Geoscience Reference
In-Depth Information
Fig. 4.7 Mathematical for-
mulation of the problem of
tsunami generation by an un-
derwater eruption
Fig. 4.8 Perturbation of free
surface caused by underwa-
ter eruption. The calculation
is performed at the time
moment, when th
e eru
ption
finishes,
t
= 10
H
/
g
,for
various ratios of the crater
radius and the ocean depth,
R
/
H
(indicated in the fig-
ure). The
x
-axis is nor-
malized
to
the quantity
r
0
=
τ
√
g
H
,the
y
-axis to
ξ
0
=
V
0
/
(
πτ
2
g
H
)
ξ
(
r
,
t
)=
θ
ζ
(
r
,
t
)
−
θ
−
τ
ζ
(
r
,
t
−
τ
)
,
(
t
)
(
t
)
J
0
(
rk
)
J
1
(
Rk
) sin
t
g
k
tanh(
kH
)
1
/
2
cosh(
kH
)
g
k
tanh(
kH
)
1
/
2
∞
(4.10)
V
0
π
ζ
(
r
,
t
)=
d
k
.
R
τ
0
The form of the free-surface displacement at the mo
ment
, when the eruption
finishes (
t
=
= 10
H
/
g and various radii of
the crater,
R
/
H
= 0
.
1, 0.3, 1 and 3 is shown in Fig. 4.8. The curves are presented
in dimensionless coordinates. The x
-ax
is is normalized to the distance covered by
a long wave during eruption time
τ
), calculated by formula (4.10) for
τ
τ
√
g
H
,the
y
-axis is normalized to the free-surface
displacement, determined by estimation formula (4.7). From the figure it is seen, that
the form and amplitude of the free-surface perturbation depend little on the radius of
the crater, when
R
/
H
<
1. Moreover, the quantity
ξ
0
, determined by formula (4.7),
is indeed seen to represent a good estimate for the surface displacement amplitude.
Note that application of the theory of incompressible liquids imposes natural
limits on the outflow velocity of material from the crater,
w
0
<
c
, where c is
the velocity of sound in water, and on the relationship between the eruption duration
and the ocean depth,
>
4
H
/
c
.
For estimates we take advantage of the modest, as compared to the 1883 event
(Krakatau), eruption of an underwater volcano, located at a depth
H
= 1,000 m. Let
the release of material amount to
V
0
= 1km
3
, and the eruption duration
τ
= 100 s.
For the indicated duration of the process the perturbation (elevation) radius of
the water surface amounts to
r
τ
10 km. Its height, calculated in accordance with
formula (4.7), amounts to the significant value
≈
ξ
0
≈
3 m. And the potential energy,
10
13
J. A tsunami, generated by such an initial ele-
vation will evidently represent a serious threat.
calculated by (4.8) is
W
p
= 1
.
7
·
