Turbulent fluxes

Vertical turbulent flux of pollutant mass

Turbulent stress Vertical fluxes of momentum, heat, water vapor, and pollutant

Gradient-transport (K-theory)

K m , K h , and K z

coefficients

are called the

eddy diffusivities

or

exchange

of momentum, heat, and mss, respectively. (K m is also called eddy or turbulent viscosity).

The gradient-transport (K-theory) relations are not based on any rigorous theory, but only on an intuitive analogy between molecular and turbulent exchange processes.

Prandtl’s mixing length theory From dimensional analysis, we recognizing that eddy diffusivity must be a product of appropriate length and velocity scales. In the surface layer (constant flux layer) k= 0.40 is the

von Karman constant

is

the friction velocity

that is related to

surface stress

Finally, we have

In the surface layer, under neutral condition, we have: We can define a surface roughness parameter such that at

, .

Therefore, we can obtain the well-known

logarithmic velocity profile

law, that is

FIRST-ORDER PARAMETERIZATION OF TURBULENT FLUX

•

Observed mean turbulent dispersion of pollutants is near Gaussian

e

parameterize it by analogy with molecular diffusion: Instantaneous plume Time-averaged envelope

z

Near-Gaussian profile Source

Turbulent flux = 

K z n a

 

C



z

Turbulent diffusion coefficient



•

Typical values of

K z

: 10 2 cm 2 s -1 (very stable) to 10 mean value for troposphere is ~ 10 5 cm 2 s -1 7 cm 2 s -1 (very unstable);

•

Same parameterization (with different

K x , K y

) is also applicable in horizontal direction but is less important (mean winds are stronger)

Mass conservation and diffusion equation

If

U

=0, the diffusion equation can be simplified to For an instantaneous point source, the solution of the above equation is Q is the total mass of pollutant in the puff

TYPICAL TIME SCALES FOR VERTICAL MIXING

•

Estimate time

D

t

to travel

D

z

by turbulent diffusion:

  2

with

K z

2

K z

2 -1

tropopause (10 km) 10 years “planetary 2 km boundary layer” 0 km 5 km 1 week 1 month 1 day