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where orbital velocity is
u
orb
is the kinematic viscosity of the ocean water.
Re
wave
indicates the transition from laminar orbital motion to turbulent. It is interesting to
notice that WRN can have the velocity scale eliminated if the dispersion relationship
(2.17)
is used, and can be expressed in terms of the two length scales:
=
a
ω
and
ν
a
2
√
gk
ν
2
g
a
2
ν
√
λ
.
Re
wave
=
=
π
(7.71)
Thus, according to
(7.69)
, for a given wavelength
λ
(wave frequency
ω
) Reynolds number
Re
wave
decays rapidly as a function of depth:
2
Re
wave
λ
(
z
)
∼
a
(
z
)
∼
exp
(
−
2
kz
).
(7.72)
At the surface (
z
=
0
,
exp
(
−
2
kz
)
=
1), longer waves of the same amplitude
a
0
will produce
smaller Reynolds numbers.
Let Re
wave
critical
be the critical value of WRN
(7.70)
-
(7.71)
. If near the surface Re
wave
λ
>
Re
wave
critical
, then the corresponding wave orbital motion will be turbulent. At some depth
z
critical
, dependence
(7.72)
will lead to Re
wave
λ
=
Re
wave
critical
, and from that depth down the
orbiting will become laminar, or, in the turbulent ocean, will stop generating turbulence.
Depth
z
critical
, therefore, will define a mixed layer depth - the depth of the upper-ocean
layer mixed due to the turbulence generated by orbital movement produced by the surface
waves. Obviously, in reality the convection, advection, heating and other processes can
alter this value. Also, the background ocean waters are nearly always turbulent, and there-
fore
z
critical
is not the depth below which turbulence is absent, but is rather a depth below
which we do not expect the presence of wave-induced turbulence.
From
(7.70)
-
(7.71)
, the Reynolds number at a given
λ(ω)
, as a function of
z
,is
a
0
exp
g
z
2
Re
wave
=
ω
ν
)
=
ω
ν
2
ω
a
0
exp
(
−
2
kz
−
.
(7.73)
Therefore, if the critical Reynolds number Re
wave
critical
were known, the critical depth,
which is also wave-induced MLD, would be readily available:
2
k
ln
Re
wave
critical
ν
2
ln
a
0
ω
Re
wave
critical
ν
1
g
z
critical
=−
=
.
(7.74)
a
0
ω
2
ω
ω
As seen, for a wave frequency
, if the wave height grows, MLD will increase. If a few
waves of the same height but different scales are present at a time, the mixed layer will
mostly be determined by the lowest frequency
ω
(longest length
λ
) as its
z
critical
will be the
largest.
The real wind-generated waves are spectral, and, apart from the cases of pure swell,
multiple wave scales are superposed at the ocean surface. The wave spectrum, however,
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