Geoscience Reference
In-Depth Information
Unstable
Stable
Neutral
2.0
2.0
2.0
Both
buoyant and
frictional
production
of TKE
Frictional
production
and buoyant
destruction
of TKE
Frictional
production
of TKE
1.5
1.5
1.5
1.0
1.0
1.0
0.5
0.5
0.5
0
0
0
0
123
TKE per m (m
2
s
−
2
)
4
0
12 3
TKE per m (m
2
s
−
2
)
4
0
12 3
TKE per m (m
2
s
−
2
)
4
Figure 15.6
Typical examples of the variation of the linear density of TKE per unit mass with height in different conditions
of atmospheric stability.
Consequently, if
A
and
B
are expressed in mean and fluctuating parts, the
covariance is:
(15.20)
C
=+′ −
((
A a
)
A B b
)((
+′ −=′ ′
)
B
)
(
a b
)
AB
,
In practice, covariance is most commonly used in the form of the
Linear Correlation
Coefficient, r
A
,
B
, which is the covariance normalized by the standard deviation of
the two variables, thus:
(
ab
′′
)
r
(15.21)
=
AB
,
σσ
ab
The value of
r
A
,
B
always lies in the range from −1 to +1 and indicates the degree of
commonality between variations in the two variables. Thus, if the two variables are
perfectly correlated (i.e., they vary together in the same direction) then
r
A
,
B
= 1,
and if they varied together but in opposite directions
r
A
,
B
= −1. In fact perfect
correlation between variations in the values of atmospheric variables is rare, but
significant and important correlations can and do occur in the atmospheric
boundary layer. For example, if one of the variables is the vertical wind speed,
w
,
and the second is the virtual temperature of the air,
q
v
, then if air that is warmer
than average tends to move upward and air that is colder than average tends to
move downward, their covariance will likely be greater than zero. In this situation,
hotter air is moved upward in the upward fluctuations, to be replaced by cold air
that is moved downward in downward fluctuations, so there is a net flow of energy
upward. Consequently, although on average there is no net vertical motion of the
air and
_
= 0, there is a flow of heat away from the surface that is associated solely
