Geoscience Reference
In-Depth Information
∂
ww
∂′
∂
w
∂′
w
∂
w
∂′
w
∂
w
∂′
w
∂
w
∂′
w
+ +
u
+
u
+′
u
+′
u
+
+
+′
v
+′
v
v
v
∂
t
∂
t
∂
x
∂
x
∂
x
∂
x
∂
y
∂
y
∂
y
∂
y
∂
ww
∂
′
∂
w w
∂
′
θ
θ
′
1
∂
1
∂
P
′
P
+
w
+
w
+
w
′
+
w
′
= −
g
+
g
v
−
−
(18.4)
ρ
ρ
∂
z
∂
z
∂
z
∂
z
∂
z
∂
z
v
a
a
⎛
⎞
∂
2
ww ww ww
x
∂
2
′
∂
2
∂
2
′
∂
2
∂
2
′
+
υ
+ ++ ++
⎜
⎟
2
2
2
2
2
2
∂
∂
x
∂
y
∂
y
∂
z
∂
z
⎝
⎠
Subtracting from this last equation the prognostic equation for mean vertical wind
speed given in Table 17.5 gives:
⎛
⎞
∂′
w
⎛
∂′
w
∂′
w
∂′
w
⎞
∂
w
∂
w
∂
w
⎛
∂′
w
∂′
w
∂′
w
⎞
u
w
u
v
w
u
v
w
+
+
v
+
+′
+′
+′
+′
+′
+′
⎜
⎟
⎜
⎟
⎜
⎟
∂
t
⎝
∂
x
∂
y
∂
z
⎠
∂
x
∂
y
∂
z
⎝
∂
x
∂
y
∂
z
⎠
⎝
⎠
(
)
(
)
(
)
⎛
⎞
(18.5)
∂′′ ∂′′ ∂ ′′
uw
vw
ww
q
′
1
∂′
P
⎛
∂′
2
www
∂′
2
∂′
2
⎞
=
g
v
−
+
u
+
+
+
+
+
⎜
⎟
⎜
⎟
r
∂
z
∂
x
2
∂
y
2
∂
z
2
∂
x
∂
y
∂
z
⎝
⎠
q
⎝
⎠
a
v
Using essentially the same procedure as above, similar prognostic equations can
be derived for turbulent fluctuations in other velocity components.
Equation (18.5) is the short-lived prognostic equation for the fluctuation
w
. In
principle, it might be used in a short time-step model of turbulence. However, in
such basic form prognostic equations of turbulent fluctuations have limited value
because their descriptive ability is limited to the time of existence of a turbulent
eddy. However, the equations can be used to derive prognostic equations for
turbulent variance. Again their derivation is illustrated by example for the case of
variance in vertical velocity.
The first step is to multiply Equation (18.5) by 2
w
′
′
and then to collect terms
using the relationships:
2
2
2
2
∂ ′
()
w
∂′
w
∂ ′
()
w
∂′
w
∂ ′
()
w
∂′
w
∂ ′
()
w
∂′
w
=′
2
w
;
=′
2
w
;
=′
2
w
;
=′
2
w
(18.6)
∂
t
∂
t
∂
x
∂
x
∂
y
∂
y
∂
z
∂
z
to give the result:
2
⎛
2
2
2
⎞
⎛
⎞
∂′
()
w
∂′
()
w
∂′
()
w
∂′
()
w
∂
w
∂
w
∂
w
u
+
+
v
+
w
+
2
wu
′ ′
+
2
wv
′ ′
+
2
ww
′
′
⎜
⎟
⎜
⎟
∂
t
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
⎝
⎠
⎝
⎠
⎛
∂′
()
w
2
∂′
()
w
2
∂′
()
w
2
⎞
q
′
1
∂′
P
+′
u
+′
v
+′
w
= ′
2
w g
v
− ′
2
w
⎜
⎟
r
⎝
∂
x
∂
y
∂
z
⎠
∂
z
q
(18.7)
a
v
⎛
⎞
⎛
∂′
2
www
∂′
2
∂′
2
⎞
∂′
(
u w v w
′
)
∂′
(
′
)
∂′
(
ww
′
)
+′
2
w
u
+
+
+′
2
w
+
+
⎜
⎟
⎜
⎟
2
2
2
x
y
z
⎝
∂
x
∂
y
∂
z
⎠
∂
∂
∂
⎝
⎠
