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Jupiter’s deep ’s to Jupiter’s the about made prominent much are more to reveal measurements are due the which closer (5) (4) perturbations, the field field These this (6). gravity in dynamics Jupiter’s exist above in perturbations far small and measured However, when dipolar surface. dominantly its is It R system. of solar 1% vis- than the less this below is Although extend depth (3). Jupiter 21-bar surface. bars on the 21 winds surface, below about zonal ible until bars, at the 4 constant lost that of approximately was suggests pressure signal remained a at probe’s speed s the / wind m of 170 the pressure about which a of at radius, maximum top s a / cloud m to Jupiter’s 80 (roughly zonal about bar measured 1 from about 1995, increased in that NASA’s equator speeds (2)]. wind into its Orbiter near latitude Galileo the atmosphere in from Jupiter’s dropped alternating [that was which (1) streams Probe, winds Galileo jet zonal clouds banded east–west banded latitudinally these is, by that sheared being shown are have observations spacecraft J Jupiter core. rocky Jupiter’s of size for evidence and provide existence surface, model’s Jupiter’s would the frequency on the oscillation detected of was and size wave latitudes a its the such on If core. depend rocky that small surface with wave, the inertial at axisymmetric properties deep a for years, maintained, simulated also couple has simulations a our deep of One extending surface. winds the zonal below Jupiter’s (2017) for al. evidence et provide space- S, would orbiting NASA’s [Bolton Juno, Jupiter by at winds magnetic detected craft were near-surface zonal fields global Jupiter’s Jupiter’s gravity if in and that, long- do patterns suggest A simulations depth banded interior. Our latitudinally the what surface. of its to part below extend upper is generation the question field to and debated confined deep convection be very the to the in seems of dynamics much the its although including interior, is, of importance that the high- interior: and simulations light different deep three convection from Jupiter’s Olson) Results L. thermal dynamo. in Peter convective and of generation Jones Chris field simulations by reviewed magnetic 2010. computer 2018; in 13, elected February describe Sciences review of for We Academy (sent National 2018 the 21, of May Glatzmaier, members A. by Gary Articles by Inaugural Contributed of series special the of part is contribution This a sees Glatzmaier A. Juno Gary what interpret help internal dynamics deep Jupiter’s of simulations Computer at n lntr cecsDprmn,Uiest fClfri,SnaCu,C 95064 CA Cruz, Santa California, of University Department, Sciences Planetary and lhuhmn hoeia tde fJptrsitro have interior Jupiter’s of studies theoretical many Although our in field magnetic planetary intense most the has Jupiter ptrslttdnlybne lu atr Fg )has 1) and Ground-based (Fig. years. of pattern hundreds cloud for people banded intrigued latitudinally upiter’s | oa winds zonal | antcfield magnetic a,1 | rvt field gravity Science | 5:2–2] they 356:821–825], uoMission Juno R J =7 J eo the below ⇥ 10 7 m) oee,a rsueicesswt et,tefli becomes liquid. fluid dense the a depth, into with depth approximately increases with elements is pressure compresses layer heavier as it atmospheric but However, of gas, shallow amounts ideal the smaller an minor in a and , Fluid mainly (13). , of planet of rotating amount giant a is Jupiter convective is Model Jupiter’s fields of gravity studies and computational magnetic of dynamo. its goal and major winds a maintenance zonal down and structure Jupiter’s exist the of R of flows descriptions of fluid physical 4% detailed strong least that at approach to suggesting orbital closest results R its exciting of (i.e., 5% passes Jupiter, perijove to Juno’s of two hsoe cesatcei itiue under distributed is article 1 BY-NC-ND) (CC 4.0 access License NonCommercial-NoDerivatives open benchmark performance report. computational that This a on of collaborators research report not 2016 were Chris a reviewers They on including exercise. Olson, coauthors, L. 37 Peter of and one was Jones G.A.G. statement: University. interest of Hopkins Conflict Johns P.L.O., and Leeds; data, of University analyzed C.J., paper. Reviewers: research, the wrote performed and code, research, computer the designed wrote G.A.G. contributions: Author the below surface well Jupiter’s structures at and observed intensities fields their attempts and predict in to flows (16–18) and the interior deep explain Jupiter’s to in roughly of generation radius field a with material there rocky Jupiter, of of core R center 0.1 small the a At be (14). R metal may 0.9 liquid roughly a of becomes radius gen a at poor until, a conductivity increase diatomic electrical which