For a purely barotropic flow, the CSP analysis reduces ∂Q/∂y to β-∂2U/∂y2, leading to stability criterion (i) taking the familiar Rayleigh–Kuo form for the stability of a two-dimensional flow. In effect this represents the need for an inflection point (or related feature) in a zonal shear flow as a necessary but insufficient criterion for its instability. Such a criterion is necessary to enable the possibility of supporting a pair of zonally propagating Rossby-like waves that can phase-lock across the inflection point and lead to the growth of an instability e.g. via over-reflection e.g. see. The combination of local vorticity gradients and Doppler shifting of Rossby-like wave trains located in different parts of the flow allow for the possibility of pairs of such wave trains to propagate at the same velocity relative to their common frame of reference. Provided that their lateral extent is sufficient for them to interact, these pairs of wave trains can remain mutually coherent and stationary in a common frame of reference moving with the waves, allowing them to interact strongly and draw kinetic energy from the background flow if their phase relationship is favourable.

This generally entails a phase tilt across the channel in such a way that the waves “lean into” the sheared zonal flow so that the momentum flux u∗v∗‾ acts to reduce the shear of the background zonal flow. This correlation between the eddy momentum flux and zonal shear is quantified by the CK term in the Lorenz energy budget, 2CK=d[u]‾dy[u∗v∗]‾, where u∗ and v∗ are perturbation velocities with respect to the zonal mean, indicated by the overbar, angle brackets denote a vertically integrated areal average and square brackets denote a time average. This leads naturally to CK>0 such that kinetic energy is transferred from KZ to KE.

It turns out that generating a flow in the laboratory with such an inflection point favourable for supporting phase-locked Rossby wave pairs is reasonably straightforward because such a flow structure occurs spontaneously in a viscous Stewartson shear layer. A detached shear layer may be produced in a rotating, homogeneous fluid by the use of differentially rotating horizontal boundaries, e.g. by the use of a differentially rotating disk or ring at the centre of a cylindrical tank. This was originally investigated by and later explored in experiments by , , and . A schematic diagram of the apparatus is shown in Fig. a–b, and a typical apparatus is illustrated in Fig. c.

The resulting directly forced zonal flow has a clear inflection point in its radial velocity profile in such a way that the radial vorticity gradient clearly changes sign. A typical example is illustrated in Fig.  for a jet-like flow, driven by a differentially rotating ring. The azimuthal velocity is shown in Fig. a, while the corresponding vorticity gradient is shown in Fig. b.

This would appear to suggest that the flow is unstable, yet it is observed in this state in a time average under conditions in which a regular azimuthal wavenumber m=6 flow is found. The CSP condition, however, is only a necessary condition for instability, not a sufficient one. An alternative condition for instability was considered by and is often referenced as “Arnol'd's second stability theorem” e.g.. This approach considers the instability that could arise from the interaction of two (or more) parallel wave trains (commonly, as herein, taken to be Rossby waves, although this may be generalised to other interacting wave modes; e.g. see ), each centred on a local maximum or minimum of the potential vorticity gradient. The existence or otherwise of a steadily moving reference frame in which both wave trains, Doppler shifted by the zonal flows in their vicinity, can be at rest relative to each other provides a criterion for their sustained mutual interaction and hence exchange of momentum to gain energy at the expense of the background zonal flow. The resulting condition for perturbations to be stable is of the form 3u‾-α∂Q‾/∂y≥L02, where L0 is a constant length scale which, depending on the model, may represent the Rossby radius of deformation (LD) and α is a constant representing the speed of the waves in a shifted inertial reference frame. The marginal stability condition is set by replacing the inequality in Eq. () with an equality such that u‾ is linearly related to ∂Q‾/∂y with an offset α at the intercept where ∂Q‾/∂y=0 e.g..

