As a demonstration of the power of Monin-Obukhov
thus implies that asymptotically in the free-convection
Similarity Theory (MOST), let me consider the variances
limit
of vertical velocity, w2 ≡ σ2 , and temperature, θ2 ≡ σ2 .
σw
∝ (-ζ)1 / 3
w
t
Theory predicts that when these statistics are properly
(65)
u*
scaled, they can be functions only of ζ. That is
and
σw
= f (z / L)
σt
(59)
∝ (-ζ)1 / 3.
u*
(66)
t*
σt
= g(z / L)
(60)
t*
Again, although MOST was the basis for our deriving
these asymptotic expressions, it tells us nothing about
since u* is our primary velocity scale in the ASL and t*
the values of the implied proportionality constants. These
is our primary temperature scale.
must be found experimentally.
While MOST predicts that f(ζ) and g(ζ) should exist,
The opposite extreme from free convection is very
it does not predict their functional forms. Ultimately,
stable stratification--in which the vertical density gra-
these must be evaluated experimentally. MOST does,
dient is actually steep enough to suppress vertical ex-
however, provide some guidance as to the asymptotic
change. Thus, the eddies reaching a sensor placed at
behavior of f(ζ) and g(ζ). Let me elaborate.
height z are compacted in their vertical dimension and
In the so-called free-convection limit, the buoyancy
may never have been in contact with the surface. In other
production term in eq 33 swamps the mechanical pro-
words, contrary to near-neutral or unstable conditions, in
duction term; thus, u* loses its significance as the fun-
very stable conditions, z is no longer a meaningful scale;
damental velocity scale. From the remaining fundamen-
a turbulence sensor placed at height z can make no meas-
tal parameters, wtv (or tv ), g / Tv and z, however, it is
urement that tells it where the surface is. Thus, z drops
*
possible to define a new velocity scale appropriate for
out of our list of surface-layer scales.
the free-convection limit (e.g., Hess 1992)
Without z, we cannot form the stability parameter
z/L. Therefore, the nondimensional standard deviations
1/ 3
z g wtv
uf ≡
in vertical velocity and temperature must be indepen-
.
(61)
Tv
dent of stability in very stable conditions
σw
= constant
As in eq 55, we can, in turn, define a new temperature
(67)
u*
scale for the free-convection limit
σt
1/ 3
wt Tv wt
3
= constant.
(68)
tf ≡
=
uf z g wtv
.
(62)
t*
As usual, MOST does not tell us what these constants
should be--only that they should exist.
If these are the proper scales in the free-convection
Figure 1 demonstrates the veracity of MOST. Figure
limit, making σw and σt nondimensional with them
1a shows σw / u* as a function of stability from meas-
should yield universal functions. But now, without u* ,
urements I made with three-axis and vertical sonic an-
we have used all the scales available to us. In other words,
emometers at the Sevilleta Long-Term Ecological Ref-
without u* , L cannot be defined; therefore, σw/uf and
uge near Socorro, New Mexico, in August 1991. Figure
σt/tf can depend on no other variables. The only conclu-
1b shows σ t/ t* and σq / q , where σq is the standard
sion is that they are constants in the free-convection limit
*
deviation in specific humidity, measured during the same
σf
experiment. Some of the σt values came from the tem-
= constant
(63)
uf
perature fluctuations measured by the three-axis sonic
anemometerthermometer; some came from a 76-m
σt
chromel-constantan thermocouple. The humidity data
= constant.
(64)
came from a krypton hygrometer.
tf
In Figure 1a, σw / u* goes as (ζ)1/3 for ζ < 0.4, as
eq 65 predicts. For ζ > 0, σw / u* is almost independent
Since u* must lose its dynamical significance gradu-
ally as ζ decreases, a matching region must exist where
of ζ, as eq 67 suggests.
both eq 59 and 63 and where both eq 60 and 64 are ap-
The data in Figure 1b are more scattered than those
in Figure 1a because t* and q* also reflect the uncer-
proximately true. Using the definitions eq 58, 61 and 62
8
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