Geoscience Reference
In-Depth Information
The decomposition of the equations is carried out as follows. Phase averaging
the momentum equation,
p
∂x
i
−
∂
u
i
∂t
+˜
˜
u
j
∂
u
i
∂x
j
=−
˜
1
ρ
0
∂
˜
g
θ
0
θ
δ
3
i
+
2
2
ij k
j
˜
u
k
+
ν
∇
u
i
,
˜
(
8
.
57
)
u
i
. Ensemble averaging
Eq. (8.57)
produces an
equation for
U
i
. Subtracting that equation from the equation for
U
i
+
produces an equation for
U
i
+
u
i
produces
an equation for
u
i
u
i
from
Eq. (8.57)
gives an
equation for
u
i
.
The same sequence is applied to the potential temperature equation
. Subtracting the equation for
U
i
+
∂ θ
∂t
+˜
u
i
∂ θ
α θ
,jj
.
∂x
i
=
(12.26)
The equations for the wave and turbulent components of the fluctuating velocity
field are
u
i
u
j
−
u
i
u
j
,j
1
g
u
i,t
+
u
i,j
U
j
+
U
i,j
u
j
+
r
ij,j
+
ρ
0
p
,i
+
θ
0
θ
w
δ
3
i
,
(12.27)
=−
u
i
u
j
−
u
i
u
j
,j
u
i,t
+
u
i,j
U
j
+
U
i,j
u
j
−
r
ij,j
+
u
i
u
j
+
u
i
u
j
+
1
g
ρ
0
p
,i
+
θ
0
θ
t
δ
3
i
+
νu
i,jj
.
=−
(12.28)
Here
(u
i
u
j
)
p
u
i
u
j
.
r
ij
=
−
(12.29)
We have neglected the molecular diffusion term in the equation for the wave com-
ponent.
Equations (12.27)
and
(12.28)
sum to the usual equation for fluctuating
velocity
u
i
=˜
u
i
−
U
i
, as required:
1
ρ
0
p
,i
+
g
θ
0
θδ
3
i
+
u
i,t
+
u
i,j
U
j
+
U
i,j
u
j
+
(u
i
u
j
)
,j
−
(u
i
u
j
)
,j
=−
νu
i,jj
.
(12.30)
The wave and turbulent parts of
θ
satisfy
u
j
θ
w
u
j
θ
w
,j
θ
,t
θ
,j
U
j
+
,j
u
j
r
jθ,j
+
+
+
−
=
0
,
(12.31)
u
j
θ
t
u
j
θ
w
,j
θ
,t
+
θ
,j
U
j
+
,j
u
j
−
r
jθ,j
+
u
j
θ
t
u
j
θ
t
γθ
,jj
.
−
+
+
=
(12.32)
