Geoscience Reference
In-Depth Information
8.4.1 The general case
The momentum, continuity, thermodynamics, and vapor-conservation equations
are
p
∂x
i
−
∂
u
i
∂t
+˜
˜
u
j
∂
u
i
∂x
j
=−
˜
1
ρ
0
∂
˜
g
θ
0
θ
v
δ
3
i
+
2
2
ij k
j
˜
u
k
+
ν
∇
u
i
,
˜
(8.57)
∂
u
i
˜
∂x
i
=
0
,
(1.18)
∂ θ
∂t
+˜
u
j
∂ θ
θ
ρ
0
c
p
T
∂ R
j
∂x
j
,
2
θ
∂x
j
=
α
∇
−
(8.35)
∂
2
∂
q
∂t
+˜
˜
u
i
∂
q
∂x
i
=
˜
q
∂x
i
∂x
i
.
˜
γ
(8.36)
Here the mixing ratio for water vapor,
ρ
, also called the
specific humid-
ity
, is the conserved water-vapor variable.
ρ
0
and
θ
0
are the background density
and potential temperature profiles,
q
˜
≡˜
ρ
v
/
˜
p
is the deviation in pressure from its back-
ground profile
p
0
,and
θ
v
is the deviation in virtual potential temperature from its
background value
θ
0
:
˜
θ
v
=
θ
v
−
θ
0
=
θ(
1
+
0
.
61
q)
˜
−
θ
0
.
(8.58)
R
j
is the radiative heat flux;
α
is the thermal diffusivity of moist air.
In atmospheric applications these equations have too large a range of scales to
be solved numerically. Thus before attempting numerical solutions we reduce the
scale range by averaging the equations. Denoting the average, ensemble or space,
with an overbar, this gives the set
p
∂x
i
−
∂
u
i
∂t
+ ˜
˜
∂
u
i
∂x
j
=−
˜
1
ρ
0
∂
˜
g
1
ρ
0
∂τ
ij
∂x
j
,
θ
0
θ
v
δ
3
i
+
u
j
2
ij k
j
˜
u
k
+
(8.59)
τ
ij
ρ
0
= ˜
u
i
u
j
− ˜
˜
u
i
˜
u
j
,
∂
u
j
∂x
j
=
˜
0
,
(8.60)
θ
T
,f
θ
j
∂ R
j
∂x
j
∂ θ
∂t
∂ θ
∂x
j
=−
∂f
θ
j
1
ρ
0
c
p
u
j
θ
θ,
+ ˜
∂x
j
−
= ˜
− ˜
u
j
u
j
(8.61)
∂f
q
j
∂x
j
,f
q
j
∂
q
∂t
+ ˜
˜
q
∂x
j
=−
∂
˜
u
j
= ˜
u
j
˜
q
− ˜
u
j
q.
˜
(8.62)