In practice, however, marginally stable real flows with both forcing and dissipation will not obey this condition exactly but may approximate to it if the forcing and dissipation are relatively weak compared to the advective transport of Q. For the laboratory flow presented in Fig. a and b, this is illustrated in Fig. c, which shows a plot of u‾ against -∂Q‾/∂r (=∂Q‾/∂y with y directed towards the rotation axis), where Q is now the shallow-water potential vorticity 4Q=ζ+2Ωh(r), ζ is the relative vorticity and h(r) is the radius-dependent depth of the fluid layer. Figure c clearly shows a strong and nearly linear correlation between ∂Q‾/∂y and u‾ where u‾>0.2 cms-1, corresponding to the core of the zonal jet in Fig. a. Thus, even though the equilibrated time-averaged flow continues to satisfy the CSP condition for instability, it has effectively stabilised the flow to a configuration that is close to marginal stability with regard to Arnol'd's second stability condition (hereafter “Arnol'd II”), as a result of a process which can be regarded as a form of barotropic adjustment.

The resulting flows thus equilibrate to a marginally stable state in which the original jet meanders in the form of a train of azimuthally travelling waves, on either side of which are patterns of closed vortices whose vorticity matches in sign with that of the flanks of the zonal jet. The flow is typically found to select just one dominant zonal wavenumber in many of the published experiments (e.g. see Fig. ), the wavelength of which depends mainly upon a combination of the Rossby Ro and Ekman E numbers, commonly defined e.g. as 5Ro=Rω2Ω‾d,6E=νΩ‾d2, where R is the radius and ω is the relative angular velocity of the shear layer or jet, ν is the kinematic viscosity of the fluid, and d is the depth of the fluid layer. Ω‾ is the mean angular velocity of the fluid, representing the average of the turntable rotation rate Ω and the differential rotation rate ω.

As shown by and , although the inviscid CSP criterion (i) or (more realistically) Arnol'd II criterion needs to be satisfied for the driven jet to be unstable, the observed stability boundary in a real, viscous fluid is also consistent with the existence of a critical Reynolds number Rec, defined with respect to a length scale of the same order as that of the E1/4 Stewartson layer such that 7Rec=ULSν≃Ro⋅E-3/4≃27, where 8LS=(E/4)1/4d is the width of the viscous Stewartson layer. This expression provides a reasonably accurate description of the observed stability boundary (e.g. see Fig. ) for both the jet and detached shear layer flows, such that for Re<Rec the flow observed is essentially axisymmetric with no discernible wave perturbation.

The corresponding boundaries between different flow regimes dominated by different wavenumbers are observed empirically to lie almost parallel to the main stability boundary, suggesting that the wavenumber selected is largely determined by the effective Reynolds number and hence the effective supercriticality of the barotropic instability (Re-Rec). In fully developed flows this may be consistent with other work on scale selection in barotropic instability, which associates the most unstable wavelength with a scale comparable to the width of the background jet or shear layer e.g.. In these barotropic instability experiments, however, the equilibrated amplitude of unstable waves grows with supercriticality, effectively broadening the width of the original jet beyond its initial Stewartson layer width. At its equilibrated finite amplitude, therefore, strongly supercritical flows will tend to favour longer wavelength disturbances, qualitatively consistent with the observed ordering of different wavenumber regimes. This can be clearly seen in Fig. , in which the lower wavenumber states are seen to fill the entire tank with large meanders and vortices, whereas high wavenumber flows tend to be more tightly confined in radius to the vicinity of the unstable jet beneath the rotating ring. At large supercritical Reynolds numbers, however, the equilibrated flow may no longer be steady but may become chaotic, as observed at the lowest Ekman numbers (see Fig. ), although this has not been explored in much detail as of yet.

This method of forcing and maintaining a barotropically unstable flow is relatively strong and fast, so even at large equilibrated amplitudes the zonally averaged flow maintains a reversal in the lateral gradient of potential vorticity. The forcing therefore would seem to be too strong to allow the zonally symmetric component of the flow to maintain itself close to a marginally unstable state. The resulting equilibrated flow is therefore strongly nonlinear, especially at relatively large effective Reynolds numbers.

Although from an energetic viewpoint it is reasonably clear why the basic state maintained by the differential heating in the rotating annulus experiment can lead to the release of potential energy to energise the growth of wave-like perturbations, it is less clear how the flow might satisfy the CSP criteria for instability. Early work by compared the conditions under which active sloping convection was observed in the laboratory with various extensions of the linear model of baroclinic instability by . Provided that account was taken of the dissipative effects of viscous Ekman layers adjacent to the upper and lower boundaries, and these comparisons found generally good quantitative agreement for the onset of the instability around a Burger number, defined as 11Bu=N2d24Ω2L2, of the order RoT/4≃0.5. Such an analogy, however, presumes that potential vorticity gradients in the interior are negligible compared with the thermal gradients at the horizontal boundaries, satisfying the CSP instability condition through criterion (iv).