is electrons, depth, conducting surface with become electrons increases Jupiter’s hydrogen pressure at as atmosphere however, gas conductor; The rapidly depth. stud- increases with conductivity to (15) electrical insensitive Jupiter’s computational that relatively “quantum indicate and ies being a (14) pressure becomes Experimental and it temperature. density is, with that fluid” degenerate; electron partially mi:[email protected]. Email: epitrrtadepantemitnneo hs global these measure. to of continues maintenance Juno will that the patterns that explain near-surface interior and deep interpret Jupiter’s help and in the time- convection generation 3D us field thermal describe tell magnetic of we could simulations Here, par- core. computer that rocky dependent a properties small detect Jupiter’s with of may wave size monitoring Juno of by surface, type addition, Jupiter’s ticular of In on type interior. motions the deep cloud into sur- Jupiter’s insight Jupiter’s in providing above winds started fields already gravity have and resulting face The magnetic surface. of near patterns its of collecting Jupiter, measurements orbiting high-fidelity currently is spacecraft Juno NASA’s Significance optrmdl iuaetemlcneto n magnetic and convection thermal simulate models Computer J (13). y J J bv h ufc)hv led produced already have surface) the above eo uie’ lu ufc.Providing surface. cloud Jupiter’s below raieCmosAttribution- Commons Creative NSLts Articles Latest PNAS . 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EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES at the surface that are also Jupiter like, we have to compensate for these enhanced diffusivities by driving thermal convection with a luminosity that is larger than Jupiter’s. Jupiter’s magnetic diffusivity is inversely proportional to its electrical conductivity and is larger than its viscous and ther- mal diffusivities. We prescribe an electrical conductivity for the model that increases (approximately) exponentially (21) by four orders of magnitude from the model’s surface down to 0.9 RJ, where (Fig. 2, Right) it transitions to a constant in the deeper interior. It is important for the simulated dynamics to have the amplitudes of these three model diffusivities be in the same order as they are for Jupiter. Therefore, since we are forced to prescribe enhanced values for the viscous and thermal diffusivi- ties, the maximum electrical conductivity that we prescribe below 0.9 RJ (Methods and Fig. 2, Right) is several orders of magnitude less than predicted values (14, 15). A required feature of our study is the self-consistent calcula- Fig. 1. (Left) An image of Jupiter in natural color taken by the Hubble tion (in 3D and at every numerical time step) of the perturbation Space Telescope [Image courtesy of NASA, ESA, and A. Simon (Goddard in the gravitational potential by solving a Poisson equation (Eq. Space Flight Center)]. (Right) Jupiter’s surface zonal wind speed in meters 3) forced by the local density perturbation. We also monitor the per second relative to its (rotating) magnetic field (1) (Reprinted from ref. local mass flux in and out of the model’s (fixed but permeable) 1, with permission from AAAS). spherical upper boundary and approximate the resulting local time-dependent accumulation or depletion of mass above this boundary as a surface mass density perturbation in hydrostatic surface. These models are defined by a coupled set of non- balance on this upper boundary surface. Then, assuming no other linear partial differential equations based on the classical laws density perturbations above this boundary, this procedure allows of conservation of mass, momentum, and energy that describe us to easily calculate the gravity perturbations anywhere above fluid flow, heat flow, and magnetic field generation in a rotat- the upper boundary as a function of time. This provides a con- ing, self-gravitating, density-stratified fluid sphere as an analogue nection between centrifugal accelerations due to, for example, of Jupiter’s dynamic interior. Massively parallel supercomput- the surface zonal winds and gravity perturbations at Jupiter’s per- ers produce the 3D time-dependent numerical solutions to these ijove. A similar method (16) allows us to calculate the magnetic equations. We have published (16, 19) detailed descriptions of field at perijove, assuming that electric currents are negligible our basic computer model used to produce the computer sim- above the surface. ulations presented here. The set of equations and the model The simulations presented here were initially run with a spa- prescriptions are presented in Methods. However, before dis- tial resolution in grid space (radial latitudinal longitudinal cussing the results, we briefly mention some of the model details levels) of 241 