Scale ratios in decaying quasi-geostrophic turbulence

The fundamental nature of rapidly rotating, stably stratified (i.e. quasi-geostrophic) turbulence has been debated since Charney (CHARNEY J. G., Geostrophic turbulence, J. Atmos. Sci., 28 (1971) 1087-1095) proposed an isotropic theory for it. That theory implies that the (total) energy is partitioned, in spectral space, equally between the x, y and z wave number components after scaling z in the governing equations by Prandtl's ratio f/N, or the ratio of the rotational and buoyancy frequencies. In (scaled) physical space, this implies that structures, i.e. vortices, would have equal dimensions, i.e. they would be spheres on average. Recent numerical results (DRITSCHEL D. G., DE LA TORRE JUAREZ M. and AMBAUM M. H. P., The three-dimensional vortical nature of atmospheric and oceanic turbulent flow, Phys. Fluids, 11 (1998) 1512-1520), obtained at exceptionally high resolution, have confirmed for the first time Charney's prediction that vertical- to horizontal-scale ratios in quasi-geostrophic turbulence are equal to f/N. These results were largely unanticipated because the simulations analysed in Dritschel, de la Torre Juarez and Ambaum (1998) began with two-dimensional, height-independent vortices, and previous numerical simulations (MCWILLIAMS J. C., WEISS J. B. and YAVNEH I., Anisotropy and coherent vortex structures in planetary turbulence, Science, 264 (1994) 410-413) showed the emergence of approximately two-dimensional vortices at late times from isotropic, three-dimensional initial conditions. One key difference between Dritschel, de la Torre Juarez and Ambaum (1998) and McWilliams et al., 1994 is the vertical boundary conditions used; in the former rigid, isothermal boundaries were used, while in the latter periodic boundaries were used. This difference was explored in (DRITSCHEL D. G. and MACASKILL C., The role of boundary conditions in the simulation of rotating, stratified turbulence, submitted to Fuid Dyn. Res., which revealed that periodic boundaries tend to suppress three-dimensional behaviour by constraining the range of interactions between vortices, all other things being equal. Using a sufficiently wide domain, for fixed domain height, eventually reduces this constraint, allowing continued, three-dimensional behaviour. Here, we re-examine quasi-geostrophic turbulence in vertically periodic boundaries, at times well before two-dimensional vortices might emerge. We focus specifically on the characteristic scale ratio of structures, and how it compares to that found for rigid, isothermal boundaries (Dritschel, de la Torre Juarez and Ambaum, 1998).