- 1. Nature | Vol 629 | 23 May 2024 | 769 Article Thesolardynamobeginsnearthesurface Geoffrey M. Vasil1✉, Daniel Lecoanet2,3 , Kyle Augustson2,3 , Keaton J. Burns4,5 , Jeffrey S. Oishi6 , Benjamin P. Brown7 , Nicholas Brummell8 & Keith Julien9,10 ThemagneticdynamocycleoftheSunfeaturesadistinctpattern:apropagating regionofsunspotemergenceappearsaround30°latitudeandvanishesnearthe equatorevery11 years(ref. 1).Moreover,longitudinalflowscalledtorsionaloscillations closelyshadowsunspotmigration,undoubtedlysharingacommoncause2 .Contrary totheoriessuggestingdeeporiginsofthesephenomena,helioseismologypinpoints low-latitudetorsionaloscillationstotheouter5–10%oftheSun,thenear-surface shearlayer3,4 .Withinthiszone,inwardlyincreasingdifferentialrotationcoupledwith apoloidalmagneticfieldstronglyimplicatesthemagneto-rotationalinstability5,6 , prominentinaccretion-disktheoryandobservedinlaboratoryexperiments7 . Together,thesetwofactspromptthegeneralquestion:whetherthesolardynamois possiblyanear-surfaceinstability.Herewereportstrongaffirmativeevidenceinstark contrasttotraditionalmodels8 focusingonthedeepertachocline.Simpleanalytic estimatesshowthatthenear-surfacemagneto-rotationalinstabilitybetterexplains thespatiotemporalscalesofthetorsionaloscillationsandinferredsubsurface magneticfieldamplitudes9 .State-of-the-artnumericalsimulationscorroboratethese estimatesandreproducehemisphericalmagneticcurrenthelicitylaws10 .Thedynamo resultingfromawell-understoodnear-surfacephenomenonimprovesprospects foraccuratepredictionsoffullmagneticcyclesandspaceweather,affectingthe electromagneticinfrastructureofEarth. Keyobservationsthatanymodelmusttakeintoaccountinclude(1)the solarbutterflydiagram,adecadalmigrationpatternofsunspotemer- gence1,4 withstronglatitudedependence;(2)thetorsionaloscillations constituting local rotation variations corresponding with magnetic activity2–4 ;(3)thepoloidalfield,anapproximately1 Gphotosphericfield witha1/4-cyclephaselagrelativetosunspots11 ,andapproximately100 G subsurface amplitudes9 ; (4) the hemispherical helicity sign rule com- prisinganempiricallyobservednegativecurrenthelicityinthenorthern hemisphere and positive current helicity in the south10 ; (5) the tacho- cline at the base of the convection zone, the traditionally proposed seat of the solar dynamo; and (6) the near-surface-shear layer (NSSL) withintheouter5–10%oftheSuncontainingstronginwardlyincreasing angular velocity fostering the magneto-rotational instability (MRI). Despiteprogress,prevailingtheorieshavedistinctlimitations.Inter- facedynamos(proposedwithinthetachocline8 )preferentiallygener- ate high-latitude fields12 and produce severe shear disruptions13 that are not observed14 . Mean-field dynamos offer qualitative insights but suffer from the absence of first principles15 and are contradicted by observed meridional circulations16 . Global convection-zone models oftenmisalignwithimportantsolarobservations,requireconditions divergingfromsolarreality17–19 andfailtoprovideatheoreticaldynami- cal understanding. Borrowing from well-established ideas in accretion-disk physics5,6 , weproposeanalternativehypothesisthatproducesclearpredictions and quantitatively matches key observations. ForelectricallyconductingplasmasuchastheSun,thelocalaxisym- metric linear instability criterion for the MRI is5,6 ΩS ω 2 > , (1) A 2 where the local Alfvén frequency and shear are ω B k ρ S r Ω r = 4π and = − d d . (2) r A 0 0 The system control parameters are the background poloidal mag- netic field strength (B0 in cgs units), the atmospheric density (ρ0), the smallest non-trivial radial wavenumber that will fit in the domain (kr ≈ π/Hr, where Hr is a relevant layer depth or density-scale height), bulkrotationrate(Ω)andthedifferentialrotation,orshear(S > 0inthe NSSL).Anadiabaticdensitystratificationholdstoagoodapproxima- tionwithinthesolarconvectionzone,eliminatingbuoyancymodifica- tions to the stability condition in equation (1). The MRI is essential for generating turbulent angular momentum transport in magnetized astrophysical disks6 . Previous work20 pos- tulated the NSSL as the possible seat of the global dynamo without invokingtheMRI.Akinematicdynamostudy21 dismissedtherelevance ofNSSLbutdidnotallowforfullmagnetohydrodynamic(MHD)insta- bilities(suchastheMRI).Modernbreakthroughsinourunderstanding oflarge-scaleMRIphysics22,23 havenotbeenappliedinasolarcontext, https://doi.org/10.1038/s41586-024-07315-1 Received: 19 August 2023 Accepted: 14 March 2024 Published online: 22 May 2024 Open access Check for updates 1 School of Mathematics and the Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, UK. 2 Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, USA. 3 CIERA, Northwestern University, Evanston, IL, USA. 4 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA. 5 Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA. 6 Department of Physics and Astronomy, Bates College, Lewiston, ME, USA. 7 Department of Astrophysical and Planetary Sciences, University of Colorado Boulder, Boulder, CO, USA. 8 Department of Applied Mathematics, Jack Baskin School of Engineering, University of California Santa Cruz, Santa Cruz, CA, USA. 9 Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO, USA. 10 Deceased: Keith Julien. ✉e-mail: gvasil@ed.ac.uk
- 2. 770 | Nature | Vol 629 | 23 May 2024 Article andlocalMRIstudiesoftheSun24 haveconsideredonlysmallscales.To ourknowledge,noworkhasyetconsideredlarge-scaleMRIdynamics relevant to the observed features of the global dynamo. We therefore describe here a potential MRI-driven solar dynamo cycle. The start of the solar cycle is the period surrounding the sunspot minimum when there is no significant toroidal field above the equa- tor and a maximal poloidal field below the photosphere. This con- figuration is unstable to the axisymmetric MRI, which generates a dynamically active toroidal field in the outer convection zone. The observed torsional oscillations are the longitudinal flow perturba- tionsarisingfromtheMRI.Therelativeenergeticsareconsistentwith nonlinear dynamo estimates (Methods). As the cycle progresses, the toroidalfieldcanundergoseveralpossibleMHDinstabilitiescontrib- uting to poloidal-field regeneration, for example, the helical MRI, non-axis-symmetric MRI, the clamshell instability and several more, including a surface Babcock–Leighton process. We propose that the axisymmetric subsurface field and torsional oscillations constitute a nonlinear MRI travelling wave. The instability saturates by radial transportof(globallyconserved)meanmagneticflux(B0)andangular momentum (Ω, S), which neutralize the instability criterion in equa- tion (1) (Methods). Empiricaltimescalesofthetorsionaloscillationsimplyanapproxi- mate growth rate, γ, for the MRI and, therefore, a relevant poloidal fieldstrength.Toagoodapproximation,S ≈ Ω ≈ 2π/monthintheNSSL (Fig.1a–c).Theearly-phasetorsionaloscillationschangeonatimescale of 2–12 months, implying a growth rate of γ/Ω ≈ 0.01–0.1 (Methods). A modest growth rate and the regularity of the solar cycle over long intervalstogethersuggestthattheglobaldynamicsoperateinamildly nonlinear regime. Altogether, we predict roughly ωA ≈ S ≈ Ω. The torsional oscillation pattern shows an early-phase mode-like structurewithanapproximately4:1horizontalaspectratiooccupying adepthofroughlyr/R⊙ ≈ 5%,or ⊙ k R ≈ 70 r −1 (Fig.1d).Usingequation(2), the approximate background Alfvén speed vA ≈ 200–2,000 cm s−1 ≈ 0.1–1.0 R⊙/year. Alfvén-speed estimates