Numerically consistent budgets of potential temperature, momentum, and moisture in Cartesian coordinates: Application to the WRF model

Abstract

Numerically accurate budgeting of the forcing terms in the governing equations of a numerical weather prediction model is hard to achieve. Because individual budget terms are generally 2 to 3 orders of magnitude larger than the resulting tendency, exact closure of the budget can only be achieved if the contributing terms are calculated consistently with the model numerics. We present WRFlux, an open-source software that allows precise budget evaluation for the WRF model and, in comparison to existing similar tools, incorporates new capabilities. WRFlux transforms the budget equations from the terrain-following grid of the model to the Cartesian coordinate system, permitting a simplified interpretation of budgets obtained from simulations over non-uniform orography. WRFlux also decomposes the resolved advection into mean advective and resolved turbulence components, which is useful in the analysis of large-eddy simulation output. The theoretical framework of the numerically consistent coordinate transformation is also applicable to other models. We demonstrate the performance and a possible application of WRFlux with an idealized simulation of convective boundary layer growth over a mountain range. We illustrate the effect of inconsistent approximations by comparing the results of WRFlux with budget calculations using a lower-order advection operator and two alternative formulations of the coordinate transformation. With WRFlux, the sum of all forcing terms for potential temperature, water vapor mixing ratio, and momentum agrees with the respective model tendencies to high precision. In contrast, the approximations lead to large residuals: the root mean square error between the sum of the diagnosed forcing terms and the actual tendency is 1 to 3 orders of magnitude larger than with WRFlux.