Linear potential theory for tsunami generation and propagation

Transcription

1 Linear potential theory for tsunami generation and propagation 2 3 Tatsuhiko Saito National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Ibaraki, Japan Running Title: Tsunami generation and propagation Corresponding Author: Tatsuhiko Saito Keywords: tsunami, theory, generation and propagation Original version was submitted on November 5, 202

2 Linear potential theory A general framework We formulate the tsunami generation and propagation from the sea-bottom deformation in a constant water depth based on a linear potential theory [eg, Takahashi 942; Hammack 973] We use the Cartesian coordinates shown in Figure, where the z-axis is vertically upward, and the x- and y-axes in a horizontal plane The sea surface is located at z = 0, and the sea bottom is flat and located at z = h 0 We assume that the 23 height of the water surface η( x, y,t) at time t is small enough compared to the water 24 depth η << h 0 and the viscosity is neglected The velocity in the fluid is given by a vector v( x,t) = v x e x + v y e y + v z e z, where x = xe x + ye y + ze z, and e x, e y, and e z are the basis vectors in the x, y, and z axes, respectively We also assume an incompressible and irrotational flow, v = 0, in which the velocity vector is given as v( x,t) = gradφ ( x,t) using a velocity potential φ ( x,t) The velocity potential satisfies the Laplace equation 30 3 Δφ ( x,t) = 0, () and the boundary conditions at the surface (z = 0) are given by φ ( x,t) + g 0 η( x, y,t) = 0 t, (2) z=0 φ ( x,t) η( x, y,t) = 0 z t, (3) z=0 2

3 where g 0 is the gravitational constant Assuming the final sea-bottom deformation, or permanent vertical displacement at the sea bottom, to be d ( x, y), we give the vertical component of the velocity as the boundary condition at the sea bottom, as follows: 4 42 v z ( x,t) z= h0 = d x, y χ ( t), (4) where the function χ ( t) depends only on time, which satisfies the following: χ ( t)dt = (5) The function of χ ( t) has the dimension of the inverse of time and is referred to hereinafter as the source time function We obtained the velocity potential that satisfies (), (2), (3), and (4) as follows [eg, Saito and Furmura 2009]: φ ( x,t) = 2π dω exp iωt [ ] ˆχ ( ω) dk ( 2π ) 2 x exp$% ik x x + ik y y& ω 2 sinhkz + g 0 k coshkz ' k ω 2 coshkh 0 g 0 k sinhkh 0 d ( kx, k y ), (6) 54 where k = k 2 x + k 2 y, ˆχ ( ω) is the Fourier transform in the time-frequency domain 3

4 55 defined as follows: = dτ exp[ iωτ ] χ ( τ ) ˆχ ω (7) and d ( kx, k y ) is the 2-D Fourier transform in space-wavenumber domain defined as follows: 6 62 d ( k x, k y ) = dx dyd ( x, y)exp$ % i k x x + k y y & ' (8) Eq (6) represents a formal expression of the velocity potential with the inverse Fourier transform with respect to the time-frequency domain This was derived by Takahashi [942] in cylindrical coordinates Similar equations were obtained by Kervella et al [2007] and Levin and Nosov [2009] in the Cartesian coordinates using the inverse Laplace transform It is necessary to conduct an integration over the angular frequency ω in order to obtain a solution in the time domain Takahashi [942] and Kervell et al [2007] calculated the integral for the sea surface z = 0, but not for an arbitrary depth, z Levin and Nosov [2009] performed the inverse Laplace transform for the velocity potential for any depth z However, they assumed a very special case with the boundary condition at the sea bottom given by a linearly increasing sea-bottom deformation for -D sea-bottom deformation (Eqs (267) and (268) in Levin and Nosov [2009]) A solution for the integration over the angular frequency ω in Eq (6) has not yet been obtained for the sea-bottom deformation generally given by Eq (4) 4

5 The main difficulty with respect to the integration is that the residue theory is not applicable to the sea-bottom deformation that is given by the arbitrary function of χ ( t) or χ ( t) = δ ( t) In the following, we theoretically derive the solution in the time domain for the sea-bottom deformation that is given by the arbitrary function of χ ( t) A general solution: impulse response We obtained the solution for the instantaneous sea-bottom deformation or for the impulse response of the source time function given by χ ( t) = δ ( t) The solution is obtained as follows: φ impulse ( x,t) = 2π 2 dk x exp$% ik x x + ik y y& ' d kx, k y ) cosh kz, * δ ( t) + coshkz + sinhkz / ω 0 sinω 0 t H t k + sinhkh 0 -sinhkh 0 coshkh 0 0, coshkz + sinhkz / cosω 0 t δ t -sinhkh 0 coshkh d kx, k = dk ( 2π ) 2 x exp$% ik x x + ik y y& y ' coshkh 0 ) g 0 * + ω 0 ( coshkz + tanhkh 0 sinhkz)sinω 0 t H ( t) + sinhkz δ ( t) k 3 4 6, (7) 88 where " $ H ( t)= # %$ for 0 t T 0 for t < 0, t > T, 9 5

