A numerical model of the balance equations in a periodic domain and an example of balanced turbulence

• Published in: Journal of computational physics Vol. 67; no. 2; pp. 439 - 471
• Main Authors: Norton, N.J, McWilliams, J.C, Gent, P.R
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.1986
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/0021-9991(86)90271-8

The Balance Equations represent rotating, stratified fluid motions and therefore are of potential utility in studies of geophysical phenomena. In this paper we present numerical algorithms for this model. The most novel aspect resolves the implicit structure of the time-derivative terms in the Balance Equations with iterative, predictor-corrector methods which combine time-stepping and iteration. The discrete spatial operations are low-order, finite-differences which exactly conserve certain integral properties of the model. The numerical performance of the model is evaluated, and comparisons with other geophysical models are made—the more highly approximated Quasigeostrophic Equations (QG) and the more fundamental Primitive Equations (PE). We find that the Balance Equations are an excellent Intermediate Model: their computational cost is closer to that of the less expensive QG, but they quite accurately represent the evolution of the vorticity field of the more accurate PE, even when fluid advection rates are comparable to the Coriolis frequency (i.e., the Rossby number is order one) and a substantial gravity wave component is generated in the PE. In addition, the existence of a solvability barrier at large Rossby number is demonstrated. As an illustration of a physical phenomenon for which the Balance Equations are well suited, solutions are examined for the slow viscous decay of broad-band, large-scale, baroclinic initial conditions at finite Rossby number (i.e., balanced turbulence). Several departures from the more familiar geostrophic turbulence are found: the enstrophy cascade is stronger, the baroclinic energy conversion is stronger, and there is a tendency to form isolated frontal structures in the vorticity field.