Comparison of the deep atmospheric dynamics of Jupiter and Saturn in light of the Juno and Cassini gravity measurements
Yohai Kaspi, Eli Galanti, Adam P. Showman, David J. Stevenson, Tristan Guillot, Luciano Iess, Scott J. Bolton
PPreprint for ISSI/SSRv - Comments welcome. Manuscript No. (will be inserted by the editor)
Comparison of the deep atmospheric dynamics of Jupiter and Saturnin light of the Juno and Cassini gravity measurements
Yohai Kaspi · Eli Galanti · Adam P. Showman · David J. Stevenson · Tristan Guillot · Luciano Iess · Scott J. Bolton
Submitted: 22-Aug-2019
Abstract
The nature and structure of the observed east-west flows on Jupiter and Saturn has beenone of the longest-lasting mysteries in planetary science. This mystery has been recently unraveled dueto the accurate gravity measurements provided by the Juno mission to Jupiter and the Grand Finaleof the Cassini mission to Saturn. These two experiments, which coincidentally happened around thesame time, allowed determination of the vertical and meridional profiles of the zonal flows on bothplanets. This paper reviews the topic of zonal jets on the gas giants in light of the new data from thesetwo experiments. The gravity measurements not only allow the depth of the jets to be constrained,yielding the inference that the jets extend roughly 3000 and 9000 km below the observed clouds onJupiter and Saturn, respectively, but also provide insights into the mechanisms controlling these zonalflows. Specifically, for both planets this depth corresponds to the depth where electrical conductivityis within an order of magnitude of 1 S m − , implying that the magnetic field likely plays a key role indamping the zonal flows. Keywords
Jupiter · Saturn · Juno · Cassini · Planetary Atmospheres · Gravity Science
Y. KaspiCorresponding authorDept. of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, 76100, IsraelTel.: +972-8-9344238Fax: +972-8-9344124E-mail: [email protected]. GalantiDept. of Earth and Planetary Sciences, Weizmann Institute of Science, Rehovot, 76100, IsraelA. P. ShowmanLunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721-0092, USAD. J. StevensonDivision of Geological and Planetary Sciences, California Institute of Technology. Pasadena, CA, 91125, USAT. GuillotUniversit´e Cˆote d’Azur, OCA, Lagrange CNRS, 06304 Nice, FranceL. IessSapienza Universit´a di Roma, 00184, Rome, ItalyS. J. BoltonSouthwest Research Institute, San Antonio, TX 78238, USA a r X i v : . [ a s t r o - ph . E P ] A ug Yohai Kaspi et al.
The most prominent features in the appearance of Jupiter and Saturn are their east-west banding,which have been observed ever since the invention of the first telescopes in the 17th century. Theseso called ’zones’ (bright regions) and ’belts’ (dark regions) are related to the two gas giants’ east-westjet-streams. The exact interplay between these zonal flows and the banded structure of the clouds isnot completely understood (see recent review by Fletcher et al. 2019), yet the eastward (westward) jetsare typically accompanied by a zone to the south (north) and a belt to the north (south). Jupiter hasabout six distinct jets in each hemisphere (Fig. 1), including a wide superrotating eastward jet aroundthe equator and narrower jets poleward. The strongest jet, reaching 140 m s − is at latitude 23 ◦ N, andis not accompanied by a similar jet in the southern hemisphere, creating a hemispherical asymmetry(Fig. 1). On Saturn, the winds are stronger, with a wider equatorial eastward flow (up to latitude ∼ ◦ ) reaching velocities of nearly 400 m s − . Poleward of that, Saturn has 3-4 distinct jets in eachhemisphere (Fig. 1). These jet velocities, measured by cloud tracking (e.g., Garc´ıa-Melendo et al. 2011;Tollefson et al. 2017) and typically quoted relative to Jupiter and Saturn’s magnetic field rotation, havebeen overall very consistent since the first spacecraft observations in the 1970s. As Saturn’s magneticfield is almost perfectly axisymmetric, this reference frame has a larger uncertainty for the case ofSaturn, although recent measurements and theoretical calculations have limited the rotation perioduncertainty to within a few minutes (Read et al. 2009; Helled et al. 2015; Mankovich et al. 2019).Prior to the recent Juno and Cassini missions there have been very little data regarding the flowsbeneath the cloud tops. The only in-situ measurements come from the Galileo probe, which descendedin 1995 into Jupiter’s atmosphere around latitude 6 . ◦ N, and found that the zonal wind velocityincreased from around 80 m s − at the cloud level, where the probe entered, to ∼
160 m s − ata depth of 4 bars, and from there downward the zonal velocity remained nearly constant down to21 bars (130 km) where the probe was lost (Atkinson et al. 1996). This indicated that the zonal flowwas not restricted to the cloud-level, although the depth where it was lost is only a mere fraction ofthe planetary radius, and thus this measurement did not provide definitive tests to separate betweentheories suggesting the flows are a shallow atmospheric phenomena (e.g., Williams 1978, 1979; Choand Polvani 1996), and theories suggesting the surface flows are just a surface manifestation of deepcylindrical columns extending deep into the planetary abyss (e.g., Busse 1976; Heimpel et al. 2005).On Saturn, Cassini observations indicate that low-latitude winds seem to be stronger at the 2-3 barlevel than at the cloud-level (0.5 bar), while mid-latitude winds seem to be nearly constant or weakerwith depth (Choi et al. 2009; Studwell et al. 2018). Above the cloud layer, tracking wind velocities isdifficult as the wind shear can only be indirectly inferred based on temperature measurements (Simon-Miller et al. 2006; Fletcher et al. 2007). Generally for both planets it seems that winds decay abovethe cloud-level but these measurements are very uncertain (S´anchez-Lavega et al. 2019).The question of how deep the observed jets extend has been debated extensively in the literaturesince the early observations by the Pioneer and Voyager missions in the 1970s. Particularly, the researchhas split into two different approaches for how to explain the jets. According to the first approach, thejets are suggested to be a shallow atmospheric feature, as appears on a terrestrial planet, thus assumingall the dynamics are limited to a shallow weather-layer. Geostrophic turbulence theory provides goodunderstanding to what sets the jet width and overall number of jets (Rhines 1975; Held and Larichev1996; Chemke and Kaspi 2015), and matches the numbers observed on Jupiter and Saturn. There havebeen many shallow-type models which have showed formation of jets similar to those on Jupiter andSaturn beginning with the models of Williams (1978, 1979), and over the years evolved to more complexmodels showing formation of multiple jets (e.g., Panetta 1993; Vallis and Maltrud 1993; Cho andPolvani 1996; Huang and Robinson 1998; Lee 2004; Smith 2004; Showman 2007; Kaspi and Flierl 2007;Scott and Polvani 2007). These shallow-type models most commonly do not exhibit superrotation, butwith particular configurations of bottom drag, internal heating, moist convection or thermal dampingthey can produce an equatorial superrotating jet and multiple high latitude jets (Scott and Polvani2008; Lian and Showman 2008, 2010; Liu and Schneider 2010; Warneford and Dellar 2014; Younget al. 2019; Spiga et al. 2020). The second approach considers deep convection models, in which thesource of the jets is suggested to be internal convection columns that interact to form the jets seen he deep atmospheres of Jupiter and Saturn 3 Jupiter
Zonal wind (m s -1 )-100 0 100 200 300 400 Zonal wind (m s -1 ) -80-4004080 La t i t ude SymmetricAntisymmetric
Saturn
Zonal wind (m s -1 )-100 0 100 200 300 400 Zonal wind (m s -1 ) Fig. 1
