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Figure 5.9. Stability of zonal structure of Jupiter and Saturn assuming: (1) a shallow layer and barotropic instability; and (2) assuming deep Taylor-Proudman columns (Ingersoll and Pollard, 1982). The left-hand figure shows the curvature of the zonal winds of Jupiter derived from both Voyager 1 and 2 data. The barotropic stability curves are the smooth curves for positive values of d2u/dy2, and it can be seen that the winds on Jupiter clearly exceed this curve at several points. The winds do seem stable with respect to the deep Taylor-Proudman model, however, since the curvature is rarely more negative than the second stability curve shown, derived by Ingersoll and Pollard (1982). Similar results are shown on the right-hand side of the figure for the zonal winds measured on Saturn by Voyagers 1 and 2. Reprinted with permission of Elsevier.

boundary was predicted to intersect the surface spheroid. This critical latitude is at approximately 40°-45° for Jupiter and 65° for Saturn and it is interesting to note from Figures 5.2-5.4 that both Jupiter and Saturn have three eastward jets between the equator and these latitudes (Smith et al., 1982). Polewards of this latitude, less organized motion would be expected since Taylor-Proudman columns could not pass right through the planet, but instead would intersect the metallic-hydrogen/ molecular-hydrogen phase boundary. An additional advantage of this model is that the stability criterion of zonal flow is different from the barotropic instability criterion mentioned earlier and is found to be better satisfied by the zonal flows of all the giant planets (Ingersoll and Pollard, 1982) as can be seen in Figure 5.9 for the case of Jupiter and Saturn. It should also be mentioned that the deep zonal flow need not be forced by deep convection. Showman et al. (2006) show that surface forcing can also lead to deep zonal flow if the interior has low static stability, which it should do. Hence, eddy pumping can accelerate the jets, which can accelerate the interior also.

However, this model, while very elegant, has suffered a setback following the recent Jupiter flyby by CassinijHuygens. The Cassini ISS camera has now found organized, long-lived zonal motion extending all the way to Jupiter's poles and high-latitude organized polar motion has also been observed on Saturn by Voyager,

Tulie shortens with latitude

Figure 5.10. Vortex tube-stretching associated with Taylor-Proudman columns. Columns intersecting at latitudes less than the critical latitude pass right through from north to south. Motion of the column towards the rotation axis stretches the column. Columns at latitudes poleward of the critical latitude do not pass through to the other hemisphere. Furthermore, motion of the column towards the rotation axis compresses the column. Hence, this model would suggest very different atmospheric flow on either side of the critical latitude, which is not actually seen on any of the giant planets.

Tube lengthens v with latitude

Figure 5.10. Vortex tube-stretching associated with Taylor-Proudman columns. Columns intersecting at latitudes less than the critical latitude pass right through from north to south. Motion of the column towards the rotation axis stretches the column. Columns at latitudes poleward of the critical latitude do not pass through to the other hemisphere. Furthermore, motion of the column towards the rotation axis compresses the column. Hence, this model would suggest very different atmospheric flow on either side of the critical latitude, which is not actually seen on any of the giant planets.

Cassini, and ground-based observations. There is also a problem that very different behavior is expected either side of the critical latitude in that on the equatorward side, northern motion stretches the columns, but on the poleward side, northern motion compresses the columns (Figure 5.10). Such differential behavior in atmospheric flow is not observed.

To address these problems, computer modeling of the dynamics in the molecular-hydrogen envelopes in giant planets has been performed (Aurnou and Olson, 2001; Christensen, 2001, 2002; Grote et al., 2000). Such models solve the Navier-Stokes momentum equations, subject to the Boussinesq approximation that density variations are negligible except where coupled with buoyancy terms. Key parameters in the resulting equations are the Ekman number (ratio of frictional to Coriolis forces), the Prandtl number (ratio of kinematic viscosity to thermal diffusivity), and the Rayleigh number, which is a measure of convective vigor (Vasavada and Showman, 2005). Many solutions exist, depending on these parameters, but a general feature is that convective overturning in the molecular-hydrogen region gives rise to deep, symmetrical zonal flows from the equator to the pole. Although such models do produce an eastward-flowing jet at the interior, as in seen on Jupiter and Saturn, a major drawback is that the jets are generally too wide with just the equatorial eastward and half of the adjacent westward jets appearing at latitudes less than the critical latitude, where a cylinder just touching the metallic-molecular boundary intersects the surface spheroid. If it can be argued that the metallic-hydrogen region is actually rather larger than has previously been estimated, or that vigorous convection is limited to the outer layers of the molecular-hydrogen region then the critical latitude moves closer to the equator and the flow resembles the gas giants much more closely (Heimpel and Aurnou, 2007; Heimpel et al., 2005), with wider jets near the equator and narrower ones at latitudes greater than the critical latitude. Recent estimates of Guillot et al. (2004) put the metallic-hydrogen/molecular-hydrogen boundary at a pressure level of 1 Mbar to 3 Mbar, placing it somewhere in the range 0.7 to 0.9 Jupiter radii, which would be consistent with these models, although it is also possible that a barrier to convection lies within the molecular-hydrogen mantle itself, since so little is currently known about the interior structures of these planets. However, a drawback of these models is their use of the Boussinesq approximation, which leads to an incompressible continuity equation. Since the actual density of the Jovian interior increases by a factor of ^ 1,000 from the cloud decks down to the molecular-metallic boundary, the real circulation in Jupiter's interior is likely to be more complicated than these models suggest. Two-dimensional studies solving the anelastic equations, which are similar to the Boussinesq Navier-Stokes equations, but which have a more realistic continuity function containing an assumed variation of density with radius, have shown that convection is indeed altered when density variations are included (Evonuk and Glatzmaier, 2004). Another drawback of these deep models is that none can produce the multitude of vortices seen in giant planet atmospheres, which are easily produced by shallow-layer models.

Although there are many problems with the Taylor-Proudman column theory, it remains an attractive and intriguing possibility and thus it is of great interest to determine how deep the winds in the giant planets actually extend. Coherently organized internal motion of the kind just described will create perturbations in the gravitational equipotential "surface" of the planets, which will affect gravitational /-coefficients. For Neptune, which has a very broad and rapidly rotating zonal structure, the measured J4-coefficients have been found to be inconsistent with Taylor-Proudman columns extending throughout the planet and instead the winds are concluded to be limited at most to the outer hydrogen-helium shell (Hubbard, 1997d). However, this does not discount the Taylor-Proudman theory since for Neptune (and Uranus), the transition to the icy interior, which should present the same boundary to Taylor-Proudman columns as the transition from molecular hydrogen to metallic hydrogen does in Jupiter and Saturn, occurs just a few thousand kilometers below the cloud tops. Planets such as Jupiter and Saturn have a much finer zonal structure and thus the effects of such columns (if they exist) will only be apparent in the higher order /-coefficients, which have not yet been accurately measured. However, the Juno mission (Section 8.6.1), due for launch in 2011, will be placed in a close polar orbit to Jupiter and close tracking of its motion should allow the determination of Jupiter's higher order /-coefficients.

One-and-a-half-layer models

We have seen that the apparently deep nature of the zonal flow argues against shallow-layer models of the giant planets in favor of perhaps a system of co-rotating cylinders. However, this deep-atmosphere theory is apparently cast into doubt by the absence of a difference in atmospheric flow at the latitude of the cylinder tangential to the metallic-hydrogen/molecular-hydrogen boundary and the observation of zonal jets extending polewards of the critical latitudes. Hence, the real flow of these planets would appear to be more complicated than either of these more simple approaches suggests and ideally a fully three-dimensional model of the atmospheric flow needs to be constructed to investigate the dynamics of the giant planets. While there has been some progress in this area, as we have seen in the previous section ("Deep-layer models''), simplifying assumptions have to be made in order to make the calculations computationally achievable, whose validity calls into question the predictions of such models. However, since the deep interiors of the giant planets are almost certainly barotropic, and since the surface layers exhibit many similarities with shallow-layer models, it may well be that these planets can be represented by models that have deep barotropic flows up to near the observable levels, capped by a statically stable layer that is driven by turbulent energy, injected from below (Leovy, 1986). One way of representing this idea is to use "one-and-a-half''-layer models, where deep atmospheric flow is represented as latitudinal variations in the height of a lower boundary layer. These height variations may be determined by tracking the vorticity of features on the edge of anticyclones such as the GRS, which have significant north/south motion and assuming that the potential vorticity q = ( + f )/h is conserved. Hence, the effective height h of the weather layer may be determined by observing variation in the vorticity of the flow (Dowling and Ingersoll, 1988, 1989). One-and-a-half-layer models with lower topography determined in this way are able to simulate the zonal flow of Jupiter (and other planets) and mimic the spontaneous formation and growth of large oval circulations. However, like shallow-layer models, they have the disadvantage that they need to be continuously forced in order to maintain the flow and thus seem to lack the principal energy source that maintains the zonal circulations of these planets.

