Computer simulations of ’s deep internal dynamics help interpret what Juno sees

Gary A. Glatzmaiera,1

aEarth and Planetary Sciences Department, University of California, Santa Cruz, CA 95064

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2010.

Contributed by Gary A. Glatzmaier, May 21, 2018 (sent for review February 13, 2018; reviewed by Chris Jones and Peter L. Olson)

We describe computer simulations of thermal convection and Juno’s perijove passes (i.e., its closest orbital approach to Jupiter, magnetic field generation in Jupiter’s deep interior: that is, its 5% of RJ above the surface) have already produced exciting convective dynamo. Results from three different simulations high- results suggesting that strong fluid flows exist down to at least light the importance of including the dynamics in the very deep 4% of RJ below Jupiter’s cloud surface. Providing detailed phys- interior, although much of the convection and field generation ical descriptions of the structure and maintenance of Jupiter’s seems to be confined to the upper part of the interior. A long- zonal and its magnetic and fields is a major goal of debated question is to what depth do Jupiter’s zonal winds computational studies of Jupiter’s convective dynamo. extend below its surface. Our simulations suggest that, if global latitudinally banded patterns in Jupiter’s near-surface magnetic Model and gravity fields were detected by Juno, NASA’s orbiting space- Jupiter is a giant rotating of mainly , a smaller craft at Jupiter [Bolton S, et al. (2017) Science 356:821–825], they amount of , and minor amounts of heavier elements would provide evidence for Jupiter’s zonal winds extending deep (13). Fluid in the shallow atmospheric layer is approximately below the surface. One of our simulations has also maintained, for an ideal gas, but it compresses with depth into a dense liquid. a couple simulated years, a deep axisymmetric inertial wave, with However, as pressure increases with depth, the fluid becomes properties at the surface that depend on the size of the model’s partially electron degenerate; that is, it becomes a “quantum small rocky core. If such a wave was detected on Jupiter’s surface, fluid” with density and pressure being relatively insensitive to its latitudes and oscillation frequency would provide evidence for temperature. Experimental (14) and computational (15) stud- the existence and size of Jupiter’s rocky core. ies indicate that Jupiter’s electrical conductivity increases rapidly with depth. The gas atmosphere at Jupiter’s surface is a poor Jupiter | zonal winds | magnetic field | gravity field | Juno Mission conductor; however, as pressure increases with depth, diatomic hydrogen electrons become conducting electrons, which increase electrical conductivity until, at a radius of roughly 0.9 RJ, hydro- upiter’s latitudinally banded cloud pattern (Fig. 1) has gen becomes a liquid metal (14). At the center of Jupiter, there Jintrigued people for hundreds of years. Ground-based and may be a small core of rocky material with a radius of roughly spacecraft observations have shown that these banded clouds 0.1 RJ (13). are being sheared by latitudinally banded zonal winds (1) [that Computer models simulate thermal convection and magnetic is, east–west jet streams alternating in latitude (2)]. NASA’s field generation in Jupiter’s deep interior (16–18) in attempts Galileo Probe, which was dropped from the Galileo Orbiter into to explain the flows and fields observed at Jupiter’s surface Jupiter’s atmosphere near its equator in 1995, measured zonal and to predict their intensities and structures well below the speeds that increased from about 80 m / s at a pressure of 7 about 1 bar (roughly Jupiter’s cloud top radius, RJ = 7 × 10 m) Significance to a maximum of about 170 m / s at a pressure of 4 bars, below which the wind speed remained approximately constant until NASA’s Juno spacecraft is currently orbiting Jupiter, collecting the probe’s signal was lost at about 21 bars (3). Although this high-fidelity measurements of its near surface. The resulting suggests that the zonal winds on Jupiter extend below the vis- patterns of magnetic and gravity fields above Jupiter’s sur- ible surface, the 21-bar depth is less than 1% of RJ below the face have already started providing insight into the type of surface. winds in Jupiter’s deep interior. In addition, by monitoring Jupiter has the most intense planetary magnetic field in our cloud motions on Jupiter’s surface, Juno may detect a par- . It is dominantly dipolar when measured far above ticular type of wave with properties that could tell us the its surface. However, small perturbations exist in this field (4) size of Jupiter’s small rocky core. Here, we describe 3D time- and in Jupiter’s gravity field (5) due to the planet’s internal dependent computer simulations of thermal convection and dynamics (6). These perturbations, which are more prominent magnetic field generation in Jupiter’s deep