Geoscience Reference
In-Depth Information
10
−
1
-10
0
m is significantly smaller than the ocean depth
H
10
3
m,
amplitude
A
∼
∼
10
4
-
10
6
m. These two facts permit to apply with success the simplest linear theory of
long waves. At any rate, the manifestations of amplitude and phase dispersion will
be insignificant. To describe the run-up of waves is already a more complicated
problem pertaining to the class of non-linear problems in a region with moving
boundaries. Indeed, as one approaches the coast, the ocean depth decreases, while
the tsunami amplitude increases, so the non-linearity parameter
A
/
H
is no longer
a small quantity. Moreover, currents associated with the wave become turbulent,
the influence of friction on the sea floor increases, processes resulting in suspension
of bottom sediments are activated. A problem of no less complexity is presented
by the fact that as the wave propagates over the land, the boundaries of the re-
gion, in which the hydrodynamic problem is resolved, alter quite essentially. First
of all, this naturally concerns the advancement of the shoreline, but the 'water-air'
and the 'water-bottom' (owing to erosion) boundaries also move. In this chapter,
the main physical regularities, determining tsunami propagation in the open ocean
and the run-up of waves on a shore, will be dealt with. Significant attention will be
devoted to mathematical models, applied in numerical tsunami simulation.
=
T
(g
H
)
1
/
2
while the depth, in turn, is much smaller than the wavelength
λ
∼
5.1 Traditional Ideas Concerning the Problem
of Tsunami Propagation
Right up till the last quarter of the twentieth century all measurements of tsunami
waves were performed exclusively by coastal stations. Only during the past decades
has the development of engineering reached a level that provides for the pos-
sibility of reliable tsunami registration in the open ocean and even at the very
source during generation. Measurements of wave parameters, done with the aid of
pressure sensors at the ocean bottom [Jacques, Soloviev (1971); Gonzalez et al.
(1987); Kulikov, Gonzalez (1995); Milburn et al. (1996)], and satellite radio-
altimeters [Okal et al. (1999); Kulikov et al. (2005)] permit to claim with certainty
that the amplitude of a tsunami in open ocean, as a rule, lies between several
centimeters and several tens of centimeters. In the most strong cases the amplitude
of the free water surface displacement in the vicinity of the source may, apparently,
reach several meters.
In any case, at great distances from the coast the tsunami amplitude
A
turns out to
be essentially smaller than the ocean depth
H
.Thevalueof
H
, in turn, is essentially
inferior to the wave length
. These two facts permit, in a first approximation, to
consider tsunami as long (not subject to dispersi
on)
linear waves, the velocity of
λ
which is determined by the simple formula
c
=
√
g
H
, where g is the acceleration of
gravity. The period of tsunami w
ave
s,
T
, lies within the range 10
2
-10
4
s. With ac-
cou
nt of
the relationship
=
T
√
g
H
it is possible to rewrite the condition
λ
λ
H
as
T
g
H
1. It is easy to verify that for the range of periods indicated this condition
is always quite satisfied at small (shelf) depths. But for short-period tsunamis, prop-
agating in the open ocean, fulfilment of this condition is not so evident.
