NRC DISPERSION MODELING SEMINAR

                        Roland Draxler

                             1994

 

BASIC EQUATIONS

 

 

 

Equivalent to that for heat conduction:

 

The solution, with K equal to atmospheric diffusivity:

 

Since the exponential term is a Gaussian distribution:

 

Therefore after substitution we have the well known form:

 

The functions for Gaussian distributions will be:

 

In other situations, such as continuous emissions:

 

Or when the vertical layer is well mixed:

 

    PARAMETERS REQUIRED TO COMPUTE DISPERSION AND TRANSPORT

 

                       Emission Rate - Q

 

                    Wind Speed (vector) - U

 

                       Mixing Depth - Z_i

 

                Dispersion Parameters - σ_i or K_i

 

 

 

 

 

 

                     DISPERSION PARAMETERS

 

 

Eddy Diffusivity

 

 

                              or

 

 

 

Horizontal diffusivity is very difficult to obtain exactly.

 

 

Statistical Theory (Taylor's):

 

 

 

The functions f_x and f_y are usually obtained empirically by relating dispersion experiment results and measured turbulence values. The functions can also be obtained theoretically from the "spectrum" of the horizontal and vertical turbulence.

 

 

 

                  TURBULENCE CHARACTERIZATION

 

 

PASQUILL -  Measured wind fluctuations

 

PASQUILL/GIFFORD - Pre-determined curves and

corresponding categories (solar elev, cloud, U)

 

SLADE -  Categories from wind direction fluctuation

 

NRC - Categories from temperature difference

 

GOLDER - Categories from roughness and Monin-Obukhov L

 

 

 

 

Category  Pasquill  Slade     NRC (deg C/100m)

 

A         60        25        -1.9

B         45        20        -1.7

C         30        15        -1.5

D         20        10        -0.5

E         15        5        +1.5

F         10        2.5      +4.0

 

 

 

                         MIXING DEPTH

 

 

 

 

Turbulent mixing during the daytime reflects the amount of sensible heat that is absorbed by the atmosphere at the earth's surface.

 

It is a function of the solar insolation, cloud cover, wind speed, and soil moisture.

 

It can be computed from the intersection of the morning temperature sounding and the maximum surface temperature (Holzworth).

 

At night the mixing depth represents the height to which mechanical/roughness induced turbulence penetrates the stability induced by surface radiational cooling.

 

 

 

 

                      WINDS FOR ADVECTION

 

 

In the vertical:

 

 

 

In the horizontal:

 

 

 

 

 

The effect of the vertical and horizontal "shears" of the wind is that depending upon the degree of vertical mixing a single wind may not adequately represent pollutant transport.

 

 

 

          MODELING POLLUTANT TRANSPORT AND DISPERSION

 

 

Lagrangian and Eulerian Models (from eq 1):

 

 

We expand the total derivative so that:

 

 

Gaussian Models tend to be Lagrangian Models, however Eulerian models may or may not be Gaussian, depending upon the turbulence closure scheme selected.

 

Some other modeling terms:

 

CONTINUOUS GAUSSIAN PLUME

 

SEGMENTED GAUSSIAN PLUME

 

GAUSSIAN PUFF

 

NON-GAUSSIAN PUFF (multi-layer modeling approach)

 

PARTICLE-IN-CELL MODEL

 

 

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