interior that will the closer the measurements are made to Jupiter’s surface, can help interpret and explain the maintenance of these global reveal much about Jupiter’s deep interior. near-surface patterns that Juno continues to measure. Although many theoretical studies of Jupiter’s interior have been conducted using a variety of methods, many questions still Author contributions: G.A.G. designed research, performed research, analyzed data, remain about the structures of its deep fluid flows and mag- wrote the computer code, and wrote the paper. netic fields and about the physical mechanisms that maintain Reviewers: C.J., University of Leeds; and P.L.O., Johns Hopkins University. them. Do Jupiter’s banded zonal winds at the surface extend Conflict of interest statement: G.A.G. was one of 37 coauthors, including reviewers Chris deep enough below the surface to where electrical conductivity Jones and Peter L. Olson, on a 2016 report of a computational performance benchmark and zonal wind shear are high enough to generate a latitudinally exercise. They were not research collaborators on that report. banded magnetic field that could be detected by Juno, NASA’s This open access article is distributed under Creative Commons Attribution- orbiting spacecraft (7–9)? Would strong zonal winds that extend NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). to such a depth perturb enough to produce latitudinally See QnAs on page 6880. banded perturbations in the gravity field that Juno could detect? 1 Email: [email protected] The recent analyses (10–12) of the gravity field from two of Published online June 25, 2018.

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EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES simulates thermal convection and magnetic field generation in a that are coaxial with the rotation axis. The total kinetic energy deep rotating density-stratified fluid shell with an upper bound- of the zonal wind, integrated over the entire convection zone, ary radius of 0.98 RJ and a lower boundary radius of 0.10 RJ. is at least an order of magnitude greater than the total kinetic Below this model’s lower boundary is a solid, electrically con- energy of the 3D convection, which maintains the zonal wind. ducting, rocky core that rotates in response to the viscous and The local kinetic energy density of convection peaks in the upper magnetic stresses on it and according to its Jupiter-like moment part of the interior where the temperature gradient is the most of inertia. Internal heating maintains a mean temperature pro- unstable (superadiabatic) and the relative drop in density with file in the radius that is convectively unstable; that is, convection radius is greatest. However, for this Case 1, convection is sig- occurs throughout the spherical fluid shell for Case 1. The lower nificant enough throughout the entire fluid shell to maintain boundary for Case 2 is defined as an impermeable stress-free zonal winds constant on cylinders down to the model’s rocky core spherical interface at a radius of 0.8 RJ. Thermal convection (Fig. 3, Right). is also driven throughout this (artificially shallower) spherical The surface manifestation of this deep zonal wind is an fluid shell by internal heating. Case 3 uses the same rocky eastward equatorial jet, with a maximum amplitude of about core as that of Case 1; that is, the spherical fluid shell extends 120 m / s, slightly greater than Jupiter’s maximum eastward wind from radius 0.10 to 0.98 RJ. However, its mean temperature at cloud tops (about 100 m / s). The maximum convective velocity profile is constantly “nudged” by internal heating toward a tar- near the surface is about 10 m / s. However, instead of multiple, get mean temperature profile that is convectively stable below alternating, narrow zonal jets at higher latitude as observed on 0.90 RJ and above 0.96 RJ and convectively unstable between Jupiter’s surface (Fig. 1), only one latitudinally broad westward these two radii. jet exists at middle latitude and one broad eastward jet exists at high latitude in each hemisphere. Case 1: Convective Dynamo in a Deep Shell Note that, long before this steady zonal wind structure sets in, Rotating thermal convection produces bands of longitudinal flow banded zonal winds first develop in a shallow layer just below the that alternate eastward and westward with latitude relative to surface, where the steep temperature gradient there drives vig- the rotating frame of reference (Fig. 3, Left). After roughly 3 orous convection. A strong steady eastward jet develops in the million numerical time steps, the average in longitude of this equatorial region. At higher latitude, multiple narrow alternat- flow, the zonal wind, becomes virtually steady in time (Fig. 3, ing jets develop in this shallow layer that are not constant on Right). This is also called differential rotation, since in the iner- cylinders and do not extend from the northern to southern hemi- tia frame of reference, it