Geoscience Reference
In-Depth Information
Fig. 4.11 Amplitude (range)
of waves on water surface ver-
sus distance
L
=
Vt
, covered
by perturbation of atmo-
spheric pressure of amplitude
p
0
and with horizontal di-
mension
a
. The calculation
is performed for different
propagation velocities of
the atmospheric perturbation.
The numbers indicate values
of di
me
nsionless velocity
V
/
√
g
H
for respective curves
increase is possible of the wave perturbation amplitude as compared with the value
determined by the inverse barometer law.
One of the most important properties of meteotsunamis is the proportionality of
the wave amplification coefficient to the ratio of the length of the 'resonance' area
of water and the horizontal size of the atmospheric perturbation. Taking advantage
of this property, it is possible to determine in advance the sections of the coast,
potentially endangered by meteotsunamis. To this end, it is necessary to analyse
the littoral bathymetry and to reveal extended shelf zones, within which the reso-
nance conditions can be fulfilled. For this work it is, naturally, necessary to know
the characteristic propagation velocities of atmospheric perturbations.
Since typical propagation velocities of atmospheric perturbations amount are
from units to tens of meters per second, fulfilment of the Proudman resonance
conditions is most probable in shallow-water areas of the ocean. But, when me-
teotsunamis of significant amplitude are excited within shallow-water areas, linear
theory is no longer applicable. Therefore, it is expedient to consider the problem of
wave generation by atmospheric perturbations within the framework of non-linear
theory of long waves. We shall now assume that the displacement amplitude of
the free water surface may be comparable to the basin depth, i.e. the main parame-
ters of the problem are related as follows:
a
. We shall write the equations
of non-linear theory of long waves in dimensionless variables, bearing mind the for-
mulae (4.19),
|
ξ
| ∼
H
∂
u
+
u
∂
u
x
+
∂ξ
−
∂
p
atm
∂
=
,
(4.27)
∂
t
∂
∂
x
x
x
(1 +
)
u
= 0
.
∂ξ
∂
+
∂
∂
ξ
(4.28)
t
It is not possible to resolve the complete non-linear problem (4.27) and (4.28)
analytically. But, when movement of the atmospheric perturbation is not limited
in time (
), it is possible to obtain an analytical relationship between
the free-surface displacement in the induced wave and the perturbation of atmo-
spheric pressure.
−
∞
<
t
<
+
∞
