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**1 - 1**of**1**### Enstrophy bounds and the range of space-time scales in the hydrostatic primitive equations

"... 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 (NuRaRe)1/4 for the range of horizontal spatial sizes in HPE solutions, which is much broad ..."

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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 (NuRaRe)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. The hydrostatic primitive equations (HPE) have been the foundation of most numerical weather, climate and global ocean circulation calculations for many decades [1, 2, 3, 4, 5]. In practice, modern computational power can handle integrations of these on global horizontal grids ranging in size between 15km and 60km, which correspond respectively to one-eighth degree and one-half degree in latitude and longitude at the equator. This limitation raises the long-standing question, “Can numerical simulations at these grid sizes adequately predict climate and other natural phenomena that occur on the much wider range of scales observed in Nature? ” See Figure 1. Figure 1: A NASA image [6] illustrates the large range of fluid scales that exist in atmospheric circu-lation. The oceanic range of scales is similar, but is not so easily observed. ar