384 768 and then⇥ after reaching⇥ a statistical that define these particular Jupiter simulations. equilibrium, continued⇥ ⇥ at 241 768 1,536. In spectral space, We fit our 1D reference-state density (Fig. 2, Left), pressure, all spherical harmonics are used⇥ up⇥ to degree and order 255 and temperature to a 1D evolutionary interior model (5, 15) initially and later 511 (for the latitudinal and longitudinal expan- using a polytropic relationship of pressure proportional to den- sions) and in Chebyshev polynomials up to degree 241 (for the sity squared (20). The 1D evolutionary model solves only for radial expansions). The typical numerical time step is 100 (sim- the spherically averaged radial profiles of the thermodynamic ulated) seconds based on a 10-h rotation period. Each of the variables and simply parametrizes the effects of 3D convection. simulations presented here has run at least 20 simulated years However, this 1D model can afford to incorporate a more realis- (i.e., about 6 million numerical time steps). The details of this tic equation of state with a more detailed treatment of chemical computational procedure are described in ref. 16. composition and radiation transfer than can current 3D global We discuss three cases that illustrate how the prescribed depth convective dynamo models. It can also afford the much higher of the model’s fluid interior affects the patterns and intensi- resolution in radius required to model the very shallow atmo- ties of the winds and fields at the surface, which are compared spheric layer, which current dynamo models ignore. Our code with those observed on Jupiter to estimate how realistic the sim- then numerically solves the system of equations (Methods) that ulated interior dynamics may be. In addition, different mean describes the 3D time-dependent perturbations relative to our temperature profiles in radius are forced in these three cases 1D time-independent rotating reference state [that is, the ther- via thermal boundary conditions and prescribed internal heating modynamic and gravitational perturbations and the fluid velocity that determine if and where thermal convection occurs. Case 1 and magnetic field vectors (Methods) (16)]. Our model prescriptions for the (dimensional) planetary mass, radius, and rotation rate are set to Jupiter’s. Based on the cal- culations of French et al. (15), we choose constant values for the viscous and thermal diffusivities. However, the estimated val- ues of these molecular diffusivities would result in a cascade of energy down to eddies too small to resolve with any 3D global simulation of convection in Jupiter’s interior. Moreover, this unresolved small-scale turbulence transports much more heat and momentum than molecular diffusion. Therefore, we numer- ically resolve as much of the energy spectrum as we practically can and then parametrize the transport of heat and momentum by the remaining unresolved eddies as a diffusion process using enhanced values for the (turbulent) viscous and thermal diffusiv- Fig. 2. The model’s prescribed reference-state density (Left) and electrical ities. However, to obtain peak amplitudes of the winds and fields conductivity (Right).

2 of 9 www.pnas.org/cgi/doi/10.1073/pnas.1709125115 Glatzmaier | Glatzmaier represent reds and reference). of Yellows frame rotating plane. the (relative to winds meridian westward-directed represent a blues fluid winds; in the eastward-directed plotted of surface winds) upper tudinal spherical entire the ( ( of model. projection 1. area Case equal for an winds) (i.e., velocity 3. Fig. cylinders” on “constant is wind that zonal axis; this rotation of planetary amplitude the from the R distance is, the of of 4% function least a at to down agree exist data, surface. flows gravity Jupiter’s Juno fluid early the strong which to 12), balance. that models (11, of a studies types two such different these indicate fit that encouraging not shallow is convective do it Earth’s deep Jupiter However, the our of for but simulations insolation, approximation dynamo solar bad by a driven not atmosphere is This ance. Eq. ( equations in of forces set “thermal full a the solving as assumption ods of maintained the instead is, is on That based rotation wind.” are differential however, Jupiter’s Some 12), that 22). (11, (16, studies interior recent sink density-stratified or vor- the (expand) when rise through they zone as (contract) forces convection Coriolis by rotating generated a are tices in occurs redistri- transport, the naturally advective by which (nonlinear) simulations by momentum our angular in of dependent maintained bution latitudinally is and 3, It radially a rate. (Fig. is rotation it time reference, in this of frame steady of tia virtually longitude becomes in wind, Right average zonal the the 3, steps, flow, to (Fig. time relative reference numerical latitude of million with frame westward rotating and the eastward alternate flow longitudinal that of bands produces Shell convection Deep thermal Rotating a in Dynamo Convective 1: Case below radii. stable two convectively these is tar- a R that toward profile 0.90 rocky extends heating temperature shell internal same mean by fluid R get the “nudged” spherical 0.98 constantly the uses is to is, profile 3 that 0.10 1; Case radius Case from of heating. spherical that internal shallower) as core by (artificially R this shell 0.8 stress-free throughout of fluid impermeable driven radius an also as a is defined at is interface lower 2 The spherical 1. 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EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES their surface magnetic fields are dominantly dipolar with differ- ential rotation in the deep interior suppressed by magnetic drag. The differences between their results and the banded results pre- sented here could be due to the different choices for the model parameters, boundary conditions, internal heating, centrifugal force, or our relatively weak electrical conductivity below 0.9 RJ. Analysis of Juno’s first perijove pass (7) shows two intense mag- netic flux patches on opposite sides of the equator indicating that Jupiter’s near-surface magnetic field is more complicated than previously expected. It will be interesting to see, after more Juno perijove passes have been analyzed, if Jupiter’s near-surface magnetic field pattern is globally banded. These same fluid flows determine the 3D distribution of local low and high mass densities, which slightly perturb the nearly spherically symmetric (reference state) gravity field below and above the surface. Consider the radial component of the grav- ity perturbation at Juno’s perijove in Fig. 5, Upper Left. Blue bands (downward-directed gravity perturbations) in this image correspond to faster rotating (eastward) jets at the surface (Fig. 3), which due to greater centrifugal forces, slightly ele- vate mass through our permeable upper boundary, enhancing the (downward-directed) gravitational acceleration at perijove. Juno data from two perijove passes (10) already show evidence of local gravity variations banded in latitude a few times smaller in ampli- tude compared with those in Fig. 5, Upper Left. When Juno data Fig. 4. A snapshot of the simulated magnetic field for Case 1 plotted in 3D from many more than two orbital passes are analyzed, we will see as magnetic lines of force. The axis of planetary rotation is centered and ver- if Jupiter’s near-surface gravity field is globally banded. tical. The model’s insulating near-surface region starts at the depth where Spherical harmonic spectra of the simulated magnetic field the east–west magnetic lines begin to bend upward. Lines are gold where patterns (measured at perijove) provide additional information the field is outward directed and blue where the field is inward directed. to be compared with Juno’s data. Fig. 6, Upper is a plot of the amplitudes of the axisymmetric spherical harmonic (Gauss) coef- 0 ficients, gl , as a function of spherical harmonic degree, l, for the more rapidly with distance from Jupiter than the dipole mode Fig. 5 snapshot. Our simulated data are calculated every numer- (degree l =1); therefore, the banded magnetic patterns seen in ical time step for all spherical harmonic degrees and orders up Fig. 5, Right would be virtually nonexistent along most of Juno’s to 511 but plotted here (for order 0) out to degree 30. They highly elliptic (53-d) polar orbit far from Jupiter, where the field show how the intensity of the longitudinally averaged part of the appears dominantly dipolar. simulated magnetic field decreases with increasing spherical har- Other 3D (nearly full-sphere) convective dynamo simulation monic degree (that is, as the length scale of the mode decreases). studies of Jupiter (17, 18) have not developed latitudinally The relatively broad spectrum at low degrees reflects the banded magnetic structures like those in Fig. 5, Right. Instead, relatively large contribution from the banded (nondipolar) part

Fig. 5. A snapshot of the simulated gravity field perturbation (Left) and magnetic field (Right) for Case 1 plotted in a spherical surface with a radius equal to Juno’s perijove. Upper shows the radial components of the fields (yellows and reds are outward directed, and blues are inward); Lower shows the colatitudinal components (yellows and reds are southward directed, and blues are northward). The peak amplitudes of the gravity perturbation for the radial and colatitudinal components are 18 and 14 mgal, respectively. The peak amplitudes of the magnetic field for the radial and colatitudinal components are 5 and 4 gauss, respectively.

