Geoscience Reference
In-Depth Information
The leading terms here are of order
s/τ
t
,where
s
is the intensity scale of
θ
and
τ
t
∼
/u
is the turbulence time scale. The fluctuating radiative term is of order
r/τ
r
,
with
τ
r
a radiative time scale. Following
Townsend
(
1958
) we assume that
τ
r
τ
t
,
so the fluctuating radiative term in
Eq. (8.69)
can be neglected.
The equation for the components of the kinematic Reynolds-stress tensor is
∂u
i
u
k
∂t
U
j
∂u
i
u
k
∂x
j
+
=
u
j
u
k
∂U
i
u
j
u
i
∂U
k
∂x
j
−
∂x
j
−
(
mean-gradient production
)
∂u
i
u
k
u
j
∂x
j
−
(
turbulent transport
)
u
k
∂p
(
pressure-gradient interaction
)
1
ρ
0
u
i
∂p
−
∂x
i
+
∂x
k
−
2
ij m
j
u
m
u
k
−
2
kj m
j
u
m
u
i
(
Coriolis
)
θ
0
θ
v
u
k
δ
3
i
+
θ
v
u
i
δ
3
k
(
buoyant production
)
g
+
2
3
δ
ik
(
viscous dissipation
).
−
(8.70)
The equation for the flux of a conserved scalar is
∂cu
i
∂t
U
j
∂cu
i
+
∂x
j
=
u
j
u
i
∂C
cu
j
∂U
i
∂x
j
−
∂x
j
−
(
mean-gradient production
)
∂cu
i
u
j
∂x
j
−
(
turbulent transport
)
c
∂p
∂x
i
(
pressure-gradient interaction
)
1
ρ
0
−
−
2
ij k
j
u
k
c(
Coriolis
)
g
θ
0
cθ
v
δ
i
3
(
buoyant production
).
−
(8.71)
The equation for conserved-scalar variance is
∂c
2
∂t
+
U
j
∂c
2
∂x
j
(
time derivative following the mean motion
)
