Enstrophy bounds and the range of space-time scales in the hydrostatic primitive equations

TL;DR

This paper derives analytical bounds on enstrophy in the hydrostatic primitive equations, revealing a broad range of spatial scales that challenge current computational capabilities and impact the understanding of atmospheric and oceanic dynamics.

Contribution

It introduces an analytical method to bound enstrophy in HPE, establishing a link between enstrophy and the range of space-time scales in climate models.

Findings

The scale range is proportional to (Nu Ra Re)^{1/4}.

Enstrophy bounds allow very small spatial scales.

Small scales could excite unphysical oscillations.

Abstract

The hydrostatic primitive equations (HPE) form the basis of most numerical weather, climate and global ocean circulation models. Analytical (not statistical) methods are used to find a scaling proportional to $(Nu\,Ra\,Re)^{1/4}$ for the range of horizontal spatial sizes in HPE solutions, which is much broader than currently achievable computationally. The range of scales for the HPE is determined from an analytical bound on the time-averaged enstrophy of the horizontal circulation. This bound allows the formation of very small spatial scales, whose existence would excite unphysically large linear oscillation frequencies and gravity wave speeds.