Potential Vorticity Inversion on a Hemisphere

Abstract

Several different kinds of accurate potential vorticity (PV) inversion operators, and the associated balanced models, are tested for the shallow water equations on a hemisphere in an attempt to approach the ultimate limitations of the balance, inversion, and slow-manifold concepts. The accuracies achieved 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| = flow speed; and c = gravity wave speed). Numerical inversions, and corresponding balanced-model integrations testing cumulative accuracy, are carried out for cases that include substantial PV anomalies in the Tropics. The balanced models in question are constructed so as to be exactly PV conserving and to have unique velocity fields (implying, incidentally, that they cannot be Hamiltonian). Mean layer depths of 1 and 2 km are tested. The results show that, in the cases studied, the dynamical information contained in PV distributions is remarkably close to being complete even though R = ∞ at the equator and even though local maximum Froude numbers, Fmax, approach unity in some cases. For example, in a 10-day integration of the balanced 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 Fmax ≃ 0.7, hardly small in comparison with unity. At the end of 10 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” refers to a comparison between the results of a balanced integration and those of a corresponding primitive equation integration initialized to have low gravity wave activity on day 0. Contour maps of the PV distributions remained almost indistinguishable by eye over the 10-day period. This remarkable cumulative accuracy, far beyond anything that could have been expected from standard scale analysis, is probably related to the weakness of the spontaneous-adjustment emission or “Lighthill radiation” studied in the companion paper by