Geoscience Reference
In-Depth Information
bottom oscillation amplitude
η
1
(
x
). From the figure it is seen that the terms
X
(
x
)
and
Z
(
x
), as a rule, exhibit differing signs. This means the structure of the force
field is such that the contribution of the horizontal force component to the gravi-
tational wave formation is always partly compensated by the vertical component.
In the case of a source of small size (
a
H
) this effect is capable of significantly
reducing the wave amplitude. However, in the case of a large horizontal extension
of the source (
a
∼
H
) the action of the horizontal component turns out to pre-
vail (
). The dimensions of real tsunami sources are always significantly
greater than the ocean depth, therefore, the contribution of the vertical component
of the force field can be neglected.
Neglecting the contribution of the vertical component of the force field,
Z
(
x
),we
write equation (3.53) in a dimensionless form (in accordance with formulae (3.60)):
|
X
| |
Z
|
2
2
∂
x
2
−
∂
ξ
ξ
=
H
g
∂ Φ
x
,
(3.72)
∂
∂
t
2
∂
x
0
where
1
Φ
x
d
z
is the horizontal component of the force field, averaged along
the vertical direction,
Φ
x
=
−
is the displacement of the free surface of the liquid from
its equilibrium position, corresponding to the mean movement. We recall that there
are, also, present above the oscillating ocean bottom fast oscillations of the surface,
which are related to induced oscillations.
The solution of equation (3.72) is well known [Tikhonov, Samarsky (1999)]:
ξ
t
)
x
+(
t
−
t
(
x
,
t
)=
H
2g
∂ Φ
x
∂
d
t
ξ
d
x
.
(3.73)
x
0
t
)
x
−
(
t
−
Oscillations of the ocean bottom (3.56) take place during a finite period of time
τ
and exhibit fixed amplitude and frequency. Therefore we can write
Φ
x
(
x
)
θ
)
.
Φ
x
(
x
,
t
)=
(
t
)
−
θ
(
t
−
τ
(3.74)
Substituting expression (3.74) into formula (3.73) and performing integration
over the space variable, we obtain
t
θ
)
Φ
x
(
x
+(
t
t
))
d
t
.
(3.75)
H
2g
(
t
)
(
t
t
))
ξ
(
x
,
t
)=
−
−
θ
−
τ
−
−
Φ
x
(
x
−
(
t
−
0
The process of tsunami formation by the non-linear mechanism is shown in
Fig. 3.21. Shifts of the surface of the liquid,
, were calculated by formula (3.75) as
functions of the horizontal coordinate
x
for consecutive moments of time. A com-
pletely formed wave is sure to consist of a hump and depression, which have a zero
total volume. The perturbation always starts with a positive phase. The wave length
approximately corresponds to the size of the source.
ξ
