Accurate Column Moist Static Energy Budget in Climate Models. Part 1: Conservation Equation Formulation, Methodology, and Primary Results Demonstrated Using GISS ModelE3
Column-integrated moist static energy (MSE) budgets underpin theories of tropical convection and circulation, yet in reanalyses and climate models the budget rarely closes; residuals routinely match the leading terms and mask physical insights. This study derives an MSE conservation law that is strictly consistent with GISS ModelE3 and elucidates why conventional diagnostics fail. Multiple intertwined factors -- the breakdown of the product rule upon discretization, effects of mass-filtering, mismatched flux and advective forms, numerical noise in diagnosed vertical velocity, asynchronous model output timing, and postprocessing including vertical interpolation and temporal averaging -- leave significant residuals in both annual means and daily variability, even when raw 30-min model output is used. Residuals are even larger over land and along coastlines. To tackle this obstacle, this study implements the "process increment method," which accurately computes the column MSE flux divergence by calculating the change in column-integrated internal energy, geopotential energy, and latent heats before and after applying the dynamics scheme. Furthermore, the calculated column flux divergence is decomposed into horizontal and vertical advective components. The most crucial finding is that vertical interpolation into pressure coordinates can introduce errors substantial enough to reverse the sign of vertical MSE advection in the warm-pool regions. In ModelE3, native-grid values show MSE import via vertical circulations, while values after interpolation into pressure coordinates indicate export. This discrepancy may prompt a reevaluation of vertical advection as an exporting mechanism and underscores the importance of precise MSE budget calculations.