Katabatic Flow: A Closed-Form Solution with Spatially-Varying Eddy Diffusivities
The Nieuwstadt closed-form solution for the stationary Ekman layer is generalized for katabatic flows within the conceptual framework of the Prandtl model. The proposed solution is valid for spatially-varying eddy viscosity and diffusivity (O’Brien type) and constant Prandtl number (Pr). Variations in the velocity and buoyancy profiles are discussed as a function of the dimensionless model parameters z0 ≡ ẑ0 N̂ 2 Pr sin(α) / |b̂ s| and λ ≡ ûref N̂ √Pr / |b̂ s| , where ẑ0 is the hydrodynamic roughness length, N̂ is the Brunt-Väisälä frequency, α is the surface sloping angle, b̂s is the imposed surface buoyancy, and ûref is a reference velocity scale used to define eddy diffusivities. Velocity and buoyancy profiles show significant variations in both phase and amplitude of extrema with respect to the classic constant K model and with respect to a recent approximate analytic solution based on the Wentzel-Kramers-Brillouin theory. Near-wall regions are characterized by relatively stronger surface momentum and buoyancy gradients, whose magnitude is proportional to z0 and to λ. In addition, slope-parallel momentum and buoyancy fluxes are reduced, the low-level jet is further displaced toward the wall, and its peak velocity depends on both z0 and λ. .