Post-Onset Behaviour of the Pure Katabatic Flow

An analytic improvement of the classic Prandtl model for pure katabatic flows is obtained. The one-dimensional unsteady problem employs gradually-varying eddy diffusivity K(z) fixed in time. A new solution is found for thefourth-order governing equation that couples the momentum and heat transfer in an approximate but still systematic way. The solution for wind and temperature perturbations is a generalization of the Prandtl solution allowing for: (1) Local andcumulative K(z) effects, (2) gradual evolution from the initial, discontinuous- towardthe steady-state profiles and (3) the given two-dimensional background potential temperature gradient, the surface slope and its potential temperature deficit atthe surface. The solution that is based on a relaxation equation compares adequatelywith its numerical counterpart soon after the estimated flow onset. It is a product ofthe steady-state solution with a spatio-temporal transfer function. The results can beuseful for data analyses, especially for the scale estimations of inclined stable boundarylayers and for surface flux calculations.