Accurate Column Moist Static Energy Budget in Climate Models. Part 1: Conservation Equation Formulation, Methodology, and Primary Results Demonstrated Using GISS ModelE3

TL;DR

This study derives a column-integrated moist static energy ($MSE$) budget that is fully consistent with ModelE3 physics and implements an inline, physically consistent diagnostic—the process increment method—to achieve exact budget closure. It identifies seven intertwined factors that yield large residuals in postprocessed, a posteriori budget analyses and demonstrates that conventional flux-divergence or advective reconstructions can be biased or incomplete, particularly due to errors in the vertical wind field ($\omega_p$) and vertical-coordinate changes. The authors show that vertical-interpolation to pressure coordinates can even reverse the sign of vertical $MSE$ advection in warm-pool regions, highlighting the importance of coordinate choices and inline diagnostics. The work argues for broader adoption of inline $MSE$ budgeting and provides a foundation for Part 2, which will quantify each budget term’s role and explore implications for tropical climate dynamics and model interpretation.

Abstract

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.

Figures (7)

• Figure 1: Annual mean (a, b) and standard deviation (c, d) of the column MSE budget residual and the vertical advection, $-\left \langle \omega_p\partial h/\partial p \right \rangle$, calculated using daily averaged variables in the $p$-coordinate system. Contour intervals are specified in each panel, with regions of saturated contours masked in gray.
• Figure 2: Annual mean of $\omega_p$ at 500 hPa from ModelE3 for the year 2000.
• Figure 3: Annual mean and daily standard deviation of the column MSE budget residual, calculated using the raw data in the m-coordinate system, in the flux-divergence form (Eq. \ref{['eq:column_mse3']}) (a, c), and in the advective form (Eq. \ref{['eq:column_mse_p']}) (b, d). Contour intervals are specified in each panel, with regions of saturated contours masked in gray.
• Figure 4: Flowchart illustrating the integration scheme in ModelE3. A prognostic variable, denoted as $a_0^n$ at time level $n$, undergoes forward integration to the next time level $n+1$ through a series of four distinct schemes: turbulence, cloud, radiation, and dynamics. In each step of this sequence, the scheme updates $a_0^n$ by adding its specific tendency. Notably, the variable is harvested for output, labeled as $a^n$, immediately following the completion of the cloud scheme.
• Figure 5: Annual mean of ${-\nabla_z\cdot\left \langle h\mathbf{v} \right \rangle}$ derived using two distinct methods and their difference: (a) the process increment method, (b) the direct computation using the raw data in the m-coordinate system at a 30-minute resolution, and (c) the difference between (a) and (b). For the direct computation in panel (b), ${-\nabla_z\cdot\left \langle h\mathbf{v} \right \rangle}$ is calculated in the advective form as specified in Eq. \ref{['eq:flux_advective_mse']}, following the same approach as shown in Fig. \ref{['fig:res_m']}(b). The contour interval for each panel is 20 W m$^{-2}$.