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Linear potential theory for tsunami generation and propagation 1

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Tatsuhiko Saito 3

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National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Ibaraki, 5

Japan 6

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Running Title: Tsunami generation and propagation 8

Corresponding Author: Tatsuhiko Saito 9

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Keywords: tsunami, theory, generation and propagation 12

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Original version was submitted on November 5, 2012 14

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Linear potential theory 16

A general framework 17

We formulate the tsunami generation and propagation from the sea-bottom deformation 18

in a constant water depth based on a linear potential theory [e.g., Takahashi 1942; 19

Hammack 1973]. We use the Cartesian coordinates shown in Figure 1, where the z-axis 20

is vertically upward, and the x- and y-axes in a horizontal plane. The sea surface is 21

located at z = 0 , and the sea bottom is flat and located at z = −h0 . We assume that the 22

height of the water surface η x, y, t( ) at time t is small enough compared to the water 23

depth η << h0 and the viscosity is neglected. The velocity in the fluid is given by a 24

vector v x, t( ) = vxex + vyey + vzez , where x = xex + yey + zez , and ex , ey , and ez are 25

the basis vectors in the x, y, and z axes, respectively. We also assume an incompressible 26

and irrotational flow, ∇⋅v = 0 , in which the velocity vector is given as 27

v x, t( ) = gradφ x, t( ) using a velocity potential φ x, t( ) . 28

The velocity potential satisfies the Laplace equation 29

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Δφ x, t( ) = 0 , (1) 31

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and the boundary conditions at the surface (z = 0) are given by 33

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∂φ x, t( )∂t

z=0

+ g0η x, y, t( ) = 0 , (2) 35

∂φ x, t( )∂z z=0

−∂η x, y, t( )

∂t= 0 , (3) 36

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where g0 is the gravitational constant. Assuming the final sea-bottom deformation, or 38

permanent vertical displacement at the sea bottom, to be d x, y( ) , we give the vertical 39

component of the velocity as the boundary condition at the sea bottom, as follows: 40

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vz x, t( ) z=−h0 = d x, y( )χ t( ) , (4) 42

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where the function χ t( ) depends only on time, which satisfies the following: 44

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χ t( )dt =1−∞

∞

∫ . (5)

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The function of χ t( ) has the dimension of the inverse of time and is referred to 48

hereinafter as the source time function. We obtained the velocity potential that satisfies 49

(1), (2), (3), and (4) as follows [e.g., Saito and Furmura 2009]: 50

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φ x, t( ) =12π

dω−∞

∞

∫ exp −iωt[ ] χ̂ ω( )

12π( )2 −∞

∞

∫−∞

∞

∫ dkxdky exp ikxx + ikyy$% &'1kω 2 sinhkz+ g0k coshkzω 2 coshkh0 − g0k sinhkh0

d kx,ky( ), (6)

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where k = kx2 + ky

2 , χ̂ ω( ) is the Fourier transform in the time-frequency domain 54

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defined as follows: 55

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χ̂ ω( ) = dτ exp iωτ[ ]χ τ( )−∞

∞

∫

(7) 57

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and d kx,ky( ) is the 2-D Fourier transform in space-wavenumber domain defined as 59

follows: 60

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d kx,ky( ) =−∞

∞

∫ dxdyd x, y( )exp −i kxx + kyy( )$% &'−∞

∞

∫ . (8) 62

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Eq. (6) represents a formal expression of the velocity potential with the inverse 64

Fourier transform with respect to the time-frequency domain. This was derived by 65

Takahashi [1942] in cylindrical coordinates. Similar equations were obtained by 66

Kervella et al. [2007] and Levin and Nosov [2009] in the Cartesian coordinates using 67

the inverse Laplace transform. It is necessary to conduct an integration over the angular 68

frequency ω in order to obtain a solution in the time domain. Takahashi [1942] and 69

Kervell et al. [2007] calculated the integral for the sea surface z = 0, but not for an 70

arbitrary depth, z. Levin and Nosov [2009] performed the inverse Laplace transform for 71

the velocity potential for any depth z. However, they assumed a very special case with 72

the boundary condition at the sea bottom given by a linearly increasing sea-bottom 73

deformation for 1-D sea-bottom deformation (Eqs. (2.67) and (2.68) in Levin and 74

Nosov [2009]). A solution for the integration over the angular frequency ω in Eq. (6) 75

has not yet been obtained for the sea-bottom deformation generally given by Eq. (4). 76

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The main difficulty with respect to the integration is that the residue theory is not 77

applicable to the sea-bottom deformation that is given by the arbitrary function of χ t( ) 78

or χ t( ) = δ t( ) . In the following, we theoretically derive the solution in the time domain 79

for the sea-bottom deformation that is given by the arbitrary function of χ t( ) . 80

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A general solution: impulse response 82

We obtained the solution for the instantaneous sea-bottom deformation or for the 83

impulse response of the source time function given by χ t( ) = δ t( ) . The solution is 84

obtained as follows: 85

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φimpulse x, t( ) = −12π( )2 −∞

∞

∫−∞

∞

∫ dkxdky exp ikxx + ikyy$% &' d kx,ky( )

×1kcoshkzsinhkh0

δ t( ))*+

+coshkzsinhkh0

+sinhkzcoshkh0

,

-.

