Geoscience Reference
In-Depth Information
bottom to form at the site, must result in the formation of a certain vortical structure.
Usually, bipolar deformation of the ocean bottom occurs at real tsunami sources,
therefore, it may be assumed that several vortical structures are formed with differ-
ent directions of rotation.
Let us estimate the energy of the vortical structure formed by the circular residual
deformation. To this end we integrate expression (2.14) over the area of a circle of
radius
r
R
, the centre of which coincides with the centre of the source. Applying
the known Stokes formula, we pass in the left-hand part of the obtained expression
to circulation of the velocity. With account of the radial symmetry of the problem
we obtain for the velocity of vortical motion at a distance
r
from the centre,
f
2
H
η
−
0
r
.
V
(
r
)=
(2.15)
Note that, when
r
>
R
, the velocity
V
= 0. Knowledge of the velocity distribution
readily permits to calculate the kinetic energy of the vortex,
0
R
4
16
H
f
2
2
W
k
=
πρ
η
.
(2.16)
Let us, now, compare the energy of the vortex with the energy of the tsunami
wave, which we estimate as the potential energy of the initial elevation, similar in
shape to the residual deformation of the ocean bottom (a circular area of radius
R
and height
η
0
),
g
R
2
2
0
W
p
=
πρ
η
.
(2.17)
2
Comparison of formulae (2.16) and (2.17) reveals the ratio of the energy of
the vortex, formed at the tsunami source and due to rotation of the Earth, and the en-
ergy of the tsunami wave itself to be given by the following expression:
f
2
R
2
8 g
H
∼
W
k
W
p
=
10
−
2
10
−
4
.
−
(2.18)
The part of the energy due to vortical motion is seen to increase quadratically
with the horizontal dimension of the source and to decrease as the ocean depth
increases. But, in any case, the contribution of this energy does not exceed 1% of
the energy of the tsunami wave. Note that such an estimate is correct for medium or
high latitudes; for equatorial regions, where the Coriolis parameter is small, it will
be significantly overestimated.
Let us, now, estimate the energy contribution of internal waves that are due to
ocean bottom displacements. We shall consider the model of an ocean consisting of
two layers: the upper layer of thickness
h
1
with a free surface, and the lower layer
of thickness
h
2
. The density of the upper layer is
ρ
1
and of the lower layer is
ρ
2
(
ρ
1
). In this case it is convenient to base estimations on the one-dimensional
(1D) (along the horizontal coordinate) model, constructed within the framework
of the linear theory of long waves. We shall consider a segment of the ocean
ρ
2
>
