Geoscience Reference
In-Depth Information
bottom of length
L
to undergo a vertical displacement
η
0
during a time interval
L
(g(
h
1
+
h
2
))
−
1
/
2
. Such a displacement represents an impulse not only for sur-
face waves, but also for internal waves, since the propagation velocity of the latter is
significantly smaller. The displacement results in the formation of initial elevations
both on the water surface and on the boundary surface separating the two layers; we
shall consider these elevations to be similar in shape to the deformation of the ocean
bottom. In principle, it should be possible already at this stage of reasoning to com-
pare the energies of internal,
W
int
and surface,
W
sur
tsunami waves by comparison
of the potential energies of the initial elevations. This ratio is evidently given by
the formula
τ
W
int
W
sur
≈
ρ
2
−
ρ
1
10
−
3
.
2
∼
(2.19)
ρ
But such a value is actually strongly overestimated. The point is that the evolution
of initial elevations gives rise to two sets of waves, each of which consists of pertur-
bations on the water surface and on the jump of density [Hammack (1980)]. One of
the sets of waves propagates rapidly with the velocity of surface waves, the other one
is essentially slower and propagates with the velocity of internal waves. As the ini-
tial elevation evolves, the water particles on the free water surface in the vicinity
of the source are shifted downward. The maximum of this displacement, equal to
η
0
, corresponds to the free surface. At the ocean bottom, owing to there being no
flow, the displacement equals zero. Assuming the displacement to depend linearly
on the vertical coordinate, we obtain the displacement at the level of the density
jump,
η
0
h
2
(
h
1
+
h
2
). The evolution of the elevation on the free surface is
seen to result in the initial elevation at the density jump being reduced by the quan-
tity
∆η
=
η
0
h
1
(
h
1
+
h
2
). Naturally, the po-
tential energy that is proportional to the square height of the initial elevation, also,
decreases here. A more correct estimation yields the following relationship between
the energies of the internal and surface tsunami waves:
∆η
, while its height becomes equal to
η
int
=
h
1
h
1
+
h
2
2
W
int
W
sur
≈
ρ
2
−
ρ
1
10
−
5
.
∼
(2.20)
ρ
2
Estimations reveal that stratification of the ocean and rotation of the Earth can-
not significantly influence the process of tsunami generation by an earthquake. But
a small part of the earthquake's energy is transferred both to baroclinic motions and
to vortical fields.
A complete physical formulation of the problem of tsunami generation by an
earthquake should, generally speaking, consider a layer of viscous compressible
stratified liquid on an elastic semispace in the gravitational field with account of
the Earth's rotation. The above reasoning makes it possible to essentially simplify
formulation of the problem. As a first approximation, we shall consider the pro-
cess of tsunami generation by an earthquake to be a phenomenon occurring in a
homogeneous (nonstratified) perfect incompressible liquid in the gravitational field
in an inertial (without rotation) reference frame. Deformations of an absolutely rigid
ocean bottom of finite duration and small amplitude (
A
H
) serve as the source