In practice, however, the strong radial flow within the Ekman layers in typical annulus experiments rapidly transfers hot or cold fluid across the domain, maintaining relatively weak horizontal thermal gradients at the top of each Ekman layer. This would suggest that the potential vorticity dynamics is controlled much less by the structure of the flow close to the horizontal boundaries and is more strongly influenced by a change of sign of ∂Q/∂y in the interior flow. This would indicate that the basic zonal flow maintained by the differential heating at the side boundaries is closer in character to an internal jet e.g.. Baroclinic or barotropic instability would therefore occur through satisfying the CSP conditions directly through criterion (i). The consequences of this in the context of rotating annulus experiments were explored by , who noted that idealised, inviscid, baroclinic internal jets without any lateral shear would become unstable at around the same value of the Burger or thermal Rossby number as (or slightly larger than) the classical Eady problem. This was broadly in agreement with experiments, which did show a tendency for the onset of instability to occur at slightly larger values of RoT than the Eady problem would suggest. However, the critical value of Bu or RoT was found to be quite strongly sensitive to the addition of lateral shear to a baroclinic internal jet .

Numerical analyses, using more realistic linearised models based on the full Boussinesq Navier–Stokes equations with molecular viscosity and thermal diffusion e.g.and references therein, however, show close agreement with experimental measurements and also reveal sensitivity of the instability to other factors such as the Prandtl number, especially at low values of both RoT and T along the boundary of the so-called “lower-symmetric” regime. At a high Prandtl number, the boundary roughly follows a line of constant ΔT, indicative of a critical Rayleigh (Rac) or Grashof number (Grc) in a similar way to the criterion for Rayleigh–Bénard convection. The product RoT⋅T actually has the form of a Grashof number such that RoT⋅T=Grc is consistent with the shape of the lower-left instability boundary shown in Fig. .

As instabilities develop and equilibrate, azimuthally travelling waves emerge with characteristic three-dimensional structures and azimuthal phase tilts in radius and height. These result in eddy fluxes of heat and momentum that lead to energy conversions between potential and kinetic energies in the zonally averaged and perturbation fields. A typical example of the Lorenz energy cycle for an equilibrated baroclinic wave developed from a baroclinically unstable initial state is shown in Fig. . This was computed by from a numerical simulation of flow in a rotating annulus experiment from solutions of the Boussinesq Navier–Stokes equations in cylindrical annular geometry. The boundary conditions represent non-slip, rigid sidewalls and horizontal end walls, thermally insulating end walls, and isothermal sidewalls at temperatures Ta and Tb at inner and outer radii ra and rb, with ΔT=Tb-Ta=4 K, ra=2.5 cm, rb=8.0 cm, d=14 cm and rotation rate Ω=3.0 rads-1, corresponding to RoT=0.056.

The values shown in Fig.  clearly indicate that the conversion terms implicated in energising baroclinic instability (CA and CE; see Fig. b) are dominant in the exchanges between eddies and the zonal-mean flow. CK is at least a factor of 10 smaller than CA though in the same sense for the generation of KE, indicating that the dominant instability has a somewhat mixed baroclinic–barotropic character. The direct zonal-mean exchanges (CZ) are also of a similar magnitude to CA and CE and are consistent with a thermally direct overturning circulation in parallel with the conversion of available potential energy to KE.

As mentioned above, baroclinic adjustment is commonly associated with a tendency for baroclinic instability to equilibrate at a finite amplitude in such a way as to modify the initially unstable basic state towards a less unstable configuration. But it is less clear how this tendency might manifest itself in practice. However, one possible way this might be observable could be in the way heat transport by baroclinic disturbances varies with external parameters as the instability threshold is exceeded.

