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Fig. 3.3 Relation between the geostrophic wind speed, G and the friction velocity,u*using the
simplified geostrophic drag law ( 3.4 ) with A* = 3.8 for onshore and A* = 4.7 for offshore
conditions
order of about 100 km. Unfortunately, Eq. 3.3 is an implicit relation, because the
friction velocity appears on both sides of the equal sign. Therefore, simplifications
of this drag law have been suggested, e.g., by Jensen ( 1978 ). Here we suggest a
similar simplification which has also been used in Emeis and Frandsen ( 1993 ).
Neglecting B and forming a new parameter A* = A - ln C D gives:
u
G ¼
j
ð 3 : 4 Þ
G
ln
fz 0 A
Equation 3.4 can easily be solved for the friction velocity if the modulus of the
geostrophic wind speed, G and the parameter A* are known. Due to the given
choice of A*, the parameter A* depends on stability and on surface roughness.
A*, A and B are empirical parameters which have to be estimated from mea-
surement data. Hess and Garratt ( 2002 ) have listed several estimations. They
suggest, as the best approximation to steady, homogeneous, neutral, barotropic (no
thermal wind) atmospheric conditions that they could find, i.e., the near-neutral,
near-barotropic ABL in middle and high latitudes, to choose A = 1.3 and B = 4.4.
Using these two values, we get A* = 3.7 for a roughness length of 0.1 m (onshore)
and A* = 4.5 for a roughness length of 0.0001 m (offshore). Peña et al. ( 2010a , b )
choose A = 1.7 and B = 5 to be close to the values used by the wind atlas
program WAsP (Troen and Petersen 1989 ). This gives A* = 3.8 for onshore and
A* = 4.7 for offshore conditions. The difference between onshore and offshore
conditions using the simplified drag law ( 3.4 ) is illustrated in Fig. 3.3 .
Please note that the parameters G and f are external parameters in the drag law
( 3.3 ) and its simplification ( 3.4 ). This means, that neither the drag law ( 3.3 ) or its
simplification ( 3.4 ) can be used to compute a roughness length-dependent modulus
of the geostrophic wind speed. As already stated in Sect. 2.3 , the geostrophic wind
solely depends on the large-scale horizontal pressure gradient and the latitude-
dependent Coriolis parameter, but not on surface properties.
The friction velocity obtained from ( 3.2 )or( 3.3 ) is the usual scaling velocity
for the wind speeds and the vertical wind shear in the atmospheric surface layer. In
cases with strong convective vertical motions, the convective velocity scale [see
( 3.20 )] should be used as scaling velocity.
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