The zonal wind profile of Jupiter (left) and Saturn (right), divided in the bottom panels into the north-southsymmetric (black) and asymmetric (orange) parts. In the top panels the zonal wind profile for Jupiter (Tollefson et al.2017) is overlaid on a image taken by the Hubble Wide Field Camera in 2014, and for the Saturn case the zonal windprofile (Garc´ıa-Melendo et al. 2011) is overlaid on an image created by Bjrn Jnsson by combining Cassini and Voyagerimages and removing the rings. The grid in both images has a 20 ◦ latitudinal spread and a 45 ◦ longitudinal spread. Thescale of the zonal flows for Jupiter is the same as the longitudinal grid on the sphere and for Saturn it is triple. at the surface. These ideas have also emerged in the 1970s with the seminal papers of Busse (1970,1976), and evolved to more complex interior convention 3D simulations (e.g., Busse 1994; Sun et al.1993; Christensen 2001; Aurnou and Olson 2001; Wicht et al. 2002; Heimpel et al. 2005; Kaspi et al.2009; Jones and Kuzanyan 2009; Heimpel et al. 2016). These models naturally exhibit superrotationdriven by the convergence of convectively driven momentum near the equator, but do not naturallyproduce the multiple jet structure that appears at the higher latitudes. These two approaches havebeen debated greatly over the past several decades, but due to the lack of observational evidence, thedebate has remained unresolved (see reviews by Vasavada and Showman 2005 and Showman et al.2018). Now, following the Juno and Cassini gravity measurements (Iess et al. 2018, 2019), which arereviewed here, the discussion about the source and structure of the jets can be reinvigorated by thesenew evidence.The Juno gravity experiment is one of the key objectives of the Juno mission (Bolton 2005), withthe purpose of measuring Jupiter’s gravity spectra to high accuracy, and thereby providing informationabout Jupiter’s interior and atmospheric flows (Hubbard 1999; Kaspi et al. 2010). Juno has been inorbit around Jupiter since July 2016, orbiting Jupiter every 53 days with X and Ka-band radio links toEarth allowing measurements of Jupiter’s gravity field via Doppler shifts in the radio frequencies sentto Earth (Bolton et al. 2017). The measurements are obtained around the time of closest approach(perijove) at 4000 km above the cloud level (Iess et al. 2018). The perijoves are designed to give an Yohai Kaspi et al. overall 360 ◦ longitudinal coverage of Jupiter’s as the planet rotates underneath the orbiting spacecraft.In addition, due to the oblateness of Jupiter the perijoves drift about 1 ◦ in latitude poleward everyorbit, with the first being at latitude 3 ◦ N. As the Juno microwave radiometer and the Ka-band radioexperiment can not operate in tandem only a subset of the orbits have been devoted to gravity.Nonetheless, there have been enough gravity orbits to date that the error estimate of the measuredzonal harmonics has reached saturation.Motivated by the Juno mission polar orbital configuration, it was decided that during the CassiniGrand Finale (the final Cassini orbits before terminating the mission by a decent into Saturn), thespacecraft would be sent into a polar orbit similar to that of Juno, diving between the planet and theinnermost ring with close, 3500-km flybys (Edgington and Spilker 2016). Between May and August2017, Cassini performed 22 such flybys (every 6 days), out of which six were devoted for gravity science.Similar to the case of Jupiter, these gravity measurements have allowed the measurement of Saturn’sgravity spectrum up to J , and increased the accuracy of the known harmonics by more than twoorders of magnitude (Iess et al. 2019).In light of these two monumental new experiments, this paper provides a comparative review ofwhat was learned from the gravity measurements regarding the atmospheric and interior dynamics onJupiter and Saturn. In section 2 we briefly review the dynamical relations connecting the momentumand gravity fields. In section 3 we review the gravity measurements and compare the measured fieldson both Jupiter an Saturn. The interpretation of these results in terms of the resulting vertical andmeridional profile of the zonal flows that best matches the gravity measurements is shown in section4. In section 5 we discuss the implications of the commonalities between the Juno and Cassini results,and on what these imply about the possible mechanisms that affect the flow at depth and how theflow might interact with the magnetic field. In section 6 we discuss similar gravity constraints for thezonal flows on Uranus and Neptune, and we conclude in section 7. The theoretical starting point for understanding the zonal jet dynamics are the Euler equations in therotating frame ( u · ∇ ) u + 2 Ω × u + Ω × ( Ω × r ) = − ρ ∇ p + ∇ V, (1)where u is the 3D velocity vector, ρ is density, p is pressure and V is the body force. The rotationrate of Jupiter is given by the System III rotation (Riddle and Warwick 1976; May et al. 1979),with Ω = 1 . × − corresponding to a period of 9 .
92 hours. For Saturn there has been significantuncertainty in its rotation rate due to the axisymmetric nature of the planets’s magnetic field, butrecent studies, using gravity measurements, have constrained the rotation period to 10 . ± .
03 hours(Helled et al. 2015; Mankovich et al. 2019). As both giant planets are rapid rotators, for the purpose ofstudying the large scale zonal flows, the Rossby number, which is the ratio of the inertial accelerations(first term on lhs in Eq. 1) and the Coriolis accelerations (second term on lhs in Eq. 1), is small. Thus,in the limit of small Rossby number, the fluid is in geostrophic balance (Pedlosky 1987), meaning:2 Ω × ρ u = −∇ p − ρ g ∗ , (2)where g ∗ is the effective gravitational field, g ∗ = −∇ V + Ω × ( Ω × r ), with the second term being thecentrifugal acceleration. Multiplying equation 2 by the density ρ , and taking its curl gives2 Ω · ∇ ( ρ u ) = ∇ ρ × g ∗ , (3)where the left hand side (lhs) has been simplified using mass conservation ∇ · ( ρ u ) = 0 and sincethe rotation rate vector is constant. This also implies that g ∗ can be expressed as a scalar potentialmeaning that ∇ × g ∗ = 0, which has been used for the rhs of Eq. 3. This thermal-wind like relation he deep atmospheres of Jupiter and Saturn 5
2 4 6 8 10 12 14-8-6-4-2
Jupiter
Zonal harmonic degree
Rigid body modelJuno (evens)Juno (odds)3 uncertainty
2 4 6 8 10 12 14-8-6-4-2
Saturn
Zonal harmonic degree
Rigid body modelCassini (evens)Cassini (odds)3 uncertainty
Fig. 2
The measured zonal gravity harmonics of Jupiter and Saturn divided between the even harmonics (red) andodd harmonics (green). For reference the rigid-body harmonics calculated by the CMS model (Hubbard 1999, 2012) areshown as well (gray). Full circles denote positive values on the log scale and open circles denote negative values. Thelines are the Juno and Cassini measurement 3 σ uncertainty (Iess et al. 2018). (Kaspi et al. 2009) is different from the standard thermal-wind used in atmospheric science for ashallow atmosphere (e.g., Vallis 2006), in that the derivatives on the lhs are in the direction of the spinaxis and not in the radial direction (the latter is an approximation that holds when the planetaryaspect ratio between the vertical and horizontal scales is small), and the rhs involves the full densityand effective gravity . Thus, this is a general expression applicable for a rotating atmosphere at anydepth as long as the Rossby number is small.A considerable simplification to this equation can be taken by assuming spherical symmetry. With-out this assumption the rhs will involve several terms coming from the deviation of gravity from radialsymmetry (both due to the planetary oblateness and dynamical contributions to the gravity vector)and the centrifugal terms (see Eq. 7). In Galanti et al. (2017) and Kaspi et al. (2018) a careful treatmentof all these terms is taken, and it is shown that to leading order Eq. 3 is given by2 Ω · ∇ ( ρ s u ) = ∇ ρ (cid:48) × g s , (4)where ρ has been split into static ρ s ( r ) and dynamical ρ (cid:48) ( r, θ ) components, r is the radial directionand θ is latitude. Here g s is the radial gravitational acceleration coming from integrating ρ s . It isimportant to note that if the spherical assumption is not taken in Eq. 3 the rhs evolves into severaldifferent terms of equal magnitude (Galanti et al. 2017), and using only part of them (Zhang et al.2015) leads to an inconsistent expansion (see more detail below). Since the flows on the giant planetsare predominantly in the zonal direction, taking the zonal components of Eq. 4 allows integrating theflow induced density gradient to give the dynamical contribution to the gravity harmonics given by ∆J n = − πM a n (cid:90) − dµ R ( µ ) (cid:90) r n +2 P n ( µ ) ρ (cid:48) ( r, µ ) dr, (5)where M is the planetary mass, a is the planetary mean radius, R is the 1-bar radius, P n are theassociated Legendre polynomials and µ = sin ( θ ) . Note that when integrating ∂ρ (cid:48) ∂θ from the zonalcomponent of Eq. 4, for use in Eq. 5, an undetermined radially dependent integration function arrises( ρ (cid:48) ( r )). However such a function will not project onto the gravity harmonics when multiplied by the Note that 2 Ω · ∇ = 2 Ω ∂∂z , where z is the direction parallel to the spin vector ( Ω ) Note that the barotropic limit is not simply when the rhs of Eq. 3 vanishes, but rather when the lhs changes as well,resulting in 2 Ω · ∇ u − Ω ∇ · u = 0. See full derivation in Kaspi et al. (2016). Yohai Kaspi et al. P n in Eq. 5, since (cid:90) − dµ a (cid:90) r n +2 P n ( µ ) ρ (cid:48) ( r ) dr = 0 , (6)because the latitudinally dependent associated Legendre polynomials P n have a zero mean. Thereforein spherical geometry the dynamical gravity anomalies can be uniquely determined, despite the densityanomaly itself being determined only up to an unknown constant of integration (Kaspi et al. 2016).There has been debate in the literature whether an additional term, namely ∇ ρ s × g (cid:48) which appearsto be of the same order as the rhs of Eq. 4 should be included in that equation (termed the thermal-gravity wind equation by Zhang et al. 2015). However, this additional term contains a deviation fromradial symmetry and therefore it was dropped going from Eq. 3 to Eq. 4. If this term is retained, thenfor consistency, other terms that involve deviation from radial symmetry, and are of the same orderfrom Eq. 3, must to be retained as well (Galanti et al. 2017). Then the azimuthal component of Eq. 3will take the form:2 Ω ∂∂z ( ρ s u ) = g ( r ) s r ∂ρ (cid:48) ∂θ − g ( θ ) s ∂ρ (cid:48) ∂r + g (cid:48) ( r ) r ∂ρ s ∂θ − g (cid:48) ( θ ) ∂ρ s ∂r + Ω (cid:20) ∂ρ (cid:48) ∂θ cos θ + ∂ρ (cid:48) ∂r r cos θ sin θ (cid:21) , (7)where u is the velocity component in the azimuthal direction, and the notation ∂∂z ≡ cos θ r ∂∂θ + sin θ ∂∂r denotes the derivative along the direction of the axis of rotation. Note that in the radial symmetriclimit the rhs reduces to only the first term on the rhs which is exactly the azimuthal component of Eq. 4giving thermal-wind balance. Eq. 7 is an integro-differential equation since both the gravity g s and g (cid:48) ,are calculated by integrating ρ s and ρ (cid:48) , respectively. Although this equation can be solved numerically(Galanti et al. 2017), the additional terms (terms 2-6 on the rhs) are all small and contribute very littleto the gravity solution. The individual contribution of each of the terms in Eq. 7 is shown in Kaspiet al. (2018) for the case of Jupiter, demonstrating that the first term on the rhs is indeed the leadingorder term. All other terms in this equation are at least an order of magnitude smaller, meaning thattaking g = g ( r ) and neglecting the centrifugal terms gives the leading order solution. Galanti et al.