is a radially and latitudinally dependent spheres. However, these shallow high-latitude jets are relatively rotation rate. It is maintained in our simulations by the redistri- weak and very intermittent, unlike those observed on Jupiter. bution of angular momentum by (nonlinear) advective transport, Eventually, this pattern evolves into a deep, slowly changing, which naturally occurs in a rotating convection zone when vor- approximately constant on cylinders pattern of differential rota- tices are generated by Coriolis forces as they rise (expand) or sink tion that then takes a couple million more time steps to evolve (contract) through the density-stratified interior (16, 22). Some into the steady, cylindrically symmetric structure that extends recent studies (11, 12), however, are based on the assumption throughout the deep interior (Fig. 3). that Jupiter’s differential rotation is maintained as a “thermal These convective flows and zonal winds also generate mag- wind.” That is, instead of solving the full set of equations (Meth- netic field in the electrically conducting fluid deep below the ods), they consider only two terms, the buoyancy and Coriolis surface, and as seen in Fig. 4, this field extends up through the forces in Eq. 5, and assume that the curls of these exactly bal- electrically insulating surface layer, where the field is unaffected ance. This is not a bad approximation for the ’s shallow by fluid flows. For example, Fig. 5, Right shows the latitudinally atmosphere driven by solar insolation, but our deep convective banded magnetic field seen in Fig. 4 in a spherical surface at dynamo simulations of Jupiter do not indicate such a balance. a distance above the model’s upper boundary equal to Juno’s However, it is encouraging that these two studies (11, 12), which closest orbital approach (perijove) to Jupiter’s cloud surface. At fit different types of models to the early Juno gravity data, agree a depth where electrical conductivity is large, a pair of deep that strong fluid flows exist down to at least 4% of RJ below zonal winds in opposite directions would shear poloidal (vertical Jupiter’s surface. and north–south directed) field into toroidal (east–west directed) The zonal wind for Case 1 (Fig. 3, Right) is (very nearly) only field. Also, convective vortices twist toroidal field into latitudinal a function of the distance from the planetary rotation axis; that bands of poloidal magnetic field loops. This “dynamo” process is, the amplitude of this zonal wind is “constant on cylinders” converts kinetic energy of the fluid flow into magnetic energy. The greatest magnetic energy density occurs where the product of the fluid flow amplitude (which decreases with depth) and the electrical conductivity (which rapidly increases with depth) is maximum; this occurs in Case 1 near 0.9 RJ. Note also that, for this particular simulation, the magnetic dipole axis is tilted (relative to the planetary rotation axis) much less than it is on Jupiter. This is probably due to the constant on cylinders zonal wind structure that dominates the entire fluid interior. To limit the distortion of the poloidal field and the resulting ohmic heating within the deep electrically conducting interior, poloidal field tends to align itself on surfaces of constant angu- lar velocity (in this case, cylindrical surfaces of constant zonal wind speed) as suggested by Ferraro’s isorotation law (23). The resulting banded (poloidal) magnetic structure, formed deep Fig. 3. A snapshot of the longitudinal component of the simulated fluid below the surface, is retained in the magnetic field that extends velocity (i.e., winds) for Case 1. (Left) The longitudinal winds plotted in up through the insulating upper layer and out to Juno’s per- an equal area projection of the entire spherical upper surface of the fluid model. (Right) The “zonal wind” (i.e., longitudinal average of the longi- ijove. A similar magnetic effect has been produced (24) with tudinal winds) plotted in a meridian plane. Yellows and reds represent a prescribed differential rotation using a 2D time-independent eastward-directed winds; blues represent westward-directed winds (relative kinematic mean-field model of the semiconducting region above to the rotating frame of reference). 0.9 RJ. Small magnetic length scales above the surface decay