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EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES of the convection. The magnetic and gravity fields at perijove for produce “Alfv´en waves.” Coriolis forces, which occur (in the this case also have latitudinally banded patterns, although not as rotating frame of reference) when fluid flows perpendicular to well-defined as those for Case 1. An earlier simulation of Saturn’s the axis of rotation, serve as yet another restoring force that dynamics (16) in its outer region produced latitudinally banded drive “inertial waves.” Alfv´en waves and inertial waves exist in magnetic fields qualitatively similar to Case 2. Although Case both the convectively stable and unstable regions; internal grav- 2 maintains surface winds that better match those of Jupiter, ity waves get excited only in the convectively stable regions. All the flows, fields, and thermodynamics that should exist below three types of waves are transverse waves; that is, the oscil- the (artificial) impermeable lower boundary at 0.8 RJ have been lating fluid motion is perpendicular to the direction of phase ignored. propagation. Internal gravity wave frequency is maximum when the wave propagates horizontally (i.e., when oscillating fluid Case 3: Convective Dynamo in a Shallow Shell Above a Deep motions are parallel to gravity). Alfv´en wave frequency is max- Convectively Stable Interior imum when phase propagation is parallel to the local magnetic As discussed for Case 2, a more Jupiter-like surface zonal wind field direction (i.e., when oscillating fluid motions are perpen- pattern can be simulated when an impermeable lower bound- dicular to the local magnetic field). Inertial wave frequency is ary to the convection zone is imposed at radius 0.8 RJ (or maximum when the wave is propagating parallel to the rota- above) instead of at 0.1 RJ, which is what 1D evolutionary tion axis (i.e., when oscillating fluid motions are perpendicular models predict. There may be important physical mechanisms to the rotation axis). When all three types of waves are excited operating in Jupiter, but missing in our model (in addition to with comparable frequencies, the resulting wave motion can be higher electrical conductivity), that tend to confine convection complicated. to a relatively shallow upper region, while allowing only less As in the two previous cases, the eastward zonal jet in the energetic motions in the deeper part of the fluid interior. There- equatorial region for Case 3, seen in Fig. 8, Top, is main- fore, we ask what modifications to the model could produce tained by the nonlinear convergence of angular momentum flux a more Jupiter-like latitudinally banded pattern at the surface that is driven by 3D rotating convection within the density- with multiple alternating zonal wind jets up to high latitude stratified unstable region. Weak, relatively shallow zonal wind while simulating the internal magnetohydrodynamics down to jets, mainly westward directed relative to the rotating frame of radius 0.1 RJ. reference, also exist at high latitude. In addition, for Case 3, an Case 3 is a simple test simulation of such a scenario. Con- vectively stable fluid regions are maintained from the lower boundary at radius 0.10 RJ up to 0.90 RJ and from 0.96 RJ up to the upper boundary at 0.98 RJ; a convectively unstable region is maintained between these two stable regions. Internal grav- ity waves are excited in the lower and upper (stable) regions, where the mean temperature decreases more slowly with radius than an adiabatic temperature profile would; that is, in these two regions, the mean temperature profiles are “subadiabatic,” and the mean (specific) entropy increases with radius. Thermal convection occurs in the middle (unstable) region, where the mean temperature decreases faster with radius than an adiabatic temperature profile would; that is, the mean temperature pro- file is “superadiabatic,” and the mean entropy decreases with radius. The shallow upper layer serves as a crude stratosphere, which provides some resistance to convective penetration from below, albeit without realistic atmospheric physics. Our tests show that this upper layer has little effect on the large-scale flow patterns at the surface, likely because this layer is not stable enough. The model maintains these three regimes using a time- dependent and radially dependent heat source that continually “nudges” the mean entropy perturbation toward a prescribed target profile (16). This heat source is proportional to the difference between the current (local) mean entropy and the target value and inversely proportional to a time constant that determines how closely the entropy perturbation tracks the target profile. For Case 3, the target entropy increases lin- early with radius in the lower (stable) region by a total of 250 J / (kg K), it decreases linearly with radius in the middle (convectively unstable) region by 22 J / (kg K), and it increases linearly with radius in the upper (stable) region by 22 J / (kg K). These changes in entropy are relatively small compared with the model’s specific heat capacity at constant pressure (cP = 15,000 J / (kg K)) as required for codes like ours that use the “anelastic approximation” to the fully compressible system of equations (16). Downwelling plumes in the convecting region (between 0.90 Fig. 8. A snapshot of the longitudinal and radial components of fluid veloc- ity and the radial component of magnetic field for Case 3. (Left) Plotted in and 0.96 RJ) penetrate slightly into the lower stable region a constant radius surface slightly below the upper boundary. (Right) Lon- exciting 3D “internal gravity waves” that propagate through the gitudinally averaged and plotted in a meridian plane. Reds and yellows interior due to buoyancy restoring forces. Magnetic fields, dis- represent eastward (Top) or upward (Middle and Bottom); blues represent torted by fluid flows, provide magnetic restoring forces that westward or downward.

