Power‐law velocity profile in a turbulent Ekman layer on a transitional rough surface

Abstract

AbstractA two‐layer asymptotic theory of mean momentum in a turbulent Ekman layer without any closure model (such as eddy viscosity, mixing length, or k − ϵ) for large Rossby numbers is proposed. The flow in the inner wall layer and the outer wake layer are matched, using the Izakson–Millikan–Kolmogorov hypothesis; this leads to an open functional equation. Another open functional equation is obtained from the ratio of two successive derivatives of the basic functional equation; this admits two functional solutions, with power‐law and log‐law velocity profiles. The envelope of the geostrophic‐drag power law leads to the log law, and determines the power‐law index and prefactor as a function of the surface Rossby number or the drag coefficient. The log laws and power laws for velocity and friction velocity, including the power‐law constants, are universal, and independent of the wall roughness. This universality is well supported by extensive experimental and laboratory data. In traditional smooth‐wall variables, there is no universality of scalings, and different expressions are needed for different types of roughness. Approximate solutions of the power‐law geostrophic drag and cross‐isobaric angle are also obtained. The power‐law geostrophic‐drag solution for each prescribed value of the power‐law index is valid for a limited domain of Rossby numbers. Copyright © 2008 Royal Meteorological Society