Figure  shows a series of calorimetric measurements of total heat transport in a differentially heating rotating annulus experiment for fixed ΔT as Ω is varied see. In this case, heat transport is non-dimensionalised with respect to thermal conduction, either by the Nusselt number, Nu, defined by 12Nu=Hln⁡(rb/ra)2πκdρcHΔT, where H is the total heat transport in W, κ is the thermal diffusivity of the working fluid and cH is its specific heat capacity, or by the Péclet number Pe=Nu-1. Thus the solid symbols in Fig.  show little variation with Ω or RoT until the highest rotation rates. This corresponds to where the regular wave regime in the laboratory experiments begins to break down towards small-scale irregular flows and geostrophic turbulence. The solid line in Fig. , on the other hand, shows the variation of Nu for a purely axisymmetric flow, obtained from numerical simulations in which non-axisymmetric eddies were suppressed seefor details. This shows that the axisymmetric flow under similar conditions decays much more quickly with Ω than when baroclinic waves are allowed to develop. The corresponding contribution to the dimensionless total heat transport by baroclinic waves is indicated by the open symbols, which are obtained from the difference between total heat transport and that of the pure axisymmetric flow. This clearly shows that the eddy contribution increases systematically from zero as the rotation rate is increased beyond the critical value for the onset of instability, until it begins to saturate and even begin to turn over for RoT<0.1.

The observation that total heat transport may be almost independent of rotation rate over a wide range of Ω, from the axisymmetric regime into the fully developed baroclinic wave regime, has been noted since the early work of , who was the first to carry out calorimetric measurements of heat transport in rotating annulus experiments. This tendency for the heat transport to remain close to its value at Ω=0, even though the (mainly axisymmetric) boundary layer contribution should reduce substantially over the same range of parameters, is unlikely to be coincidental, but it more likely reflects a systematic result of equilibrated baroclinic instability in modifying the overall flow to transport almost as much heat (and therefore release almost as much potential energy) as the non-rotating flow. The non-rotating flow represents a fully relaxed state, in which isotherms and isopycnals are as nearly horizontal as possible (except in conductive boundary layers) and is therefore a state that would be energetically stable to further baroclinic instabilities. In this sense, these results demonstrate that rotating annulus flows are capable of exhibiting a relatively strong form of baroclinic adjustment throughout the regular baroclinic wave regime. It is only when the regular regime begins itself to become unstable at values of RoT≪0.1, where the dominant energetic length scale of the baroclinic waves becomes significantly smaller than the width of the imposed baroclinic zone, that the eddies are no longer able to sustain the level of heat transfer required to maintain the fully relaxed flow state and that the baroclinically adjusted state breaks down to a more strongly supercritical flow.

Such a tendency would seem to be quite general, and so it is of interest to see whether similar trends might be observable in a planetary atmosphere in which differential heating is maintained by (radiative) relaxation towards a baroclinically unstable radiative equilibrium state.

Its similarities with Earth are also reflected in its potential vorticity structure, as can be seen e.g. in Fig. , which shows the zonal-mean, zonal wind and QGPV gradient for a typical atmospheric state during winter in the northern hemisphere on Mars, from data obtained from the Mars Climate Database (e.g. ; see also http://www-mars.lmd.jussieu.fr, last access: 30 March 2020). As can be seen in Fig. a, zonal winds are typically concentrated into deep, mid-latitude jet streams during winter and the adjacent seasons, with strong vertical and horizontal shears and a pronounced latitudinal tilt with altitude. The corresponding structure in QGPV (see Fig. b) shows ∂Q/∂y (∂Q/∂ϕ) positive in the jet itself (cf. Fig. a) but with weaker reversals in sign on either side. Together with the positive equatorward temperature gradient at the surface in the north, this would indicate that the CSP conditions can be satisfied in this season either by criterion (i) or (iii), much as on Earth. The atmosphere might then be expected to exhibit either baroclinic or barotropic instabilities, or a mixture of both, although the linear instability analysis of suggests that the baroclinic instability mechanism dominates for Martian conditions, while the barotropic shear mechanism acts to damp the baroclinic instability.

It is interesting to note that, at this time during southern summer, the corresponding zonal jet stream becomes much weaker and even reverses direction in places. The QGPV gradient does exhibit a weak reversal in sign, though the equatorward thermal gradient near the surface also reverses in sign during summer so that conditions are much less favourable for baroclinic instability via CSP criterion (iii). This is consistent with the observed near suppression of baroclinic instability in Martian summers at mid-latitudes e.g., except for occasional very shallow disturbances that are sometimes observed on small horizontal scales close to the polar cap edge e.g..

