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.

翻译：本文通过数学论证补充了本研究第一部分中修订方案的实证依据，利用半离散分析框架评估过程耦合方法的分裂误差。该框架的创新之处在于将分裂误差与过程时间积分误差（即单个过程离散时间积分导致的误差）区分开来，从而能够利用现有对各项所代表过程的物理理解，更直观地解释相关表达式。将本框架应用于EAMv1中的沙尘生命周期，展示了这种解释方式：利用框架得出的主导阶分裂误差，证实了（i）原始EAMv1方案人为增强了干清除过程的效果，（ii）修订后的分裂方案减少了这种人为增强。尽管误差分析框架是在EAMv1沙尘生命周期背景下提出的，但该框架可广泛应用于评估过程耦合方案，既适用于其他物理问题，也可用于任意数量的过程。当不同过程实现支持多种时间积分方法时，该框架将尤为强大。传统局部截断误差方法需要对每种时间积分方法的组合进行单独考量，而本框架能够独立于各过程的特定时间积分方法评估耦合方案，同时仍允许在需要时纳入这些特定时间积分误差。该框架还解释了分裂误差项如何源自：（i）各过程在与其他过程隔离条件下的独立积分，（ii）过程独立积分时输入状态和时间步长的选择。