Validity of the pressure vertical-scale assumption in deep convection

Establish whether the non-hydrostatic pressure in a convecting region is vertically coupled over the full depth of the weakly stratified layer, so that its vertical scale is set by the layer depth rather than by the plume width.

Background

The paper estimates the AC/DC column-solve error in deep convection by taking the vertical wavenumber of the non-hydrostatic pressure to be controlled by the depth D of the weakly stratified convecting layer, while the horizontal wavenumber is controlled by the plume width L. This yields the predicted error factor (D/L)2.

The authors explicitly acknowledge that this identification is only a scaling assumption and is not established analytically. They propose measuring the layer depth, plume width, and residual pressure fraction in simulations to test whether the assumed pressure scale is correct.

References

The reasoning there is a scaling argument, and the identification of the pressure's vertical scale with the layer depth is not proved, so the point of these diagnostics is to measure what cannot be predicted.

— A Computational Model for Global Ocean Dynamics at all Scales  (2608.25679 - Korn, 26 Aug 2026) in Section 5.3, “Experiment 3: Open-ocean deep convection,” paragraph “Diagnostics bearing on the approximation itself”