(2017) solves the full equation 7 and shows that the resulting gravity harmonics are very close to thoseresulting from using thermal wind balance. Other solutions, such as retaining only the first and thirdterms on the rhs of Eq. 7 (Zhang et al. 2015; Kong et al. 2018), are thus inconsistent and invalid. The close orbits of Juno and Cassini yielded determination of the gravity harmonics of Jupiter and Sat-urn to unprecedented accuracy (Iess et al. 2018, 2019). Prior to these missions, the only known gravityharmonics were J , J and J (Jacobson 2003; Jacobson et al. 2006). Supplementary Fig. 1 illustrateshow much these have been improved over the last few decades showing the significant reduction inthe uncertainty going from the Voyager era to the Juno and Cassini measurements. In addition, thehigher-order even harmonics J and J have been now determined with high accuracy as well (Tables 1and 2). These even harmonics are mostly affected by the interior density distribution and shape of theplanet, and only to second order by the flow ( ∆J n , Eq. 5), although the relative contribution from theflow grows for the higher harmonics and becomes of similar order to that associated to the rotationalflattening beyond J (Hubbard 1999). Conversely, the odd gravity harmonics ( J , J , J etc.) haveno contribution from the interior static density distribution and shape as these are purely north-southsymmetric for such gas planets. The only possible contribution to the odd gravity harmonics comesfrom asymmetries in the dynamics (see the asymmetry in the wind profiles of both Jupiter and Saturnin Fig. 1). Therefore, in terms of probing the dynamics using gravity measurements, the odd harmonicsprovide a more direct way of determining the depth of the flows (Kaspi 2013).The values of the even harmonics are to leading order powers of q n , where q is the ratio of thegravity to centrifugal terms in Eq. 1 (Hubbard 1984). The rotation therefore is dominant is determiningthe values in the rigid-body limit (no dynamics) in addition to the internal density distribution. Thedependence on the density distribution is more complex, and can be calculated by internal models he deep atmospheres of Jupiter and Saturn 7 Jupitergravityharmonics ⇥ Juno measurement NoDynamics Dynamicalcontribution( J n ) Cloud-level flowwith simpleexponential decay( H = 1471 km) Cloud-level flowwith complexdecay function Adjustedcloud-level flowwith complexdecay function Adjustedcloud-level flowwith complexdecay function.Optimizing bothevens and odds. J . ± .
40 1469657 .
22 0 40 .
92 54 .
62 60 .
11 55 . J . ± . . . . . . J . ± . .
92 0 . . . . J . ± . . . . . . J . ± .
90 3418 .
80 1 .
00 1 .
76 0 .
33 2 .
06 1 . J . ± . .
39 7 .
47 12 .
77 12 .
39 12 . J . ± . .
08 3 .
50 2 .
49 5 .
41 4 .
02 3 . J . ± . . . . . . J . ± .
94 20 . . . . . . rms of fit T ABLE : Values of Jupiter’s J J gravity harmonics. Measured values are the Juno measured values (Iess et al., 2018). Theinterior model values with no dynamics are averages from the ensemble of rigid-body interior models presented in Guillot et al.(2018). The dynamical contribution is the difference between the measured values and the rigid-body models with no dynamics.The fifth column shows the dynamical solutions when using the cloud-level profile of the zonal winds (Fig. 1), extend down alongthe direction of the spin axis with a best fit e-folding decay value of H = 1471 km (Eq. 9). The sixth column shows the solutionswhen using the cloud level winds but with a more complex decay function as described in Kaspi et al. (2018). The seventh columnshows the solutions when optimizing both the meridional and vertical profile of zonal wind (as shown in Fig. 5, blue) with theoptimization done for the odd gravity harmonics only (as for columns 5 and 6), and the last column shows the same but optimizingfor the even gravity harmonics as well (the profiles are shown in Fig. 5, green). The last row shows the normalized rms of thedifference between the measurements and the solution for all cases, giving a relative value for how close the solution is to themeasurements. planet’s radius. These results, however, heavily dependon the choice of EOS. Recently, Debras and Chabrier(2019) presented a new model for the interior of giantplanets, and were able to match with a new EOS theJuno measurements as well as the abundance of heavyelements measured by Galileo. Their results also sup-port the existence of an extended diluted core enrichedby heavy elements.Despite the dynamical contribution to the even gravityharmonics being small relative to the rotation and inte-rior mass distribution, due to the high accuracy of theJuno and Cassini gravity measurements the dynamicaleffects on the even harmonics turned to be larger than theformal measurement uncertainty (supplementary Fig. 1).Therefore, without any knowledge of how much mass isinvolved in the flow (i.e., how deep the flows are), thedynamical effects can be regarded as the effective uncer-tainty of the gravity harmonics (Kaspi et al., 2017; De-bras and Chabrier, 2019). In Supplementary Fig. 1 weshow this effective uncertainty considering a wide rangeof possible flow profiles calculated using the method de-scribed in section 2 and assuming no a priori knowl-edge about the internal dynamics (i.e., before the Junoand Cassini measurements). However, given the currentknowledge about how deep the cloud-level flows are, and assuming there are no other significant internal flows thataffect the even harmonics, allows more accurate esti-mates of the effective uncertainty where the dynamicalcorrection to the measurements of the even harmonics isalready taken into account. In Supplementary Fig. 1 weshow both the effective uncertainty assuming no knowl-edge on the dynamical contribution (yellow shading),and the dynamical contribution given the knowledge ofthe flow depth (green dots). For Jupiter, the odd harmon-ics allowed getting an independent measure of the flowdepth without need to use the even harmonics; while forSaturn, as discussed in section 4, the even harmonics areneeded to determine the flow depth which makes this ef-fective uncertainty more ambiguous.The measured even gravity harmonics by Juno andCassini, as well as theoretical estimates for the gravityvalues if Jupiter and Saturn were rotating as a rigid body(equivalent to a case where the dynamics are very shal-low and have no influence on the gravity field) are pre-sented in Fig. 2 (red and gray dots, respectively). Thenumerical values for Jupiter and Saturn are presentedas well in Tables 1 and 2, respectively. As expected,the low order even harmonics match previous estimatesas they are mostly dominated by non-dynamical effects,and thus very close to the rigid body values. For Jupiter, Table 1
