engleski

Weakly-nonlinear Prandtl model for simple slope flows

Prandtl model couples, perhaps in the most succinct way, basic boundary-layer dynamics and thermodynamics for simple anabatic or katabatic flows over inclined surfaces. This 1D analytic model assumes steady-state balance between buoyancy (positive for anabatic, negative for katabatic flows) and turbulent friction. While the classical Prandtl model is linear with a priori assigned vertically constant eddy diffusivity and heat conductivity, in this analytic work we partly relax both of these assumptions. Focusing on katabatic flows, one of the main weaknesses of Prandtl model is that strong, near-surface flow- induced finite-amplitude potential temperature gradient does not feed back to the assigned environmental potential temperature gradient in the thermodynamic equation, even though the induced gradient below the katabatic jet can be 20 to 50 times stronger than the related environmental gradient. This issue is tackled by using a weakly-nonlinear approach where a small parameter controls feeding of the flow-induced potential temperature gradient back to the environmental potential temperature gradient. An appropriate range of values for the small parameter controlling the weak nonlinearity is provided. In this way, the near-surface potential temperature gradient becomes stronger and the corresponding katabatic jet somewhat weaker (at a slightly lower height) than that in the classical Prandtl solution. The other issue of constant eddy coefficient K in the classical Prandtl model is treated by a given gradually varying K with the height above the surface, all within the validity of the zero-order WKB approximation. The new model is compared to data from PASTEX-94, Austria. Besides other potential applications, this modified Prandtl model may help in parameterizing inclined boundary layers in climate models since there is only little confidence on the projected changes in future wind fields and the corresponding extremes.

Katabatic flows; Prandtl model; WKB method

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