Gaussian Plume Ground-Level Concentration

This is the equation regulatory air quality is built on. Take an emission rate, spread it into a normal distribution in both crosswind directions, carry it downwind at the wind speed, and evaluate it at the ground on the plume centreline. A source emitting 100 g/s into a 5 m/s wind from an effective height of 100 m, at a distance where $\sigma_y = 200$ m and $\sigma_z = 100$ m, gives $\pi \sigma_y \sigma_z u = 314159$, so $Q/(\pi\sigma_y\sigma_z u) = 0.1/314159 = 3.18\times10^{-7}$ kg/m³, and the exponential $e^{-100^2/(2\times100^2)} = e^{-0.5} = 0.607$ brings it to $1.93\times10^{-7}$ kg/m³, which is 193 micrograms per cubic metre.

The $\pi$ in the denominator hides something, and readers who have seen the full form notice it immediately. The complete equation carries $2\pi$ and a bracket with two exponential terms, one for the real plume and one for an image plume reflected in the ground, since pollutant that reaches the surface does not vanish, it bounces. At $z = 0$ those two terms are identical, the bracket doubles, and the 2 cancels to leave $\pi$. So the reflection is already included in the form above. Writing $2\pi$ here and calling it the ground-level concentration halves the answer, and it is one of the most common errors in a hand calculation.

The exponential is where the physics lives, and it is brutal. Hold the same source and walk in toward the stack, where $\sigma_z$ is smaller. At $\sigma_z = 100$ m the exponential is 0.607. At 50 m it is $e^{-2} = 0.135$. At 25 m it is $e^{-8} = 3.4\times10^{-4}$. The plume is overhead and the ground beneath it is essentially clean, no matter how large Q is, which is why the maximum ground-level concentration always lies well downwind of an elevated stack and why the nearest neighbour is rarely the worst-affected one. It is also why an effective height error is so expensive: H enters squared and inside an exponential.

Now the honest part. This model assumes a steady wind in speed and direction, a steady emission, flat terrain, no chemistry, no deposition, no wet removal, total reflection at the ground, and a concentration profile that is exactly normal in both crosswind directions. Not one of those is true of a real afternoon. It survives because its errors were characterised against decades of tracer releases and are roughly conservative, and the accepted standard of agreement is a factor of two on an hourly average in flat terrain, degrading badly in complex terrain, in light winds, and for averaging times under an hour. Treat a single hand calculation as an order-of-magnitude screen. A real assessment runs a model such as AERMOD over a full year of hourly meteorology, because the number a permit turns on is a rank-ordered statistic over 8760 hours, not one arithmetic result.