Geoscience Reference
In-Depth Information
al. (2003)], demonstrated good agreement between the model and observed ar-
rival times of waves at various points of the World Ocean distant from the source.
This result testifies that the w
ave
front indeed propagated with a velocity close to
the velocity of long waves,
√
g
H
, i.e. the waves were sufficiently long. Note that,
in the case of abyssal depths, strongly dispersive waves are formed in the area of
the caldera collapse, which rapidly die out with the distance from the source.
Further, we shall deal with the most 'original' of the tsunami generation mecha-
nisms peculiar to volcanoes. What is meant is the release of a large volume of matter
in the case of an underwater eruption. First, consider the case of a slow outflow of
matter. An adequate model for describing the tsunami generation process will con-
sist of a set of hydrodynamic equations with a source of mass (volume). Assume
eruption of the underwater volcano to proceed slowly: a volume
V
0
is released in
time
from the crater. On the basis of general physical arguments it is not diffi-
cult to estimate the amplitude and energy of surface gravitational waves, caused by
such an underwater 'eruption'. Consider the ocea
n
depth
H
to be fixed, and the area
τ
S
of the crater to be small, and the condition
√
S
H
to be satisfied. Of course,
the model of an ocean of constant depth is limited (the crater of the volcano is usu-
ally situated on top of a cone), but for presenting general physical regularities of
the process such a simplified model is quite applicable.
The volume thrown out will oust an identical volume of water. This volume will
spread over the area of a circle of radius, equal
to
the distance, which a long wave has
time to cover during the eruption time
r
=
τ
√
g
H
. As a result, we have the amplitude
of the initial water elevation
V
0
πτ
ξ
0
=
2
g
H
.
(4.7)
The potential energy of such an initial elevation, calculated by formula (2.2),
V
0
ρ
W
p
=
2
H
.
(4.8)
2
πτ
From formulae (4.7) and (4.8) the amplitude and especially the energy are
seen to increase with the rate
V
0
/
, at which volcanogenic material is released
from the crater. An increase of the ocean depth reduces the efficiency of tsunami
excitation.
For more accurate description of the waves caused by a flow of material from
a hole of radius
R
in the ocean bottom, it is possible to apply the general solution
of the problem, (2.43), (2.30), (2.31), obtained in Sect. 2.2.2 within the framework
of linear potential wave theory. Formulation of the axially symmetric problem is
schematically presented in Fig. 4.7. In the case dealt with the boundary condition
on the bottom, (2.31), assumes the following form:
∂
τ
=
w
(
r
,
t
)=
w
0
1
R
)
θ
)
,
F
−
θ
(
r
−
(
t
)
−
θ
(
t
−
τ
z
=
−
H
,
(4.9)
∂
z
R
2
) is the outflow velocity of material from the crater. Displace-
ment of the free surface, caused by the flow, released from the ocean bottom, is
determined by formula
where
w
0
=
V
0
/
(
τπ
