On Adding Thermodynamic Damping Mechanisms to Refine Two Classical Models of Katabatic Winds

Abstract

The Prandtl and layer-averaged models of katabatic winds contain some nonphysical singularities in the analytical solutions, which give unbounded steady flow anomalies at zero slope angles or adiabatic lapse rates. This study presents some simple refinements of these two classical models, in which the aforementioned singularities are removed when Newtonian cooling and Rayleigh friction are included in the system. It is pointed out that, in the limit of zero slope angles or adiabatic lapse rates, the along-slope buoyancy force and the adiabatic heating caused by air descending approach zero. Under such circumstances, a bounded steady solution for the katabatic winds is impossible unless some damping mechanisms are included to retard the anomalies induced by the radiative cooling effect in the boundary layer. Newtonian cooling and Rayleigh friction are the two simplest thermodynamic damping mechanisms that can be included to balance the effects of eddy viscosity and eddy thermal conductivity in the katabatic-flow model. Physically speaking, the Newtonian cooling term represents a small partition of the radiative effect and the Rayleigh friction term represents an approximation of the bottom drag effect in a turbulent boundary layer.

Weakly nonlinear Prandtl model for simple slope flows

“…1984, Pielke); however, if Eq. is extended so that it contains additional damping mechanisms, as in Mo (), this zero‐slope singularity is removed. As we are interested here in slopes from, say, a half of a degree and more, the zero‐slope issue is irrelevant here.…”

“…van den Broeke, ; Oerlemans and Grisogono, ; Parmhed et al , ; Axelsen and van Dop, ). Further improvements in treating certain shortcomings of the classic Prandtl model are in Mo (). Overall, one of the main drawbacks for most analytic models of katabatic flows, including the classic Prandtl model, is the assumed linearity and prescribed constancy of eddy diffusivity and conductivity (all for the sake of mathematical tractability).…”

Oscillations in Prandtl slope flow started from rest

“…Grisogono () suggested that the assumed constancy of the eddy viscosity in the original Prandtl model produced near‐surface wind gradients that were generally too weak and obtained an unsteady approximate solution of the Prandtl equations for a prescribed slowly varying (in slope‐normal coordinate) eddy viscosity. Mo () extended the McNider () study of oscillations in katabatic flows by including a Newtonian cooling term in the layer‐averaged thermal energy equation.…”