6 of 9 www.pnas.org/cgi/doi/10.1073/pnas.1709125115 Glatzmaier | Glatzmaier simulated 20 about only years. span simulations how- No the simulations; mentioned, influence. Jupiter as these ever, global in occurred less have have reversals dipole therefore and shallow the region is to confined structure convecting more the are wind scales interior; zonal length fluid dynamic entire the dominant the because throughout cylinders occurs space on time in 3 constant more variability not Case greater little for This a time 1. is and Case more for and little that planetary Jupiter’s) a than the is like dependent that 3 as dipole; more Case same for axial (somewhat the axis an tilted dipole nearly nearly the is magnetic However, is axis axis. 1 the rotation dipole Case that magnetic in also the field the is, Note the exactly of patterns. for part thermal equations dipole and differential flow of also set same would coupled polarity reversed initial the exactly arbitrary satisfy of the the reverse result- on because the the depends and likewise, because simulations conditions issue and these an 5) in not and polarity is ing 4 This (Figs. polarity. 1 dipole Case Jupiter’s in that of 8, ity of Fig. in interior regions polar the deep (check the for predicted than 15). con- (14, less electrical Jupiter much prescribed is model’s downwellings. the ductivity convective by mentioned, region no as stable see convecting However, lower the We the from field. field into magnetic magnetic region generating the in of “pumping” efficient significant less there- and is helical, less it and by fore, energetic excited less is is which above, interior, convection (stable) the deep radius. the this in peak near motion occurs The wave gauss, The maximum. 500 its about intensity, at field is magnetic conductivity electrical and nificant 8, R (Fig. 0.9 field near magnetic banded globally Bottom a of somewhat period generate short or likely flows years, convective simulated waves. with and 13 gravity interference disappeared additional eventually an of simulation after because iner- 3 returned the Case not complications, the has in these wave without tial model’s Even our off boundary. those than upper efficient less would and was surface complicated boundary Jupiter’s more core be at rocky reflections Jupiter’s Likewise, if well-defined. difficult not be wave could inertial Alfv Jupiter dominant long- simulated in large-scale a the of much such of maintaining and periods periods However, simulation, than characteristic this than less in less waves times gravity 10-h six internal the wavelength with roughly comparable was period, period rotation Its years. simulated 2 least planet’s the of size and core. existence rocky example, the for determine surface and could by, frequency and latitudes its surface (28), zonal Jupiter’s Junocam rocky from of on imagery model’s pattern time-lapse observed axisymmetric was the oscillating flows of surface similar radial diameter the a the 27). at If 26, equals rays core. 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Jupiter-like support and more least what deep early are at The that (10) atmosphere. winds Juno gaseous from shallow inso- very results solar by Jupiter’s of maintained dynamics in winds zonal the lation near not by interior, fields fluid maintained deep and the winds are dynamics flows cases zonal three the all “deep” the in considered affect winds how zonal simulated can The illustrate Jupiter. surface. interior Jupiter’s here in deep generation presented Jupiter’s field in cases magnetic three the and when The investigate model flows a to in fluid is interior fluid studying One entire the objectives. including main of importance two has project This Discussion w eioepse 1–2 rvd norgn vdnefor evidence from encouraging gravity provide peri- near-surface (10–12) Juno’s Juno’s passes of at perijove of analyses variations two amount recent gravity sufficient The detectable a argue jove. shear produce would that to winds perturbations mass zonal field latitudinally banded gravity global deep Juno’s the a for to of in out Juno