Values of Jupiter’s J − J gravity harmonics. Measured values are the Juno measured values with theuncertainty being three times the formal uncertainty (Iess et al. 2018). The interior model values with no dynamics areaverages from the ensemble of rigid-body interior models presented in Guillot et al. (2018). The dynamical contributionis the difference between the measured values and the rigid-body models with no dynamics. The fifth column shows thedynamical solutions when using the cloud-level profile of the zonal winds (Fig. 1), extend down along the direction ofthe spin axis with a best fit e-folding decay value of H = 1471 km (Eq. 8). The sixth column shows the solutions whenusing the cloud level winds, but with a more complex decay function as described in Kaspi et al. (2018). The seventhcolumn shows the solutions when optimizing both the meridional and vertical profile of zonal wind (as shown in Fig. 5,blue), with the optimization done for the odd gravity harmonics only (as for columns 5 and 6). The last column showsthe same but optimizing for the even gravity harmonics as well (the profiles are shown in Fig. 5, green). The last rowshows the normalized rms of the difference between the measurements and the solution for all cases, giving a relativevalue for how close the solution is to the measurements. (e.g., Hubbard 1975; Hubbard and Marley 1989; Hubbard 1999, 2012; Nettelmann et al. 2012; Miguelet al. 2016; Hubbard and Militzer 2016), and depends on the equation of state (EOS) as well (e.g., Mil-itzer and Hubbard 2013; Chabrier et al. 2019). Both the interior models and the EOS are topics ofintense research and will not be reviewed here, as the focus is on the dynamics. Most pre Juno/Cassinipublished interior structure models for Jupiter and Saturn gave gravity harmonics outside of the narrowrange of the Juno and Cassini measurements (Supplementary Fig. 1), resulting in a need for improvingthe interior models and EOSs. The first to match the Juno measurements to an internal model wasWahl et al. (2017), who found that the Juno measured harmonics can only be matched if Jupiter has adilute core that expands to a significant fraction of the planet’s radius. These results, however, heavilydepend on the choice of EOS. Recently, Debras and Chabrier (2019) presented a new model for theinterior of giant planets, and were able to match with a new EOS the Juno measurements as well asthe abundance of heavy elements measured by Galileo. Their results also support the existence of anextended diluted core enriched by heavy elements.Despite the dynamical contribution to the even gravity harmonics being small relative to the rota-tion and interior mass distribution, due to the high accuracy of the Juno and Cassini gravity measure-ments the dynamical effects on the even harmonics turned to be larger than the formal measurementuncertainty (supplementary Fig. 1). Therefore, without any knowledge of how much mass is involved inthe flow (i.e., how deep the flows are), the dynamical effects can be regarded as the effective uncertaintyof the gravity harmonics (Kaspi et al. 2017; Debras and Chabrier 2019). In Supplementary Fig. 1 weshow this effective uncertainty considering a wide range of possible flow profiles calculated using themethod described in section 2 and assuming no a priori knowledge about the internal dynamics (i.e.,before the Juno and Cassini measurements). However, given the current knowledge about how deepthe cloud-level flows are, and assuming there are no other significant internal flows that affect the evenharmonics, allows more accurate estimates of the effective uncertainty where the dynamical correctionto the measurements of the even harmonics is already taken into account. In Supplementary Fig. 1we show both the effective uncertainty assuming no knowledge on the dynamical contribution (yellow Yohai Kaspi et al. Saturngravityharmonics ⇥ Cassinimeasurement NoDynamics Dynamicalcontribution( J n ) Cloud-level flowwith simpleexponential decay( H = 12955 km) Cloud-level flowwith complexdecay function Adjustedcloud-level flowwith complexdecay function Adjustedcloud-level flowwith complexdecay function.Optimizing withodds only J . ± . .
67 8249 .
40 8660 .
38 5053 .
98 4471 . J . ± . . .
92 164 .
77 8 .
73 6 . J . ± . . .
56 49 .
04 758 . . J . ± . .
41 72 .
47 42 . . . J . ± . . .
39 347 .
23 480 .
68 409 .
32 473 . J . ± . .
77 97 .
73 127 .
47 11 .
04 10 . J . ± . . . . . . . J . ± . .
91 15 .
33 45 .
00 38 .
06 36 . J . ± . .
79 348 .
45 85 .
69 147 .
76 366 .
00 60 . rms of fit .
09 28 .
90 0 .
610 0 . T ABLE : Values of Saturn’s J J gravity harmonics. Measured values are the Cassini Grand Finale measured values withthe uncertainty being three times the formal uncertainty (Iess et al., 2019). The model values without dynamics are taken from theensemble of models presented in Galanti et al. (2019). The dynamical contribution is the difference between the measured valuesand the rigid-body models with no dynamics. The fifth column shows the solutions when using the cloud-level profile of the zonalwinds (Fig. 1), and extending them along the direction of the spin axis with a best-fit e-folding decay value of H = 12995 km(Eq. 9). The sixth column shows the solutions when using the cloud level winds but with a more complex decay function asdescribed in Galanti et al. (2019). The seventh column shows solutions when optimizing both the meridional and vertical profile ofzonal wind (as shown in Fig. 5, green) with the optimization done for the even and odd gravity harmonics (as for columns 5 and 6),and the last column shows the same but optimizing for the odd harmonics only (the profiles are shown in Fig. 5, blue). The last rowshows the normalized standard deviation of the difference between the measurements solution for all cases, giving a relative valuefor how close the solution is to the measurements. also J and J are relatively close to the rigid-bodyvalues and therefore the measured even harmonics andthe rigid-body values are virtually indistinguishable inFig. 2. However, for the Saturn case, these values differsubstantially, indicating that the flows are deeper thanon Jupiter. This separation between the measurementsand the rigid-body values matches Hubbard (1999)’s pre-diction that if a planet is differentially rotating the evengravity harmonics beyond n = 8 will differ from therigid-body theoretical values. Hubbard’s solution al-lowed only for cases of full differential rotation, mean-ing the surface flows extend throughout the whole planet(i.e., following the Busse, 1976 barotropic model). Inter-mediate cases for which the surface flows halt at a cer-tain depth can be obtained using methods as presentedin section 2 (Kaspi et al., 2010). Quantitative solutionsshowing which vertical decay profiles best match thesemeasurements are presented in section 4.A key result of the gravity measurement was that themeasured odd gravity harmonics vary significantly fromzero. The measured values (green dots in Fig. 2), matchpredicted theoretical values (Kaspi, 2013) calculated byextending the observed cloud-level wind inward alongthe direction of the spin axis and calculating their af-fect on the gravity field (see section 4). The fact that the expected gravity signal of zonal flows was in factdetected, confirmed that dynamics play a role in redis-tributing mass inside the planet and that the dynamicsare deep enough to affect the measured gravity field. Dif-ferent from the even gravity harmonics, the odd harmon-ics have no contribution from the non-dynamical interiormass distribution as it should not have any north-southasymmetries on a gas planet. The observed jets on theother hand do have north-south asymmetries (Fig. 1) andare the only considerable source of north-south asymme-tries on the gravity field. Other sources of north-southasymmetries can be internal oscillations (Durante et al.,2017) and the known north-south asymmetry in the mag-netic field (Connerney et al., 2018; Moore et al., 2018).However, internal oscillations can be expected to givefluctuating contributions from orbit to orbit whereas themeasured odd harmonics are steady. The magnetic effectcan be expected to scale as the ratio of magnetic pres-sure to total pressure. For a field of Gauss (plausiblythe unobserved toroidal field) in a region of total pres-sure of ⇠ kilobars (at ⇠ . Jupiter radii), thisis of order ⇥ , likely too small to be important,although it cannot be excluded with complete certaintybecause this field (unlike the observed poloidal field) isnot known. For the case of Jupiter, J , J , J and J Table 2
Values of Saturn’s J − J gravity harmonics. Measured values are the Cassini Grand Finale measured valueswith the uncertainty being three times the formal uncertainty (Iess et al. 2019). The model values without dynamicsare the average values from the ensemble of models presented in Galanti et al. (2019). The dynamical contribution isthe difference between the measured values and the rigid-body models with no dynamics. The fifth column shows thesolutions when using the cloud-level profile of the zonal winds (Fig. 1), and extending them along the direction of thespin axis with a best-fit e-folding decay value of H = 12995 km (Eq. 8). The sixth column shows the solutions when usingthe cloud level winds, but with a more complex decay function as described in Galanti et al. (2019). The seventh columnshows solutions when optimizing both the meridional and vertical profile of zonal wind (as shown in Fig. 5, green) withthe optimization done for the even and odd gravity harmonics together (as for columns 5 and 6). The last column showsthe same, but optimizing for the odd harmonics only (the profiles are shown in Fig. 5, blue). The last row shows thenormalized standard deviation of the difference between the measurements solution for all cases, giving a relative valuefor how close the solution is to the measurements. shading), and the dynamical contribution given the knowledge of the flow depth (green dots). ForJupiter, the odd harmonics allowed getting an independent measure of the flow depth without need touse the even harmonics; while for Saturn, as discussed in section 4, the even harmonics are needed todetermine the flow depth which