Geoscience Reference
In-Depth Information
Linear
(Hz)
Fig. 3.23 Amplitude of gravitational waves, excited by oscillations of ocean bottom, versus os-
cillation frequency: linear and non-linear responses. Calculations are performed for exponential
distribution of amplitude of bottom oscillations for
a
= 10 km and ocean depth of 1 km
formation of a gravitational wave. The amplitude of such a wave does not de-
pend on the space law, governing variations in the amplitude of the ocean bottom
oscillations (providing the law is sufficiently smooth), but depends on the veloc-
ity amplitude of oscillations,
η
0
ω
, their duration
τ
and the horizontal size of
the oscillating area.
The data presented in Fig. 3.22 permit to estimate the amplitude of a tsunami
wave caused by the non-linear mechanism considered. Thus, for example, when
the ocean depth is 1 km, oscillations of an area of the ocean bottom of the charac-
teristic size of 20 km (the space distribution of
1
,
a
= 10), amplitude of oscillatory
velocity of 10 m/s, lasting for 60 s, gives rise to a wave of amplitude 0.8 m.
For illustrative estimation of the contribution of the non-linear effect to
the tsunami amplitude, Fig. 3.23 presents the dependence of the gravitational wave
amplitude upon the frequency of ocean bottom oscillations. The wave amplitude is
normalized to the amplitude of bottom oscillations. Calculation of the dependence
is performed for the case of
η
η
1
(
x
) for
a
= 10 km and
H
= 1 km. The oscillations
of the ocean bottom, having started at a certain moment of time, are assumed to
continue sufficiently long for the amplitude of the wave, formed by the non-linear
mechanism, to reach the maximum value. The linear response (dotted line) is cal-
culated using formula (3.59). Owing to the auxiliary problem being linear, this
dependence is the same for any amplitude of bottom oscillations. The contribution
of the non-linear effect is proportional to the square velocity of bottom oscillations;
therefore, it depends on both the amplitude and the frequency of oscillations. Within
the range 0.1-1 Hz this contribution is already capable of competing with the linear
response and even of exceeding it.
