Vertical Profiles Over Flat Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 44

the wind speed as well as the wind shear are continuous at the height z = z p
(Emeis et al. 2007 b). Equating the first two equations of the wind profile Eq. ( 3.55 )
for z = z p gives an equation for the friction velocity:

u ¼ ju g ð sin a 0 þ cos a 0 Þ
ln ð z p = z 0 Þ

ð 3 : 56 Þ

and from equating the respective equations for the vertical wind shear at the same
height z = z p we get a second equation for u * :

cjz p sin a 0

u ¼ 2 u g ð 3 : 57 Þ

These two equations must be valid simultaneously. Equating the right hand
sides of these two Eqs. ( 3.56 ) and ( 3.57 ) yields the desired relation for the turning
angle, a 0 :

1
1 þ 2cz p ln ð z p = z 0 Þ

a 0 ¼ arctg ð 3 : 58 Þ

Unfortunately, Eq. ( 3.58 ) still depends on the friction velocity u * via the def-
inition of c: For the height z = z p we have from the definition of the inverse length
scale, c ( 3.51 ):

s

f
2ju z p

c ¼ ð 3 : 59 Þ

Thus, the friction velocity u * has to be determined iteratively starting with a
first guess for u * in ( 3.59 ), subsequently computing a 0 from ( 3.58 ), and then re-
computing u * from ( 3.56 )or( 3.57 ).
Inversely the system of Eqs. ( 3.56 )-( 3.59 ) can be used to determine the height
of the Prandtl layer, z p if the friction velocity, u * is known from other sources.
The second idea is to modify the dependence of the mixing length on height and
has been proposed by Gryning et al. ( 2007 ). They reformulated the height-
dependent mixing length l (which is denoted jz = L L in the Prandtl layer in the
following equations) in order to limit its growth with height and thus to extend the
validity of the logarithmic law ( 3.6 ) to above the surface layer. They have chosen:

1

l ¼ 1

þ 1
L M

þ 1
L U

ð 3 : 60 Þ

L L

A modified mixing length is formed in ( 3.60 ) by introducing a length scale for
the middle part of the boundary layer, L M = u * /f (-2 ln(u * /(fz 0 )) ? 55 -1 and a
length scale for the upper part of the boundary layer, L U = (z i - z). This results in
the following wind profile alternative to ( 3.6 )or( 3.55 ):

u ð z Þ¼ u
j

z
z 0

þ z
L M

z
z i

z
2L M

ln ð 3 : 61 Þ