Logarithmic Wind Profile (Neutral Surface Layer)

Close to the ground, the wind speed rises with height in a regular way when the air is neither heating nor cooling the surface (neutral stability, as on an overcast, windy day). Over flat, uniform ground the speed is proportional to the logarithm of the height over the roughness length, $M(z) = (u_*/k)\ln(z/z_0)$, for $z$ at or above $z_0$. It is the way to carry a measurement to another height in the lowest tens of metres, for example a 10 m anemometer reading to the 2 m wind that another calculation needs.

Reading the terms. The friction velocity $u_*$ is the turbulent velocity scale of the surface layer, $k = 0.4$ is the von Karman constant, and the aerodynamic roughness length $z_0$ is the height at which the profile reaches zero. Rougher ground has a larger $z_0$: Stull's Table 18-1 gives about 0.0002 m for the open sea and 0.5 m for an orchard or a forest edge. When you compare a wind at one height with a wind at another, $u_*$ cancels, and $M_2 = M_1\ln(z_2/z_0)/\ln(z_1/z_0)$ carries the wind across using only the roughness length.

Worked examples. From Stull's sample applications: a wind of 20 m/s at 10 m over an orchard ($z_0$ = 0.5 m) gives $u_* = 0.4\times20/\ln 20$ = 2.67 m/s. A wind of 5 m/s at 10 m over the same orchard has $u_*$ = 0.66762 m/s and is 6.53 m/s at 25 m. With $u_*$ = 0.3 m/s and $z_0$ = 0.02 m, the wind at 50 m is (0.3/0.4) ln(50/0.02) = 5.9 m/s. Over open sea with $u_*$ = 0.4 m/s and $z_0$ = 0.0002 m, the wind at 10 m is ln(50 000) = 10.82 m/s.

Source and limits. The relation is equation 18.14a of Stull's Practical Meteorology (2017), chapter 18, with the friction velocity from equation 18.13. It holds for neutral air in the surface layer, a few tens of metres deep. In stable air at night the wind rises faster than this, in unstable air more slowly, and above the surface layer it no longer holds. For a hub-height wind above 100 m the library's power-law page is the usual alternative.