Geoscience Reference
In-Depth Information
From formula (5.4) it is seen that, even when the tsunami wave height in open
ocean is quite significant,
A
= 1 m, in the case of a typical depth
H
= 4 km and
wavelength
= 100 km, the value of
L
cn
will amount to about 400,000 km, which
exceeds the length of the Earth's equator by an order of magnitude. Therefore,
non-linear effects, in the case of tsunami propagation in the open ocean, can indeed
be neglected.
The ratio of the quantities
L
cd
and
L
cn
, determined in accordance with the approx-
imate formulae (5.2) and (5.4), gives the Ursell number
Ur
=
A
λ
2
/
H
3
[Pelinovsky
(1996)], known in the theory of non-linear-dispersion waves on water. In open
ocean, as a rule,
Ur
λ
1, which means that phase dispersion prevails over non-
linear effects. Near the coast (in shallow water), if microtsunamis are not consid-
ered, the parameter
Ur
1, i.e. non-linear effects become predominant. Estimation
of the distances of dispersive and non-linear destruction, yielding formulae similar
to (5.2) and (5.4), can be found in the topic [Pelinovsky (1982)],
From the above analysis it follows that in simulating tsunamis, even along ex-
tended routes, the application of linear theory is quite justified. Moreover, long-wave
theory is also quite appropriate for long-period waves. In this connection, it will be
expedient to dwell upon certain partial results, following from the linear theory of
long waves.
The ocean depth is the o
nly
variable quantity entering into the formula for the ve-
locity of long waves,
c
=
√
g
H
. Therefore, many effects of tsunami propagation and
run-up are related to the relief of the ocean bottom.
Consider the one-dimensional problem of the propagation of a long wave in
a basin, the depth of which varies along the horizontal coordinate. We consider depth
variations to be sufficiently smooth, so the reflection of waves from inclined sections
of the bottom can be neglected. For definiteness, we shall consider a sine wave of
length
. Within the linear model, the kinetic and potential energies of the wave are
equal to each other, therefore, the total energy attributed to a single space period
(and to unit front length) can be calculated as twice the potential energy,
λ
g
λ
2
d
x
,
W
=
ρ
ξ
(5.5)
0
where
ξ
is the free water surface displacement from the equilibrium position and
ρ
is the density of water.
Since in a linear system the perturbation frequence remains unchanged, while
the wavelength may change during propagation, it is worthwhile to perform in for-
mula (5.5) integration over time, instead of space,
T
g
g
H
0
√
H
.
2
d
t
= const
2
W
=
ρ
ξ
·
ξ
(5.6)
0
From
en
ergy conservation (we neglect dissipation, here) it follows that the quan-
tity
0
√
H
must be conserved along the route of the wave propagation. In other
words, if the ocean depth decreases, as the wave propagates, then the wave
ξ
