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The hydrostatic primitive equations (HPE) form the basis of most numericalweather, climate and global ocean circulation models. Analytical (notstatistical) methods are used to find a scaling proportional to$(Nu\,Ra\,Re)^{1/4}$ for the range of horizontal spatial sizes in HPEsolutions, which is much broader than currently achievable computationally. Therange of scales for the HPE is determined from an analytical bound on thetime-averaged enstrophy of the horizontal circulation. This bound allows theformation of very small spatial scales, whose existence would exciteunphysically large linear oscillation frequencies and gravity wave speeds.

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