Geoscience Reference
In-Depth Information
∂
u
+ g
∂ξ
∂
1
ρ
∂
p
atm
∂
=
−
,
(4.16)
∂
t
x
x
∂ξ
∂
+
H
∂
u
x
= 0
.
(4.17)
t
∂
If the atmospheric pressure is constant in time, but depends on the space co-
ordinate (
t
= 0), then from equation (4.16) immediately follows the so-called
'inverse barometer law'
∂
/
∂
p
atm
(
x
)
ρ
ξ
(
x
)=
−
.
(4.18)
g
In accordance with this law, the local enhancement of atmospheric pressure 'presses
down' the free sea surface, forcing the water to occupy those regions, where the at-
mospheric pressure is lower. And, contrariwise, in the region of local reduction of
pressure, for example, in cyclones, an enhancement of the water level should be
observed. Extreme variations of atmospheric pressure are observed in tropical cy-
clones. The pressure at the centre of such a gigantic whirlwind can drop by a value
of
100 hPa, which amounts to about 10% of normal atmospheric pressure. A local
elevation of the level by
∼
1 m corresponds to such a depression. But in the case of
most tropical cyclones and of other atmospheric processes the amplitude of pres-
sure perturbations and, consequently, the amplitude of the free water surface devia-
tion will be by 1-3 orders of magnitude smaller. Variations of atmospheric pressure
with amplitudes exceeding 10% can, most likely, arise only in the case of powerful
explosions of natural (volcanoes, meteorites) or of artificial origin. In such cases
the pressure perturbation will, naturally, not be motionless, but will propagate in
the atmosphere, most probably, like a shock wave.
We, now, introduce the dimensionless variables (the asterisk * will be further
omitted)
∼
t
∗
=
t
g
x
H
,
V
√
g
H
,
x
∗
=
V
∗
=
H
,
(4.19)
H
.
p
atm
ρ
u
√
gH
p
atm
=
ξ
∗
=
u
∗
=
g
H
,
Note that the dimensionless velocity is the well-known Froud number Fr =
V
/
√
g
H
.
With account of transformations (4.19) the set of equations (4.16) and (4.17) is
easily reduced to the inhomogeneous wave equation
2
2
2
p
atm
∂
∂
t
2
−
∂
ξ
ξ
=
∂
.
(4.20)
∂
∂
x
2
x
2
If motion exists only at times
t
>
0, then for zero initial conditions the solution
of equation (4.20) is determined by the formula [Tikhonov, Samarsky (1999)]
x
+(
t
−
T
)
t
2
p
atm
∂
(
x
,
t
)=
1
2
d
X
∂
ξ
.
d
T
(4.21)
X
2
0
x
−
(
t
−
T
)
