Substep Vertical Pressure Decomposition - #3881
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Extensive testing should be completed before merging this PR.
Summary
This PR modifies the coefficients for the tridiagonal solve in the substepping algorithm. The motivation for this change is that it yields a consistent vertical pressure gradient decomposition and it makes the substep solution independent of the base state.
KSD Formulation
The pressure gradient relation and its relation to potential temperature
The linearized approximation
The concern with this formulation is that it formally replaces perturbations from the base state,
\phi^{'}, with perturbations from the RK stage,\phi^{''}. By doing so, the pressure perturbation in the substep is not with respect to the RK stage and the ensuing coefficients depend upon the base state.Proposed Change
Formally decompose the pressure gradient into a slow and fast component. The slow has not changed and is with respect to the base state, consistent with the RK integrator on its own. The fast now considers
\partial_z P^{''}as a starting point.