Geoscience Reference
In-Depth Information
the tsunami period and
λ
is the wavelength. With account of the obvious relation-
ship
T
(g
H
)
1
/
2
=
and of the fact that the tsunami period is usually tens and even
hundreds of minutes long, fulfilment of the second condition is also doubtless.
It is interesting to compare the energies of acoustic and gravitational waves, ex-
cited by one and the same mechanism: by vertical displacement of part of the ocean
bottom. We shall consider a column of an ideal homogeneous compressible (or in-
compressible) liquid with a free upper surface of thickness
H
located on an abso-
lutely hard bottom in the field of gravity exhibiting the free-fall acceleration g.At
a certain moment of time an area
S
of the ocean bottom starts to move vertically
with a constant velocity
v
. The motion will take place during a time interval
λ
, upon
which the ocean bottom stops. Such a process results in a residual displacement of
the ocean bottom of height
τ
η
0
=
v
τ
over an area
S
. It is known that in an incompress-
S
1
/
2
(g
H
)
−
1
/
2
, at the time moment
t
=
ible liquid, when
the shape of the sur-
face perturbation is close to the shape of the residual displacement of the bottom,
so the energy transferred to the ocean by the moving bottom can be estimated (from
above) as the potential energy of the initial elevation of area
S
and height
τ
τ
ξ
0
=
η
0
:
2
0
.
W
1
= 0
,
5
ρ
g
S
ξ
(3.1)
Within the framework of the model of compressible liquids, the energy of acous-
tic waves [Landau, Lifshitz (1987)], excited by the motion of the ocean bottom,
described above, has the following form:
0
τ
−
1
.
W
2
=
c
ρ
S
η
(3.2)
It is readily verified that, within the range of
τ
values peculiar to real seismic
events, the ratio
W
2
W
1
= 2
c
(g
)
>
1. In other words, a significant part of the en-
ergy transferred from the moving bottom to the ocean exists in the form of acoustic
waves. As time passes, this energy can be transferred to seismic waves or to other
forms of motion of the water column. In any case, elastic oscillations represent an
energetically significant effect, which must be taken into account.
The obtained estimates are expediently compared with natural data. Taking ad-
vantage of the empirical relationships (2.3)-(2.5) and of formulae (3.1) and (3.2), it
is possible to calculate the energy of a gravitational tsunami wave and the energy of
elastic waves in water depending on the earthquake magnitude. We shall compare
these quantities with the earthquake energy
E
[J], estimated by the formula [Puzyrev
(1997)]
τ
lg
E
= 1
.
8
M
+ 4
.
(3.3)
In calculations we assumed the duration of the ocean bottom to be half the du-
ration of the process at the earthquake source,
T
hd
, while the area of the tsunami
source is calculated via its radius by the elementary formula
S
=
R
TS
.
Figure 3.1 presents the dependences of the earthquake energy (1), the energy
of a gravitational tsunami wave (2) and of the energy of elastic waves (3) upon
the earthquake magnitude. It can be seen that the formation of a gravitational
tsunami wave requires
<
1% of the earthquake energy, which is in good agreement
with the data of [Levin (1981)] and [Voight (1987)]. But tens of times more—up to
π
