### Turbulent fluxes

```Turbulent
fluxes
Vertical turbulent flux
of pollutant mass
Turbulent stress
Vertical fluxes of
momentum, heat, water
vapor, and pollutant
Km, Kh, and Kz are called the eddy diffusivities or exchange
coefficients of momentum, heat, and mss, respectively. (Km
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 nearGaussian eparameterize it by analogy with molecular diffusion:
Instantaneous
plume
Time-averaged
envelope
z
Near-Gaussian
profile
Source
Turbulent flux =  K z na
 C 
z
<C>
Turbulent diffusion
coefficient
• Typical values of Kz: 102 cm2s-1 (very stable) to 107 cm2 s-1 (very unstable);
mean value for troposphere is ~ 105 cm2 s-1
• Same parameterization (with different Kx, Ky) 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 Dt to travel Dz by turbulent diffusion:
Dz 

Dt 
2
2K z
with K z
105 cm2s-1
tropopause
(10 km)
10 years
5 km
“planetary 2 km
boundary layer”
0 km
1 month
1 week
1 day
```