and for the unstable region, the bulk coefficient ratio can be written as

*c*m

= 1 + ln (1 - *aRi*b ) and

7

*c*m,n

*a*

(46a,b)

*c*h

= 1 + ln (1 - *bRi*b ) ,

11

*c*h,n

*b*

where *a *= 0.83 (*c*m,n)0.62 and *b *= 0.25 (*c*m,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 */ ρ*c*p z

*L*

*z * -*n*

*kzu** q

= 1- γ

,

-*E */ ρ *z *

*L*

the ratio of *K*h/*K*m for the unstable case is

*z * 1/ 4

*K*h

= 1- γ

,

(48)

Km

*L*

and for the stable case it is

(

)

*z*

1 + βm

*K*h

*L* .

=

(49)

(

)

*z*

*K*m

1 + βh

*L*

The ratio of *K*w/*K*m depends on the exponent *n*; if *n *= 1/4, then *K*w = *K*m, and if *n *= 1/2, then *K*w/*K*m

= *K*h/*K*m for the unstable case. For the stable case, the ratio of *K*w/*K*m as for *K*h/*K*m can be similarly

written as

(

)

*z*

1 + βm

*K*w

*L* .

=

(50)

(

)

*z*

*K*m

1 + βh

*L*

Therefore, if βm, βh, and βw are much greater than one, then *K*h/*K*m and *K*w/*K*m will approach

2

βh / βm = 11/49, corresponding to the value of *Ri*c at which the turbulent transfers vanish.

Deardorff (1968) has indicated that *K*m > *K*h and *K*w 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 *K*m > *K*h and *K*w.

Furthermore, the fact that the value of *K*h is usually greater than *K*w 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

11