/

01ω0 sinω0t ⋅H t( )

−coshkzsinhkh0

+sinhkzcoshkh0

,

-.

/

01cosω0t ⋅δ t( )

345

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=12π( )2 −∞

∞

∫−∞

∞

∫ dkxdky exp ikxx + ikyy$% &'d kx,ky( )coshkh0

×−g0ω0

coshkz+ tanhkh0 sinhkz( )sinω0t ⋅H t( )+ 1ksinhkz ⋅δ t( )

)*+

346

, (17) 87

where 88

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H t( )=1 for 0 ≤ t ≤ T0 for t < 0, t > T

"#$

%$, 90

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ω0 ≡ g0k tanhkh0 . 92

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By convoluting an arbitrary function of χ t( ) with Eq. (17), we now obtain the 94

solution of the velocity potential with respect to the function generally given by Eq. (4), 95

as follows: 96

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φ x, t( ) = φimpulse x, t −τ( )χ τ( )dτ−∞

∞

∫. (18) 98

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Using Eq. (17), we obtain the horizontal components of the velocity field 100

vH = vxex + vyey , the vertical component of the velocity vz , and the height of the water 101

surface η for an instantaneous sea-bottom deformation, or the velocity at the bottom 102

is given by the delta function, vz (x, t) z=−h0 = d x, y( )δ t( ) as follows: 103

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vH x, t( ) =∇Hφimpulse x, t( )

=12π( )2 −∞

∞

∫−∞

∞

∫ dkxdky exp ikxx + ikyy%& '(d kx,ky( )coshkh0

×−ig0kHω0

fH k, z,h0( )sinω0t ⋅H t( )+ ikHksinhkz ⋅δ t( )

+,-

./0 , (20)

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vz x, t( ) =∂φimpulse x, t( )

∂z

=12π( )2 −∞

∞

∫−∞

∞

∫ dkxdky exp ikxx + ikyy%& '(d kx,ky( )coshkh0

× −ω0 fz k, z,h0( )sinω0t ⋅H t( )+ coshkz ⋅δ t( ){ } , (21)

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η x, y, t( ) = − 1g0

∂φimpulse x, t( )∂t z=0

=12π( )2 −∞

∞

∫−∞

∞

∫ dkxdky exp ikxx + ikyy%& '(d kx,ky( )coshkh0

cosω0t ⋅H t( ).

(22) 109

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where ∇H is the gradient in the horizontal plane given by ∇H = ∂ ∂x ex +∂ ∂yey , and 111

kH is the wavenumber vector in the horizontal plane given by kH = kxex + kyey . Here, 112

we introduced the distribution functions of the horizontal and vertical components of 113

the velocity, as follows: 114

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fH k, z,h0( ) = coshkz+ tanhkh0 sinhkz , (23)

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fz k, z,h0( ) = sinhkztanhkh0

+ coshkz . (24) 117

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We discuss the meaning of these distribution functions (Eqs. (23) and (24)) together 119

with the interpretations of Eqs. (20) through (22) in the following section. We can 120

confirm that Eqs. (20) through (22) satisfy ∇⋅v = 0 and the boundary conditions (Eqs. 121

(2) through (4)). 122

It is also beneficial to provide a representation for the pressure at the sea bottom 123

because an ocean-bottom pressure gauge has been often used for recording tsunami 124

generation and propagation [e.g., Tang et al. 2011; Tsushima et al. 2012]. The pressure 125

in the ocean is given by the sum of the hydrostatic pressure and an excess pressure due 126

to the wave motion, i.e., −ρ0g0z+ pe , where the excess pressure is given by the velocity 127

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potential as pe = −ρ0 ∂φ ∂t . Using the velocity potential of Eq. (19), we obtain the 128

excess pressure brought about by the tsunami at the sea bottom as follows: 129

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pe x, t( ) z=−h0 = −ρ0∂φ x, t( )∂t

z=−h0

=12π( )2 −∞

∞

∫ dkx dky exp i kxx + kyy( )%& '(−∞

∞

∫

×ρ0

coshkh0g0cosω0tcoshkh0

⋅H t( )+ 1ksinhkh0 ⋅

dδ t( )dt

+,-

./0

,

(25) 131

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for a point impulse response. For the source given by (4), we obtain, 133

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pe x, t( ) z=−h0 =12π( )2 −∞

∞

∫ dkx dky exp i kxx + kyy( )$% &'−∞

∞

∫

×ρ0 d kx,ky( )coshkh0

g0coshkh0

cos ω0 t −τ( )$% &'−∞

t

∫ χ τ( )dτ + 1ksinhkh0 ⋅

dχ t( )dt

*+,

-./

. (26) 135

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This equation is similar but not identical to that obtained by Kervell et al. [2007]. 137

Kervella et al. [2007] derived the pressure at the sea bottom after an instantaneous 138

sea-bottom uplift. However, they did not consider a source term, so that the pressure 139

change during the source process time cannot be inclusive. On the other hand, Eq. (26) 140

includes an additional term (the second term in brackets { } in Eq. (26)) as a source 141

term, which represents the contribution from the source. This term leads to an increase 142

in excess pressure at the sea bottom when the sea-bottom uplifts with an increasing rate 143

dχ dt > 0 . 144

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O

z

x, yz = !(x,y,t)

z = -h0

z = 0Surface

Bottom 210

Figure 1. Coordinates used for the formulation. 211