Winds in Complex Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 82

Fig. 4.5 Depth of the inner layer, l as function of the half-width, L and the surface roughness
length z 0 , found as an iterative solution of ( 4.9b )(left)or( 4.9a )(right). Lowest curve:
z 0 = 0.02 m, second curve: z 0 = 0.1 m, third curve: z 0 = 0.5 m and upper curve: z 0 = 2.5 m

Equation ( 4.10 ) fulfills the non-slip condition at the surface. Hoff ( 1987 ) gives
the following relation which considers also the surface pressure gradient across the
ridge by an additional term:

u ð x ; z\l Þ¼ u 1 ð z Þþ Du ð x ; z\l Þþ du ð x ; z\l Þ ð 4 : 11 Þ

with the pressure gradient-related term:

ln

du ð x ; z\l Þ¼ 1

x
L

z
z 0

ð 4 : 12 Þ

j du

which requires a modified formulation for the friction velocity:

¼

L
z 0

ln

x
L

l
2qu 1

o p
ox ¼ u 1

H
L Dr

x
L

du ð 4 : 13 Þ

l
z 0

ln

The increment Dr in Eq. ( 4.13 ) is the horizontal difference of the form function
r in the range between x/L—D and x/L ? D, where D is supposed to be small
compared to L:

¼ 1
2D

r

x
L

x

L þ D ; z L D ; z

x

Dr r

L ¼ 0 L ¼ 0

ð 4 : 14 Þ

Smooth vertical wind profile functions which cover both the inner and the outer
layer can be formulated as follows (Hoff 1987 ):

P d ð z Þ

P 0 ð z Þþ 1 ln

u ð x ; z Þ¼ u 1 ð z Þþ u 1 ð L Þ H

x

L ; z

x
L

l
z 0 L r j du

ð 4 : 15 Þ

L

with: