The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere

Abstract

Results from a series of simulations of unforced turbulence evolving within a shallow layer of fluid on a rotating sphere are presented. Simulations show that the turbulent evolution in the spherical domain is strongly dependent on numerical and physical conditions. The independent effects of (1) (hyper)dissipation and initial spectrum, (2) rotation rate, and (3) Rossby deformation radius are carefully isolated and studied in detail. In the nondivergent and nonrotating case, an initially turbulent flow evolves into a vorticity quadrupole at long times, a direct consequence of angular momentum conservation. In the presence of sufficiently strong rotation, the nondivergent long-time behavior yields a field dominated by polar vortices—as previously reported by Yoden and Yamada. In contrast, the case with a finite deformation radius (i.e., the full spherical shallow-water system) spontaneously evolves toward a banded configuration, the number of bands increasing with the rotation rate. A direct application of this shallow-water model to the Jovian atmosphere is discussed. Using standard values for the planetary radius and rotation, we show how the initially turbulent flow self-organizes into a potential vorticity field containing zonal structures, where regions of steep potential vorticity gradients (jets) separate relatively homogenized bands. Moreover, Jovian parameter values in our simulations lead to a strong vorticity asymmetry, favoring anticyclonic vortices—in further agreement with observations.

The Changing Face of the Extrasolar Giant Planet HD 209458b

“…An alternate model of observed cloud level circulation on Jupiter as a surface expression of deep convective columns, oriented parallel to the planetary rotation axis, also exists (Sun, Schubert, & Glatzmeier 1993;Schubert & Zhang 2000). However, only the shallow-layer model has thus far qualitatively reproduced the essential features (bands, zonal winds, persistent spots, and anticyclonic dominance) on all of the giant planets in our solar system using only the observed values of physical parameters (Cho & Polvani 1996a).…”

“…High-resolution, shallow-layer barotropic models in spherical geometry have been successfully used to model the atmospheric dynamics of the solar system gaseous giant planets (Williams 1978;Cho & Polvani 1996a) as well as stratospheric (stable region) phenomena on the Earth (see, e.g., Juckes & McIntyre 1987). An alternate model of observed cloud level circulation on Jupiter as a surface expression of deep convective columns, oriented parallel to the planetary rotation axis, also exists (Sun, Schubert, & Glatzmeier 1993;Schubert & Zhang 2000).…”

Statistics of an Unstable Barotropic Jet from a Cumulant Expansion

“…Though not crucial for this paper, we prefer to work on the sphere and not in the ␤-plane approximation, as the sphere can support interesting phenomena not found on the plane (e.g., Cho and Polvani 1996). The absolute vorticity q is given by…”

Brightness power spectral distribution and waves in Jupiter's upper cloud and hazes

“…On rapidly rotating planets, like Jupiter, the wind system outside the Equator is under quasi-geostrophic equilibrium (Ingersoll et al 2004). The theoretical study of the quasi-geostrophic turbulence in a sphere (Rhines 1975) combined with numerical studies with or without forcing and dissipation (Williams, 1978(Williams, , 1979Cho and Polvani, 1996;Scott et al 2007), showed that zonal winds could result from the combination of planetary rotation with the so-called inverse cascade of energy, characteristic of twodimensional turbulence (Vallis 2006). The measurement of turbulent motions (turbulent wind speeds, u', v') is not an easy task since typical values are similar to the standard measurement errors.…”