Katabatic flow with Coriolis effect and gradually varying eddy diffusivity

Katabatic flows over high-latitude long glaciers experience the Coriolis force. A sloped atmospheric boundary-layer (ABL) flow is addressed which partly diffuses upwards, and hence, becomes progressively less local. We present the analytical and numerical solutions for (U ,V, θ) depending on (z, t) in the katabatic flow, where U and V are the downslope and cross-slope wind components and θ is the potential temperature perturbation. A Prandtl model that accounts for the Coriolis effect, via f, does not approach a steady state, because V diffuses upwards in time; the rest, i.e., (U, θ), are similar to that in the classic Prandtl model. The V component behaves in a similar manner as the solution to the 1st Stokes (but inhomogeneous) problem. A WKB approach to the problem of the sloped ABL winds is outlined in the light of a modified Ekman-Prandtl model with gradually varying eddy diffusivity K(z). Ideas for parameterizing these high-latitude persistent flows in climate models are revealed. After Wentzel, Kramers and Brillouin, who popularized the method in theoretical physics.