Pasquill–Gifford Dispersion Coefficient

The Gaussian plume model needs a width, and this power law supplies it: $\sigma = a x^{b}$, one pair of coefficients for the crosswind spread and another for the vertical, each pair read from the stability class. With a vertical fit of $a = 0.113$ and $b = 0.911$, a receptor 1000 m downwind sees $\sigma_z = 0.113 \times 1000^{0.911} = 0.113 \times 540.8 = 61.1$ m. Exponents cluster near 0.9, which is a real physical statement: plume width grows very nearly in proportion to distance, so a plume is close to a cone, and the ratio of width to distance is roughly fixed for a given stability.

These curves come from a specific field campaign and it is worth knowing which one. The Prairie Grass experiment, run in Nebraska in 1956, released tracer near the ground over flat open grassland and sampled arcs out to 800 m. The Pasquill classes A through F were built on top of that and a few similar sets, sorted by surface wind speed, daytime insolation and night-time cloud cover. Gifford turned them into the curves that carry both names, and Briggs later published smooth interpolation formulas for rural and urban terrain that most software uses today. Everything past 800 m is extrapolation, and the site was flat, unobstructed, and open. If your problem is a valley, a shoreline, a refinery, or a downtown, the curves are being asked a question they were never posed.

Averaging time is the part that gets forgotten. The crosswind coefficient was fitted to samples of roughly 10 minutes, and over a longer period the wind direction wanders, so the plume sweeps a wider arc and the average concentration on the centreline drops. The usual scaling is $\sigma_y \propto t^{0.2}$, so going from 10 minutes to an hour widens the plume by $(60/10)^{0.2} = 1.43$, and centreline concentration falls by about 30 percent. Quoting a 10 minute coefficient against a one hour standard, or the reverse, is an error of that size and it is invisible in the arithmetic. The vertical coefficient is much less sensitive to averaging time, because vertical meander is bounded by the ground and the inversion in a way horizontal meander is not.

Three limits, then. Below about 100 m these curves are unreliable and near-field concentrations need a building-downwash treatment instead. Beyond about 10 km the steady-wind assumption behind the whole model has usually failed anyway, so precision in the coefficients is misplaced. And the vertical coefficient has a lid: once $\sigma_z$ reaches roughly 0.8 times the mixing height the plume has filled the mixed layer, further growth is impossible, and the Gaussian vertical term must be replaced by uniform mixing, $C = Q/(\sqrt{2\pi}\,\sigma_y u L)$ with L the mixing depth. Applying an unbounded power law past that point predicts a plume politely dispersing into the stratosphere.