Katabatic Flow: A Closed-Form Solution with Spatially-Varying Eddy Diffusivities

Abstract

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 $z_0 \equiv \hat{z}_0 \hat{N}^2 Pr \sin {(\alpha )} |\hat{b}_\mathrm{s} |^{-1}$ and $\lambda \equiv \hat{u}_{\mathrm{ref}}\hat{N} \sqrt{Pr} |\hat{b}_\mathrm{s} |^{-1}$, where $\hat{z}_0$ is the hydrodynamic roughness length, $\hat{N}$ is the Brunt-Väisälä frequency, $\alpha$ is the surface sloping angle, $\hat{b}_\mathrm{s}$ is the imposed surface buoyancy, and $\hat{u}_{\mathrm{ref}}$ 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 $\textit{K}$ 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 $z_0$ and to $\lambda$. 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 $z_0$ and $\lambda$.