On the Laws of Geostrophic Drag and Heat Transfer over a Slightly Inclined Terrain

Abstract

Analytical solutions of a set of equations that couples the Ekman boundary layer and the Prandtl slope wind equations are presented for terrain inclinations with an upper limit of ?0.2. With the aid of the logarithmic asymptotes for wind and temperature profiles in the dynamic sublayer, a system of transcendental equations connecting the internal and external parameters of the boundary layer is derived. For terrain sloping at an angle much less than a critical value of 0.01, this system gives a generalization of the well-known Monin and Zilitinkevich formulas, valid for the Ekman boundary layer, for the geostrophic drag and heat exchange coefficients. On the other hand, for terrain at an angle much larger than the critical value, and in the absence of geostrophic wind, the system yields simple expressions for the momentum and heat exchange coefficients as a function of only one nondimensional external parameter. For the latter case a graphical representation is also given which can be used to calculate, for given external parameters, the corresponding internal parameters of the boundary layer. Furthermore. from the complete solution of the problem, analytical expressions for the universal functions of the internal stability parameter, h/L, namely, A, B and C, are obtained where h is a characteristic thickness of the boundary layer over the slope and L the Monin-Obukhoy length scale. It is shown that 1) the shape of these functions depends only on the model of vertical turbulent diffusivity and the form of the upper boundary conditions and that 2) the terrain slope affects only the argument of these functions and not the shape. An interesting conclusion, therefore, is that the universal functions found in this study are the very same functions as those for the case when the underlying surface is horizontal and which are currently the object of much theoretical and experimental research.