Diffusion Model

Turbulent mixing results in the dispersion (diffusion) of pollutants throughout the volume. We have treated vertical dispersion by assuming that pollutants mix uniformly only up to the boundary layer. We also introduce a small leakage into the boundary layer over the entire region.

Horizontal dispersion occurs across the vertical boundary between two boxes when there is a difference in pollutant mass between two boxes. It tends to even out the difference. The amount of matter transferred is (1) directly proportional to the rate of change of pollutant mass with distance, (2) proportional to the time elapsed and (3) inversely proportional to the distance traveled. Consider two boxes $x,y$ and $x+a,y$. The changes in mass due to matter transferred across the boundary between these boxes are then given by

$$\Delta m(x,y,t) = K_h [m(x+a,y,t) - m(x,y,t)]dt/a^2.$$ (6)

$$\Delta m(x+a,y,t) = -K_h [m(x+a,y,t) - m(x,y,t)]dt/a^2.$$

The positive constant $K_h$ is the horizontal diffusion constant. The $a^2$ comes from a factor of $a$ for the gradient and another factor of $a$ for the distance traveled.

We require

$$4 K_h dt/a^2 < 1 \nonumber$$

or the pollutant mass could go negative. In this case the smaller this quantity, the better.

Diffusion in the negative $x$ direction follows the same rule. If we apply the algorithm above to each cell $x,h$, the second equation already takes into account backward diffusion.

A similar set of equations applies in the $y$ direction:

$$\Delta m(x,y,t) = K_h [m(x,y+a,t) - m(x,y,t)]dt/a^2.$$ (7)

$$\Delta m(xa,y+a,t) = -K_h [m(x,y+a,t) - m(x,y,t)]dt/a^2.$$

All contributions must be summed up to get the net change in each cell due to diffusion in all four directions. (In the limit of small $a$ and $dt$ the combined equations reduce to the diffusion equation $\partial m/\partial t = K_h \nabla^2 m$.)