Geoscience Reference
In-Depth Information
the left side of
Eq. (14.32)
depends on
R
λ
. As a result the two dominant terms,
which are of order
us
2
/λ
3
, are to first approximation in balance at large
R
λ
:
∂u
j
∂x
i
g
i
g
j
γ
∂g
i
∂x
j
∂g
i
∂x
j
.
=−
(14.34)
This says that to leading order the mean-squared intensity of scalar gradients is in
equilibrium, its rate of production through the turbulent strain rate balanced by its
rate of removal through molecular diffusion.
To help grasp the meaning of
Eq. (14.34)
, consider the effect of a
n im
balance
I
in its tw
o ter
ms. From
Eq. (14.32)
the time scale of the response in
g
i
g
i
would be
of order
g
i
g
i
/I
/uR
−
λ
- much less than the eddy-turnover time
/u
.So
Eq. (14.34)
says that on the time scale of the large eddies these two leading terms
keep themselves in balance.
The other terms in
Eq. (14.32)
represent finite-
R
λ
corrections to the balance of the
leading pair. Thus we see a fundamental difference in the nature of second-moment
budgets for energy-containing-range variables, which we discussed in detail in
Part I
, and those for dissipative-range variables. In the former, all terms can be
of the same order; some involve mean field-turbulence interactions and others
represent turbulence-turbulence interactions. By contrast, budgets for dissipative-
range quantities are inherently
R
λ
dependent, so at large
R
λ
some terms are much
larger than others. The leading terms represent turbulence-turbulence interactions.
With that background, we'll now consider th
e budg
ets of the thirdmoment of spa-
tial derivatives of a conserved scalar. That for
(θ
,
1
)
3
in a quasi-steady, horizontally
homogeneous ABL
(Problem 14.23)
is
∼
λ/u
∼
∂(θ
,
1
)
3
∂t
=−
3
,
3
u
3
,
1
θ
,
1
θ
,
1
−
(θ
,
1
θ
,
1
θ
,
1
u
3
)
,
3
−
3
θ
,
1
θ
,
1
θ
,j
u
j,
1
−
6
γ θ
,
1
j
θ
,
1
j
θ
,
1
.
(14.35)
s
3
λ
3
s
3
λ
3
s
3
λ
3
u
u
u
λ
The terms on the right are, in order, mean-gradient production, turbulent transport,
turbulent production, and molecular destruction; below each of the first three terms
is its scaling estimate. The third term is the largest by a factor
/λ
R
λ
. Thus,
as in the budget of gradient variance,
Eq. (14.34)
, to lowest order the quasi-steady
balance in
Eq. (14.35)
is between turbulent production and molecular destruction.
However, this lowest-order formof
Eq. (14.35)
involves only derivatives of veloc-
ity and temperature, which qualifies it for the local isotropy assumption. But each
of its two leading terms involves an odd-order tensor that vanishes under isotropy.
∼
