Geoscience Reference
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growth rate, which includes both linear growth and damping, and the nonlinear
flow of energy from eddy to eddy through the spectrum. In a steady state the
input of energy at long wavelengths equals the dissipation at short wavelengths.
The eddy process forms a mechanism to transfer energy from the growth por-
tion of the spectrum to the region where damping is strongest. The argument
goes as follows. In a range of wave numbers,
k
, the total density fluctuation
strength is
2
(
n
/
n
)
=
k
I
k
2
π
kdk
2
π
I
k
k
k
k
If we consider the range
k
=
k
/
2
π
, that is, a bandwidth in
k
space equal to
k
/
2
π
, then
2
k
2
I
k
(
/
)
n
n
k
Since from (4.44)
δ
V
/
V
D
=
[
ν
i
/
i
(
1
+
0
)
]
(δ
n
/
n
)
, we also see that in this range
of
k
the velocity fluctuation strength is given by
V
2
V
D
k
2
I
k
δ
k
∝
Now the classical turbulence argument is that the eddy decay rate
(
k
)
is given
by the inverse of the eddy turnover time
τ
k
, where
k
=
(τ
k
)
−
1
V
D
k
2
1
/
2
=
k
δ
V
∝
(
I
k
)
k
This is the time it takes for the material in the eddy to move one eddy scale
size (
k
−
1
). The total energy
V
2
ε
k
in a given eddy is proportional to the
δ
k
given
above, so the rate of energy loss is given by
V
D
k
4
I
3
/
2
(
ε
k
/τ
k
)
k
=
ε
k
k
∝
(4.56)
k
The rate of energy gain or loss in the same wavelength band from the linear
growth and damping processes is determined by the linear growth/damping rate
γ
k
averaged over all angles at wavelength
k
. In a time-stationary steady state it
must be the case that the spectrum
ε
k
(
k
)
has the property that
kd
/
dk
(ε
k
k
)
=
γ
k
ε
k
(4.57)
so the steady-state energy spectral density remains the same in any
k
interval.
From linear theory, for the primary gradient drift process,
γ
k
is of the form
Bk
2
γ
k
=
A
−
(4.58)
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