pattern detectable by banded be detection to a enough Similarly, where perijove. field to shear to magnetic surface winds deep these cloud for the high the sufficiently below is banded in conductivity deep Jupiter’s electrical pattern for extending banded evidence winds latitudinally provide zonal global would a field might of magnetic that Juno the mission suggest by Juno here detection presented the simulations a that The Jupiter. patterns at global discover certain of strati- tations weight molecular thermal mean (superadiabatic) composi- destabilizing stabilizing (31). element a fication a with heavy and region molecular and stratification a different temperature very in to of tion instabili- due rates double-diffusive layer be this diffusion also below fluid may occurring region, heavy There ties with this layer fluid. in stable light falls a is, gravitationally below produces which That and fluid, hydrogen region. the 30) in through same (29, insoluble this rainout” becomes in “helium helium that occurring proposed subadia- be been possibly may also and has layer It this occur batic. in continually R constant 0.9 to and nearly thought 0.8 temperature is roughly between which Jupiter change, in hydro- phase molecular This into combines gen. it as in emitted hydrogen is monatomic heat rising latent dissoci- Likewise, it hydrogen. monatomic sinking, of into while absorption ates when, be hydrogen molecular could by Jupiter heat con- within latent a R deep maintaining 0.9 layer for below stable mechanism interior vectively thermal deep the plausible of a to model example, physical enough more be a of might it time, in far insight. to useful Case afford very provide not simulation continue could the spa- one to Although increasing extend be resolutions. while temporal could diffusivities and investigation these tial results, decreasing this 3 gradually in Case 3, step by next encouraged However, the fields magnetic values. and Jupiter-like winds zonal to to of Jupiter’s amplitudes than surface greater the evolve luminosity a require which diffusivities, oee,a icse,orcmue oe sfrfo being from far is model computer our discussed, as However, u te betv o hspoeti opoieinterpre- provide to is project this for objective other Our development the be should investigation this in step Another ´ nwvsadieta ae hogotthe throughout waves inertial and waves en NSLts Articles Latest PNAS J 1) hudke the keep should (14), | J For . f9 of 7

EARTH, ATMOSPHERIC, INAUGURAL ARTICLE AND PLANETARY SCIENCES strong fluid flow well below Jupiter’s cloud surface. Soon, Juno (unbarred) perturbation relative to its reference state value that depends will have gathered sufficient global coverage of Jupiter’s near on time and 3D space. surface to reveal how globally banded its magnetic and gravity This coupled system of nonlinear partial differential equations describes fields are. In addition, although much less likely, a detection by mass conservation (Eq. 1), magnetic flux conservation (Eq. 2), the gravita- Junocam of an axisymmetric inertial wave, manifested at the sur- tional potential perturbation (Eq. 3), the perturbation equation of state (Eq. 4), momentum conservation (Eq. 5), magnetic induction (Eq. 6), and energy face as a pair of zonal wind jets oscillating in sync with radial conservation (Eq. 7). The numerical solution to these equations updates, flows and occurring at the same latitudes in both hemispheres, at each numerical time step, the 3D time-dependent fluid flow v, mag- could provide exciting evidence for the existence and size of netic field B, and perturbations in density ⇢, pressure p, specific entropy Jupiter’s rocky core. S, and gravitational potential U. The rate of strain tensor is eij; µ0 is the magnetic permeability; ⌫¯ and ¯ are the turbulent viscous and thermal dif- Methods fusivities, respectively; ⌘¯ is the magnetic diffusivity; ij is the unit tensor; The details of the anelastic approximation, boundary conditions, numerical and ˆr and ✓ˆ are unit vectors in the radial and colatitudinal directions, method, parallel programing, and computer graphics are described in refs. respectively. 