The corresponding Lorenz energy budget is needed to establish which is the dominant mechanism to energise the Martian eddy circulation. This has been computed recently by , based on an assimilated analysis of orbiting spacecraft measurements of Mars over a period of at least 3 Mars years . The data were assimilated into the UK version of the Laboratoire de Météorologie Dynamique (LMD) Mars GCM to produce a daily record of Martian meteorology over the entire period, from which the energy budget could be computed (see Fig. b).

The results, illustrated e.g. in Fig. b, show a qualitatively similar pattern to that of Earth, with the majority of dynamically active energy in the atmosphere stored as zonal potential energy AZ. The dominant conversion terms are the CA and CZ conversions from AZ to AE and KZ respectively (at 50 mWm-2) and especially the CE term (at 110 mWm-2). This indicates a strong energetic role for the thermally direct axisymmetric (Hadley cell) overturning circulation as well as the principal baroclinic conversions CA and CE maintaining synoptic eddies. This confirms the role for baroclinic instability as suggested from the configuration of potential vorticity gradients in Fig. b. The barotropic eddy-zonal conversion CK, however, is positive and around half the amplitude of CA, suggesting a role for exchanges consistent with a mixed baroclinic–barotropic instability. The GE source term for AE is also quite substantial and of the same order as CA and CZ. This is because the day–night contrast in solar heating on Mars generates a strong thermal tide response which contributes to the AE and KE reservoirs, although this varies significantly with season and the amount of dust loading in the atmosphere. As discussed by , this budget does exhibit fairly significant seasonal variations, also between the northern and southern hemispheres, but the overall structure of the energy budget remains largely unchanged.

A key difficulty in determining the PV structure of Venus's atmosphere arises from a lack of detailed observations of winds and temperature within and beneath its main cloud decks e.g. see. It is necessary, therefore, to fall back on numerical models of Venus's atmospheric circulation for the kind of information needed to compute quantities such as ∂Q/∂y. But the numerical simulation of Venus's atmospheric circulation is still at a relatively immature state, with realistically forced models appearing only recently e.g.. Simpler models have been studied more extensively but have struggled to reproduce key features of the circulation and have exhibited significant divergence among different model formulations e.g.. This may reflect important differences e.g. in the computation of static stability (and hence the ratio f/N) and in how well different dynamical cores are able to conserve and transport angular momentum.

For the present purpose, therefore, we follow and in examining a somewhat idealised form of the zonal-mean state of a Venus-like atmosphere. This is illustrated in Fig. , taken from the work of , and shows a zonal-mean section of the zonal wind (Fig. a) and corresponding field of ∂Q/∂ϕ (in the average frame of the zonal wind at an altitude of 56 km; Fig. b). The zonal wind section (in the planetary reference frame) captures the main features of the observed zonal winds on Venus, with a monotonic growth of zonal wind strength with altitude at all latitudes towards maximum values of 100–120 ms-1 at around 70 km altitude but with the formation of jet-like features at mid-latitudes in both hemispheres. This idealisation then neglects any vertical shear above the cloud tops, although this may not be particularly realistic. However, the flow structure around the cloud top altitudes are of most interest here. The winds are constructed to be in gradient wind balance (between horizontal pressure gradients and a combination of Coriolis and centrifugal “forces”) with the corresponding pressure and temperature fields.

The distribution of ∂Q/∂ϕ in Fig. b indicates a concentration of the PV gradient at high latitudes, especially within the cloudy layers between altitudes of 55 and 65 km (although the clouds are not represented explicitly in this model). Moreover, this gradient clearly changes sign in the vertical direction while remaining positive over most of the domain at lower levels. This would immediately suggest the possibility of a baroclinic instability localised within the cloud deck levels by satisfying the CSP conditions for instability via criterion (i). A change of sign in the horizontal is only discernible in the layer around 56 km, however, suggesting that barotropic instabilities might also be possible, though again centred mainly within the cloud decks. Nearer the surface, ∂Q/∂ϕ>0, but surface temperature gradients are very weak, suggesting that Venus may be able to satisfy the CSP condition via criterion (iv) only marginally. Besides, the flow is far from being in geostrophic balance at these levels, although that would not necessarily rule out the possibility of a non-geostrophic form of baroclinic instability.