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EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES harmonic coefficients of Jupiter’s gravity field out to degree 12. In addition to the even degrees due to Jupiter’s oblateness, odd degrees have been measured indicating that strong wind jets exist on Jupiter well below the depth to which the Galileo probe measured. The amplitudes of the corresponding odd degrees produced in Case 1 (Fig. 6, Lower) are slightly smaller than those reported in ref. 10. Case 2: Convective Dynamo in a Shallow Shell Here, we examine an approximation similar to what has been made in many previously published global simulations of Jupiter (25). Instead of simulating the fluid dynamics down to a small rocky core, an impermeable lower boundary of the convec- tion zone is prescribed at a radius of 0.8 RJ. The main reason for ignoring convection below this artificial lower boundary is to focus all of one’s computational resources in the region where magnetic field generation is expected to be the most efficient, which as in Case 1, is the upper part of the convec- tion zone. Another reason has been based on the assumption that differential rotation in Jupiter is severely suppressed by magnetic drag below a depth where the electrical conductiv- ity is sufficiently large. This is supported by 2D models of Jupiter’s interior fitted to the recent Juno gravity data (11, 12). It is also supported by some deep 3D dynamo simulations of Jupiter (17, 18) that maintain sufficiently vigorous convection and consequently intense magnetic fields. However, the evolved differential rotation in Case 1 (Fig. 3, Right) is not severely sup- pressed in the deep interior, likely because (as mentioned above) the magnetic field lines there tend to be in surfaces of con- stant angular velocity, which reduces magnetic drag and ohmic Fig. 6. A snapshot of the longitudinally averaged magnetic and gravity heating. fields as a function of spherical harmonic degree measured at what would The radial profiles for Case 2 reference state density, temper- be perijove for Juno’s orbits around Jupiter. (Upper) The amplitude of the ature, pressure, and electrical conductivity are the same as those magnetic Gauss coefficients g0 in units of one billionth of a Tesla (i.e., l in the upper region of Case 1. Thermal boundary conditions are 10−5gauss). (Lower) The amplitude of the perturbation in the gravity field relative to the spherically averaged value. The relatively straight line out- prescribed for Case 2 that drive convection throughout this (shal- lines the (discrete) even degree values that represent just the effect of the lower) fluid shell. However, since Jupiter’s rocky core would be planet’s oblateness (due to the planet’s rotation) (6). The jagged curve out- much deeper than this artificial lower boundary, here (unlike lines the (discrete) degree values that represent the remaining effects due for Case 1) we apply viscous stress-free and magnetic stress-free to the simulated fluid flows within the modeled planet. conditions on this lower boundary. As for Case 1, a shallow zonal wind layer initially develops in the upper part of this fluid shell, with multiple narrow weak intermittent jets at high latitude. However, after a couple mil- of the field (Fig. 4). However, this simulated surface field is some- lion more numerical time steps, the zonal wind evolves into a what more intense than Jupiter’s; for example, the dipole part, steady constant on cylinders structure throughout the fluid shell l = 1 , for Case 1 is four times larger than Jupiter’s (4). Of course, (Fig. 7). Several high-latitude jets are maintained on the surface, even if the model was near perfect, we would only expect qualita- a pattern slightly more like Jupiter’s (Fig. 1) than that for Case tive agreement with Jupiter’s magnetic spectra, because Jupiter 1. The maximum surface zonal wind speed occurs in the east- and the simulation are both time dependent. ward equatorial jet with an amplitude of 150 m / s, which is 50% Fig. 6, Lower is a plot of the amplitudes of the axisymmetric 1 higher than the maximum measured at Jupiter’s cloud tops but spherical harmonic coefficients, (2l + 1) 2 |∆Jl |, for the non- less than what the Galileo probe measured on Jupiter below the spherically symmetric part of the simulated gravity field (mea- cloud tops. As for Case 1, the total kinetic energy in these Case 2 sured at perijove) as a function of spherical harmonic degree zonal winds is typically an order of magnitude greater than that l. It shows that this small aspherical perturbation in the sim- ulated gravity field is dominated, for even degrees up to 10, by the planetary oblateness (due to centrifugal force) (6). For degrees higher than 10 (i.e., smaller length scales), fluid flows, especially the strong surface zonal winds, have a greater effect (than oblateness) on the structure of the aspherical part of the gravity field near the surface. Note that, because oblateness is symmetric across the equator, it makes no odd degree contribu- tion to the gravity field. Therefore, the contributions from odd degrees, including those less than degree 10, are totally due to deep axisymmetric fluid flows that are asymmetric with respect to the equator. Juno is able to detect these spherical harmonic gravity coeffi- cients down to an amplitude of roughly 10−8 (6, 10). An exciting recent analysis (10) of two of Juno’s near-surface perijove passes Fig. 7. Snapshots of the longitudinal wind for Case 2 in the same has already produced estimates of the axisymmetric spherical projections as those in Fig. 3 for Case 1.

