Numerical coupling of aerosol emissions, dry removal, and turbulent mixing in the E3SM Atmosphere Model version 1 (EAMv1), part II: a semi-discrete error analysis framework for assessing coupling schemes

TL;DR

This work develops a semi-discrete error analysis framework to separate splitting error from process-time integration error in multiprocess models, enabling a clear interpretation of how coupling schemes affect cross-process dynamics. It formalizes two splitting-error sources—isolation-induced and input-induced—and derives their leading-order contributions for two-process parallel and sequential splitting, then extends the approach to a three-process problem reflecting the dust life cycle in EAMv1. Applying the framework to emissions, dry removal, and turbulent mixing, the authors compare the original EAMv1 coupling with a revised scheme from Part I and show the revised method reduces the dry-removal truncation error, providing a mathematical justification for empirical improvements. The results illuminate how coupling choices propagate through process rates and prognostic variables, offering a generalizable methodology for evaluating and improving process coupling across atmospheric and other multiphysics models.

Abstract

This paper complements the empirical justification of the revised scheme in Part I of this work with a mathematical justification leveraging a semi-discrete analysis framework for assessing the splitting error of process coupling methods. The novelty of the framework is that splitting error is distinguished from the process time integration errors, i.e., the errors caused by discrete time integration of individual processes, leading to expressions that are more easily interpreted utilizing existing physical understanding of the processes that the terms represent. This application of this framework to dust life cycle in EAMv1 showcases such an interpretation, using the leading-order splitting error that results from the framework to confirm (i) that the original EAMv1 scheme artificially strengthens the effect of dry removal processes, and (ii) that the revised splitting reduces that artificial strengthening. While the error analysis framework is presented in the context of the dust life cycle in EAMv1, the framework can be broadly leveraged to evaluate process coupling schemes, both in other physical problems and for any number of processes. This framework will be particularly powerful when the various process implementations support a variety of time integration approaches. Whereas traditional local truncation error approaches require separate consideration of each combination of time integration methods, this framework enables evaluation of coupling schemes independent of particular time integration approaches for each process while still allowing for the incorporation of these specific time integration errors if so desired. The framework also explains how the splitting error terms result from (i) the integration of individual processes in isolation from other processes, and (ii) the choices of input state and timestep size for the isolated integration of processes.

Figures (4)

• Figure 1: The parallel splitting (left column) and sequential splitting method (right column) for solving the two-process ODE defined in Eq. \ref{['eq:two-process-ODE']}. The top, middle, and bottom rows depict the methods in three different ways.
• Figure 2: Three different descriptions of two process coupling schemes for the three-process problem defined in Sect. \ref{['sec:error-analysis']}. The scheme depicted in the left column corresponds to the original scheme used in EAMv1 for the coupling of aerosol emissions, dry removal, and the parameterization of turbulent transport and aerosol activation-resuspension. The scheme depicted in the right column corresponds to the revised scheme proposed and evaluated in the companion paper (Part I). We note that these descriptions are simplified versions of the coupling implemented in EAMv1. Here, we focus only on the three strongest sources and sinks of the global mean dust budget presented in Sect. 3 of the companion paper, while the many other processes in EAMv1 (see Fig. 1 in the companion paper) are omitted.
• Figure 3: Dust aerosol dry removal rate (y-axis) plotted against dust aerosol mixing ratio (x-axis) in the lowest model layer in dust sources regions simulated by the original EAMv1 using a vertical grid with 72 layers. The data used in the figure included 90 days of 6-hourly instantaneous output.
• Figure 4: Comparison of key terms in the splitting truncation error associated with dry removal (process $B$) using 10-year mean interstitial dust mass mixing ratio process rates (unit: kg kg$^{-1}$ s$^{-1}$) caused by emissions (process $A$) and dry removal (process $B$) in the lowest model layer in EAMv1 simulations using the original coupling scheme (upper row) and the revised scheme (lower row).