Wind Shear Power Law

The ground drags on the air above it, so wind speed rises with height, and the power law $v_2 = v_1 (h_2/h_1)^{\alpha}$ is the engineering shorthand for that profile. It is empirical rather than derived — the theoretically better-founded log law uses a roughness length instead — but the power law needs one parameter rather than two and fits measured profiles well enough over the heights a turbine occupies.

The exponent carries the terrain. Roughly 0.10 over open water, 0.14 over smooth open country (the famous "one-seventh law"), 0.20 over crops and hedgerows, and 0.25 to 0.40 over woodland and built-up ground. Whenever two measurement heights are available, fit $\alpha$ from them instead of reading a table: $\alpha = \ln(v_2/v_1)/\ln(h_2/h_1)$, and the fitted value carries the site's real roughness and stability rather than a category someone chose for it.

The cube law is what makes this equation matter. Going from a 10 m mast to an 80 m hub at $\alpha = 0.14$ raises the speed by a factor of 1.338 — and the power by $1.338^3 = 2.39$. A modest gain in wind speed buys a very large gain in energy, and the tower is usually the cheapest place to buy it. It also means an error in $\alpha$ is amplified threefold by the time it reaches a yield forecast, which is why extrapolating from a short mast across a large height ratio is treated with suspicion.

One caution the single-number form hides: $\alpha$ is not constant through the day. On a clear night the surface cools, the boundary layer stabilises, the upper air decouples from the ground, and shear rises sharply — exponents above 0.4 are common over land before dawn. Strong afternoon convection mixes the layer and flattens the profile toward 0.10. An annual-average $\alpha$ is a useful fiction standing in for something that changes hourly, and near-surface data extrapolated across a large ratio can be badly wrong for individual hours even when the annual mean comes out right.