16 and 19. Here, we briefly described the set of equations that are solved For Cases 1 and 2, the radial gradient of the spherically symmetric part of and the model details for the cases presented in this paper: the entropy perturbation is determined by applying a zero radial gradient lower boundary condition on the entropy perturbation and a fixed entropy ⇢¯v = 0 [1] perturbation condition on the upper boundary. Convection is driven by a r · prescribed internal heating source (⇢¯TQ in Eq. 7), which represents the slow B = 0 [2] cooling rate of the planet (17, 32). For these two cases, Q is a constant in r · space and time. For Case 3, both the lower and upper thermal boundary 2U = 4⇡G⇢ [3] r conditions are fixed entropy, equal to the target entropy values at these @⇢ @⇢ boundaries. The internal heating source for this case constantly nudges ⇢ = S + p [4] @S @p the spherically averaged perturbation entropy toward the target profile as ✓ ◆p ✓ ◆S described above. The pressure perturbation part of the buoyancy term in the momen- @v ⇢¯ = (⇢¯vv) ⇢¯ (p/⇢¯+ U) [5] tum equation (Eq. 5) has been combined with the gradient of the pressure @t r · r perturbation using the Lantz–Braginsky–Roberts formulation within the 2 ˆ @⇢ anelastic approximation (33, 34). Here, g¯ˆr is the reference state grav- g¯ˆr + ⌦ r sin ✓(ˆr sin ✓ + ✓ cos ✓) S 2 @S p itational acceleration, and the ⌦ term is centrifugal acceleration. Near h i✓ ◆ 1 Jupiter’s surface at low latitude, the latter is nearly one-tenth of the former +2⇢¯v ⌦ + ( B) B and therefore, is not neglected. ⇥ µ r⇥ ⇥ 0 For the simulations presented here, the total mass of the model planet 1 is 2 1027 kg, and the average planetary rotation rate is ⌦ = 1.77 + (2⇢¯⌫¯(eij ( v)ij)) 4⇥ ⇥ r · 3 r · 10 rad/s (that is, a 10-h rotation period). The reference state density, tem- perature, and pressure at the lower boundary are 4.4 103 kg/m3,1.9 @B ⇥ ⇥ = (v B) (⌘¯ B) [6] 104 K, and 4.15 1012 N/m2, respectively. The viscous and thermal diffusiv- @t r⇥ ⇥ r⇥ r⇥ 6 ⇥ 7 2 ities are 10 and 10 m /s, respectively. Magnetic diffusivity, ⌘¯ = 1/(µ0¯), where ¯ is electrical conductivity (Fig. 2, Right), is 105 m2/s at the lower @S 9 2 ⇢¯T = T (⇢¯Sv) + (¯⇢¯T S) +¯⇢TQ boundary and 10 m /s at the upper boundary. @t r · r · r

1 2 ⌘¯ 2 +2⌫¯⇢¯ eijeij ( v) + B . [7] ACKNOWLEDGMENTS. We thank C. Jones and P. Olson for their helpful 3 r · µ |r⇥ | ✓ ◆ 0 suggestions for improving this manuscript. Support for this project was pro- vided by NASA Grant OPR NNX13AK94G. Computational resources were Each dependent variable in these equations is written as the sum of a pre- provided by the NASA Ames Research Center and by National Science scribed time-independent (hydrostatic and nonmagnetic) reference state Foundation funds that supported parallel computing at the University of that only depends on radius (the “barred” variables in Eqs. 1–7) plus a small California Santa Cruz.

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8 of 9 www.pnas.org/cgi/doi/10.1073/pnas.1709125115 Glatzmaier | 31. 30. Glatzmaier olR aadP tlmc 21)Anwmdlfrmxn ydouble-diffusive by mixing for model new A (2016) grav- S Juno Stellmach P, to Garaud models R, structure Moll interior Jupiter Comparing (2017) al. et S, Wahl ovcin(eicneto) I.Temladcmoiinltasottruhnon- through transport ODDC. compositional layered and Thermal III. (semi-convection). convection core. expanded an of role the 4659. and measurements ity srpy J Astrophys 823:33. epy e Lett Res Geophys 44:4649– 34. 33. 32. rgnk ,RbrsP(95 qain oenn ovcini at’ oeadthe and core earth’s in convection governing Equations (1995) fluid P Roberts convecting S, stratified, Braginsky a in fields simulation magnetic of geodynamo behavior evolutionary Dynamical (1992) anelastic S Lantz An (1996) P Roberts G, Glatzmaier geodynamo. NY). Ithaca, University, (Cornell thesis PhD layer. convection. thermal and compositional by driven epy srpy li Dynam Fluid Astrophys Geophys 79:1–97. hsc D Physica NSLts Articles Latest PNAS 97:81–94. | f9 of 9

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