required for MRI dynamics are consistent withobservedinternalmagneticfieldstrengths.Measurementssuggest 100–200 Ginternalpoloidalfield9 ,agreeingwiththeaboveestimates using NSSL densities ρ0 ≈ 3 × 10−2 g cm−3 to 3 × 10−4 g cm−3 . The same min+1yr – min min+2yr – min min+3yr – min min+4yr – min –1.0 –0.5 0 nHz 0.5 1.0 0 0.2 0.4 0.6 0.8 1.0 r/R( 0 0.2 0.4 0.6 0.8 1.0 Ω 300 350 400 450 nHz 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 sin(T)w T Ω −100 −50 0 50 100 nHz 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 −500 0 500 nHz a b c d r/R( r/R( r sin(T)w r Ω Fig.1|Measuredinternalsolarrotationprofiles.a,Heliosesimicdifferential rotationprofile, Ω(θ, r) usingpublicly available datafromref. 29.b,c,The respectivelatitudinaland radialsheargradients r θ Ω θ r ∇ ∇ sin( ) ( , )computedbya non-uniformfourth-ordercentredfinite-differencescheme.Thelatitudinal meanoftachoclineshearisabout200 nHzandpeakamplitudesarebelow about350 nHz.Conversely,thenear-surfaceshearaveragesabout400–600 nHz (withrapidvariationindepth)andpeakvaluesaveragearound1,200 nHz. d,Helioseismicmeasurementsofsolartorsionaloscillations.Theredshows positiveresidualrotationratesandblueshowsnegativeresidualrotationrates afterremovingthe1996annualmeanofΩ(r, θ).Eachsliceshowstherotational perturbations1,2,3and4yearsaftertheapproximatesolarminimum.The notation‘min+1yr–min’meanstakingtheprofileat1yearpastsolarminimum andsubtractingtheprofileatsolarminimum.Thecolourtablesaturatesat ±1 nHz,correspondingtoabout400 cm s−1 surfaceflowamplitude.Further contourlinesshow1 nHzincrementswithinthesaturatedregions.Diagramind reproducedwithpermissionfromfigure2inref.3,AAAS.
- 3. Nature | Vol 629 | 23 May 2024 | 771 studies found roughly similar (300–1,000 G) internal toroidal field strengthconfinedwithintheNSSL.Givensolar-likeinputparameters,a detailedcalculationshowsthattheMRIshouldoperatewithlatitudinal field strengths up to about 1,000 G (Methods). Background shear modification dominates the MRI saturation mechanism (Methods), roughly Ω H R ΩS ω S ω Ω S Ω S Ω ω ′ ≈ (2 − )( + ) 2 ( + 2 )( + (2 ) + 2 ) , (3) r 2 2 2 A 2 2 A 2 2 2 2 A 2 ∣ ∣ ⊙ whereΩ′representsthedynamicchangesindifferentialrotation.For S ≈ Ω ≈ ωA,|Ω′| ≈ 7 nHz,roughlyconsistentwiththeobservedtorsional oscillation amplitude (Fig. 1d). WecomputeasuiteofgrowingglobalperturbationsusingDedalus25 to model the initial phase of the solar cycle with quasi-realistic solar inputparameters(Methods).Figure2showsrepresentativesolutions. Wefindtwodistinctcases:(1)afastbranchwithdirectgrowthrates, γ, comparable to a priori estimates and (2) a slow branch with longer butrelevantgrowthtimesandoscillationperiods.Theeigenmodesare confinedtotheNSSL,reachingfromthesurfacetor/R⊙ ≈ 0.90–0.95,at which point the background shear becomes MRI stable. For case 1, γ/Ω0 ≈ 6 × 10−2 (given Ω0 = 466 nHz) with corresponding e-folding time, te ≈ 60 days and no discernible oscillation frequency. Thepatterncomprisesroughlyonewaveperiodbetweentheequator and about 20° latitude, similar to the rotation perturbations seen in the torsional oscillations. Forcase2,γ/Ω0 ≈ 6 × 10−3 withte ≈ 600 daysandoscillationfrequency ω/Ω0 ≈ 7 × 10−3 , corresponding to a period P ≈ 5 years. The pattern comprises roughly one wave period between the equator and about 20°–30° latitude. Apart from cases 1 and 2, we find 34 additional purely growing fast-branch modes, two additional oscillatory modes and one inter- mediate exceptional mode (Extended Data Figs. 1–3). Using the full numerical MHD eigenstates, we compute a system- atic estimate for the saturation amplitude using quasi-linear theory (Methods):|Ω′|≈6nHzforcase1and|Ω′|≈3nHzforcase2;bothcom- parabletotheobservedtorsionaloscillationamplitudeandthesimple