Geoscience Reference
In-Depth Information
Fig. 3.29 Ratio between ener-
gies of tsunami waves formed
by the non-linear (
A
N
) and
the linear (
A
L
) mechanisms
versus the displacement du-
ration. Curves 1-3 are drawn
for
L
/
H
= 20, 10 and 5
Taking advantage of the data presented in Figs. 3.28 and 3.29 one can readily
perform the following estimations. For an ocean depth of 1.5 km, displacement du-
ration and amplitude of 1 s and 1 m, respectively, the contribution of the non-linear
mechanism to the tsunami amplitude will be at a level of 10%, and to its energy of
1%. The contribution of the non-linear mechanism may increase as the amplitude
of the ocean bottom displacement increases or the displacement duration decreases,
but, most likely, the linear mechanism will continue to prevail in the case of a piston-
like displacement.
The non-linear mechanism may provide for an essential contribution to the
amplitude of a tsunami wave in the case of ocean bottom oscillations at one of
the normal frequencies,
ν
k
=
c
(1 + 2
k
)
/
4
H
,
k
= 0
,
1
,
2
,...
(resonance pumping
of energy). According to linear theor
y, oc
ean bottom oscillations without residual
displacements at frequencies
>
g
/
H
do not pr
oduce
gravitational waves (see
Sect. 2.3.4). In conditions of the planet Earth
ν
ν
k
>
g
/
H
, consequently, in the case
of ocean bottom oscillations with frequencies
ν
k
only the non-linear mechanism can
give rise to tsunamis.
Calculations carried out for
U
(
x
,
t
)=
U
osc
(
x
,
t
) have revealed the following. If an
area of the ocean bottom of dimension
L
= 40 km at a depth of
H
= 4 km under-
goes
N
= 10 oscillations of frequency
0
.
094 Hz and amplitude 0.3 m,
then the non-linear mechanism produces a tsunami of amplitude
ν
0
=
c
/
4
H
≈
∼
0
.
5 m. In similar
conditions, but at a higher frequency
ν
3
= 7
c
/
4
H
≈
0
.
65 Hz, the tsunami amplitude
will already amount to
1
.
2 m. If the frequency of ocean bottom oscillations differs
noticeably from the normal frequency, then the efficiency of the non-linear mech-
anism decreases significantly. Thus, for example, if
∼
ν
= 0
.
55 Hz (
ν
2
<
ν
<
ν
3
),
the tsunami amplitude will only be of the order of 6 cm.
In conclusion, we note that the frequencies of seismic oscillations of the ocean
bottom lie within the range of several first normal frequencies of the water column,
ν
k
, which creates favourable conditions for realization of the non-linear mechanism
of tsunami generation.