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EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES axisymmetric inertial wave developed (Fig. 8) at midlatitude in Discussion both hemispheres throughout the interior. This snapshot illus- This project has two main objectives. One is to investigate the trates the ray-like pattern of shear layers in the axisymmetric importance of including the entire fluid interior in a model when longitudinal and radial components of fluid velocity on conical studying fluid flows and magnetic field generation in Jupiter. surfaces. Although several waves like this but propagating at dif- The three cases presented here illustrate how the dynamics ferent angles were initially excited, this one survived, because it in Jupiter’s deep interior can affect the flows and fields near is continually reinforced by closing on itself after “reflecting” off Jupiter’s surface. The simulated zonal winds in all three cases are the upper boundary at four latitudes and being tangent to the considered “deep” zonal winds maintained by the dynamics of lower (rocky core) boundary at two latitudes. A dominant iner- the deep fluid interior, not zonal winds maintained by solar inso- tial wave would likely be difficult to maintain in a full sphere with lation in Jupiter’s very shallow gaseous atmosphere. The early no rocky core to force nodes in the transverse oscillation. All legs results from Juno (10) at least support the existence of local of this wave oscillate at the same frequency and propagate at the winds that are deep and strong. Cases 2 and 3 maintain some- same angle relative to the planetary rotation axis (16, 26, 27). what more Jupiter-like zonal winds at the surface than does Case The latitudinal separation between the two rays at the surface 1 by completely ignoring the very deep interior or by suppress- in each hemisphere equals the diameter of the model’s rocky ing convection there, respectively. Case 3 crudely explores such core. If a similar oscillating axisymmetric pattern of zonal and a situation by forcing, via internal heating, a subadiabatic (con- radial flows was observed on Jupiter’s surface by, for example, vectively stable) fluid interior below a relatively shallow upper time-lapse imagery from Junocam (28), its frequency and surface convecting region that excites internal gravity waves in the lower latitudes could determine the existence and size of the planet’s stable region and Alfv´en waves and inertial waves throughout the rocky core. entire fluid interior. This inertial wave (Fig. 8) was a dominant feature for at However, as discussed, our computer model is far from being least 2 simulated years. Its period was comparable with the 10-h perfect. No one has been able to produce a 3D global convective rotation period, roughly six times less than periods of long- dynamo simulation with all model prescriptions being Jupiter- wavelength internal gravity waves in this simulation, and much like because of the huge computational resources that would be less than characteristic periods of the simulated Alfv´en waves. required. Although we set the planet size, mass, and rotation rate However, maintaining such a large-scale dominant inertial wave to Jupiter values, we are forced to use enhanced values for the in Jupiter could be difficult if Jupiter’s rocky core boundary was diffusivities, which require a luminosity greater than Jupiter’s to not well-defined. Likewise, reflections at Jupiter’s surface would evolve the surface amplitudes of zonal winds and magnetic fields be more complicated and less efficient than those off our model’s to Jupiter-like values. However, encouraged by Case 3 results, upper boundary. Even without these complications, the iner- the next step in this investigation could be to continue Case tial wave in the Case 3 simulation eventually disappeared and 3, gradually decreasing these diffusivities while increasing spa- has not returned after an additional 13 simulated years, likely tial and temporal resolutions. Although one could not afford to because of interference with convective flows or short period extend the simulation very far in time, it might be enough to gravity waves. provide useful insight. The shearing and twisting fluid flows in Case 3 self-consistently Another step in this investigation should be the development generate somewhat of a globally banded magnetic field (Fig. 8, of a more physical model of the deep interior below 0.9 RJ. For Bottom). The convective dynamo mechanism is most efficient example, a plausible thermal mechanism for maintaining a con- near 0.9 RJ, where convective eddies and zonal winds are still sig- vectively stable layer deep within Jupiter could be absorption of nificant and