Stability Dependence of Height Scales and Effective Roughness Lengths of Momentum and Heat Transfer Over Roughness Changes

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It is widely accepted that the correct formulation of an effective roughness length, defined as the area average of the roughness length in heterogeneous terrain, relies upon the appropriate de-termination of a height scale. At this height a meteorological quantity is approximately in equilibrium with local surface conditions and independent of horizontal position. This research note determines explicitly the different height scales from the perturbation solutions of flow velocity and temperature, as well as the fluxes of momentum and heat, in a stratified boundary layer. These solutions are derived from the asymptotic approximation theory and shown to capture major characteristics of momentum and heat transfer over heterogeneous terrain with changes of the underlying roughness lengths. The effective roughness lengths can then be computed by use of these height scales. The dependence of height scales and effective roughness lengths upon stratification is also discussed briefly.