Geoscience Reference
In-Depth Information
certain plane—the result of averaging either over the entire area of the tsunami
source, or over a part of it. We shall consider the differences between this plane
and the actual surface of the bottom to be irregularities.
We shall show that, for the excitation of motions in a water layer, normal dis-
placements of the ocean bottom are essentially more effective than tangential ones.
Let each point of the bottom surface at the tsunami source of area
S
undergo dis-
placement over a distance
: once in the tangential direction and
then in the normal direction. The normal to the bottom surface is at an angle
η
0
during a time
τ
to
the vertical direction. The slope of the surface of the oceanic bottom rarely exceeds
0.1, therefore the angle
α
can be considered small.
During tangential shifts the ocean bottom exerts a force on the water layer, equal
α
(
u
∗
)
2
S
, where
u
∗
is the friction velocity and
to
is the density of water. The
energy transferred to the water layer by the ocean bottom undergoing motion can be
estimated as the work performed by this force along the path
ρ
ρ
η
0
:
(
u
∗
)
2
S
W
t
=
ρ
η
0
.
(2.6)
If one passes to the reference frame related to the moving ocean bottom, then one
obtains the traditional problem of a logarithmic boundary layer, in which the quan-
tity
η
0
/
τ
plays the part of the velocity of the average flow far from the boundary.
The friction velocity is known to be essentially smaller than the velocity of the av-
erage flow, therefore, it is possible to write
0
S
η
W
t
ρ
2
.
(2.7)
τ
We shall estimate the energy transferred to an incompressible layer of water
by a normal displacement as the potential energy of the initial elevation above
the water surface. We shall assume the horizontal dimensions of the source to es-
sentially exceed the ocean depth
S
1
/
2
H
and the displacement to be quite rapid,
S
1
/
2
(g
H
)
−
1
/
2
. In this case the entire volume of water dislodged by the dis-
placement,
τ
η
0
S
, will be distributed over an area
S
cos
α
of the ocean surface. Thus,
the amplitude of the initial elevation will amount to
η
0
/
cos
α
. Taking into account
α
the smallness of the angle
we obtain the following estimate for the potential en-
ergy of the intial elevation:
0
2
g
S
η
W
n
=
ρ
.
(2.8)
Let us find the ratio between the energies transferred to the water layer by the nor-
mal and tangential displacements,
2
η
0
.
W
n
W
t
g
τ
(2.9)
η
ξ
τ
If one assumes
0
=
0
,
=
T
hd
and applies formulae (2.4) and (2.5), then one
2
can readily show that
g
1. Hence it follows that tangential motions
of the ocean bottom can be neglected in the problem of tsunami generation.
τ
η
≈
800
0