The energy budget for Venus is similarly not available from observations, and so we must rely on model simulations to compute the likely energies and exchanges. To the authors' knowledge, this has not so far been done for any realistically forced, comprehensive GCM of Venus that faithfully reproduces the observed winds and circulation. However, did compute Lorenz energy budgets for a set of somewhat simplified GCM simulations of Venus-like circulations, and so this is what is presented here. However, these model simulations did not fully represent the radiative forcing of the atmosphere, including its diurnal cycle, nor did they fully capture the observed distribution of angular momentum in the simulations. So the budget shown here is likely to be inaccurate in some degree and missing some key processes. Nevertheless, the results are worthy of attention as a first step towards a more accurate calculation.

Figure c presents the energy budget calculations for the spectral core case of with full damping as being reasonably representative of this class of Venus model (without an explicit diurnal cycle). The energy reservoirs show a very different distribution of energy from either Earth or Mars, with most of the dynamically active energy in the atmosphere residing in the zonal and eddy kinetic energies KZ and KE. The potential energy reservoirs, however, are certainly not negligible and suggest a significant concentration of available potential energy in the AE reservoir. AZ is relatively small, though, as would be expected for an atmosphere with relatively weak meridional temperature gradients. Among the conversion terms, CZ is the largest, indicative of a strong, thermally direct Hadley overturning circulation to maintain the very large KZ reservoir. The next largest conversion is CK, which evidently acts in the positive direction to convert from KZ to KE. This would suggest a dominance of barotropic processes, much as found in the idealised Earth-like simulations of Sect.  at low values of Ω∗.

CA, on the other hand, also acts in a positive direction (from AZ to AE), indicating some alignment with baroclinically active processes and perhaps suggesting a mixed baroclinic/barotropic character to the dominant eddy processes. CE, however, acts in a negative direction, from KE to AE, which would suggest that mechanical exchanges between eddy potential and kinetic energies dominate over buoyancy-driven instabilities. The one term that may be anomalous when compared with Venus is the potential energy source/sink GE. This appears to be acting to remove energy from AE at a fairly strong rate. But this model takes no account of the diurnal cycle and hence the input of thermal energy to the thermal tides. More realistic model simulations suggest that the thermal tides in and above the main cloud decks play an important role in maintaining the observed very strong atmospheric super-rotation e.g.. It seems more likely, therefore, that a more complete simulation of the Venus circulation would include a large positive contribution to GE, much as indicated for Mars in Fig. b. This is a question that should be followed up urgently through analysis of more realistic model simulations.

The occurrence of such instabilities, however, depends crucially on the distribution of PV within the weather layers of these planets. As with the other planets under discussion here, Jupiter's atmosphere needs to satisfy the CSP necessary condition for instability somehow. Without a solid surface, however, CSP criterion (iii) cannot be satisfied, unlike on Earth or Mars. Such an argument was used by to suggest only a minor role at most for baroclinic instabilities on Jupiter. But as subsequently noted by , the possibilities of satisfying the CSP condition through criteria (i) or (ii) remain, especially if Jupiter or Saturn possess a strong tropopause which can act as an internal interface that supports a horizontal thermal gradient. The analysis of noted that, in the absence of a significant horizontal shear and with a lower boundary that was essentially isentropic and weakly stratified, westward jets could satisfy the CSP condition through criterion (i) (or (ii) if the tropopause is considered as an upper boundary or interface). Their calculations indicated that such an “inverted Charney” form of baroclinic instability could have relatively fast growth rates and even lead to up-gradient momentum fluxes for instabilities on a small enough horizontal scale.

The vorticity structure of the zonal-mean flow at the cloud tops has been measured since the time of the Voyager spacecraft encounters with Jupiter in 1979. By tracking features in the ubiquitous ammonia cloud decks in successive images taken by the spacecraft, the pattern of eastward and westward jet streams could be derived as a profile in latitude from which the vorticity and its northward gradient could be obtained e.g.. These profiles indicate that the northward gradient of absolute vorticity robustly changes sign with latitude in a number of places, typically around the peaks of westward jets, leading to some suggestions that these jets may be prone to barotropic instabilities, since they would appear to satisfy criterion (i) for the CSP instability condition . The stretching term associated with the thermal structure of the flow may also be important, leading to some suggestions e.g. that vigorous lateral mixing of PV could result in a monotonic, staircase-like variation of Q with latitude which would be unconditionally stable to shear instabilities. Such a process that might hold the atmosphere close to a state of marginal instability would effectively constitute a form of barotropic adjustment by analogy with Stone's baroclinic adjustment hypothesis .

