Geoscience Reference
In-Depth Information
the problem should, evidently, not be based on the Navier—Stokes equations, but on
Reynolds equations. The existence of a vertical flow structure complicates transition
from the general non-linear equations of hydrodynamics to the long-wave equations.
But, if the non-linear term (v
,
) v is neglected, then the Reynolds equations can be
integrated over the vertical coordinate from the bottom,
z
=
∇
−
H
,uptothefreewater
surface,
z
=
. As a result, a set of equations will be obtained, which will contain
flow velocities averaged over the depth, while the term, describing the vertical tur-
bulent momentum transfer, will be expressed as the difference between tensions on
the bottom and on the free surface,
ξ
K
z
∂
d
z
=
ξ
∂
∂
v
1
ρ
(
T
S
−
T
B
)
.
z
∂
z
−
H
Without going into the details of obtaining the equations, expounded, for exam-
ple, in the monograph of [Murty (1984)], we present a version of the set of equa-
tions applied in practice for calculating meteotsunami generation and propagation
[Vilibic et al. (2004)],
∂
u
+
u
∂
u
x
+
v
∂
u
y
−
fv
∂
t
∂
∂
)
+
K
L
∂
,
2
u
2
u
−
g
∂ξ
∂
1
ρ
∂
p
atm
∂
+
(
T
S
T
B
)
x
x
2
+
∂
=
−
x
−
x
ρ
(
H
+
ξ
∂
∂
y
2
∂
v
t
+
u
∂
v
x
+
v
∂
v
y
+
fu
∂
∂
∂
)
+
K
L
∂
y
2
,
(
T
S
−
T
B
)
y
2
v
2
v
g
∂ξ
∂
1
ρ
∂
p
atm
∂
x
2
+
∂
−
y
−
=
+
y
ρ
(
H
+
ξ
∂
∂
x
(
H
+
)
u
+
y
(
H
+
)
v
= 0
,
∂ξ
∂
∂
∂
∂
∂
+
ξ
ξ
t
where
u
,
v
are velocity components averaged over the depth,
ξ
is the free-surface
displacement from equilibrium position,
f
= 2
is the Coriolis parameter and
K
L
is the constant horizontal turbulence viscosity coefficient. In principle, the quan-
tity
K
L
may be variable, and then it should be present under the derivative sign. In
[Vilibic et al. (2004)] the assumption was made that
K
L
= 15 m
2
/s.
Now, consider the main physical regularities of the meteotsunami generation
process, taking advantage of the example of waves, caused by moving perturba-
tions of atmospheric pressure. For clarity the problem will be considered within
the framework of the simple one-dimensional model. Let
ω
sin
ϕ
be the absolute value
of the free-surface displacement from equilibrium, and consider the ocean depth
H
= const and the horizontal scale of atmospheric perturbation
a
to be related as
follows:
|
ξ
|
a
. With account of such assumptions the meteotsunami forma-
tion process can be described by linear equations of long-wave theory,
|
ξ
|
H
