Geoscience Reference
In-Depth Information
be manifested in a noticeable manner the wave must cover a distance exceeding
the length of the Earth's equator, which is not possible in practice. From a purely
theoretical point of view it is interesting that at large depths the quantities
L
1
and
L
2
become closer. Anyhow, this fact rather reflects the correct choice of coefficient
C
B
. The point is that viscous dissipation is not taken into account in tsunami mod-
els, applied in practice. At the same time non-linear dissipation is present in model
equations. As it is seen, it provides approximately the same (albeit tiny) contribution
to wave damping, which could have been provided by viscous dissipation.
Essential manifestations of dissipation are only possible at small depths
H
<
10 m. Here, the role of viscous linear wave dissipation turns out to be in-
significant. Most likely, in shallow water the dissipation lengths will be related as
L
1
/
L
2
>
10. Therefore, only taking into account non-linear dissipation, like it is
presently done in practical models, can be considered justified. The role of classical
linear dissipation can indeed be neglected. We stress, that one can speak of a no-
ticeable influence of dissipation on a tsunami wave only in the case of very small
depths. If we were to consider, for example, typical shelf depths
H
∼
100 m, then
the dissipation length would, most likely, exceed 1,000 km.
The wave amplitude decreasing as it propagates can be related not only to dissi-
pation, but also to waves being scattered on small-scale irregularities of the ocean
bottom. For estimating the significance of the scattering effect, we shall make use
of the aforementioned results, concerning the transformation of long waves above
a rectangular obstacle. Consider a 'comb' on the ocean bottom, consisting of rectan-
gular obstacles with a repetition period in space of 2
D
. Then, along a route of length
x
, the number of obstacles encountered by a wave will be
N
=
x
/
2
D
. Each time in-
teraction with an obstacle takes place, a decrease in the wave amplitude will occur
determined by the transmission coefficient
T
. The law by which the wave amplitude
A
decreases with distance is written as
A
(
x
)=
A
0
T
N
.
(5.27)
Formula (5.27) is expediently represented in a more customary exponential form,
A
(
x
)=
A
0
exp
,
x
L
3
−
where
L
3
=
2
D
/
ln
T
is the characteristic distance, along which the wave ampli-
tude is reduced by
e
times, owing to scattering on irregularities of the ocean bottom.
In the case of small relative heights of the roughnesses
−
α
and a small phase differ-
ence
β
we obtain the simple formula
16
D
L
3
=
)
2
.
(
αβ
Expressing parameter
β
through the wave length and the obstacle width, we
ultimately obtain
