Geoscience Reference
In-Depth Information
that need not be directly related to
u
a
nd
. Analogously, for the time-change term
we allow a scale
τ
e
of time changes in
c
2
that can differ from
/u
:
s
2
τ
e
.
∂c
2
∂t
∼
(5.23)
The order of magnitude of the mean-advection term in
Eq. (5.7)
is then
U
uL
x
s
2
u
U
s
2
mean advection
∼
L
x
=
.
(5.24)
U
is typically of the order of but larger than
u
.Ifso,andif
L
x
, then the
parameter
(U /uL
x
)
1 and mean advection is negligible, as in a homogeneous
flow. We call this
local homogeneity
. Similarly, the time-change term scales as
/u
τ
e
s
2
u
s
2
τ
e
=
time change
∼
.
(5.25)
If
(/u)/τ
e
1, meaning that the large-eddy turnover time
/u
is much less than
the time scale
τ
e
of the changing boundary conditions, then the time-change term
is negligible. It is as if the mean flow is steady; we call it
quasi-steady
.
As we shall show in
Part II
, in homogeneous terrain and away from the morn-
ing and evening transitions in the surface heat flux this can allow a quasi-steady,
locally homogeneous interpretation of the second-moment budgets. For
Eq. (5.20)
this is
∂c
2
∂t
∂c
2
w
∂z
2
wc
∂C
0
=−
∂z
−
−
χ
c
.
(5.26)
5.3.4 Interpreting the molecular destruction term
We can scale the derivative covariance in
χ
c
,
Eq. (5.12)
, through the power spectral
density that we introduced in
Chapter 2
. There we represented a real, statistically
homogeneous, zero-mean, one-dimensional random scalar function
f(x
;
α)
on an
interval 0
≤
x
≤
L
as a complex Fourier series. We'll express this as
N
f(κ
n
;
α)e
iκ
n
x
.
f(x
;
α)
=
(5.27)
n
=−
N
