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Stratosphere
Troposphere
1 km
Planetary Boundary Layer
10 m
Roughness Layer
height
Surface Layer
0.1m
The Atmospheric (or Planetary) Boundary Layer is formed as a
consequence of the interactions between the atmosphere and the
underlying surface over time scales of a few hours to about one
day (Arya)
Surface Layer. About 1/10 of PBL. Intense small scale turbulence
generated by surface roughness and convection. Effects of Earth
Rotation negligible. Significant exchanges of momentum, heat,
mass. Layer in which human beings, animal and vegetation live.
(Arya and Oke)
Roughness Layer. Flow highly irregular strongly affected by the
nature of the individual roughness features (grass, trees,
buildings). (Oke)
Neutral surface boundary Layer
Neglecting Buoyancy forces.
Turbulent case (High Reynolds number)
At the surface there is a sink of momentum. The
momentum flux density (or the force per unit
area) is called surface shear stress (t). (Units
N/m2, or Pa).
The sink of momentum at the surface can also
be seen as an horizontal force exerted by the
flow on the surface (drag force) in the
direction of the mean flow. The opposing
force exerted by the surface on the fluid
retards the flow.
The shear stress is generated within the lowest
layers and transmitted downwards as a vertical
flux of horizontal momentum.
Since velocity is zero at the surface (no-zero
boundary condition), there is a vertical gradient
of momentum.
Computation of the vertical wind profile for
a horizontally homogeneous neutral surface
boundary layer, by similarity arguments
(Landau, 1944).

 0   u, v, p,  ....
x
z
Lev Landau,
1908-1968,
Russia
x
The flow is parallel to the surface
For turbulent flows, (very large Reynolds numbers),
we suppose that the vertical gradient of horizontal
mean wind will not explicitly involve fluid viscosity.
Re  1

t
z
The value of the velocity gradient (dU/dx) at any
distance from the surface must be determined by
the parameters t (surface shear stress),  (air
density) and the distance itself (z).
or,
U is the MEAN horizontal wind speed
U
 f (t ,  , z 
z
The simplest possibility is
U
 t z 
z
How to determine ,, ?
Dimensional analysis.
U  LT 1   1
 T


z  L  


t   ML1T  2 


   ML3 


z  L
Fundamental dimensions :
M  Mass
L  Length
T  Time


T 1   ML1T 2   ML3  L

 
 

 1  2

0    
0    3  


1
1
,   ,   1
2
2
As a consequence
U
t1

z
 z
t
2
if u* 

U u*

z
z
friction velocity
Integrating
u*
z
U( z ) 
ln
k
zo
k  0.4 von Karman constant
U(zo )  0
Similarly, it is possible to show that
turbulent fluxes are constant in the surface
layer.
Below zo viscous (or
laminar) sublayer
zo
Viscosity is important, Re of the
order of unity
An alternative way to derive
wind vertical profile in the
neutral layer. Mixing length
approach (Prandtl, 1925)
zl
z
Ludwig Prandtl
(1875-1953),
Germany
U(z  l )
U(z)
The downward flux of momentum is
accomplished by random vertical
motions.
Molecular analogy. Eddies break away from the
main body of the fluid and travel a certain
distance, called the mixing length (analogous to
the mean free path in kinetic theory of gases)
zl
z
U(z  l )
U(z)
If a “blob” of fluid moves from level z+l to level z, it
brings its momentum.
So , if
u(z)  istantaneus velocity at level z
U(z)  mean velocity at level z
u(z)  fluctuation
u(z)  U(z  l)  u(z)  U(z  l) - U(z)  l
U
z z
The turbulent vertical flux of momentum is
zl
U(z  l )
z
U(z)
U
uw   w l
z z
Assuming that in a turbulent flow velocity fluctuations
are of the same order in all the directions
w   u  l
U
z z
Negative since we’re
considering the case of
a fluid moving from z+l
to z
 U
uw   l
 z




z
2
In a surface layer turbulent fluxes are (nearly)
constants with height.
Defining the friction velocity as:
uw    u* 2
We have
U u*

z
l
Close to the ground the size of the eddies
is influenced by the distance from the
ground
l  kz
U u*

z kz
integrating
u*
z
U( z ) 
ln
k
zo
It works !
Wind speed
Moreover, atmospheric measurements show
that the variations of turbulent fluxes in the
surface layer are less than 10% (nearly
constant with height)