  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.
