Geoscience Reference
In-Depth Information
In the model, described above, we assumed the eruption to be a slow process.
This provided grounds for applying linear theory and considering water to be in-
compressible. But the eruption of an underwater volcano may exhibit an explosive
character. In such a case the products of eruption form a gaseous bubble in the water,
which contains high-temperature gases and water vapour at high pressures. The ex-
pansion and floating-up of the bubble leads to the formation of a cupola or plume—
an elevation on the water surface. An analogue of this process is the formation of
a plume in the case of an underwater explosion. It must be stressed that the forma-
tion of a gaseous bubble at large depths is not always possible, owing to the colossal
hydrostatical pressure.
In this case description of the wave generation process is a difficult task. But one
can select an equivalent source and use it as the initial perturbation in calculating
tsunami waves. Reasonable agreement with reality (explosions in water for energies
within the range of 2
10
10
J) is achieved for the following form of the initial
displacement of the water surface [Kurkin, Pelinovsky (2004)]:
10
6
-3
·
·
2
r
R
s
1
2
1
R
s
)
,
ξ
0
(
r
)=
H
S
−
−
θ
(
r
−
(4.11)
where
R
s
is the source radius,
H
S
is the amplitude of the water level displace-
ment at the source. Both parameters, characterizing the source, can be estimated
via the equivalent energy of the explosion (or volcanic eruption) [Le Mehaute, Wang
(1996)].
In the case of an ocean of constant depth
H
, evolution of the initial free-surface
perturbation, exhibiting radial symmetry, is described by the following expression
(see general theory in Sect. 2.2.2):
∞
k
d
kA
(
k
)
J
0
(
kr
) cos
ω
(
k
)
t
,
ξ
(
r
,
t
)=
(4.12)
0
A
(
k
)=
∞
r
d
r
ξ
0
(
r
)
J
0
(
kr
)
,
(4.13)
0
where
ξ
0
(
r
) is a function describing the form of the initial perturbation,
J
n
is
the Bessel function of the first kind of
n
th order. The relationship between the cyclic
frequency and the wave number is determined by the known dispersion relation for
gravitational waves on water,
2
= g
k
tanh(
kH
). For an initial elevation, exhibiting
the form, determined by formula (4.11), we have
ω
H
S
R
S
J
3
(
kR
)
k
A
(
k
)=
−
.
(4.14)
For large times we represent the integral in expression (4.12) with the aid of
the stationary-phase method,
