Table 2. Values of the coefficients in the poly-
Physically, what these distinctions mean are that
nomial (eq 113) that predicts zT/z0 and zQ/z0.
when the surface is aerodynamically smooth, the rough-
ness elements are embedded in the viscous sublayer;
R* ≤ 0.135
0.135 < R* < 2.5
2.5 ≤ R* ≤ 1000
as a result, momentum is transferred to the surface
Temperature (zT/z0)
strictly by molecular processes. When the surface is
b0
1.250
0.149
0.317
aerodynamically rough, the roughness elements pro-
b1
--
0.550
0.565
trude through the viscous sublayer; turbulent eddies can
b2
--
--
0.183
therefore transfer momentum through pressure forces
Water vapor (zQ/z0)
on the roughness elements as well as through molecu-
b0
1.610
0.351
0.396
lar processes. In eq 33, term IVb reflects this explicit
b1
--
0.628
0.512
b2
--
--
0.180
dependence of the TKE budget on transfers resulting
from the covariance of velocity and pressure fluctua-
tions. We find no similar pressure transport terms
Figure 20 shows not only CHN10 and CEN10 but also
in either the temperature variance budget, eq 39, or the
CDN10 computed from eq 111, which is independent of
humidity variance budget, eq 42. Scalars cannot be
wind speed. Only at very low wind speeds, when the
transported by pressure forces; scalar transfer at a
flow is aerodynamically smooth or in transition, are
surface is always molecular. Consequently, over aero-
CHN10 and CEN10 near CDN10--the Reynolds analogy.
dynamically rough surfaces, at least, the Reynolds
For higher winds, CHN10 and CEN10 are clearly less
analogy is inherently inappropriate (cf., Joffre 1982).
than CDN10. Figure 19 shows the same behavior; zT
Measurements have not yielded conclusive results
on the behavior of the scalar transfer coefficients over
flow gets more aerodynamically rough.
snow-covered surfaces. Hicks and Martin (1972) and
Munro (1989) verified some of the predictions of
Thorpe et al. (1973) measured both CHN10 and CEN10
this model with data collected over a Canadian glacier.
over lake ice and sea ice, respectively. Joffre (1982)
Kondo and Yamazawa's (1986) data collected over
measured CHN10 over snow-covered sea ice. Kondo and
snow-covered ground provide another test of the
Yamazawa (1986) measured CHN1 (1-m value) and
model's prediction of CHN10 (Fig. 20a). Because Kondo
Barry and Munn (1967) measured CEN0.3 (30-cm
and Yamazawa reported only CHN1 and CDN1 (both at
value) over snow-covered ground. The inferred CHN10
and CEN10 values are quite scattered because, over snow,
the potential temperature and humidity gradients nec-
10 1
essary to compute the transfer coefficients (see eq 92
and 93) are generally small and, thus, difficult to meas-
ure precisely. The CHN10 values from the measurements
10 0
are typically about 1103, but CEN10 measurements
do not suggest a consensus value. Andreas (1987) re-
Water Vapor
viewed some of these observations.
10 1
In the absence of definitive experimental work, I de-
vised a model to predict CHN10 and CEN10 (Andreas
1987). My work built on Brutsaert's (1975a, b) surface-
Smooth
Transition
Rough
10 2
renewal model. Its main result is to predict the scalar
roughness, zs, as a function of z0 and R
*
ln(zs / z0 ) = b0 + b1 ln R
10 3
2
+ b2 (ln R )
(113)
*
*
Temperature
where Table 2 gives the polynomial coefficients when
zs is zT or zQ. Figure 19 shows this equation.
10 4
We see from eq 106 and 107 that knowing zT/z0 and
zQ/z0 is essential for predicting CHNr and CENr. But we
also see from these equations that both scalar transfer
10 5
10 2
10 1
10 0
10 1
10 2
10 3
coefficients also depend on CDNr. Hence, to use eq 113
R
in eq 106 and 107 to predict CHN10 and CEN10, I used
*
eq 111 to model CDN10 (Andreas 1987). As a result,
Figure 19. Model predictions of zT/z0 and zQ/z0 over
both scalar transfer coefficients depend on wind speed
snow-covered surfaces (from Andreas 1987). The three
(because zs/z0 depends on R* ) and ξ. Figure 20 shows
regions label where the flow is aerodynamically smooth,
aerodynamically rough, or in transition.
model predictions of CHN10 and CEN10.
22
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