Vertical Profiles Over Flat Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 30

H H 0 w 0 þ 0 : 61gq 0 w 0 . The ratio of these two contributing terms is called the
buoyancy ratio, BR which is inversely proportional to the Bowen ratio B:

g

BR ¼ 0 : 61Hq 0 w 0
H 0 w 0

¼ 0 : 61c p H
L v

1
B :

ð 3 : 14 Þ

This unstable type of the surface layer is usually found during daytime over
surfaces heated by insolation and over waters which are warmer than the air above.
The degree of instability is described by the non-dimensional ratio of the height
z over the Obukhov length L * . In an unstable surface layer, warm air bubbles rise
from the surface to the top of the unstable layer, which is usually marked by a
temperature inversion. The height of the unstable surface layer is designated by z i .
The temperature decreases with height according to the adiabatic lapse rate, but in
a shallow super-adiabatic layer near the surface the lapse rate is even stronger.
While z is the only scaling length scale in a neutrally stratifies surface layer, the
modulus of the Obukhov length L * [see ( 3.11 )] is an additional length scale in
the unstable surface layer. A non-dimensional parameter can be formed from these
two length scales. The ratio z/L * is used in the following as a stability parameter.
This parameter is negative for unstable stratification, positive for stable stratifi-
cation and zero for neutral stratification.
For small negative values of z/L * the vertical wind profiles in the surface layer
can be described by introducing a correction function W m (z/L * ) (Paulson 1970 ;
Högström 1988 ):

1 þ x 2
2

1 þ x
2

2arctg ð x Þþ p
2

W m ¼ 2 ln

þ ln ð 3 : 15 Þ

where x = (1-b z/L * ) 1/4 and b = 16. This leads to the following description of the
vertical wind profile which replaces Eq. ( 3.6 ):

u ð z Þ¼ u = j ln ð z = z 0 Þ W m ð z = L Þ ð Þ ð 3 : 16 Þ

While the surface layer mean wind profile in the unstable surface layer depends
on the local stability parameter z/L * , turbulence partly depends on non-local
parameters as well. The non-local parameter boundary layer height z i (see also
Appendix B) is another length scale in the unstable surface layer, if the thermally
induced vertical motions extend through the whole depth of the convective
boundary layer. This allows for the formulation of a second non-dimensional
stability parameter, z i /L * . The standard deviations of the 10 Hz fluctuations of the
wind components in the unstably stratified Prandtl layer depend either on this
second parameter z i /L * or the first parameter z/L * (Panofsky et al. 1977 ; Arya
1995 ):

1 = 3 1 = 3

r u
u ¼ 15 : 625 0 : 5

z i
L

; r v

z i
L

u ¼ 6 : 859 0 : 5

ð 3 : 17 Þ