electrical conductivity is at its maximum. The peak latent heat by molecular hydrogen when, while sinking, it dissoci- magnetic field intensity, about 500 gauss, occurs near this radius. ates into monatomic hydrogen. Likewise, latent heat is emitted in The wave motion in the deep (stable) interior, which is excited by rising monatomic hydrogen as it combines into molecular hydro- the convection above, is less energetic and less helical, and there- gen. This phase change, which is thought to continually occur fore, it is less efficient in generating magnetic field. We see no in Jupiter between roughly 0.8 and 0.9 RJ (14), should keep the significant “pumping” of the magnetic field from the convecting temperature nearly constant in this layer and possibly subadia- region into the lower stable region by convective downwellings. batic. It has also been proposed that “helium rainout” (29, 30) However, as mentioned, the model’s prescribed electrical con- may be occurring in this same region. That is, in this region, ductivity is much less than predicted for the deep interior of helium becomes insoluble in hydrogen and gravitationally falls Jupiter (14, 15). through the fluid, which produces a stable layer with heavy fluid Note that the dipolar part of the magnetic field in Case 3 below light fluid. There may also be double-diffusive instabili- (check the polar regions in Fig. 8, Bottom) has the reversed polar- ties occurring below this layer due to very different molecular ity of that in Case 1 (Figs. 4 and 5) and likewise, the reverse of diffusion rates of temperature and heavy element composi- Jupiter’s dipole polarity. This is not an issue because the result- tion in a region with a destabilizing (superadiabatic) thermal ing polarity in these simulations depends on the arbitrary initial stratification and a stabilizing mean molecular weight strati- conditions and because the exactly reversed polarity would also fication (31). satisfy the coupled set of differential equations for exactly the Our other objective for this project is to provide interpre- same flow and thermal patterns. Note also that the magnetic tations of certain global patterns that the Juno mission might dipole part of the field in Case 1 is nearly an axial dipole; that discover at Jupiter. The simulations presented here suggest that is, the magnetic dipole axis is nearly the same as the planetary a detection by Juno of a global latitudinally banded pattern in rotation axis. However, the dipole axis for Case 3 is a little more the magnetic field would provide evidence for Jupiter’s banded tilted (somewhat more like Jupiter’s) and is a little more time zonal winds extending deep below the cloud surface to where dependent than that for Case 1. This greater variability in space electrical conductivity is sufficiently high for these winds to shear and time for Case 3 occurs because the zonal wind structure is the deep magnetic field enough to be detectable out to Juno’s not constant on cylinders throughout the entire fluid interior; the perijove. Similarly, a detection by Juno of a global latitudinally dominant dynamic length scales are more confined to the shallow banded pattern in the gravity field perturbations would argue convecting region and therefore have less global influence. No for deep banded zonal winds that shear a sufficient amount of dipole reversals have occurred in these Jupiter simulations; how- mass to produce detectable gravity variations at Juno’s peri- ever, as mentioned, the simulations span only about 20 simulated jove. The recent analyses of Juno’s near-surface gravity from years. two perijove passes (10–12) provide encouraging evidence for

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Glatzmaier htol eed nrdu te“ard aibe nEs –)pu small a plus 1–7) Eqs. in variables state pre- “barred” (the a reference radius of on nonmagnetic) sum depends the only and as that (hydrostatic written is time-independent equations these scribed in variable dependent Each solved are paper: that this equations in presented of cases set the the for described details model briefly refs. the we and in Here, described 19. are and graphics 16 computer numerical and conditions, programing, boundary parallel approximation, method, anelastic the of details The of size and Methods existence radial the with hemispheres, for both core. sync evidence rocky in in Jupiter’s latitudes exciting oscillating same provide jets the wind could at zonal occurring of and pair flows sur- by a the detection at as a manifested wave, face likely, gravity inertial less axisymmetric and an much magnetic of although near Junocam its addition, Jupiter’s banded In globally are. of fields how coverage reveal Juno global to Soon, sufficient surface. surface gathered cloud Jupiter’s have below will well flow fluid strong .Blo ,e l 21)Jptrsitro n epamshr:Teiiilpole-to-pole initial The atmosphere: deep and interior Jupiter’s (2017) al. et S, Bolton 9. .JnsC om 21)Acoeu iwo uie’ antcfil rmJn:New Juno: from field magnetic Jupiter’s of view close-up A (2017) R Holme C, Jones initial of 8. analysis The (2017) flows. 