f∇hh∇hφ] −h −h (3.40) (3.41) Operating the same step used to achieve equation (3.19) in this case it results 0 −h (f 2 ∇ 2 hϕ +2f∇hf∇hϕ + fϕ∇ 2 hf + k 2 f 2 ϕ)dz = 1 g (ϕtt + ω2ϕ) − 1 cosh(kh) ht − [f∇hh (f∇hϕ + ϕ∇hf)] −h (3.42) Now incorporating the first two terms of the LHS of equation (3.42) follows 0 −h ∇h (f 2∇hϕ)dz +[∇hhf 2∇hϕ] −h + ϕk2 0 −h f 2dz = − 0 −h ϕf∇2hfdz− 1 cosh(kh) ht − ϕ∇hh [f∇hf] −h + 1 g (ϕtt + ω2ϕ) (3.43) Applying the Leibniz’s rule for the first two terms on the LHS and knowing that 0 f −h 2 dz = ccg it results g (3.44) ∇h (ccg∇hϕ)+ϕk2ccg − ϕtt − ω2ϕ = g −ht cosh(kh) − gϕ{ 0 −h f∇2 (3.45) hfdz + ∇hh [f∇hf] −h } for the same assumptions of being on mild slope sea bottom (∇hh = α
Numerical modeling of waves for a tsunami early warning system ϕtt −∇h (ccg∇hϕ)+ ω 2 − k 2 g ccg ϕ = − cosh (kh) ht which again in terms of water free surface becomes −ηtt + ∇h (ccg∇hη) − ω 2 − k 2 1 ccg η = − cosh (kh) htt and in the frequency domain becomes (3.46) (3.47) ∇h (ccg∇hN)+k 2 1 ccgN = − cosh (kh) fft(htt) (3.48) Equation (3.48) is an elliptic equation, written in terms of the free surface elevation in the frequency domain, with a forcing term which accounts for the movements of the sea floor. 3.2 Model applications The model applications depend on how the waves are numerically generated. Two alternative approaches have been already discussed. The first approach makes use of the wave-maker boundary conditions, the second one is based on the source term in the model equation. Both these approaches can be used following a ‘direct’ or an ‘indirect’ procedure. The ‘direct’ one is applicable when the movement of the bottom is known, or when the properties of the waves to be generated are known at the wave-maker boundary. In this case from the proper time series of h (x, y, t), and its time derivatives, it can be easily calculated by means of the discrete Fourier Transform, the transformed variable fft(htt) to be used into the equation (3.48). Equivalently if the fluid velocity at z = 0 is known at the wave-maker it can be directly used into equation (3.38). The ‘indirect’ procedure is convenient when the surface elevation time series is known at some points of the computational domain and the position of the area (or of the boundary) where the waves are generated is known. In this case it is possible to find the source terms appearing into equations (3.48) or (3.38) by an inversion technique as follows. The first step is to solve the model equations using a unit value of the source term. The result of such preliminary computation is referred to as N ′ (x, y, ω). In view of the linearity of the problem the true solution in the frequency domain N (x, y, ω) canbe Università degli Studi di Roma Tre - DSIC 32
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
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Numerical modeling of waves for a t
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Numerical modeling of waves for a tsunami early warning system

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