76
JOOST A. RUSINGFR AND 9. P. S. ARYA
/I is a constant about 4.7 f 0.5; i ZE z / L ; L 3 - B ~ ~ / k g ( w ' O ' ) ~ ,
the Obukhov
length. The subscript zero refers to the value at the surface, and k is von
Kiirrntin's constant. When Eqs. (3) and (4) are integrated, the well-known
log-linear profile is obtained.
A simple similarity argument may bc given which leads to the above
equations. Near the surface, the height above thc surface is the appropriate
scaling length, but when z > L, the appropriate scaling length is L because
the buoyancy inhibits vertical excursions over distances larger than L. So,
for z > L, we have &/Jz cc u,,/L, which kads to a linear profile.
It is not a priori obvious that L should be chosen as the scaling length.
However, a simple argument to strengthen this notion may be given: A
parcel of air with a vertical velocity w' will travel a distance f' in the vertical
until its kinetic energy has been converted into potential energy. Thus we
have
In the stable surface layer
the Brunt-Vaissallii frequency. Therefore
(7)
Furthermore,
(8)
aojaz = -?@/K,
and since
eddy thert 1 diffusivity
as the characteristia of a mixing length, we may assume that the
(9)
Kh r~ u,l
Now substitute (9). (8). and (6) into (7), we obtain
A consequence of Eqs. (3a) and (3b) is that for large values of 4, Ri
approaches a constant
( 1 1 )
Ri -+ 1//3
T h i s constant is presumably the critical Ri number (see Obukhov, 1946;
Webb, 1970; Monin and Yaglom, 1971), which means that the profile behavior in the surface layer predicts an asymptotic approach to Ri,, and not a
transition from turbulent to laminar flow. This is a consequence of the