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Foundation the OPR by pro- Grant was provided project NASA this by for Support vided manuscript. this improving for suggestions ACKNOWLEDGMENTS. where onayad10 and boundary 10 4. are are ities boundary lower the 10 at pressure and perature, 2 is former the of neglected. one-tenth not nearly is is latter therefore, the and latitude, low at surface the Jupiter’s and the Here, acceleration, 34). within itational (33, formulation approximation Lantz–Braginsky–Roberts anelastic the using perturbation (Eq. equation tum nudges as constantly profile case target these the this above. toward at described entropy for values boundary perturbation source entropy averaged thermal spherically target heating upper the the and internal to lower The equal the boundaries. entropy, both cases, fixed 3, two are these Case conditions For For 32). time. (17, and planet a space the by of driven rate is cooling ( Convection source boundary. heating upper internal entropy prescribed fixed the a on gradient and radial condition perturbation zero entropy perturbation the a on applying condition by boundary determined lower is perturbation entropy the respectively. and respectively; fusivities, flow permeability; magnetic fluid time-dependent 3D S the step, field time netic numerical each at (Eq. conservation (Eq. conservation momentum 4), (Eq. perturbation potential tional (Eq. conservation mass depends that value space. state 3D reference and time its on to relative perturbation (unbarred) 10 n rvttoa potential gravitational and , 4 −4 o h iuain rsne ee h oa aso h oe planet model the of mass total the here, presented simulations the For momen- the in term buoyancy the of part perturbation pressure The of part symmetric spherically the of gradient radial the 2, and 1 Cases For describes equations differential partial nonlinear of system coupled This 475–506. flow? mean zonal 194501. convection. compressible planets. giant of region semi-conducting Soc Astron Roy circulation. meridional and rotation 316–330. differential Induced V. shell. uie n Saturn. and Jupiter unity. method. and model dynamics. jet equatorial and field 148–159. 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Press, Univ (Princeton adiabat. ,ad4.15 and K, × ˆ r ta s 0hrtto eid.Terfrnesaedniy tem- density, state reference The period). rotation 10-h a is, (that rad/s 10 and σ ¯ oitAto AJ Astron Soviet 27 seetia odciiy(i.2, (Fig. conductivity electrical is srpy Suppl J Astrophys n etrain ndensity in perturbations and B, 6 g n h vrg lntr oainrt is rate rotation planetary average the and kg, θ ˆ n 10 and r ntvcosi h ailadclttdnldirections, colatitudinal and radial the in vectors unit are × 97:458–472. 9 10 m .Tenmrclslto oteeeutosupdates, equations these to solution numerical The 7). epy srpy li Dynam Fluid Astrophys Geophys a encmie ihtegain ftepressure the of gradient the with combined been has 5) Icarus 2 opPhys Comp J 12 7 tteuprboundary. upper the at /s m nrdcint oeigCneto nPaesadStars and Planets in Convection Modeling to Introduction N/m 18:621–624. η ¯ 2 etakC oe n .Osnfrterhelpful their for Olson P. and Jones C. thank We ,mgei u osrain(Eq. conservation flux magnetic 1), ν ¯ ,rsetvl.Mgei diffusivity, Magnetic respectively. /s, hsErhPae Inter Planet Earth Phys 196:653–664. PNAS stemgei diffusivity; magnetic the is 202:5. and 2 epciey h icu n hra diffusiv- thermal and viscous The respectively. , κ ¯ U 55:461–484. | epy e Lett Res Geophys Ω h aeo tantno is tensor strain of rate The . r h ubln icu n hra dif- thermal and viscous turbulent the are ,mgei nuto (Eq. induction magnetic 5), ,teprubto qaino tt (Eq. state of equation perturbation the 3), 2 uy3 2018 3, July ρ ¯ emi etiua ceeain Near acceleration. centrifugal is term TQ Icarus nEq. in − Right g ¯ 296:59–72. pressure ρ, ˆ 232:36–50. r ,wihrpeet h slow the represents which 7), sterfrnesaegrav- state reference the is 109:541–566. | 41:5410–5419. ,i 10 is ), o.115 vol. Science δ ij 5 × pcfi entropy specific p, m steui tensor; unit the is Q 208:746–748. hsRvLett Rev Phys 10 pc c Rev Sci Space 2 | ttelower the at /s sacntn in constant a is ,tegravita- the 2), 3 ,adenergy and 6), o 27 no. srpy J Astrophys kg/m e η ¯ ij Ω = ; Icarus µ = 1/(µ mag- v, 3 o Not Mon 0 1.9 , | 1.77 sthe is 6903 0 213: 256: 241: σ ¯ 99: × × ),

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