Geoscience Reference
In-Depth Information
expect that above some height the direct effects of shear production are unimportant
compared to buoyant production. Put another way, we might expect the
mechanical
turbulence (that due to the shear production and characterized by
u
∗
) to become
unimportant at some
z/L
value. We can examine the implications of the declining
importance of shear production by deleting
u
∗
from the M-O governing parameter
group.
There are two ways to drop
u
∗
as a governing M-O parameter. The first is to redo
M-O similarity with only four governing parameters:
g/θ
0
,z,Q
0
,C
0
.
Those plus
one unknown give
m
−
=
5 parameters,
n
=
4 dimensions, so by the Buckingham Pi
Theorem there is
m
n
or one dimensionless parameter and it is a constant. Thus
from the four governing parameters we make the scales
−
g
θ
0
Q
0
z
1
/
3
Q
0
u
f
,
C
0
u
f
, ,
u
f
=
,
f
=
f
=
(10.34)
the subscript f denoting
free convection
, or convection in zero mean wind. Then we
write
(w
2
)
1
/
2
u
f
σ
w
u
f
=
σ
θ
T
f
=
=
C
1
,
C
2
,
(10.35)
with
C
1
and
C
2
constants. This says that in very unstable conditions
σ
w
(z)
and
σ
θ
(z)
approach their behavior in free convection.
Equation (10.35)
can be rewritten in standard M-O terms:
σ
w
u
∗
C
1
u
f
u
∗
σ
θ
T
∗
C
2
T
f
T
∗
z/L)
1
/
3
,
z/L)
−
1
/
3
.
=
∼
(
−
=
∼
(
−
(10.36)
As
Figure 10.4
shows, measurements are consistent with both of these predictions.
A second way to deduce the behavior at large
z/L
is to require that M-O
similar variables become independent of
u
∗
. In the case of
σ
w
, for example, since
we can write in general
σ
w
=
−
u
∗
f (z/L),
at large
−
z/L
the similarity function
f
must vary as 1
/u
∗
in order that
σ
w
=
fu
∗
be independent of
u
∗
. That implies that
z/L)
1
/
3
.
Similarly, in order that
σ
θ
be independent of
u
∗
,
σ
θ
/T
∗
must
f(z/L)
∼
(
−
z/L)
−
1
/
3
.
Tennekes
(
1970
) named this
local free convection
scaling.
Local
isotropy
(Part
III)
is isotropic behavior at large wavenumbers;
local
free convection is free-
convection-like behavior at large
vary as
(
−
z/L
.
While there are some exceptions toM-O similarity,
w
and
θ
statistics are observed
to follow it well and show clear indications of local-free-convection regimes.
According to
Figure 10.4
,
these can emerge at surprisingly small
−
−
z/L
; local free
convection behavior for
σ
w
appears by
z/L
=−
2 or so, and temperature variance
