Katabatic flow along a differentially cooled sloping surface in a stratified fluid

One of the early successes of atmospheric dynamics was Prandtl's discovery (1942) of a simple analytical solution for laminar flow of a viscous fluid along a uniformly cooled or heated planar surface in a stably stratified fluid. The Prandtl model and subsequent extensions of it by various investigators to include the Coriolis force, height-dependent eddy viscosities and flow unsteadiness have provided simple but surprisingly robust descriptions of katabatic and anabatic winds on mountains and valleys of constant slope. The solutions are generally of a boundary-layer character (intense shallow jet topped by a weak reversed flow), and are exact within the Boussinesq framework. However, thus far, even the extended solutions are fundamentally one-dimensional.

In this study we extend the Prandtl framework to include spatially inhomogeneous surface buoyancy forcings (differentially cooled slopes). We assume the surface buoyancy varies gradually enough locally that it can be approximated by a linearly-varying function of the along-slope coordinate. This extension is the simplest one capable of introducing two-dimensionality to the problem. Since the symmetry of the classical (one-dimensional) model is now broken, one must contend with such features as flow acceleration, convergence and associated vertical motions, and horizontal and vertical advection of both perturbation and base-state temperature fields. Introduction of a simple scaling hypothesis appropriate for this problem reduces the steady state Boussinesq equations of motion, mass conservation and thermodynamic energy to a set of eight coupled nonlinear ordinary differential equations. The governing parameters include the slope angle, Coriolis parameter, Brunt-Väisälä frequency, and magnitude of the surface buoyancy forcing. Analytical results are obtained for the asymptotic structure of the flow above the boundary jet. The analytical results are complemented with numerical solutions.