Geoscience Reference
In-Depth Information
The real ocean is always stratified, and, moreover, owing to rotation of the Earth
each moving particle of the water is under the influence of a Coriolis force. There-
fore, tsunami generation is, generally speaking, accompanied by the formation of
internal waves and vortical motions.
Let us estimate the effect due to rotation of the Earth, when vertical displace-
ments of the ocean bottom are generated by a tsunami. We shall apply the linearized
equations of shallow water written with account of the Coriolis force for a horizon-
tally infinite ocean of depth
H
.
∂
u
g
∂ξ
∂
=
−
x
+
fv
,
(2.10)
∂
t
∂
v
g
∂ξ
∂
=
−
y
−
fu
,
(2.11)
∂
t
H
∂
+
∂ξ
∂
u
x
+
∂
v
t
−
∂η
= 0
,
(2.12)
∂
∂
y
∂
t
where
u
,
v
are the components of the horizontal flow velocity,
f
= 2
ω
sin
ϕ
is
the Coriolis parameter,
represents small vertical deformations of the ocean bottom
(deviations from the initial position),
η
is the displacement of the free surface from
the equilibrium position. We differentiate equation (2.10) with respect to the coor-
dinate
y
and equation (2.11) with respect to the coordinate
x
, and, then, we subtract
one from the other. With account of the continuity equation (2.12) we ultimately
obtain an evolution equation for the vertical curl component of the velocity
ξ
∂ξ
∂
.
∂
∂
t
(rot
z
v)=
f
t
−
∂η
(2.13)
H
∂
t
We shall assume no motion to exist in the water layer at the time moment
t
=
0 and the surfaces of the water and ocean bottom to be in an unperturbed state
(v = 0
,
= 0). We shall further assume deformation of the ocean bottom,
arbitrary in space and time, but quite rapid (
η
= 0
,
ξ
R
(g
H
)
−
1
/
2
), to take place within
a circular area of radius
R
, which will result in the formation of certain residual
displacements. For simplicity we shall consider the residual displacements to differ
from zero only inside the circular area of radius
R
, where they assume the fixed value
η
τ
0
. The ocean bottom displacement results in formation of a wave perturbation of
the surface, which after a sufficiently long period of time (
T
R
(g
H
)
−
1
/
2
) will
leave the area of the source and the water surface will return to its initial unperturbed
state.
The said assumptions make it possible to integrate equation (2.13) over time in
the time interval from 0 up to
T
.
f
H
η
0
.
(rot
z
v)
|
t
=
T
=
−
(2.14)
Expression (2.14) permits to conclude that influence of the Earth's rotation man-
ifested at the tsunami source area, considering residual displacements of the ocean