Observational studies that have combined wind measurements from cloud motions with remotely sensed temperature retrievals and a geostrophic balance assumption for both Jupiter and Saturn, however, also indicate significant reversals of ∂Q/∂y. Figure a shows an example of a section of ∂Q/∂y in latitude and height in the southern hemisphere of Jupiter, based on data from the Cassini spacecraft . This clearly shows patches where ∂Q/∂y<0 overlying westward jets but persisting over significant ranges in altitude. ∂Q/∂y even appears to change sign in the vertical direction in some locations in Jupiter's stratosphere, although these observations should be treated with caution, since the reconstruction of the flow at high altitudes requires the integration of the geostrophic thermal wind shear relation which can amplify the effects of measurement noise and uncertainties, especially in quantities that require spatial differentiation e.g. see. Being based on nadir observations, the retrieved thermal measurements also have rather limited vertical resolution. So it cannot be ruled out that structures associated with sharp thermal gradients along the lines suggested by might exist that would eliminate such reversals in the sign of ∂Q/∂y.

The distribution of Q beneath Jupiter's cloud tops, however, is virtually unknown because of a dearth of detailed measurements of winds, temperatures or composition. But there seems no reason a priori to conclude that internal changes of sign of ∂Q/∂y in the deep troposphere, or of ∂T/∂y at the tropopause, cannot exist and that therefore the CSP necessary condition for baroclinic or barotropic instability could be satisfied through criteria (i) or (ii).

A number of studies have appeared recently in which fully three-dimensional, time-dependent GCMs have been developed of Jupiter's or Saturn's weather layers which seek to capture the thermal and vorticity structures of the flow around and beneath the cloud tops e.g.. These models have included the effects of solar heating in the stratosphere and upper troposphere and upwelling internal heating from the deep interior and have been capable of capturing a number of realistic features such as the equatorial prograde jets and multiple eddy-driven zonal jets at mid-latitudes.

Figure  shows a cross section of ∂Q/∂y and u‾ from a GCM simulation of Jupiter's weather layer by (their run B, with a horizontal resolution of 512×256 points in longitude and latitude). As with the observations, this shows a generally clear correlation between u‾ and ∂Q/∂y in the upper troposphere but with some sign reversals of ∂Q/∂y in both the horizontal and vertical in relation to both westward and eastward jets in several cases. This would seem to confirm that the simulated flow sustains the necessary conditions for baroclinic instability through CSP criterion (i) as discussed above.

A preliminary Lorenz energy budget, computed using an updated, higher-resolution version of the Jupiter weather layer model described by , is shown in Fig. d. This represents the model's attempt to capture the energetics of a Jupiter-like weather layer between the pressure levels of 18 bars and 10 hPa. The energy reservoirs reflect a distribution of potential and kinetic energy typical of a rapidly rotating, stratified atmosphere, with most of the stored energy residing in the zonal-mean potential energy AZ. Smaller amounts are found in the KZ and AE reservoirs with KE being the weakest, though still containing much more energy per unit area than the corresponding reservoirs for Earth.

Energy conversions, however, are remarkably Earth-like in their relative sizes and directions, with CA and CE as the largest conversion rates in the sense expected for baroclinic instabilities. This confirms the indication from the PV structure that anticipated that baroclinic instabilities could occur and play a significant role in the circulation. CK acts to maintain the zonal-mean flow through transfers from KE, at a vertically integrated rate (0.61 Wm-2) that is not unreasonable with regard to the observed transfer rate near the cloud tops e.g.. CZ acts in the thermally direct sense but is relatively weak, reflecting the comparatively minor role played by zonally symmetric overturning circulations in the global circulation. The largest terms overall are CA, GZ and GE, indicating some very strong forcing of the zonal-mean potential energy field and conversion to eddy potential energy, much of which is then extracted by diabatic processes. A complication is that this model simulation includes both latent-heat fluxes and exchanges associated with a moist convection parameterisation, which may not be fully accounted for in the conventional Lorenz energy budget approach. But the overall structure of this cycle strongly suggests an important role for baroclinic instability, at least in this model simulation of Jupiter's weather layer.