Wind Speed at Height (Power Law)

Anemometers sit at 10 m because that is where the World Meteorological Organization put them, and stacks are anywhere but 10 m, so almost every dispersion calculation starts with an extrapolation. The power law is the cheapest one available: $u_2 = u_1 (z_2/z_1)^p$. A 5 m/s reading at 10 m, taken up to a stack top at 80 m over open country with $p = 0.15$, gives $5 \ imes 8^{0.15} = 5 \ imes 1.366 = 6.83$ m/s. Turn it around and the same equation fits an exponent from two measured levels: 4 m/s at 10 m and 8 m/s at 100 m give $p = \ln 2/\ln 10 = 0.301$, which is a stable night over rough ground.

The exponent is not a property of the terrain alone, it is terrain and stability together, and the values EPA publishes are worth carrying. Rural: 0.07 for classes A and B, 0.10 for C, 0.15 for D, 0.35 for E and 0.55 for F. Urban: 0.15 for A and B, 0.20 for C, 0.25 for D, 0.30 for E and F. Notice how far the stable rural exponent sits from the rest. On a clear calm night over farmland the wind at 100 m can be more than twice the wind at 10 m, and a calculation that assumes a neutral 0.15 will under-predict the stack-height wind by nearly half. Notice too that the urban exponents are flatter at the stable end, because a city keeps mixing mechanically and thermally long after the countryside has gone still.

The power law is a convenience fit to something better founded. Surface-layer theory gives the logarithmic profile $u(z) = (u_*/k)\ln((z-d)/z_0)$, where $z_0$ is a roughness length running from a millimetre over water to a metre or more over forest, d is the zero-plane displacement, and k is von Karman's constant near 0.40. The log law is what the physics supports, with a Monin-Obukhov correction for stability. The power law survives because it needs one number instead of three and because it is what the regulatory guidance is written in. Both are surface-layer relations, which means both stop being trustworthy above roughly a tenth of the boundary-layer depth: 100 m or so on a stable night, perhaps 150 m in neutral conditions. Extrapolating a 10 m anemometer to a 200 m stack is an act of faith, and on a stable night the nocturnal low-level jet can put a maximum in the profile that no monotonic formula can reproduce.

The consequence for dispersion is that a wind error is charged twice. Plume rise goes as $1/u$, so a wind that is too high shortens the plume, and concentration goes as $1/u$ again in the Gaussian equation, so the same error moves the answer roughly as $u^2$ once the height term is squared. Get the extrapolation wrong by 30 percent and the ground-level concentration can be wrong by a factor of two, which is the whole error budget of the model spent on one input. Two habits keep this honest: record the anemometer height alongside every wind speed, and respect the regulatory floor of about 1 m/s, below which the power law and the Gaussian model both stop meaning anything.