Vertical Profiles Over Flat Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 48

frequency dissipation range. Teunissen ( 1980 ) suggests a modification of these
formulae for rougher terrain. He puts:

nS u ð n Þ
u 2

f
c u þ 33f

¼ 105 ð 3 : 78 Þ

Þ 5 = 3 ð

nS v ð n Þ
u 2

f
c v þ 9 : 5f

¼ 17 ð 3 : 79 Þ

Þ 5 = 3

ð

nS w ð n Þ
u 2

f
c w þ 5 : 3f 5 = 3

¼ 2 ð 3 : 80 Þ

ð Þ

with c u = c w = 0.44 and c v = 0.38 for agricultural flat terrain. The values for c u ,
c v and c w less than unity lead to an increase of the spectral density in the low-
frequency range. Alternatively, the von Kármán formulation of the spectra can be
used (Teunissen 1980 ) which does not depend on a determination of the friction
velocity but rather on the variances of the velocity components and three turbulent
length scales.

4kL u
1 þ 70 : 7 ð kL u Þ 2

nS u ð n Þ
r u

¼ ð 3 : 81 Þ

5 = 6

1 þ 188 : 4 ð 2kL v Þ 2
1 þ 70 : 7 ð 2kL v Þ 2

nS v ð n Þ
r v

¼ 4kL v ð 3 : 82 Þ

11 = 6

1 þ 188 : 4 ð 2kL w Þ 2
1 þ 70 : 7 ð 2kL w Þ 2

nS w ð n Þ
r w

¼ 4kL w

ð 3 : 83 Þ

11 = 6

where k = n/U and L x , L x , L x are ''free'' scaling parameters which can be chosen
to match the data. Teunissen ( 1980 ) gives

L u ¼ 0 : 146 = k u ; L v ¼ 0 : 106 = k v ; L w ¼ 0 : 106 = k w ð 3 : 84 Þ

where the k i p are the wave numbers of the peaks in the spectrum.
The spectra look different in non-neutral conditions. Kaimal et al. ( 1972 ) give
for high frequencies f [ 4:

nS u ð n Þ
u 2 u 2 = 3

Þ 2 = 3 f 2 = 3 ¼ a 1 2pj ð ð 3 : 85 Þ

e

nS v ð n Þ
u 2 u 2 = 3

¼ nS w ð n Þ
u 2 u 2 = 3

¼ 0 : 4f 2 = 3 ð 3 : 86 Þ

e e