Geoscience Reference
In-Depth Information
The averaged virtual potential temperature deviation is
θ
v
=
θ
0
.
61
θ
+
q
˜
−
θ
0
.
(8.63)
These equations have the Reynolds terms we met in
Part I
: a kinematic Reynolds
stress
τ
ij
in
Eq. (8.59)
and Reynolds fluxes of scalars,
f
θ
j
and
f
q
j
,in
Eqs. (8.61)
and
(8.62)
. The averaging also produces a covariance of radiative flux divergence
and
θ/ T
in
Eq. (8.61)
, but scaling arguments indicate it is quite small and we
have neglected it
(Problem 8.19)
. In addition,
Eq. (8.63)
has a Reynolds term, as
does the averaged form of
Eq. (8.56)
for the conserved temperature in cloud air.
Traditionally such “thermodynamic” Reynolds terms have been neglectedwith little
or no discussion, but
Larson
et al
.
(
2001
) have shown how this can bias numerical
model results.
8.4.2 The ensemble-averaged equations
We now write the dependent variables in our equation set as the sum of ensemble-
mean and fluctuating parts, generalizing the water vapor mixing ratio
q
to a
conserved scalar
c
:
˜
θ
v
=
p
=
v
+
u
i
=
˜
U
i
+
u
i
,
˜
P
+
p,
θ
v
,
θ
ρ
0
c
p
T
∂ R
i
(8.64)
θ
=
+
θ.
∂x
i
=
R
+
r,
c
=
C
+
c,
The mean equations are then
∂U
i
U
j
∂U
i
∂
∂x
j
u
i
u
j
1
ρ
0
∂P
∂x
i
−
g
θ
0
v
δ
3
i
,
∂t
+
∂x
j
+
=−
2
ij k
j
U
k
+
(8.65)
∂U
i
∂x
i
=
0
.
(8.66)
∂
∂t
+
U
i
∂
∂θu
i
∂x
i
+
∂x
i
+
R
=
0
,
(8.67)
∂C
∂t
+
U
i
∂C
∂cu
i
∂x
i
+
∂x
i
=
0
.
(8.68)
We have dropped the molecular diffusion terms because they are very small except
at the lower surface.
The fluctuating equations are derived by the process outlined in
Part I
. That for the
fluctuating thermodynamic variable does need special mention, however. It reads
∂θ
∂t
+
U
i
∂θ
u
i
∂
u
i
∂θ
u
i
∂θ
2
θ.
∂x
i
+
∂x
i
+
∂x
i
−
∂x
i
=
r
+
α
∇
(8.69)