Geoscience Reference
In-Depth Information
further successful development it is necessary to introduce a number of essentially
novel features, one of which consists in the following: wave excitation must be
described realistically, i.e. it must be dealt with as a process extended in time. As
a rule, in describing tsunami generation impulse displacements are considered,
and, consequently, only geometric characteristics of the source, i.e. the distribution
of residual bottom displacements in space, are taken into account. Here, the ac-
tual method (the time law followed by motions of the ocean bottom), by which
the different residual displacements came about, is totally neglected. At the same
time, the duration of processes at the source may amount to 100 s, and more [Sa-
take (1995)]. For example, the process at the source of the Sumatran catastrophic
tsunami of 2004 went on for about 1,000 s. In such a long period of time, a long
wave is capable of covering a distance comparable to the size of a tsunami source,
which means that a displacement cannot be assumed to exhibit an impulse character.
Moreover, in [Dotsenko (1996); Nosov (1998)] it was established that the energy,
amplitude and, even, orientation of tsunami waves is not only related to the geo-
metric characteristics of the source, but also to the time law of motion of the ocean
bottom.
In Sect. 2.2.1 the general solution was obtained, within the framework of poten-
tial theory, for the linear response of a layer of incompressible liquid of fixed depth
to deformations of the ocean bottom,
(
x
,
y
,
t
). We shall consider the following three
model laws of ocean bottom deformation:
η
•
Piston-like displacement
τ
−
1
,
η
1
(
x
,
y
,
t
)=
η
S
(
x
,
y
)(
θ
(
t
)
t
−
θ
(
t
−
τ
)(
t
−
τ
))
(2.105)
•
Membrane-like displacement
η
2
(
x
,
y
,
t
)=
η
S
(
x
,
y
)(2
θ
(
t
)
t
−
4
θ
(
t
−
τ
/
2)(
t
−
τ
/
2)
τ
−
1
,
+ 2
θ
(
t
−
τ
)(
t
−
τ
))
(2.106)
•
Running displacement
η
3
(
x
,
y
,
t
)=
η
S
(
x
−
a
,
y
)(1
−
θ
−
vt
))
,
(
x
(2.107)
where
η
S
(
x
,
y
)=
η
0
(
θ
(
x
+
a
)
−
θ
(
x
−
a
)) (
θ
(
y
+
b
)
−
θ
(
y
−
b
)) is the space dis-
tribution of ocean bottom deformations,
(
z
) is the Heaviside step-function. The
active region has the shape of a rectangle of length 2
a
and width 2
b
. The piston-like
and membrane-like displacements are characterized by amplitude
θ
η
0
and duration
τ
,
the running displacement by its amplitude
η
0
and propagation velocity
v
. In the case
of a running displacement the area of bottom deformations is shifted in the positive
direction of axis 0
x
by the quantity
a
, so as to have motions of the ocean bottom
start at the time moment
t
= 0.
