0.1m

10 m

1 km

Roughness Layer

Surface Layer

Planetary Boundary Layer

Troposphere

Stratosphereheight

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)

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.

At the surface there is a sink of momentum. The momentum flux density (or the force per unit area) is called surface shear stress (). (Units N/m2, or Pa).

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).

....p,v,u, x

0z

x

The flow is parallel to the surface

Lev Landau, 1908-1968, Russia

For turbulent flows, (very large Reynolds numbers), we suppose that the vertical gradient of horizontal mean wind will not explicitly involve fluid viscosity.

The value of the velocity gradient (dU/dx) at any distance from the surface must be determined by the parameters surface shear stress), (air density) and the distance itself (z).

1Re

z

or,

z,,fz

U

speed windhorizontal MEANthe is U

The simplest possibility is

zz

U

How to determine ?Dimensional analysis.

Lz

ML

TML

TL

LT

z

U

3

21

11

TimeT

Length L

Mass M

:dimensions lFundamenta

LMLTMLT 3211

30

0

21

12

1

2

1 ,,

zz

U 1

As a consequence

z*u

z

U

2*u if

friction velocity

Integrating

oz

zln

k*u

)z(U

0

40

)oU(z

constant Karman von .k

Similarly, it is possible to show that turbulent fluxes are constant in the surface layer.

oz

Below zo viscous (or laminar) sublayer

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)

lz

z U(z)

)lU(z

Ludwig Prandtl (1875-1953), Germany

The downward flux of momentum is accomplished by random vertical motions.

lz

z U(z)

)lU(z

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)

If a “blob” of fluid moves from level z+l to level z,

it brings its momentum.

zz

UlU(z)-l)U(z(z)ul)U(zu(z)

nfluctuatio(z)u

z level atvelocity meanU(z)

z level atvelocity sistantaneuu(z)

if ,So

The turbulent vertical flux of momentum is

lz

z U(z)

)lU(z

zz

Ulwwu

Assuming that in a turbulent flow velocity fluctuations are of the same order in all the directions

zz

Uluw

Negative since we’re considering the case of a fluid moving from z+l to z

2

zz

Ulwu

In a surface layer turbulent fluxes are (nearly) constants with height.

2*uwu

l*u

z

U

We have

Defining the friction velocity as:

Close to the ground the size of the eddies is influenced by the distance from the ground

kzl

kz*u

z

U

integrating

oz

zln

k*u

)z(U

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)