Vertical Profiles Over Flat Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 33

Table 3.1 Typical profile law parameters for vertical wind profiles in the ABL: roughness length
z 0 , power law (Hellmann) exponent a [neutral thermal stratification, see ( 3.22 )], friction velocity
u * (neutral stratification, 10 m/s geostrophic wind) and deviation angle from the geostrophic wind
direction u. The values should be regarded as estimates only
Surface type z 0 [m] a u* [m/s] u [degree]

Water 0.001 0.11 0.2 15-25

Grass 0.01-0.05 0.16 0.3

Shrubs 0.1-0.2 0.20 0.35 25-40

Forest 0.5 0.28 0.4

Cities 1-2 0.40 0.45

Megacities 5

Mountains 1-5 0.45 35-45

From Emeis ( 2001 )

For the investigation of the possibility whether the profile laws ( 3.6 ) and ( 3.22 )
can describe the same wind profile over a larger height range, we need the
mathematical formulation of the slope and the curvature of the wind profiles
expressed by ( 3.6 ) and ( 3.22 ). The slope of the logarithmic wind profile under
neutral stratification is given by the first derivative of ( 3.6 ) with respect to the
vertical coordinate z:

u ð z Þ
z

o u

oz ¼ 1

u z

z 0 z ¼ ln 1

ð 3 : 23 Þ

j

and the curvature of the logarithmic profile follows by taking the second derivative
of ( 3.6 ) with respect to the vertical coordinate:

u ð z Þ
z 2

o 2 u

oz 2 ¼ 1

u
z 2 ¼ ln 1

z
z 0

ð 3 : 24 Þ

j

The slope of the power law by differentiating ( 3.22 ) with respect to the vertical
coordinate yields:

a 1 a

¼ a u ð z Þ
z oz ¼ u ð z r Þ

o u z

z r

z
z r

z
z r

a ¼ au ð z r Þ ð 3 : 25 Þ

z r

and the curvature of the power laws reads after computing the second derivative of
( 3.22 ) with respect to the vertical coordinate:

a 1

o 2 u
oz 2 ¼ a ð a 1 Þ u ð z r Þ

z
z r z 2 ¼ a ð a 1 Þ u ð z Þ

ð 3 : 26 Þ

z 2

Equating the slopes of the logarithmic profile ( 3.23 ) and that of the power law
( 3.25 ) delivers a relation between the Hellmann exponent and the surface rough-
ness length: