Geoscience Reference
In-Depth Information
• The molecular diffusivities are ignored because the turbulence Reynolds number
R
t
in
the surface layer is so large. Thus M-O similarity is not applicable to the dissipative-range
structure, which does depend on the molecular diffusivities.
• The characteristic length of the roughness elements on the surface, called the
roughness
length z
0
, is excluded. Thus we restrict M-O similarity to
z
z
0
.
10.2.3 The application of M-O similarity
An unknown dependent variable plus the five M-O governing parameters com-
prise
m
=
6 parameters with
n
=
4 dimensions: length, time, temperature, and
c
. Thus, there are
m
2 independent dimensionless quantities that are
functionally related. M-O similarity takes one as the dependent variable nondimen-
sionalized with
z, u
∗
,T
∗
=−
−
n
=
Q
0
/u
∗
and
c
∗
=−
C
0
/u
∗
; the other is taken as
z/L
,
u
3
θ
0
/kgQ
0
is the
Monin-Obukhov length
.
†
k
where
L
0
.
4isthevonKár-
mán constant.
‡
TheBuckinghamPi Theorem then says that this nondimensionalized
dependent variable is a function only of
z/L
.
L
is negative in unstable conditions
(Q
0
>
0
)
, positive in stable conditions
(Q
0
<
0
)
, and infinite at neutral
(Q
0
=
=−
∗
;
0
)
thus, the range of the M-O independent
−∞
∞
|
|
variable is
<z/L<
. Conditions are
near-neutral
when
z/L
is sufficiently
small, which occurs near the surface when
z
|
L
|
.
M-O similarity implies that the surface-layer gradients of meanwind speed, mean
virtual potential temperature, and mean water vapor mixing ratio behave as
φ
m
z
L
,
φ
h
z
L
,
kz
u
∗
∂U
∂z
=
kzu
∗
Q
0
∂
∂z
=
kz
T
∗
∂
∂z
=
−
(10.12)
φ
c
z
L
,
kzu
∗
C
0
∂C
∂z
=
kz
c
∗
∂C
∂z
=
−
with
φ
m
,
φ
h
,and
φ
c
being functions of
z/L
that are
universal
- the same in all locally
homogeneous, quasi-steady surface layers. The minus sign in the definitions of
T
∗
and
c
∗
makes
φ
h
and
φ
c
positive, as
φ
m
is.
It is commonly assumed that
φ
h
=
φ
c
,but
Warhaf t
(
1976
) has questioned that
because the buoyancy term in the scalar flux conservation
equation (8.70)
differs for
potential temperature and for another conserved scalar - water vapor, for example.
We'll discuss that in more detail in
Chapter 11
.
†
Businger and Yaglom
(
1971
)and
Foken
(
2006
) point out that the Monin-Obukhov length was introduced by
Obukhov (
1946
), so
Obukhov length
is more appropriate.
‡
The value of
k
was traditionally thought to be
0
.
4, was measured as 0.35 in the 1968 Kansas experiments,
and extensive measurements by Andreas
et al
.(
2006
)giveavalueof0.39.