6 92 ω 0 g 0 k tanhkh By convoluting an arbitrary function of χ ( t) with Eq (7), we now obtain the solution of the velocity potential with respect to the function generally given by Eq (4), as follows: = φ impulse ( x,t τ ) χ τ φ x,t dτ (8) Using Eq (7), we obtain the horizontal components of the velocity field v H = v x e x + v y e y, the vertical component of the velocity v z, and the height of the water surface η for an instantaneous sea-bottom deformation, or the velocity at the bottom 03 is given by the delta function, v z (x,t) z= h0 = d ( x, y)δ ( t) as follows: v H ( x,t) = H φ impulse x,t d kx, k = dk ( 2π ) 2 x exp%& ik x x + ik y y' y ( coshkh 0 +, ig 0k H f H ( k, z, h 0 )sinω 0 t H t - ω 0 + ik H k sinhkz δ t / 0, (20) v z ( x,t) = φ impulse x,t z d kx, k = dk ( 2π ) 2 x exp%& ik x x + ik y y' y ( coshkh 0 ω 0 f z ( k, z, h 0 )sinω 0 t H t { + coshkz δ ( t) }, (2) 6

7 = φ impulse ( x,t) g 0 η x, y,t t z=0 d kx, k = dk ( 2π ) 2 x exp%& ik x x + ik y y' y ( cosω 0 t H t coshkh 0 (22) where H is the gradient in the horizontal plane given by H = x e x + ye y, and k H is the wavenumber vector in the horizontal plane given by k H = k x e x + k y e y Here, we introduced the distribution functions of the horizontal and vertical components of the velocity, as follows: f H ( k, z, h 0 ) = coshkz + tanhkh 0 sinhkz, (23) f z ( k, z, h 0 ) = sinhkz + coshkz (24) tanhkh We discuss the meaning of these distribution functions (Eqs (23) and (24)) together with the interpretations of Eqs (20) through (22) in the following section We can confirm that Eqs (20) through (22) satisfy v = 0 and the boundary conditions (Eqs (2) through (4)) It is also beneficial to provide a representation for the pressure at the sea bottom because an ocean-bottom pressure gauge has been often used for recording tsunami generation and propagation [eg, Tang et al 20; Tsushima et al 202] The pressure in the ocean is given by the sum of the hydrostatic pressure and an excess pressure due to the wave motion, ie, ρ 0 g 0 z + p e, where the excess pressure is given by the velocity 7

8 28 29 potential as p e = ρ 0 φ t Using the velocity potential of Eq (9), we obtain the excess pressure brought about by the tsunami at the sea bottom as follows: 30 p e φ x,t ( x,t) z= h0 = ρ 0 t z= h 0 3 = ( 2π ) 2 dk x exp% & i k x x + k y y ' ( ρ + 0 cosω g 0 t, 0 H t coshkh 0 - coshkh 0 + k sinhkh dδ ( t ) 0 dt /, 0 (25) for a point impulse response For the source given by (4), we obtain, 35 p e ( x,t) z= h0 = dk ( 2π ) 2 x exp$ % i( k x x + k y y) & ' ρ t 0d ( k x, k y ) g 0 cos$% ω 0 ( t τ )& ' χ ( τ )dτ + coshkh 0 coshkh 0 k sinhkh 0 dχ ( t (26) * )- +, dt / This equation is similar but not identical to that obtained by Kervell et al [2007] Kervella et al [2007] derived the pressure at the sea bottom after an instantaneous sea-bottom uplift However, they did not consider a source term, so that the pressure change during the source process time cannot be inclusive On the other hand, Eq (26) includes an additional term (the second term in brackets { } in Eq (26)) as a source term, which represents the contribution from the source This term leads to an increase in excess pressure at the sea bottom when the sea-bottom uplifts with an increasing rate dχ dt > 0 8

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12 z = 0 Surface z O z = η(x,y,t) x, y z = -h 0 Bottom 20 2 Figure Coordinates used for the formulation 2