and for the unstable region, the bulk coefficient ratio can be written as
cm
= 1 + ln (1 - aRib ) and
7
cm,n
a
(46a,b)
ch
= 1 + ln (1 - bRib ) ,
11
ch,n
b
where a = 0.83 (cm,n)0.62 and b = 0.25 (cm,n)0.80.
Based on the eddy transfer coefficients defined in eq 8, 18, and 19 and the unstable wind,
temperature, and specific humidity profiles by
-1 / 4
kz u
= 1- γ
z
,
u* z
L
-1 / 2
T
= 1- γ
kzu*
z
, and
(47a,b,c)
- H / ρcp z
L
z -n
kzu* q
= 1- γ
,
-E / ρ z
L
the ratio of Kh/Km for the unstable case is
z 1/ 4
Kh
= 1- γ
,
(48)
Km
L
and for the stable case it is
(
)
z
1 + βm
Kh
L .
=
(49)
(
)
z
Km
1 + βh
L
The ratio of Kw/Km depends on the exponent n; if n = 1/4, then Kw = Km, and if n = 1/2, then Kw/Km
= Kh/Km for the unstable case. For the stable case, the ratio of Kw/Km as for Kh/Km can be similarly
written as
(
)
z
1 + βm
Kw
L .
=
(50)
(
)
z
Km
1 + βh
L
Therefore, if βm, βh, and βw are much greater than one, then Kh/Km and Kw/Km will approach
2
βh / βm = 11/49, corresponding to the value of Ric at which the turbulent transfers vanish.
Deardorff (1968) has indicated that Km > Kh and Kw in the stable case due to the importance of
pressure forces in diffusing momentum. If eddies behave somewhat like internal gravity waves, the
effect of mixing by molecular processes alone would be rather insignificant, thus Km > Kh and Kw.
Furthermore, the fact that the value of Kh is usually greater than Kw can probably be attributed to the
damping of thermal fluctuations by radiative transfer in all directions, and no such mechanism is
available to add the mixing of an eddy's excessive water vapor with the surrounding unsaturated air.
C. Empirical wind functions
The turbulent vapor transfer is predominantly a function of wind speed and the vapor pressure
gradient and can be expressed in terms of Dalton's relation as
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