Potential vorticity inversion on a hemisphere

Several different kinds of accurate potential vorticity (PV) inversion oper ators, and the associated balanced models, are tested for the shallow water equations on a hemisphere in an attempt to approach the ultimate limitatio ns of the balance, inversion, and slow-manifold concepts. The accuracies ac hieved are far higher than for standard balanced models accurate to one or two orders in Rossby number R or Froude number F (where F = /u//c; /u/ = ho w speed; and c = gravity wave speed). Numerical inversions, and correspondi ng balanced-model integrations testing cumulative accuracy, are carried out for cases that include substantial PV anomalies in the Tropics. The balanc ed models in question are constructed so as to be exactly PV conserving and to have unique velocity fields (implying, incidentally, that they cannot b e Hamiltonian). Mean layer depths of 1 and 2 km are tested. The results show that, in the cases studied, the dynamical information cont ained in PV distributions is remarkably close to being complete even though R = z at the equator and even though local maximum Froude numbers, F-max a pproach unity in some cases. For example, in a 10-day integration of the ba lanced model corresponding to one of the most accurate inversion operators, "third-order normal mode inversion," the mean depth was 1 km, the minimum depth less than 0.5 km, and F-max similar or equal to 0.7, hardly small in comparison with unity. At the end of lo days of integration, the cumulative rms error in the layer depth was less than 15 m, that is, less than 5% of the typical rms spatial variation of 310 m. At the end of the first day of integration the rms error was 5 m, that is, less than 2%. Here "error" refe rs to a comparison between the results of a balanced integration and those of a corresponding primitive equation integration initialized to have low g ravity wave activity on day 0. Contour maps of the PV distributions remaine d almost indistinguishable by eye over the 10-day period. This remarkable c umulative accuracy, far beyond anything that could have been expected from standard scale analysis, is probably related to the weakness of the spontan eous-adjustment emission or "Lighthill radiation" studied in the companion paper by Ford et al.