Wind Energy

28 J. Tambke et al.
where u∗ is the friction velocity of the air flow and w∗ the one of the water
flow while ρair and ρwater are the respective densities. This is equivalent
to assuming that the shear stress is constant across the interface between air
and water. The constant stress defines the wave boundary layer, extending
from a height zB above to −zB below the water level. This wave boundary
layer is similar to the surface layer onshore where the logarithmic wind profile
is valid.
Second, the inertial coupling relation [1] represents the shear stress in the
wave boundary layer in the form of a drag law with regard to the inertially
weighted fluid speeds at the limits of this layer, where KI is a specific drag
coefficient: τwave = KI[ √ ρair u(zB) − √ ρwater u(−zB)] 2 .
Third, analogous to the first similarity relation the turbulent viscosities
νair and νwater of the two Ekman layers are also assumed to be weighted
according to the inverse density ratio of the two fluids: νwater
νair
= ρair
ρwater .
These conditions allow the determination of the velocities of the two fluids
at certain heights as an implicit function of a given geostrophic wind. The
full profiles can be derived in the following way: Due to the constant shear
stress the wind profile in the wave boundary layer has a logarithmic shape for
neutral thermal stratification. Non-neutral conditions can be modelled with
an additional stability function Ψm and the Monin–Obukhov (MO) length
L. Expressed in a coordinate system where the horizontal component of the
stress tensor τ h is parallel to the x-axis the profile can be written as
�
u(z) = uL + u∗
κ ln
� �
z
+ u∗
κ Ψm
�
z
� �
,vL , (5.1)
L
zR
and is valid for zR ≤ z ≤ zB; κ is the von Karman constant and zR is the
theoretical height where the momentum transfer from the air to the wave
field is centred. The above mentioned drift velocities are uair = u(zR) =uL
and uwater = u(−zR) ≈ 1
29uL. As a result of the above assumptions and the
choice of the coordinate system the drift (uL,vL) at the very small height
zR is collinear to the geostrophic wind: (uL,vL) = 1
2
G, where G =(ug,vg).
This theoretical, though curious relation has no relevance in-situ, since neither
(uL,vL) close to the waves nor G can be measured directly. Note that in the
chosen coordinate system ug > 0andvg < 0.
Concluding from [2], the relation that connects the friction velocity u∗ at
the water surface to the geostrophic wind is given by
|G| =
√ r 2 +1
|r +1|
u∗
√ . (5.2)
KI
The angle of rotation of the surface stress tensor to the left hand side
of the geostrophic velocity (in the northern hemisphere) is atan(-1/r) with
−1