Geoscience Reference
In-Depth Information
In general, it is evidently not correct, from a physical point of view, to trans-
fer sea-floor deformations up to the surface. In the case of deformation of the sea-
floor, lasting for a long time, i.e. when a long wave has time to propagate over a
noticeable distance, as compared with the horizontal dimension of the source, ele-
vation of the surface will at no particular moment of time coincide with the resid-
ual displacements of the sea-floor. But this effect could still be taken into account
within the framework of the long-wave theory. If, on the other hand, the duration
of the deformation is small, then the motion of the water layer must be described
within the framework of the theory of a compressible liquid. Here, the theory of long
waves turns out to be totally inapplicable. In the case of high-speed displacement
of the sea-floor an additional contribution to the tsunami wave can also be given by
non-linear effects.
Note the paradoxical effect, manifested when tsunami generation is considered
a process proceeding in an incompre
ss
ible liquid. For definiteness we shall assume
an earthquake resulting in area S (
√
S
H
) of the sea-floor being displaced verti-
cally with a constant velocity by a quantity
. According
to the theory of an incompressible liquid, practically all the water layer immedi-
ately above the moving part of the sea-floor acquires a vertical velocity
η
0
during a time interval
τ
τ
−
1
, and,
η
0
consequently, the kinetic energy
0
W
k
=
ρ
SH
η
.
(2.1)
2
2
τ
The displacement results in a perturbation forming on the water surface (we shall
consider it identical to the deformation of the sea-floor), which contains the potential
energy
2
0
W
p
=
ρ
η
Sg
.
(2.2)
2
The paradox consists in that the kinetic energy involved in the process has a fixed
value, but immediately after its completion the kinetic energy disappears without
leaving a trace. The paradox is readily resolved, naturally, if the condition
W
p
W
k
is applied. But in reality the kinetic energy may not only be comparable to the po-
tential energy, but even significantly exceed it. Indeed, from formulae (2.1) and (2.2)
we have
2
0
W
p
=
τ
W
k
2
,
τ
τ
0
=(
H
/
g
)
1
/
2
is the propagation time of a long gravitational wave over
a distance equal to the depth of the ocean (
where
τ
0
≈
20 s for
H
= 4
,
000 m). In many cases
τ
τ
0
, and, consequently,
W
k
>
W
p
. An accurate resolution of the said paradox is
possible within the framework of the theory of compressible liquids.
For an adequate mathematical description of the processes occurring when waves
are generated it is necessary to have a clear idea of the characteristic values of
the main parameters defining the problem. The range of tsunami wave periods has
already been indicated above. The depth of the ocean in area of a tsunami source
may vary from several kilometres to zero (when the area of the sea-floor deformation
<
