Geoscience Reference
In-Depth Information
3.2 Non-linear Mechanism of Tsunami Generation
This section deals with the formation mechanism of tsunami waves due to the non-
linear transfer of energy from 'high-frequency' induced or elastic oscillations
of the water column to 'low-frequency' surface gravitational waves. Seismic
movements of the ocean bottom are considered as the source of 'fast' oscilla-
tions of the water column. The 'traditional' tsunami generation mechanism, related
to residual displacements substituting the water, naturally, remains in force, and in
most cases precisely it plays the leading part. The non-linear mechanism provides
additional contributions to the tsunami amplitude and energy. It is not excluded that
in individual cases non-linear effects can also provide a determinative contribution
to a tsunami wave.
At a first glance formulation of the problem, assuming the presence of periodic
oscillations, may seem restricted. Actually, such a restriction is important only under
the condition that the water column responds to movements of the ocean bottom like
an incompressible liquid. In this case one must indeed consider
periodic
oscillations
of a part of the bottom, which lead to corresponding induced oscillations of the in-
compressible water column. But, if the water column reacts to seismic movements
of the ocean bottom like a compressible liquid, the necessity of periodic movements
of the bottom vanishes, since any vertical displacements will be accompanied by
elastic oscillations of the water column at normal frequencies.
In substantiating the application of linear theory in the tsunami generation prob-
lem one usually quotes the condition that the amplitude of the ocean bottom
deformation be small as compared to the depth of the basin,
H
. Indeed, this
condition is quite fulfilled in reality. But even when the amplitude of the ocean
bottom displacement is small, the velocity of its movement may turn out to be suf-
ficiently high for the manifestation of non-linear effects.
Before our studies were published [Nosov, Skachko (2001); Nosov, Kolesov
(2002), (2005); Nosov et al. (2008)] there existed only a single work [Novikova,
Ostrovsky (1982)], in which the possibility was investigated of tsunami formation
resulting from a non-linear effect—the 'detection' of acoustic oscillations of the
water column.
η
3.2.1 Base Mathematical Model
Before proceeding with the construction of a model describing non-linear effects, it
is useful to present a description of the character of the linear response of a com-
pressible water column to movements of the ocean bottom without residual dis-
placement. The character of the response varies depending on the position of the
spectrum of ocean bottom movements with respect to the two characteristic fre-
quencies (g
H
)
1
/
2
and
c
/
4
H
, where
c
is the velocity of sound in water and g is
the acceleration of gravity. Further, without losing generality, we shall not speak
of a spectrum, but of a certain frequency of bottom oscillations,
ν
. Thus, if the