makes this effective uncertainty more ambiguous.The measured even gravity harmonics by Juno and Cassini, as well as theoretical estimates forthe gravity values if Jupiter and Saturn were rotating as a rigid body (equivalent to a case wherethe dynamics are very shallow and have no influence on the gravity field) are presented in Fig. 2 (redand gray dots, respectively). The numerical values for Jupiter and Saturn are presented as well inTables 1 and 2, respectively. As expected, the low order even harmonics match previous estimates asthey are mostly dominated by non-dynamical effects, and thus very close to the rigid body values.For Jupiter, also J and J are relatively close to the rigid-body values and therefore the measuredeven harmonics and the rigid-body values are virtually indistinguishable in Fig. 2. However, for theSaturn case, these values differ substantially, indicating that the flows are deeper than on Jupiter. Thisseparation between the measurements and the rigid-body values matches Hubbard (1999)’s predictionthat if a planet is differentially rotating the even gravity harmonics beyond n = 8 will differ fromthe rigid-body theoretical values. Hubbard’s solution allowed only for cases of full differential rotation,meaning the surface flows extend throughout the whole planet (i.e., following the Busse 1976 barotropicmodel). Intermediate cases for which the surface flows halt at a certain depth can be obtained usingmethods as presented in section 2 (Kaspi et al. 2010). Quantitative solutions showing which verticaldecay profiles best match these measurements are presented in section 4.A key result of the gravity measurement was that the measured odd gravity harmonics vary sig-nificantly from zero. The measured values (green dots in Fig. 2), match predicted theoretical values(Kaspi 2013) calculated by extending the observed cloud-level wind inward along the direction of thespin axis and calculating their affect on the gravity field (see section 4). The fact that the expectedgravity signal of zonal flows was in fact detected, confirmed that dynamics play a role in redistribut-ing mass inside the planet and that the dynamics are deep enough to affect the measured gravity he deep atmospheres of Jupiter and Saturn 9 field. Different from the even gravity harmonics, the odd harmonics have no contribution from thenon-dynamical interior mass distribution as it should not have any north-south asymmetries on a gasplanet. The observed jets on the other hand do have north-south asymmetries (Fig. 1) and are theonly considerable source of north-south asymmetries on the gravity field. Other sources of north-southasymmetries can be internal oscillations (Durante et al. 2017) and the known north-south asymmetryin the magnetic field (Connerney et al. 2018; Moore et al. 2018). However, internal oscillations can beexpected to give fluctuating contributions from orbit to orbit whereas the measured odd harmonics aresteady. The magnetic effect can be expected to scale as the ratio of magnetic pressure to total pressure.For a field of 100 Gauss (plausibly the unobserved toroidal field) in a region of total pressure of ∼ ∼ .
96 Jupiter radii), this is of order 3 × − , likely too small to be important, although itcannot be excluded with complete certainty because this field (unlike the observed poloidal field) is notknown. For the case of Jupiter, J , J , J and J were measured to be above the 3-sigma uncertaintylevel (black line in Fig. 2), while for Saturn only the first two are above the 3-sigma uncertainty level.The robustness of the odd harmonics measurement of Jupiter allowed therefore to uniquely determinethe depth and structure of the flow even without consideration of the even harmonics. For the case ofSaturn, this turned to be more complex, because only the first two odd gravity harmonics are abovethe uncertainty level, and as shown below those alone do not give a solution that matches the evenharmonics as well. Given the measurements from Juno and Cassini, the challenge is to translate these measurementsinto the wind fields that generate them. The challenge is both in the conversion between the gravityanomaly data and the dynamically balanced wind field, and in dealing with the non-unique nature ofsuch solutions. Given that the gravity field is described by only a finite set of values (Fig. 2), while afull wind field will require many degrees of freedom to describe properly, it is obvious that the solutionis not unique, and the more degrees of freedom the wind field has, the easier it will be to find a fit tothe gravity data. We present therefore here a hierarchal approach beginning with a simple case wherethe wind is described with only one degree of freedom, meaning a greater number of observables (thegravity harmonics), and then present cases with more degrees of freedom for the wind profile allowingbetter matches to the gravity data, but never allow more degrees of freedom for the wind than thenumber of overall observables.We begin with a simple forward model in which we assume the observed cloud-level flow at the1 bar level decays radially towards the interior with an e-folding depth defined as H . This representsthe expectation that the wind will overall decay with depth (despite possible enhancement at thehigh levels as measured by the Galileo probe), due to the compressibility of the fluid and/or Ohmicdissipation at depth due to increasing electrical conductivity. Due to the dominance of rotation, thecloud-level flow is extended inward along the direction of the spin axis, but the decay itself is radialsince the density growth inward is also radial, meaning the functional dependence of the zonal flow isgiven by u ( r, θ ) = u cyl ( r, θ ) exp [( r − a ) /H ] , (8)where u cyl is the cloud-level wind profile extended inward along the direction of the axis of rotationand H is the e-folding radial decay height of this flow (the other parameters are as defined in section2). Such simplified models for the wind profile have been used in several studies (Kaspi et al. 2010;Kaspi 2013; Liu et al. 2013; Kaspi et al. 2016; Kong et al. 2016; Guillot et al. 2018).Given such a zonal wind profile, Eq. 4 can be used to generate the density anomaly gradientsthat balance this flow profile, and the dynamical gravity harmonics (Eq. 5) can be obtained. Fig. 3(solid lines) shows such calculated gravity harmonics as function of the e-folding depth of the flow aspredicted in Kaspi (2013), for both the even gravity harmonics (top) and the odd harmonics (bottom).Note that the values of all harmonics switch sign as function of depth depending on how the integrateddensity structure that is balancing the wind projects on the different spherical harmonics. Althoughthe sign of these values is not intuitive—the overall tendency to larger values with depth is—due to -10 -8 -6 Jupiter J J J -10 -10 -10 -8 -10 -6 J n -10 -8 -6 J J J J -10 -10 -10 -8 -10 -6 J n H (km)
Saturn J J J J J J J H (km)
Fig. 3
Theoretical values of the even (top) and odd (bottom) gravity harmonics as function of the e-folding depth( H ) of the cloud-level wind profile for Jupiter (left) and Saturn (right). The solid curves are the predicted dynamicalcontributions to J n (Kaspi 2013), for winds decaying exponentially from the measured cloud-top winds given an e-foldingdepth H . The horizontal dashed lines are the measured values from Juno and Cassini, where for the even harmonics therigid-body values used in Tables 1 and 2 have been subtracted. Depth values that match the corresponding measuredgravity harmonics correspond to locations where the solid curve crosses the dashed line of the same color. having more mass involved with the flow. The measured values from Juno and Cassini are shown(dashed lines) on top of the theoretical prediction curves. For the even harmonics, ∆J n is calculatedas the difference between the measurements and the average rigid-body values from an ensemble ofinterior models (Guillot et al. 2018; Galanti et al. 2019; see Tables 1 and 2).For the Jupiter case, the measured odd harmonics are all negative except J which is positive,matching the prediction for this simplified model for depths of several thousand kilometers (indicatedby the crossing between the solid and dashed lines in Fig. 3). Note that all the four gravity harmonics,independently, match the Kaspi (2013) prediction by sign and indicate that the depth of the flow is be-tween 1000 and 3000 kilometers, with the optimized best fit e-folding depth for all harmonics combinedbeing H = 1471 km. Furthermore, the even gravity harmonics (omitting J and J where the relativecontribution of the dynamics is very small) show a similar result where all three theoretical curvescross the measurement value between depths of 1500 and 2000 km. The fact that for all seven values( J and J - J ), the theoretical calculation matches the Juno measurement in sign and value gives astrong indication that the observed cloud-level flow is related to these measured gravity anomalies andindicates their depth. Nonetheless, using an exponential decay law for the cloud-level winds does notgive an exact match to the gravity data (Table 1, column 