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

Abstract

A two-layer asymptotic theory of mean momentum in a turbulent Ekman layer without any closure model (such as eddy viscosity, mixing length, or k − ) 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 KEY WORDS power-law and log-law velocity profiles; Rossby-number similarity theory; neutral barotropic planetary boundary layer; Izakson-Millikan-Kolmogorov hypothesis

Probabilistic stability and ‘tall’ wind profiles: theory and method for use in wind resource assessment

“…[2], and implicit in the IEC standard [3]). However, these power-law forms lack any systematic or universal description of the connection or difference between short-and long-term wind profiles, and have only just begun to have useful theoretical or physical connection to geostrophic theory [4] and stability measures [5]. Swift&Dixon [6] did examine power-law exponent variation and connection to the log-law over the ocean, with some consideration of the effect of a z-dependent power law upon the Weibull parameters, but this was focused on the sea-induced speed-dependent roughness and subsequent change in Weibull shape.…”