A Boundary-Layer Scaling for Turbulent Katabatic Flow |
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| Authors: | Alan Shapiro Evgeni Fedorovich |
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| Institution: | 1. School of Meteorology, University of Oklahoma, Norman, OK, USA
2. Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, OK, USA
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| Abstract: | 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 \(\Pi \) 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-Väisälä 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. |
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