5), and indeed exponential decay with a he deep atmospheres of Jupiter and Saturn 11 uniform e-folding depth was not made based on physical reasoning but for simplicity. Below we presentmore complex decay functions which better match the measurements, yet the simple model’s overallmatch to the data gives a strong indication to the relation between the observed flows, their depth andthe gravity measurements.Optimizing for a more complex decay function we use an adjoint based inversion technique (Galantiand Kaspi 2016), where a cost function is minimized to give a best fit between the decay profile andthe gravity measurements, taking into account the uncertainties in the gravity measurements and theerror covariance between the different harmonics (Kaspi et al. 2018). Solutions for the vertical decayfunctions using this method with three degrees of freedom for the shape of the vertical profile (seeKaspi et al. 2018 for details) are shown in Fig. 4. Taking the exact observed cloud-level zonal flows andextending them into the interior with this best optimized vertical decay function gives a much bettermatch to the gravity data than the exponential decay function (Table 1, column 6). Next, allowingthe optimization procedure to include small variations to the cloud-level wind profile (assuming thezonal wind meridional profile at depth may vary somewhat from what is observed at the cloud-level)shows that in this case the solutions give an even better match (Table 1, column 7) to all 4 measuredodd gravity harmonics (note that even harmonics are not optimized here, but still give a rather goodmatch). Fig. 4 shows that in this case the variations to the observed wind profiles are very minorand well within the uncertainty (and observed variation between the Voyager and Juno eras) of thisprofile. The vertical profile in this case (blue) is very similar to the one obtained without varying themeridional structure of the wind profile indicating again the decay being at around several thousandkilometers.As the odd harmonics are a consequence of the dynamics alone we have used only them so far forthe optimization procedure. Despite this, the resulting even ∆J n for these vertical profiles match wellboth in sign and in magnitude the difference between the measurements of J , J and J and therigid body values (compare columns 4 and 7 in Table 1). Thus it is clear that if we include the evenvalues in the optimization the results will not differ substantially. In the final column of Table 1 wepresent such an optimization, where now all values of the seven gravity harmonics ( J and J - J )match exactly the measurements. Here again we allow the wind profile to vary from the observed windstructure, though as can be seen in Fig. 4 the wind profile needs very minor changes in order to matchthe gravity measurements perfectly. We emphasize though that this exercise is not unique and othermeridional and vertical profiles of the zonal flow can give an exact match to the gravity data. However,following Occam’s razor reasoning, here we have shown that taking the observed cloud-level flow, andextending it inward in a very simple fashion gives an exact match to the gravity measurements.Complementary to this analysis, Guillot et al. (2018) used a wide range of rigid-body models forJupiter (without averaging as in Table 1, column 3) to construct the possible dynamical contributionrange to the even zonal harmonics, by subtracting this range of rigid-body solutions from the Junogravity measurements. This range was used to constrain a wide range of hypothetical flow profilesderived using thermal wind balance (as in Kaspi et al. (2017)), not bounded to the observed cloud-level flows with different e-folding decay depths. This analysis, consistent with the analysis presentedabove, showed that with high likelihood the flow extends down to 2000 − < − ), otherwise it would have an influence on the measured even gravity harmonics.For the Saturn case, the cloud-level flow has shown much more variability between the Voyager andthe Cassini eras (Garc´ıa-Melendo et al. 2011), and is more uncertain. Repeating the same analysis asfor the Jupiter case and taking the cloud-level flow with a simple exponential decay gives a relativelygood match to the even harmonics (same sign and within factor of 2 in magnitude) for e-foldingdepths of ∼ km. This indicates a substantially deeper flow than for the case of Jupiter. The oddharmonics for the Saturn case, despite being not very different in magnitude than for Jupiter, are closerto the measurement uncertainty and therefore only J and J have significant values (Fig. 2). Both,however, give an opposite sign compared to the theoretical prediction when using the cloud-level wind -80 -40 0 40 80-100-50050100150 Jupiter Z ona l w i nd ( m / s ) Latitude
ObservedReconstructed with oddsReconstructed with odds & evens V e r t i c a l de c a y p r o f il e Depth (km) -80 -40 0 40 800100200300400
Saturn
Latitude Depth (km)
Fig. 4
The optimized meridional profile (top) of the zonal wind for Jupiter (left) and Saturn (right), comparing theobserved cloud-level profile (red), the optimized best fit solutions taking into account the odd harmonics only (blue),and the optimized best fit solutions taking into account both the odd and even harmonics (green). The resulting gravityharmonic values for these profiles appear in Tables 1 and 2. The bottom panel shows the corresponding vertical structuresof the zonal flow as function of depth (km) and pressure (bar). profile, indicating that a more sophisticated model is needed for Saturn. Similarly, as long as usingthe cloud-level winds, taking a more complex decay profile does not give a good match to the gravitymeasurements (Table 2, column 6). However, when allowing the cloud level wind to deviate from theobserved profile, a good match to the measurements can be found (Fig. 4, green profile).The deviation from the observed wind profile of Saturn is mainly around latitude 30 ◦ where theflow needs to be more westward than the observed cloud-level wind, with values ∼
50 m s − in orderto match the measurements for both the even and odd harmonics (Table 2, column 7). The match isobtained with a vertical profile which is nearly barotropic down to ∼ ◦ is needed in order to match the even gravity harmonics, was also reached by Militzer et al.(2019) who used a model allowing the flow to extend inward only barotropically (without changingalong the direction of the spin axis), and found that such a westward zonal flow profile (but twiceas large) is needed to match the measurements. Based on theoretical argument alone, Chachan andStevenson (2019) obtained a similar conclusion that a retrograde wind profile is necessary aroundlatitude 30 ◦ in order to match the measurements. The optimizations discussed here used both thevalues of the odd and the even harmonics and took into account all cross correlations. For the caseof Saturn, optimizing with odd harmonics only does not give a good match to the even harmonics,highlighting the difference between Jupiter and Saturn and pointing to that for Saturn the high ordereven harmonics (particularly J and J ) are key to determine the depth and profile of the deep flows.The uncertainty in rotation rate affects only the dynamical J and J and thus is not important forinterpreting the Saturn gravity measurements (Galanti and Kaspi 2017).Comparing the different columns in Tables 1 and 2 and considering the different profiles in Fig. 5shows that the gravity results not only inform us about the depth of the jets, but also about themeridional profile of the zonal flow at depth. The results show that the measurements are sensitive to theexact meridional profile, although the variations to it needed to get exact matches are not significant.To test the statistical significance of this profile, other profiles with a different meridional profile ofthe zonal flow have been tested, to investigate the possibility that the flow at depth might exhibit he deep atmospheres of Jupiter and Saturn 13 major qualitative differences from the flow observed at cloud-level. Out of a sample of a thousandzonal-wind profiles as a function of latitude with the same overall amplitude but different meridionalprofile, less than 1% had a better match to the measurements using the same optimization procedure(Kaspi et al. 2018). These few profiles had no correlation to one another, nor to the cloud-level profile.This indicates that although such random solutions can be found, it is with high confidence that thesame meridional profile of the zonal flow that is observed at the surface extends to depth. The results presented above have shown some key similarities and differences between Jupiter andSaturn. On both planets, the measured gravity harmonics indicate how deep the cloud-level flowsextend, and give a good match to a zonal wind profile at depth being very similar to the one observedat the cloud-level of both planets. Differently, on Jupiter the jets extend down to ∼ . . bar, due to the dependence of electrical conductivity ontemperature (Stevenson 2003). Strikingly, the depth where the electrical conductivity rises