Vertical Profiles Over Flat Terrain - Wind Energy Meteorology: Atmospheric Physics for Wind Power Generation - page 25

The Prandtl layer or surface layer or constant-flux layer is defined meteoro-
logically as that layer where the turbulent vertical fluxes of momentum, heat, and
moisture deviate less than 10 % from their surface values, and where the influence
of the Coriolis force is negligible. Usually this layer covers only 10 % of the whole
ABL depth. Although this definition seems to be a paradox because the turbulent
vertical fluxes have their largest vertical gradients just at the surface, the concept
of the constant-flux layer has proven to be a powerful tool to describe the prop-
erties of this layer.
We start to derive the basic wind equations for this layer by stipulating a
vertically constant momentum flux, i.e. assuming a stationary mean flow in x-
direction and horizontal homogeneity [no derivatives neither in wind (x) nor in
cross-wind (y) direction]. This simplifies the equations of motion ( 2.2 )-( 2.4 ) to:

K M o u

oz ¼ const ¼ u 2

ð 3 : 1 Þ

where u * is the friction velocity defined in ( 3.2 ) and K M is the vertical turbulent
exchange coefficient for momentum, which has the effect and the physical
dimension of a viscosity. K M appears when replacing F x in term VII of Eq. ( 2.2 )
using F x = q/qz (K M qu/qz). A specification of K M for neutral stratification is given
at the beginning of Sect. 3.1.1.1 and for non-neutral stratification in Eq. ( 6.9 ). The
friction velocity can be estimated from measured logarithmic wind profiles by
inversion of Eq. 3.4 or can be derived from high-resolution wind fluctuation
measurements with a sonic anemometer in the Prandtl layer:

4

u ¼ u 0 w 0 2 þ v 0 w 0 2

ð 3 : 2 Þ

where u 0 denotes the 10 Hz turbulent fluctuation of the West-East wind compo-

nent, v 0 the fluctuation of the South-North component, and w 0

the fluctuation of the

vertical component.
In cases where high-resolution turbulent fluctuation measurements and wind
profile data are unavailable, the friction velocity can also be inferred from the
geostrophic drag law which relates the friction velocity u * with the modulus, G of
the geostrophic wind speed [see Eqs. ( 2.5 ) and ( 2.6 )] that represents the large-scale
pressure gradient force. The geostrophic drag law reads (Zilitinkevich 1975 ):
C D ¼ u

j
ln u

j
ln

G ¼

r ¼ r ð 3 : 3 Þ

2 þ B 2 2 þ B 2

G

fz 0 A fz 0 þ ln C D A

where C D is the geostrophic drag coefficient, z 0 is the roughness length of the
surface introduced in Eq. ( 3.6 ) and A and B are two empirical parameters which
principally depend on the thermal stability of the atmosphere (see Zilitinkevich
1975 ; Hess and Garratt 2002 or Peña et al. 2010b for details). The friction velocity
computed from ( 3.3 ) is a large-scale averaged friction velocity, because the geo-
strophic wind speed is a large-scale feature representing a horizontal scale of the