Geoscience Reference
In-Depth Information
1
A
L
≈
η
0
=
v
max
τ
C
1
,
C
1
=
η
(
α
)d
α
≈
0
.
7
,
(3.80)
0
and the energy as the potential energy of the initial elevation
+
∞
2
L
C
1
C
2
2
≈
ρ
g
2
2
(
x
,
g
v
2
max
τ
ξ
τ
ρ
,
W
L
)d
x
=
−
∞
(3.81)
1
2
(
C
2
=
η
α
)d
α
≈
0
.
65
.
0
Applying formulae (3.80) and (3.81), we obtain relationships permitting to calcu-
late the relative contributions of the non-linear and the linear mechanisms to the am-
plitude and energy of tsunami waves:
=
η
g
H
2
A
∗
(
0
c
2
τ
∗
,
L
∗
)
C
1
τ
∗
2
A
N
A
L
,
(3.82)
=
η
0
c
2
g
H
2
2
2
W
∗
(
τ
∗
,
L
∗
)
C
1
C
2
τ
∗
4
L
∗
W
N
W
L
,
(3.83)
where
η
0
is the amplitude of the vertical ocean bottom deformation. From formulae
(3.82) and (3.83) the quantities
A
N
/
A
L
and
W
N
/
W
L
are seen to be determined to
a large extent by the dimensionless combination
η
0
c
2
g
−
1
H
−
2
.
Figures 3.28 and 3.29 present the dependences of quantities
A
N
/
A
L
and
W
N
/
W
L
upon the piston-like displacement duration. The calculation is performed for three
different relationships between the source size and the ocean depth. The curves be-
ing non-monotonous for
τ
∗
>
1 is due to the modal structure of elastic oscillations
of the water column (the minimum normal frequency corresponds to
τ
∗
= 4). When
τ
∗
<
1, the dependences investigated behave approximately like the power func-
tions
τ
∗−
2
. An increase in the horizontal size of the source leads to an
insignificant enhancement of the role of the non-linear mechanism.
τ
∗−
1
and
Fig. 3.28 Ratio between am-
plitudes of tsunami waves
formed by the non-linear (
A
N
)
and the linear (
A
L
) mecha-
nisms versus the displacement
duration. Curves 1-3 are
drawn for
L
/
H
= 20, 10 and 5