#### Transcript Turbulent fluxes

**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**