Mechanics of Fluids

482 Flow with a free surface
the depth of the sea decreases according to the relationship
a = a0
� �1/4 h0
where a0 and h0 are arbitrary datum values of a and h. Unlike wind-generated
waves, a tsunami does not break, but forms a front similar to a bore or
hydraulic jump.
The detailed morphology of the coastline has a considerable bearing on
the impact of the tsunami. A steep gradient of the sea bed, or coastal features
such as bays and headlands, afford some localities a measure of protection
against the most severe features of a tsunami. Conversely, the greatest damage
occurs where there is a gentle gradient to the sea bed. In the shallow
waters, and depending on the proximity to and the strength of the initiating
earthquake, the wave height can build to between 10 m and 20 m. In the
absence of features to dissipate the energy of the tsunami, it is capable of
running inland for up to 3 km, with devastating effect. Because the period
T of a tsunami is long, water associated with the crest of the leading wave
continues to move towards the shoreline for a long time, but eventually the
trough of the wave arrives. There then comes a stage when the waters run
back off the land and once again return to the sea. Thereafter, the sea withdraws
leaving exposed areas of the beach which had been under water prior
to the arrival of the tsunami. The cycle is then repeated with the arrival of
the next wave crest, of reduced amplitude. Over time the wave motion is
gradually damped out until after some 24 hours it finally disappears. Departures
from the given description of a typical tsunami do seem to occur. An
initial retreat of the sea has been reported as a precursor to the arrival of a
large wave, and eye-witness accounts suggest that, occasionally, the second
wave is stronger than the first.
Besides the 2004 Asian tsunami another notable, but localized, event was
the tsunami in Lituya Bay, Alaska, in 1958, when giant waves rose to a
height of about 520 m. Looking ahead, scientists are predicting that, within
a few thousand years, a massive tsunami will occur as a consequence of a
giant landslide on the island of La Palma, in the Canaries. This event will
be triggered by a volcanic eruption and it is anticipated that the waves will
cross the Atlantic Ocean to affect the United States and Caribbean Islands.
10.15 CONCLUSION
In this chapter we have restricted our discussion of phenomena occurring in
flow with a free surface to the case of a liquid in contact with the atmosphere,
the density of the latter being negligible in comparison with that of the liquid.
Similar phenomena can, however, take place when the interface is between
two fluids whose densities are much more nearly equal. Consideration of
these further examples of free surface flow is beyond the scope of this book,
but it is worth remarking that the phenomena discussed in this chapter have
their counterparts in the atmosphere and in lakes and oceans whenever there
h