Geoscience Reference
In-Depth Information
along the underwater eminence. Not all rays happen to be captured, but only those
the direction of which does not differ strongly from the axis of the ridge, namely, in
this case is the condition of total internal reflection realized.
Figure 5.7b shows the course of rays from a pointlike source, situated on
the 'shelf'. It is well seen that part of the rays, curved at the beginning towards
large depths, turn back to shallow water. Thus, refraction results in a significant part
of the wave energy turning out to be captured by the shelf and to propagate along
the coast. In Fig. 5.7b, one also observes a classical refraction effect: as the rays ar-
rive in shallow water, they are turned around in the direction normal to the coastline.
A striking example of the role of waves, captured by the shelf is related to
the tsunami that took place in Indonesia on December 12, 1992. This event is known
as the tsunami of Flores island. At a distance of 5 km north of the Flores island coast
there is a small island (the Babi island) of approximately circular shape and diam-
eter about 2.5 km. The tsunami source was north of Babi island, however, the max-
imum run-up (7.1 m) was observed on the southern coast of the island. This effect
is explained by the fact that the tsunami wave, having approached the island from
the north, happened to be captured by the shelf, then turned round the island on both
sides and provided maximum run-up on the coast of the back side of the island rel-
ative to the tsunami source. This effect has been studied with the use of laboratory
simulation in [Yeh et al. (1994)].
In the analysis of real events, the use of equations (5.11)-(5.14) written in
Cartesian coordinates is limited to small-scale areas of water. Beam computation
for transoceanic routes requires taking into account the Earth's sphericity. Calcula-
tion of the path of a ray on a spherical surface is performed applying the following
set of equations [Satake (1988)]:
d
d
t
=
cos
ς
nR
,
(5.15)
d
d
t
ς
nR
sin
sin
,
=
(5.16)
θ
d
d
t
sin
ς
n
2
R
∂
n
∂θ
ς
n
2
R
sin
cos
n
∂ϕ
−
∂
sin
ς
cot
θ
=
+
,
(5.17)
−
θ
nR
where
θ
is the colatitude (supplement up to the latitude),
ϕ
is the ray longitude,
t
is time and the quantity
n
=(g
H
)
−
1
/
2
is the inverse velocity of long waves,
R
is
ς
the Earth's radius,
determines the ray direction counted anticlockwise from the di-
rection toward the South. For computation of the evolution of wave rays knowledge
is required of the distribution of ocean depths over latitude and longitude. At present,
information on the global topography and bathymetry with a space resolution of
2 ang. min. (ETOPO2) is available on the site of the National Center of Geophys-
ical Data (http://www.ngdc.noaa.gov/). An example of the application of equations
(5.15)-(5.17) can be found, for example, in [Choi et al. (2003)].
Any wave motion in a nonideal (viscous) liquid is subject to dissipation. Tsunami
waves also lose part of their energy as they travel, owing to its irreversible transfor-
mation into heat. We shall estimate the influence of dissipative factors on tsunami
propagation.
