Geoscience Reference
In-Depth Information
This is the required prognostic equation for the variance in vertical wind speed.
Clearly analogous equations can be derived for the variance in horizontal wind
speeds
u
and
v
, and these three equations can be combined to describe the evolu-
tion of turbulent energy as described in the next section.
Prognostic equations for turbulent kinetic energy
The
turbulent kinetic energy
,
e
, provides a measure of the intensity of turbulence in
the ABL and is therefore strongly related to the turbulent transport of momentum,
heat, and moisture, see Equation (15.11) and is defined by the equation:
(
)
TKE
1
2
(18.15)
e
′
′
′
=
=
uvw
2
+
2
+
2
r
The prognostic equation for TKE is obtained by combining the prognostic
equations for variance along all three axes and takes the form:
⎛
⎞
(
w
′′
)
∂
e
∂
e
∂
e
∂
e
q
q
∂
(
uP
′
′
)
∂
(
vP
′
′
)
∂
(
wP
′
′
)
+
u
+
v
+
w
=
g
v
−
−
−
⎜
⎟
∂
t
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
⎝
⎠
v
⎛
⎞
⎛
⎞
u
u
u
∂′ ∂′ ∂ ′
uv
w
∂
∂
∂
+′
P
+
+
− ′′
(
u u
)
+ ′′
(
u v
)
+ ′′
(
u w
)
⎜
⎟
⎜
⎟
⎝
∂∂∂
x
y
z
⎠
∂
x
∂
y
∂
z
⎝
⎠
(18.16)
⎛
⎞
⎛
⎞
∂
v
∂
v
∂
v
∂
w
∂
w
∂
w
vu
vv
vw
wu
wv
w
w
−′ ′
(
)
+′ ′
(
)
+′
(
′
)
− ′ ′
(
)
+′ ′
(
)
+′
(
′
)
⎜
⎟
⎜
⎟
∂
x
∂
y
∂
z
∂
x
∂
y
∂
z
⎝
⎠
⎝
⎠
⎛
⎞
2
2
2
⎛
⎞
∂′
ue
∂′
ve
∂ ′
we
∂′
u
⎛
∂′
u
⎞
∂′
u
⎛
⎞
⎛
⎞
⎜
⎟
−
+
+
− υ
+
+
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
∂
x
∂
y
∂
z
⎝
∂
x
⎠
∂
y
⎝
∂
z
⎠
⎜
⎝
⎠
⎟
⎝
⎠
⎝
⎠
Equation (17.17) requires that the fifth term on the right hand side of Equation
(18.16) is zero. Moreover, if this equation is written in a coordinate system which
is aligned with the mean wind so that terms involving
−
are zero, and applied over
a flat, homogeneous area with no subsidence so that terms involving (∂/∂
x
), (∂/∂
y
)
and
−
are also zero, the equation simplifies to:
′′
′′
u
′
∂
e
(
w
q
)
1(
∂
wP
)
(
∂
∂
we
′′
=
g
v
−
−
u w
)
−
−
e
(18.17)
r
∂
t
∂
z
∂
z
∂
z
q
a
v
I II
II I IV V VI
where
e
is the turbulent dissipation of TKE defined by:
⎛
⎞
2
2
2
⎛
⎞
⎛
∂′
u
⎞
∂′
u
⎛
∂′
u
⎞
eu
=
⎜
+
+
⎟
(18.18)
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
∂
x
⎝
∂
y
⎠
∂
z
⎜
⎟
⎝
⎠
