## 4. Continuous Point Sources of Pollutant

Consider a point source at emitting pollutant continuously
in time. The pollutant will be carried down stream by the wind and will
disperse by turbulent diffusion. Suppose that the wind has speed in the
direction. The situation is sketched in Fig. 8.

**Fig. 8**. A plan view of a plume of pollutant from a
continuous point source S.

Consider a slice of air 1 m thick moving in the -direction, extending
to infinity in the and directions and moving with the mean wind .
The time taken for the slice to pass a fixed point is seconds. If the
source emits kgs of pollutant, then the amount in the slice is
.
The pollutant diffuses in the , and directions. But since
the source is continuous, about the same amount of pollutant diffuses into
the sheet in the -direction as diffuses out through the opposite side.
Diffusion in the -direction therefore has negligible effect. The diffusion
problem reduces to that of 2-dimensional diffusion in the , plane but
in a frame of reference moving in the -direction with speed . The
2-dimensional diffusion solution Eq. (22) therefore applies, but with .
In other words

This is the *Gaussian plume equation* which is the basis of much dispersion
modelling. Note that following the flow, so that we expect
and to behave like

In fact this is not the case because
and
are not constant.

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