Geoscience Reference
In-Depth Information
3.2.3 Non-linear Tsunami Generation Mechanism with Account
of the Compressibility of Water
This section deals with the tsunami generation mechanism related to the
non-linear transfer of energy from 'high-frequency' elastic oscillations of the wa-
ter column to 'low-frequency' surface gravitational waves. Elastic oscillations are
the reaction of the water column to movements of the bottom of seismic origin. In
this case movements of the ocean bottom may not be periodic, it is only important
for their frequency spectrum to correspond to frequency range III (
>
c
/
4
H
).
Our aim is to find the relationship between characteristics of the 'low-frequency'
gravitational wave and the parameters determining the ocean bottom displacement,
and, also, a comparative analysis of the efficiencies of the piston and membrane
mechanisms in tsunami generation.
In the case of a compressible liquid, a non-linear tsunami source is manifested as
the action of a force field
ν
and of a distributed source of mass,
s
. For calculation
of these quantities knowledge is required of the fields of velocity v
and of dynamic
pressure
p
, which we shall find by resolving the plane problem of the linear re-
sponse of an ideal compressible liquid to small deformations of the ocean bottom,
(3.4)-(3.6).
The problem was resolved numerically by the explicit finite difference method,
using dimensionless variables (
x
∗
=
x
/
H
,
t
∗
=
tc
/
H
). The velocity of ocean bottom
deformation was given by the following model laws:
Φ
x
L
t
τ
U
pist
(
x
,
t
)=
v
max
η
η
(piston-like displacement)
,
sin
2
θ
x
L
)
π
Nt
τ
U
osc
(
x
,
t
)=
v
max
η
(
t
)
−
θ
(
t
−
τ
(oscillations of ocean bottom)
,
)=0
.
5
tanh
20(
0
.
15)
−
tanh
20(
0
,
85)
,
v
max
is the maxi-
where
η
(
α
α
−
α
−
mum deformation velocity value,
θ
(
t
) is the Heaviside function,
L
is the horizontal
extension of the deformation area,
is the duration of the deformation process and
N
is the number of oscillation periods (an integer number). The form of function
η
τ
) is shown in Fig. 3.24. The piston-like displacement resulted in residual de-
formations of the ocean bottom, oscillations of the ocean bottom finished without
residual deformations.
(
α
Fig. 3.24 Form of function
determining the space-
time law of ocean bottom
deformation
