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hence:
2 2
() ()
A
(15.11)
σ=′
a
but note that in this case Equation (15.11) also follows directly from Equation
(15.9). From Equation (15.11), the standard deviation of A is given by:
(15.12)
2
()
σ=
a
A
Measures of the strength of turbulence
One measure of the strength of atmospheric turbulence is
s m , the square root of
the sum of the variances of the three orthogonal components of wind speed, u
parallel to the mean horizontal wind, v perpendicular to the mean horizontal
wind, and w in the vertical direction.
s m is calculated from:
2 2 2
() () ( )
σ=
u
′+′+ ′
v w
(15.13)
m
Recall how the strength of the turbulence was judged to vary with time in Fig. 15.1.
Other measures of strength of turbulence can also be defined, including the turbu-
lent intensity, I , which is defined by normalizing
s m by the magnitude of the mean
wind vector, U m , at the point where
s m was measured. Because U m is given by:
(15.14)
U
=
()
u
2
+
()
v
2
+
( )
w
2
m
the turbulent intensity is given by:
(15.15)
() () ( )
()
u
′+′+ ′
2
v w
2
2
I
=
uv w
2
++
()
2
( )
2
Mean and turbulent kinetic energy
The kinetic energy, E K , of a body of mass M moving with a speed V is given by:
1
2
k E V
=
2
(15.16)
When defining the kinetic energy of air in the atmosphere it is usual to normalize
by the density of air to give the turbulent energy per unit mass and to separate the
kinetic energy associated with the mean air flow and the turbulent fluctuations
 
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