humidity scale q , another quantity that is constant with
Thus, uw and vw are basically constant with height in
*
height.
the ASL. We usually align the x axis with the mean wind
Following arguments similar to these, Monin and
so vw = 0 and then define
Obukhov (1954) recognized u and t as fundamental
*
*
u2 = -uw.
flux scales in the ASL. In light of eq 33, they also took
(50)
*
g/Tv to be an important parameter. Lastly, they knew
that the height of the observation, z, was a fundamental
Thus, u* is a fundamental velocity scale in the surface
length scale in the surface layer. Consequently, they
layer because it is constant with height.
hypothesized that all surface-layer statistics should scale
We can apply the same scaling procedure to eq 38.
with combinations of these four quantities.
Here, in addition to eq 44, however, we also need
We now know, however, that for scaling properties
Θ+ = Θ / t
that depend on air density, rather than t alone, we need
(51)
*
*
a scale that also includes q (Zilitinkevich 1966, Busch
*
1973). This is the virtual temperature flux scale, tv* ,
where t* is a temperature scale that I will explain shortly.
In eq 38, I make wθ nondimensional with u t . The non-
defined from eq 21b as
**
dimensional form of eq 38 is thus
(
)
tv = t 1 + 0.61Q + 0.61T q
(57)
(wθ / u*t* ) =
*
*
*
Θ
Θ+
2
1
1
+
(52)
where Q and T must be layer-averaged mean values.
2
Ro+ t+
z+
PrR
z+
*
Because t and q are constant with height in the sur-
*
*
face layer, tv* is too.
where Pr (≡ ν/D) is the Prandtl number and
From u*, tv* and g / Tv , it is possible to define a fun-
u z0
damental length scale L that is also a constant in the
R* ≡ *
(53)
ν
ASL
is the roughness Reynolds number.
1 k g tv*
kg
0.61T
≡
=
t* +
q
(58)
Again, as with eq 46, Θ+ / t+ should be of order
1 + 0.61Q *
Tvu2
T u2
L
one if our scaling is accurate. But in eq 52, Θ+ / t+ is
*
*
divided by 106. In the atmosphere, Pr is about 0.7; and
where k (= 0.4) is the von Krmn constant. Many call
over sea ice R* is rarely smaller than 10, except in very
L the Monin-Obukhov length. I prefer, however, to call
light winds (Andreas and Claffey 1995). Thus, the third
it the Obukhov length, since Obukhov defined it in print
term in eq 52 is also small. Consequently, again the tur-
eight years before the Monin-Obukhov (1954) paper
bulence term is all that remains of eq 52 in the ASL
appeared (Businger and Yaglom 1971, Obukhov 1971).
On recognizing the dynamical significance of the sur-
(wθ / u*t* ) ≅ 0.
face layer scales u*, t*, z and L (we have since added
(54)
tv and q ), Monin and Obukhov (1954) speculated that
z+
*
*
all surface-layer turbulence statistics should behave
similarly when properly expressed in terms of these
As a result, wθ , the kinematic sensible heat flux, is basi-
scales (see also Businger 1973, Wyngaard 1973). In par-
cally constant with height in the ASL. We define its val-
ticular, Monin-Obukhov similarity quantifies stability
ue as
effects in the ASL with the nondimensional parameter
z/L ≡ ζ . Shortly, when we see that U / z ≅ u* / kz , it
u t ≡ - wθ.
(55)
will be evident that ζ is the ratio of the buoyant produc-
**
tion to the mechanical production (term III to term II) in
Thus, because it is independent of height, t* is the fun-
the turbulent kinetic energy equation, eq 33. Thus, roughly
damental surface-layer temperature scale.
when ζ > 1, buoyancy effects dominate mechanical pro-
The mean humidity equation, eq 41, yields to the same
cesses in the surface layer; when ζ < 1, mechanical ef-
scaling arguments that the mean temperature equation
fects dominate. When ζ < 0, the surface is heating and,
did. We thus see that wq , the kinematic latent heat flux,
thus, destabilizing the air in the ASL through the turbu-
is also independent of height in the ASL. We define its
lent exchange of sensible and latent heat. This process
value as
creates unstable stratification. When ζ > 0, the surface
is extracting heat from the surface layer and thereby
u q ≡ - wq
(56)
**
cooling it from below. This results in a stably stratified
surface layer.
which therefore introduces the fundamental surface-layer
7
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