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 diffusivities (O'Brien type) and constant Prandtl number $(Pr)$. Variations in the velocity and buoyancy profiles will be discussed as a function of the dimensionless model parameters $z_0 \equiv \hat{z}_0 \hat{N}^2 Pr \sin{(\alpha)} |\hat{b}_s |^{-1}$ and $\lambda \equiv \hat{u}_{\rm{ref}} \hat{N} \sqrt{Pr} |\hat{b}_s |^{-1}$, where $\hat{z}_0$ is the hydrodynamic roughness length, $\hat{N}$ is the buoyancy frequency, $\alpha$ is the surface sloping angle, $\hat{b}_s$ is the imposed surface buoyancy, and $\hat{u}_{\rm{ref}}$ is a reference velocity scale used to define eddy diffusivities. 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, hence shedding new light on the problem.