in bothplanets is at the same depth where the gravity measurements imply that the zonal winds decay (wherethe blue and red curves cross in Fig. 5). This strongly hints that Ohmic dissipation plays a role indamping the flow at depth, and in fact was predicted previously based on theoretical arguments (Liuet al. 2008; Cao and Stevenson 2017).As a consequence, the depth at which a particular pressure is reached in Saturn is about threetimes greater than the corresponding depth for Jupiter. The temperature at a given pressure is onlymodestly ( ∼ bars in both planets) is approximately 1 S m − (similarto that of salty water at room temperature). At this conductivity, strong zonal winds would create atoroidal magnetic field whose associated electrical currents would produce a total Ohmic dissipationthat is comparable to the observed luminosity of the planets (Liu 2006; Liu et al. 2008).The factor of three difference in zonal wind depth between Jupiter and Saturn, together witha remarkable correspondence to the theoretical argument of (Liu et al. 2008) (their prediction was2800 km for Jupiter) strongly suggests the role of magnetohydrodynamics. It should also be noted thatbecause the electrical conductivity is such an extremely strong function of temperature and thereforeradius, the results hold even given a likely order of magnitude uncertainty in the electrical conductivityand the large difference in field strengths between Jupiter and Saturn. Some cautionary comments arein order, however: first, the argument is purely kinematic; that is, there is as yet no fully dynamicalargument that explains this truncation of the flows by the magnetic field. Moreover, the Lorentz forceis not sufficient by itself to dampen the flows from large values (tens of m s − ) to zero (Cao andStevenson 2017). Clearly the role of the magnetic field is more complicated and the full solution to thezonal flow requires an understanding of the spatial structure of the deviation from constant entropythroughout the envelope (Eq. 9).Therefore, although the Ohmic dissipation might be ultimately what halts the flow at depth, itdoes not explain what diminishes the strength of the flow substantially from the cloud-level down towhere the electrical conductivity becomes large (Fig. 5). In this region the dissipation of the zonalflow needs to be due to other mechanisms. Reorganization of Eq. 4, looking at its zonal componentand assuming the background profile is adiabatic leads to a relation between the wind shear along the D ep t h [ k m ] Jupiter U ( m s - ) J R J J -100 0 100 P r e ss u r e [ B a r ] Saturn S R S S -200 0 200 Fig. 5
The vertical structure of the zonal flow on Jupiter (left) and Saturn (right) as function of depth correspondingto the best fit profile presented in column 6 of Table 1 and column 7 of Table 2, respectively. The upper panel showsthis vertical profile function (blue), its uncertainty (blue shading) and the electrical conductivity profile (red dashed) asgiven by Liu et al. (2008) for Jupiter and French et al. (2012) for Saturn. The electrical conductivity is in units of S m − with the scale going linearly from 0 to 100. The middle point in the decay profile, at depths of 1831 km and 8743 kmfor Jupiter and Saturn, respectively, is marked by the dashed horizontal line. The bottom panels show the same zonalflow profile as function of latitude and depth in the spherical projection. The middle point corresponding to that shownin the upper panels appears as the thick dashed line. The thin dashed lines contain the angle (latitude) derived fromextending the depth of the flow along the direction of the spin axis ( θ e = cos − ( x/a ), where a is the planetary radiusand x is the depth beyond the middle point of the flow profile). This latitude ( θ e ) is 13 ◦ for Jupiter and 31 ◦ for Saturn,close to the latitude where the flow is observed to turn from eastward to westward at the cloud-level. direction of the spin axis and the entropy gradients (Kaspi et al. 2009): ∂u∂z = − g Ωρ s (cid:18) ∂ρ∂s (cid:19) p r ∂s (cid:48) ∂θ (9)where s is entropy. Thus the direction of the shear is determined by the average sign of the entropygradients, and the shear is expected to be largest at the lower depths where the density is smallest.Note that in the purely barotropic limit, the rhs of equation vanishes leading to the flow being purelyaligned with the axis of rotation. This is similar to the Taylor-Proudman theorem (Pedlosky 1987),only that the Taylor-Proudman theorem requires the fluid to be incompressible, in which case all threecomponents of velocity are aligned with the rotation axis (2 Ω · ∇ u = 0 ). For a compressible flow, inthe barotropic limit, the alignment is for the zonal and meridional component of the flow (throughoutthis paper only the zonal component of the velocity is discussed). Yet, due to the entropy gradients(Eq. 9), and as evident from the gravity results, the flow is likely not fully barotropic (baroclinic). he deep atmospheres of Jupiter and Saturn 15
1% 10% 100%0.01%0.1%1%10%100% P e r c en t o f t o t a l m a ss Jupiter
Percent of total radius
1% 10% 100%
Saturn
Percent of total radius
Fig. 6
The percentage of Jupiter (left) and Saturn’s (right) mass as a function of depth beneath the 1-bar level. The greyline shows that percentage of mass contained within the depth of the zonal flows as found by the gravity measurements.
Quantitatively however, the shear on both planets is of O(100 m s − ) over thousands of kilometers,implying that the flows in effect are not very far from barotropic. If, for example, we assume thevariation from barotropic is very large and allow the observed flows to extend inward in a differentway (e.g., radially) the match to the gravity measurements would not have been as good as it is here(Tables 1, 2). For example, extending the cloud-level flow inward radially for Jupiter results in J changing sign. Only if the zonal flow meridional profile is altered substantially solutions can be found.An additional question regarding the zonal flows is the issue of their forcing mechanism. It is wellunderstood from terrestrial atmospheric dynamics that geostrophic turbulence on a rotating planetwill drive turbulent eddy momentum fluxes resulting in regions of momentum flux convergence witheastward (prograde) flows and momentum flux divergence with westward (retrograde) flows. It has beenobserved for Jupiter (Salyk et al. 2006) and Saturn (Del Genio et al. 2007) that indeed there is a strongcorrelation between the regions of eddy momentum flux convergence (divergence) and the eastward(westward) jets, implying this is a plausible mechanism for driving the zonal flows. Yet, it is not knownwhat is the source of the eddies, with candidates being barotropic or baroclinic instabilities (e.g., Kaspiand Flierl 2007) or the internal convection itself. It has also been shown that both shallow and deepforcing can drive such zonal flows (Showman et al. 2006). The driving and dissipation mechanismsdiscussed here can be seen in a single expression by taking the leading component of the momentumequation (Eq. 1) and expressing it in terms of angular momentum (Vallis 2006). Taking a zonal andvertical average and in steady state gives, u · ∇ M = − S + D , where M is angular momentum, S is theeddy momentum flux divergence and D is the Lorentz drag (Schneider and Liu 2009). In regions of lowelectrical conductivity, closer to the cloud-level, the balance will be u · ∇ M = − S with the momentumflux providing the cross angular momentum surfaces (i.e., across the direction of the spin axis) flux toforce eastward momentum flux convergence and zonal jets. In the the deep region, where the electricalconductivity is high, the drag allows for cross angular momentum flow, u · ∇ M = D , to close thecirculation. In between, u · ∇ M = 0, meaning there is no cross angular momentum flow and the flowmust be aligned with the direction of axis of rotation. Note though that this has no implication on howbarotropic is the zonal flow, and this argument puts a strong constraint on the meridional circulationbut not on the zonal flow itself. These arguments are similar to those used to explain the midlatitudeFerrel cell on Earth with surface drag taking the place of the Lorentz drag (Vallis 2006).The dynamical constraints discussed earlier suggest that the flow likely extends inward along thedirection of the spin axis as demonstrated in Fig. 5 where the full solution for the zonal wind in theradial-latitudinal plane is presented. It is evident that for the case of Jupiter, despite the jets beingdeep from an atmospheric perspective, extending down to 10 bars and advecting 1% of the mass ofthe whole planet (Fig. 6), from the point of view of the whole gaseous planet, and compared to theproposed Busse (1976) model scenario, the winds penetrate only a small fraction of the planet. ForSaturn, the fraction is larger, going down to 15% of the radius of the planet, but still containing onlya few percent of the total mass (Fig. 6). That said, on both planets, the atmospheric advection of several percent of the planetary mass is very significant, and by far extends the advection on