The idea of a thin stable surface layer on top of barotropic, but differentially rotating interior cylinders also provides one explanation for how Jupiter, Saturn, and Neptune, which have significant internal heat sources, but differential solar heating all radiate approximately equally in all directions. The solar heating will be greatest at equatorial latitudes and this will tend to increase the static stability of the air and thus reduce the amount of internal heat that is convectively transported. Conversely, solar heating is least at the poles, which will tend to make the atmosphere neutrally statically stable and thus convection is uninhibited. Hence, more internal heat is radiated at the poles than the equator, counteracting the differential absorption of sunlight. However, the fact that Uranus, which also has significant differential solar heating, but negligible internal heat, also radiates equally in all directions suggests that reality may once again be more complicated!

Eddy-mean interactions

The interaction between the zonal mean flow and waves/vortices is currently unclear and opinion is divided between two main points of view. Some scientists believe that the zonal motions are, in effect, a free mode of a low-viscosity atmospheric circulation driven directly by internal energy and absorbed sunlight and hence that the observed eddy motion is as a result of turbulence at the belt/zone boundaries, with the eddies drawing their energy from the mean flow. The alternative point of view is that it is the eddies themselves which primarily draw their energy from internal sources and absorbed sunlight, and that it is these eddies that then drive the zonal flow and the large vortices.

One feature of the atmospheres of Jupiter and Saturn that argues in favor of zonal winds being driven by eddies (which includes wave motion) is the superrotation of the equatorial zones of these planets. Another argument in favor of eddy driving comes from the analysis of the motion of eddies observed in Voyager 1 and Voyager 2 images of Jupiter by Beebe et al. (1980) and Ingersoll et al. (1981). In these studies, observation of the motion of individual clouds allowed estimation of Reynolds stress u'v' which is the average northwards transport of momentum by eddies. [NB: Here zonal and meridional winds have been split into their zonal mean and transient, or eddy, components: u = u + u', v = V + v'.] Ingersoll et al. (1981) found that the eddies were pumping momentum into the jets and thus sustaining them. This scenario was cast into doubt by Sromovsky et al. (1982) who suggested that Ingersoll's conclusion was probably caused by a biased sampling of prominent eddy cloud features and that a more uniform spatial sampling showed no evidence for eddy pumping. However, more recent cloud-tracking observations by Cassini at Jupiter (Salyk et al., 2006) and Saturn (Del Genio et al., 2007) supports the theory that it is the eddies that drive the jets, not the jets that drive the eddies. However, a note of caution is sounded by Vasavada and Showman (2005), who point out that the real situation might again be more complicated than has been previously considered and note that Read (1986) showed that momentum transfer depends not only on terms such as —d(u'v')/dy, but also many other terms, all of which need to be known before the rate of change in mean zonal velocity u with respect to time can be calculated. In some circumstances it may be sufficient to consider only the Reynolds stress u'v', but this has not yet been proven to be the case in giant planet atmospheres.

Of course, if the eddies drive the jets then that begs the question what drives the eddies! One possibility is that eddies may be produced by baroclinic instability, which releases stored potential energy set up by horizontal temperature gradients (Ingersoll, 1990). However, the only temperature variations that have been observed are associated with the jets and it is not tenable to believe that the jets sustain the eddies, which then sustain the jets! Another possibility is that the smallest eddies derive their energy by moist convection and latent heat release as suggested by Ingersoll et al. (2000).

Laboratory simulations

Modeling the dynamics of giant planet atmospheres with computer models remains a challenging exercise, even with the most modern and advanced computers. An alternative approach to computer modeling is to conduct laboratory experiments with forcings and conditions matching as closely as possible those found in giant planet atmospheres. Using a rotating annulus of fluid that was heated on the inside and cooled on the outside, Read and Hide (1983, 1984) were able to generate baroclinic eddies with many of the features of Jupiter's Great Red Spot. More recently, Read et al. (2004) and Aubert et al. (2002) have shown that in a shallow rotating tank with a sloping bottom to represent the fl effect, multiple jets can form with widths that scale with the Rhines length.

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