Weakly nonlinear Prandtl model for simple slope flows

Abstract

The Prandtl model couples, probably in the most succinct way, basic boundary‐layer dynamics and thermodynamics for pure anabatic and katabatic flows over inclined surfaces by assuming a one‐dimensional steady‐state balance between buoyancy and turbulent friction. Although the classic Prandtl model is linear, having an a priori assigned vertically constant eddy diffusivity and heat conductivity, K, in this analytic work we partly relax both of these restrictions. The first restriction is loosened by using a weakly nonlinear approach where a small parameter, ε, controls feeding of the flow‐induced potential temperature gradient back to the environmental potential temperature gradient, because the former, below the katabatic jet, can be 20–50 times stronger than the latter, background or free‐flow gradient. An appropriate range of values for ε, controlling the weak nonlinearity for pure katabatic flow, 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 classic Prandtl solution. The second restriction is partly relaxed by using a prescribed, gradually varying K with distance from the underlying surface, all within the usual validity of the zero‐order Wentzel–Kramers–Brillouin approximation to solve the coupled differential equations. The new model is compared with the glacier wind data from the Pasterze experiment (PASTEX‐94), Austria. Further discussion includes gradient Richardson number consideration and an application to simple anabatic flows. The model may be applied for estimation and interpretation of the wind affecting glacier mass balance and air pollution.