Analytical Description and Vertical Structure of the Atmospheric Boundary Layer - Surface Based Remote Sensing of the Atmospheric Boundary Layer - page 12

where p is the atmospheric pressure, is the atmospheric density, f is the Coriolis

parameter, u is the wind component in x -direction, v is the wind component in
y -direction, z is the vertical coordinate, and K M is the coefficient for vertical tur-
bulent momentum exchange. The subscript “g” denotes the respective components
of the geostrophic wind, theoretically blowing in a perfect balance between pres-
sure gradient force and Coriolis force. The introduction of the geostrophic wind
eliminates the pressure gradient from the equations of motions ( 2.1 ) and ( 2.2 ).

ρ

K M ∂ u ∂ z is a parameterization of the vertical turbulent momentum flux.

2.2.1.1 Prandtl Layer

The Prandtl layer or constant-flux layer is defined 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 paradox because the turbulent vertical fluxes have their largest vertical gra-
dients just at the surface, the concept of the constant-flux layer has proven to be a
powerful tool to describe the properties of this layer. Stipulating a vertically con-
stant momentum flux and assuming that the mean flow is in x -direction simplifies
the equations of motion ( 2.1 ) and ( 2.2 ) further to

K M ∂

u

u 2
∗

z =

const = , (2.3)

∂

where u ∗ is the friction velocity. The friction velocity is proportional to the

geostrophic wind speed and thus represents the large-scale pressure gradient force.
Dynamical considerations, which suggest formulating the exchange coefficient as
being proportional to the mixing length l

z , which in turn is proportional to the
distance to the ground, and the friction velocity ( K M = κ

= κ

z ), lead to the following
equation for the vertical wind gradient in the Prandtl layer (with the van Kármán
constant

u

∗

κ = 0.4):

∂ u u u

l = ∗

z = z .

(2.4)

∂ κ

Integration of eq. ( 2.4 ) from a lower height z 0 where the wind speed vanishes to
a height z within the Prandtl layer yields the well-known logarithmic wind profile
with the roughness length z 0 :

u z

z 0

∗
κ

u ( z ) = ln . (2.5)

Sometimes, instead of eq. ( 2.5 ), an empirical power law is used to describe the
vertical wind profile:

u ( z r ) z
z r

a

u ( z ) = , (2.5a)