Jupiter's winds and Arnol'd's 2nd stability theorem: Slowly moving waves and neutral stability
Abstract
Since the Voyager encounters in 1979, it has been known that Jupiter's cloudtop zonal winds violate the barotropic stability criterion. A vortextube stretching analysis of the Voyager wind data indicates that the more general CharneyStern stability criterion is also violated. On the other hand, the zonal winds determined by tracking cloud features in Hubble Space Telescope images taken in 1991 precisely match the zonal winds determined by tracking cloud features in Voyager images, and it is hard to understand how a complicated zonal wind profile like Jupiter's could be unstable and yet not change at all in 12 years. In fact, there are at least two known ways to violate the CharneyStern stability criterion and still have a stable flow. The better known of these is called Fjørtoft's theorem, or Arnol'd's 1st theorem for the case of largeamplitude perturbations. Although the FjørtoftArnol'd theorem has been extended from the quasigeostrophic equations to the primitive equations, the basic requirement that the potential vorticity be an increasing function of streamfunction is opposite to the case found on Jupiter, where the Voyager data indicate that the potential vorticity is a decreasing function of streamfunction. But this second case is precisely that which is covered by Arnol'd's 2nd stability theorem. In fact, the Voyager data suggest that Jupiter's zonal winds are neutrally stable with respect to Arnol'd's 2nd stability theorem. Here, we analyze the linear stability problem of a oneparameter family of sinusoidal zonal wind profiles that are close to neutral stability with respect to Arnol'd's 2nd stability theorem. We find numerically that the most unstable mode is always stationary, which may help to explain the slowly moving mode 10 waves observed on Jupiter. We find that violation of Arnol'd's 2nd stability theorem is both necessary and sufficient for instability of sinusoidal profiles. However, there appears to be no simple extension of Arnol'd's 2nd stability theorem to the primitive equations. Nevertheless, the primitive growth rates are small, and the primitive system is still governed by the quasigeostrophic neutralstability configuration.
 Publication:

Journal of Geophysical Research
 Pub Date:
 October 1993
 DOI:
 10.1029/93JE01520
 Bibcode:
 1993JGR....9818847S
 Keywords:

 Baroclinic Instability;
 Jupiter Atmosphere;
 Mathematical Models;
 Wind Profiles;
 Zonal Flow (Meteorology);
 Hubble Space Telescope;
 NavierStokes Equation;
 Voyager Project;
 Wave Interaction;
 Lunar and Planetary Exploration;
 Planetology: Fluid Planets: Meteorology