Geoscience Reference
In-Depth Information
Divergence equation for turbulent fluctuations
Expressing Equation (16.42), the equation describing the conservation of mass (or
divergence equation)
that is relevant in the ABL, in terms of mean and fluctuating
parts gives:
∂
u
∂
v
∂
(
uu vv ww
+′
)
∂
(
+′
)
∂
(
+ ′
)
∂
u
′
∂
v w w
′
∂
∂
′
+
+
=
+
+
+
+
+
=
0
(17.14)
∂
x
∂
y
∂
z
∂
x
∂
x
∂
y
∂
y
∂
z
∂
z
Averaging, this equation gives:
∂∂ ∂∂ ∂ ∂
∂∂∂∂∂ ∂
uuv
′ ′ ′
+++++ =
v ww
(17.15)
0
x
x
y
y
z
z
From Reynolds averaging rules, the second, fourth, and sixth terms in the last
equation are zero, consequently:
∂∂
∂
∂∂ ∂
uv
w
(17.16)
++=
0
x
y
z
While subtracting Equation (17.16) from Equation (17.15) gives the (obvious but
later useful) result:
∂∂∂
∂∂ ∂
uv w
x
′
′ ′
++ =
0
(17.17)
y
z
Thus, in the ABL the continuity equation holds separately for both the mean and
the fluctuating components of kinematic velocity, i.e.,
and
∇′=
.
u
.
∇=
.
u
0
Conservation of momentum in the turbulent ABL
In the following, derivation of the required mean flow equation is illustrated
for the case of the Z axis with those for the X and Y axes then written later by
analogy. Starting from Equation (16.38) and applying the Boussinesq approxima-
tion, gives:
∂
wwww
∂
∂
∂
′
1
∂
P www
⎛
∂
2
∂
2
∂
2
⎞
⎛
q
⎞
+
u
+
v w g
+
= −
1
−
−
+ υ
+
+
(17.18)
⎜
⎟
⎜
⎟
∂
t
∂
x
∂
y
∂
z
⎝
q
⎠
r
∂
z
∂
x
2
∂
y
2
∂
z
2
⎝
⎠
a