analyticalestimatesfromequation(3).Thetruesaturatedstatewould compriseaninteractingsuperpositionofthefullspectrumofmodes. Notably,theslow-branchcurrenthelicity, b b H ∇ ∇ ∝ ⋅ × ,followsthe hemisphericalsignrule10 ,with < 0 H inthenorthand > 0 H inthesouth. Theslow-branchmodesseemtoberotationallyconstrained,consist- ent with their low Rossby number26 , providing a pathway for under- standing the helicity sign rule. Further helioseismic data analyses could test our predictions. The MRI would not operate if the poloidal field is too strong, nor would it explain the torsional oscillations if it is too weak. We predict cor- relations between the flow perturbations and magnetic fields, which time-resolvedmeasurementscouldtest,constrainingjointhelioseismic inversions of flows and magnetic fields. AnMRI-drivendynamomayalsoexplaintheformationandcessation of occasional grand minima27 (for example, Maunder). As an essen- tiallynonlineardynamo,theMRIisnotatraditionalkinematicdynamo startingfromaninfinitesimalseedfieldoneachnewcycle(Methods). Rather,amoderatepoloidalfieldexistsatthesolarminimum,andthe MRI processes it into a toroidal configuration. If the self-sustaining poloidal-to-toroidal regeneration sometimes happens imperfectly, then subsequent solar cycles could partially fizzle, leading to weak subsurface fields and few sunspots. Eventually, noise could push the systembackontoitsnormalcyclicbehaviour,asintheElNinoSouthern Oscillation28 . Finally, our simulations intentionally contain reduced physics to isolatetheMRIasanimportantagentinthedynamoprocess,filtering outlarge-scalebarocliniceffects,small-scaleconvectionandnonlinear dynamo feedback. Modelling strong turbulent processes is arduous: turbulence can simultaneously act as dissipation, drive large-scale flowssuchastheNSSL,producemeanelectromotiveforcesandexcite collectiveinstabilities.Althoughsufficientlystrongturbulentdissipa- tioncouldeventuallyerasealllarge-scaledynamics,themerepresence of the solar torsional oscillations implies much can persist within the roiling background. Onlinecontent Anymethods,additionalreferences,NaturePortfolioreportingsumma- ries,sourcedata,extendeddata,supplementaryinformation,acknowl- edgements, peer review information; details of author contributions andcompetinginterests;andstatementsofdataandcodeavailability are available at https://doi.org/10.1038/s41586-024-07315-1. 1. Maunder, E. W. The Sun and sunspots, 1820–1920. (plates 13, 14, 15, 16.). Mon. Not. R. Astron. Soc. 82, 534–543 (1922). 2. Snodgrass, H. B. & Howard, R. Torsional oscillations of the Sun. Science 228, 945–952 (1985). 3. Vorontsov, S. V., Christensen-Dalsgaard, J., Schou, J., Strakhov, V. N. & Thompson, M. J. Helioseismic measurement of solar torsional oscillations. Science 296, 101–103 (2002). 4. Hathaway, D. H., Upton, L. A. & Mahajan, S. S. Variations in differential rotation and meridional flow within the Sun’s surface shear layer 1996–2022. Front. Astron. Space Sci. 9, 1007290 (2022). Ω –1 1 1 1 –2.9 9 bI –152 152 2 2 aI –1.2 1.2 2 2 –2.2 2.2 –0.8 0.8 te = 60 days P = None P = 1,835 days te = 601 days a b ′ 2.9 nHz cm s –1 R ( G G R ( –1 1 nHz –0.8 0.8 cm s –1 R ( –197 197 G –1.0 1.0 G R ( Ω bI aI ′ Fig.2|Twomeridional(r,θ)MRIeigenmodeprofiles.Longitudinalangular velocity perturbation, Ω r θ u r θ r θ ′( , ) = ( , )/( sin( )) ϕ ; momentum-density streamfunction (ϕ-directed component; Methods), ψ(r, θ); longitudinal magneticfield,bϕ(r, θ); magneticscalarpotential,aϕ(r, θ);andcurrenthelicity correlation,H r θ ( , ).Thetimescaleste andPrepresenttheinstabilitye-folding timeandoscillationperiod,respectively.a,Case1:atypicaldirectlygrowing fast-branchmodewithnooscillationandgrowthratesγ ≈ 0.06Ω0.b,Case2: atypicallarge-scaleslow-branchmodewitharoughly5-yearperiod.Ineach case,wefixtheoverallamplitudeto1 nHzfortherotationalperturbations, withallotherquantitiestakingtheircorrespondingrelativevalues.
