  This paper addresses the challenges in computing the column moist static
energy (MSE) budget in climate models. Residuals from such computations often
match other major budget terms in magnitude, obscuring their contributions.
This study introduces a methodology for accurately computing the column MSE
budget in climate models, demonstrated using the GISS ModelE3. Multiple factors
leading to significant residuals are identified, with the failure of the
continuous calculus's chain rule upon discretization being the most critical.
This failure causes the potential temperature equation to diverge from the
enthalpy equation in discretized models. Consequently, in models using
potential temperature as a prognostic variable, the MSE budget equation is
fundamentally not upheld, requiring a tailored strategy to close the budget.
This study introduces the ``process increment method'' for accurately computing
the column MSE flux divergence. This method calculates the difference in the
sum of column internal energy, geopotential, and latent heats before and after
applying the dynamics scheme. Furthermore, the calculated column flux
divergence is decomposed into its advective components. These computations
enable precise MSE budget analysis. 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, accurately computed values show MSE import via vertical circulations,
while values in 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.
