Geoscience Reference
In-Depth Information
2
2
∂
ξ
1
g
H
∂
ξ
1
g
H
Q
(
x
,
t
)
.
x
2
−
=
(3.53)
∂
∂
t
2
Thus, to calculate a long wave caused by the combined action of a force field and
distributed sources of mass it is necessary to calculate the following function:
⎛
⎝
∂Φ
x
∂
⎞
0
0
2
∂
Φ
z
d
z
∗
−
∂
s
⎠
.
Q
(
x
,
t
)=
d
z
+
(3.54)
x
2
x
∂
∂
t
−
H
z
For a constant depth of the basin the solution of equation (3.53) is given by
the well-known integral formula [Tikhonov, Samarsky (1999)]. In the general case,
when the depth is a function of the horizontal coordinate, the equation is readily
resolved numerically by the finite difference method.
3.2.2 Non-linear Mechanism of Tsunami Generation by Bottom
Oscillations in an Incompressible Ocean
Suppose that in the process of an underwater earthquake a section of the ocean
bottom oscillates with a frequency corresponding to range II. In this case the ocean
behaves like an incompressible liquid, undergoing induced oscillations following
movements of the bottom. From formulae (3.44) and (3.45) the non-linear tsunami
source is seen to be manifested only as a force field,
v
,
v
.
Φ
incompr
=
−
∇
(3.55)
In this case the linear mechanism is not capable of leading to the formation of grav-
itational waves, but they may arise as a result of the action of the force field.
Calculation of the quantity
Φ
incompr
requires knowledge of the velocity field in
the induced oscillations of the water column. The velocity field can be calculated
from the solution of the problem within the framework of linear potential theory,
(2.58)-(2.59). Let the law of motion of the ocean bottom,
η
(
x
,
t
), have the form
η
i
(
x
)
θ
)
sin(
η
(
x
,
t
)=
(
t
)
−
θ
(
t
−
τ
ω
t
)
,
i
= 1
,
2
,
x
2
a
−
2
η
1
(
x
)=
η
0
exp
{−
}
,
⎧
⎨
(3.56)
η
0
,
|
x
|
b
,
η
0
[
c
−
1
(
b
η
2
(
x
)=
−|
x
|
)+1]
,
b
<
|
x
|
b
+
c
,
⎩
0
,
|
x
|
>
b
+
c
,
η
ω
where
are the amplitude and cyclic frequency, respectively, of ocean
bottom oscillations,
a
,
b
,
c
are parameters characterizing the horizontal extension
and shape of the space distribution of the amplitudes of bottom oscillations and
θ
0
and
is the Heaviside function. The model law of motion of the ocean bottom is