everyother planet in the solar system (e.g., Earth’s atmosphere is less than one part in a million of theplanetary mass).Note that the depth for both Jupiter and Saturn obtained from the gravity measurements is alsoconsistent with the observed latitudinal extent of the equatorial eastward flow; meaning that if thedepth of the flow at the equator is extended along the direction of the axis of rotation, it interests thesurface almost exactly at the latitude where the zonal flow turns from positive to negative (eastwardto westward). Quantitatively, taking the half point of the flow depth (horizontal lines in the top panelsof Fig. 5), being 1831 km for Jupiter and 8743 km for Saturn, and calculating the latitude where thisline intersects the surface gives a latitude of 13 ◦ for Jupiter and 31 ◦ for Saturn. This is very closeto the latitude where the equatorial flow changes sign which is 13 ◦ and 35 ◦ for Jupiter and Saturn,respectively. Note that for the Saturn case if a rotation rate of 10:34 is taken instead of 10:39 asrecent publications indicate this latitude changes to 31 ◦ (Fig. 1). Thus, this gives another independentobservation which is in agreement with the conclusion from the gravity measurements regarding thedepth of the zonal jets. Such arguments regarding the depth of the equatorial superrotation regionhave been presented in the past in the context of the tangent cylinder surrounding the inner regionof deep convective models (e.g., Heimpel et al. 2005; Aurnou et al. 2008; Kaspi et al. 2009; Liu andSchneider 2010; Gastine et al. 2013). As on Jupiter and Saturn, Uranus and Neptune also have very strong east-west flows at the observedcloud-level. These flows reach ∼
200 m s − on Uranus and nearly 400 m s − on Neptune, with ameridional structure which is overall similar between the two planets, consisting of a westward broadequatorial flow and a strong and broad eastward flow at midlatitudes. The flows have an overallsimilar character despite the obliquity being very different (98 ◦ on Uranus and 29 ◦ on Neptune), andthe internal heat flux being three times stronger than the solar flux on Neptune while on Uranus theinternal heat flux appears to be negligible (Pearl et al. 1990; Pearl and Conrath 1991). As these are theonly two planets yet to host a dedicated space mission (Fletcher et al. 2019), most data comes from theVoyager encounters of the two planets in 1986 and 1989 (Smith et al. 1986, 1989). The data obtainedfrom Voyager includes the gravity harmonics up to J , although to a much lesser precision than theJuno and Cassini data discussed above (Jacobson 2007, 2009). Nonetheless, due to the broader shapeof the wind structure and its relative resemblance to the meridional structure of P (Eq. 5), it waspossible to place an upper bound on the depth of the atmospheric circulation on these planets (Kaspiet al. 2013).This was done utilizing the known values of J from Voyager and determining the difference be-tween the observed J , and the J resulting from a wide range of rigid-body models set to matchall other observational constraints besides J . Any difference in these quantities places constraints onthe dynamical contribution to J . Therefore, considering the observed J and its uncertainty and thewidest possible range of J solutions from interior models, ranging from models with no solid cores toones with massive solid cores (Helled et al. 2010, 2011; Nettelmann et al. 2013), an upper limit to thedynamic contribution to J ( ∆J , as in Eq. 5) was constructed. This revealed that the dynamics areconstrained to the outermost 0.4% of the mass on Uranus and 0.2% on Neptune, providing a muchstronger limitation to the depth of the dynamical atmosphere than previously suggested (Hubbardet al. 1991). This result implies that the dynamics must be confined to a thin weather layer of no morethan 1600 km on Uranus and 1000 km on Neptune (Kaspi et al. 2013). This is much shallower thanthe depths on Jupiter and Saturn with the pressure to which the flows extends being at most 4000 baron Uranus and 2000 bar on Neptune. he deep atmospheres of Jupiter and Saturn 17 The recent gravity measurements of Juno and Cassini have provided data accurate enough to allowinferring the effect of atmospheric dynamics on the gravity field of both planets. This allowed determin-ing the depth of the flows on both planets, and inferring that the meridional profile of the zonal flowsobserved at the cloud-level of both planets likely extends to depth. An intrinsic problem of any gravityinversion is that the contributing field is not unique, meaning in this case that the wind-induced grav-ity anomalies can not be traced uniquely to the wind field creating them. However, there are severalindependent lines of evidence supporting the solutions presented here:1. The simplest possible Jovian flow model, taking the observed cloud-level winds and extending theminward, matches all 4 measured odd gravity harmonics ( J , J , J and J ) independently both insign and in magnitude (Fig. 3).2. The same wind profile matches the dynamical component of the even gravity harmonics ( J , J and J ) as well (Fig. 3).3. There are no other likely sources of north-south asymmetries that can match in magnitude themeasured values of the odd gravity harmonics (section 5).4. A statistical analysis taking random zonal wind profiles (considering that the cloud-level winds maybe decoupled from the interior flow causing the measured gravity anomalies), shows that less than1% of such wind profiles give a match to the gravity measurements (Kaspi et al. 2018).5. The Saturnian winds, with slight modifications (within the error range of the wind measurements),match the dynamical component of the gravity measurements for both the even and odd harmonics(Fig. 4).6. For both Jupiter and Saturn, the depth of the flows inferred from the gravity measurements matchesthe depth where electrical conductivity rises abruptly. This suggests that the previously suggestedmechanism of Ohmic dissipation might play a key role in setting the flow depth (Liu et al. 2008).7. For both Jupiter and Saturn, the depth of the flows inferred from the gravity measurements matchesthe depth inferred from a tangent cylinder separating the equatorial eastward flow and the higherlatitude flows. This may explain the different latitudinal extent of the equatorial flow on bothplanets (Fig. 5).8. Temporal variation of the magnetic field of Jupiter implies that the variation is carried by the zonalflow and gives a magnitude of the zonal flow at depth consistent with the depth implied by thegravity measurements (Moore et al. 2019).Overall, although each one of these evidence separately can be perhaps challenged as being coincidental,when taken together, these consistent lines of evidence combined yield a coherent picture regardingthe extent and character of the flows beneath the cloud-level of Jupiter and Saturn. On both planets,the flows advect a substantial part of the mass of the planets (1 − ∼
4% on Jupiter and ∼
15% onSaturn), suggesting that from a planetary perspective the flows are still bound to a relatively shallowlayer.Despite this new understanding regarding the depth and structure of the flow, we are still leftwith an incomplete picture of the mechanisms driving the flow. Particularly, several open questionsremain, such as what causes the flow to decay before reaching the Ohmic dissipation level at ∼ bar? What drives the equatorial superrotation? Why are the flows on Saturn substantially stronger?What are the source of the eddies driving the jets? What are the roles of baroclinic, barotropic andconvective instabilities in driving the winds? With better constrains on the dynamical componentof the gravity fields (due to improved interior models), magnetic fields and their secular variation(future Juno orbits), temperature fields and water abundance (Juno microwave measurements) andimproved dynamical models these questions might be addressed in the coming years, to give a betterunderstanding of the fundamental physical processes driving the dynamics on the giant planets. References
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Historical (pre Juno and Cassini) measured values of J , J and J and theiruncertainty for Jupiter (top) and Saturn (bottom) and the Juno and Cassini measurements (including J , red). Voyagervalues (purple) are from Campbell and Synnott (1985) for Jupiter and Campbell and Anderson (1989) for Saturn, andCassini values (black) are from Jacobson (2003) for Jupiter and Jacobson et al. (2006) for Saturn. Recent Juno andCassini results are from Iess et al. (2018) for Jupiter and Iess et al. (2019) for Saturn. The effective uncertainty due todynamics assuming no knowledge on the flow and taking the widest possible range of internal flows (see Kaspi et al. 2017for details) centered around the Juno/Cassini-measured values appears in yellow, with the contour for H = 3000 km(Jupiter) and H = 10000 km (Saturn) in dashed. In green is the difference between the recent Juno/Cassini measurementand the best fit flow profile presented in this paper ( J n − ∆J n ), where ∆J n is taken from column 7 in Tables 1 and 2( ∆J2