A Boundary-Layer Scaling for Turbulent Katabatic Flow

Scaling relationships are proposed for the turbulent katabatic flow of a stably stratified fluid down a homogeneously cooled planar slope-the turbulent analogue of a Prandtl-type slope flow. The Theorem predicts that such flows are controlled by three non-dimensional parameters: the slope angle, the Prandtl number, and a Reynolds number defined in terms of the surface thermal forcing (surface buoyancy or surface buoyancy flux), Brunt-Vaisala frequency, slope angle, and molecular viscosity and diffusivity coefficients. However, by exploiting the structure of the governing differential equations in a boundary-layer form, scaled equations are deduced that involve only two non-dimensional parameters: the Prandtl number and a modified Reynolds number. In the proposed scaling framework, the slope angle does not appear as an independent governing parameter, but merely acts as a stretching factor in the scales for the dependent and independent variables, and appears in the Reynolds number. Based on the boundary-layer analysis, we hypothesize that the full katabatic-flow problem is largely controlled by two rather than three parameters. Preliminary tests of the scaling hypothesis using data from direct numerical simulations provide encouraging results.