The Ekman spiral

Abstract Previous workers have used an eddy viscosity varying with height linearly or according to a power law, when discussing the variation of the wind between the surface and the height at which it (supposedly) attains the geostrophic value, but they have not used the appropriate boundary conditions to obtain the familiar forms for the velocity profile very near the surface. In conditions of neutral stability this profile is given by\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{u}{{u_* }}\; = \;\frac{1}{k}\;\ln \;(z/z_0)\;.\;.\;.\;.\;.\;(1) $$\end{document}where u * = (τ) 1/2 and k = 0·40. Eq. (1) corresponds to an eddy viscosity, K M , given by\documentclass{article}\pagestyle{empty}\begin{document}$$ K_M \; = \;\;ku_* \;z\;.\;.\;.\;.\;.\;(2) $$\end{document}If this formula is assumed to hold at all heights, and the geostrophic wind is assumed to be uniform and constant, it can be shown that\documentclass{article}\pagestyle{empty}\begin{document}$$ (u_G \; - \;u)\; - \;\;i(v_G \; - \;v)\; = \;A\;H_0 ^{(1)} \;(4ifz/ku_*)^{\frac{1}{2}} .\;.\;.\;.\;.\;(3) $$\end{document}where u is the component of the wind in the direction of the surface wind and v that perpendicular to it; the suffix G denotes the geostrophic value and f is the Coriolis parameter.