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A Boundary-Layer Scaling for Turbulent Katabatic Flow

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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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Notes

  1. The Prandtl solution for wind and buoyancy, together with pressure obtained from the quasi-hydrostatic equation (the equation to which the slope-normal equation of motion reduces in the one-dimensional framework), constitute an exact solution of the viscous Boussinesq equations of motion, thermal energy and mass conservation (incompressibility condition).

  2. In Schumann (1990), as in the other LES/DNS studies cited above, the flow was generated from rest by the sudden application of a surface thermal perturbation (buoyancy or buoyancy flux).

  3. If the \(X\)-axis points downslope, the signs of the \(\sin \alpha \) factors in (2) and (5) must be changed.

  4. Results are shown at the (uncoordinated) terminal times of the individual simulations. In general, the end times correspond to different phases of the temporal oscillation. However, the amplitudes of the velocity and buoyancy oscillations decay with time and are small fractions of the mean characteristic velocity and thermal disturbances at these later times.

  5. For the purpose of the present discussion we define the boundary-layer depths for momentum and buoyancy loosely as the heights at which the magnitudes of the downslope velocity and buoyancy are reduced to some small percentage (say, 10 %) of their peak magnitudes.

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Authors and Affiliations

  1. School of Meteorology, University of Oklahoma, Norman, OK, USA

    Alan Shapiro & Evgeni Fedorovich

  2. Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, OK, USA

    Alan Shapiro

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  1. Alan Shapiro
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Correspondence to Alan Shapiro.

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Shapiro, A., Fedorovich, E. A Boundary-Layer Scaling for Turbulent Katabatic Flow. Boundary-Layer Meteorol 153, 1–17 (2014). https://doi.org/10.1007/s10546-014-9933-3

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