Challenges Associated with Adapting the Governing Flow Equations for Coordinate Systems Aligned with Steep Slopes

Turbulence measurements over complex terrain have been given increased attention toward improving parameterizations for use in modeling, when traditional parameterizations, developed for flat, statistically homogeneous terrain, are inappropriate. For example, Monin-Obukhov similarity theory often breaks down for steep, sloping terrain where vertical fluxes can vary by significantly more than 10% over as little as the lowest 10 m of the atmospheric boundary layer. Given this increased attention given specifically to measurements over steep terrain, we have found some discrepancies within the literature with regards to the use of turbulence measurements in the governing equations of motion. The root of these discrepancies is the fact that steep terrain poses additional considerations for the choice of coordinate system because shear mechanisms follow the direction of the flow (or the terrain locally), while buoyancy mechanisms always act vertically due to the directionality of the gravity vector. Commonly, a slope-parallel/slope-normal (sp/sn) coordinate system is used in place of a horizontal/vertical system because it simplifies the directionality issues for the respective directional (U, V, W) momentum budget equations. However, the first observed discrepancy arises in the terms used in the turbulence kinetic energy (TKE) budget equation, and therefore the flux Richardson number, for sloping terrain with slope angle, α. Regardless of the choice of coordinate system, either the shear or the buoyancy (or both) terms in the governing equations must include both the cos(α) and sin(α) components in the TKE equation and flux Richardson number. We show this by deriving the TKE equation for the sp/sn coordinate system from first principles, and by quantifying the relative importance of the sin(α) components (sporadically neglected) of the buoyancy term in relation to the other TKE budget terms using measurements over a steep alpine slope. The second discrepancy (more an unaddressed issue) for using the sp/sn coordinate system arises from the physical constraint that a varying wind direction must vary the slope ‘seen' by the wind. Subsequently, the coordinate system must also change, which has implications for the slope angles used in the governing equations and sensor tilt corrections. We again use the measurements to quantify the differences imposed by changing wind direction on these calculations. Finally, we make recommendations for standardizing methodologies for measurements over steep terrain, such that future comparisons between sites can be made consistently and systematically.