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- 5. Methods Numericalcalculations We solve for the eigenstates of the linearized anelastic MHD equa- tions30,31 in spherical-polar coordinates (r, θ, ϕ) = (radius, colati- tude, longitude). Using R⊙ = 6.96 × 1010 cm for the solar radius, we simulate radii between r0 ≤ r ≤ r1 where r0/R⊙ = 0.89 and r1/R⊙ = 0.99. We place the top of the domain at 99% because several complicated processes quickly increase in importance between this region and the photosphere (for example, partial ionisation, radiative trans- port and much stronger convection effects). We use the anelastic MHD equations in an adiabatic background to capture the effects of density stratification on the background Alfvén velocities (den- sity varies by roughly a factor of 100 across the NSSL, causing about a factor of 10 change in Alfvén speed) and asymmetries in velocity structures introduced by the density stratification by ∇ ⋅ (ρu). A key aspect of the anelastic approximation is that all entropy perturba- tions must be small, which is reasonable in the NSSL below 0.99R⊙. We do not use the fully compressible equations, as these linear insta- bility modes do not have acoustic components. Future MRI studies incorporatingbuoyancyeffects(forexample,thedeepMRIbranches at high latitudes) should use a fully compressible (but low Mach number) model32 . Input background parameters. We include density stratification usingalow-orderpolynomialapproximationtotheModel-Sprofile33 . Inunitsofg cm−3 ,withh = (r − r0)/(r1 − r0), ρ α α h α h α h α h = − + − + , (4) 0 0 1 2 2 3 3 4 4 α = 0.031256, (5) 0 α = 0.053193, (6) 1 α = 0.033703, (7) 2 α = 0.023766, (8) 3 α = 0.012326, (9) 4 whichfitstheModel-Sdatatobetterthan1%withinthecomputational domain.Thedensityath= 1isρ0 = 0.000326comparedwith0.031256 at h = 0. Thedensityprofileisclosetoanadiabaticpolytropewithr−2 gravity and 5/3 adiabatic index. An adiabatic background implies that buoy- ancy perturbations diffuse independently of the MHD and decouple from the system. Weusealow-degreepolynomialfittotheobservedNSSLdifferential rotation profile. For μ = cos(θ), u e Ω r θ r θ = ( , ) sin( ) , (10) ϕ 0 Ω r θ Ω R h μ ( , ) = ( ) Θ( ), (11) 0 where Ω0 = 466 nHz ≈ 2.92 × 10−6 s−1 and R h h h h ( ) = 1 + 0.02 − 0.01 − 0.03 , (12) 2 3 μ μ μ Θ( ) = 1 − 0.145 − 0.148 . (13) 2 4 We use the angular fit from ref. 34. The radial approximation results from fitting the equatorial profile from ref. 29 shown in Fig. 1a. Below 60° latitude, the low-degree approximation agrees with the full empirical profile to within 1.25%. The high-latitude differen- tial rotation profile is less constrained because of observational uncertainties. We define the background magnetic field in terms of a vector potential, ∇ ∇ = × , (14) 0 0 B A B r r θ = ( ) 2 sin( ) , (15) ϕ 0 A e where B r B r r r r ( ) = (( / ) − ( / ) ), (16) 0 1 −3 1 2 andB0 = 90 G.Ther−3 termrepresentsaglobaldipole.Ther2 termrep- resentsafieldwithasimilarstructurebutcontainingelectriccurrent, J B e B r r θ ∇ ∇ = × 4π = 5 4π sin( ) . (17) ϕ 0 0 0 1 2 The background field is in MHD force balance, ∇ ∇ × = ( ⋅ ). (18) 0 0 0 0 J B A J TheMHDforcebalancegeneratesmagneticpressure,whichinevita- bly produces entropy, s′, and enthalpy, h′, perturbations using ρ T s h ∇ ∇ ∇ ∇ ∇ ∇ ( ⋅ ) + ′= ′, (19) 0 0 0 0 A J where A J A J s Γ T ρ h Γ Γ ρ ′= 1 − 1 ⋅ , ′ = − 1 ⋅ , (20) 3 0 0 0 0 3 3 0 0 0 and Γ3 is the third adiabatic index. However, the MRI is a weak-field instability, implying magnetic buoyancy and baroclinicity effects are subdominant. For the work presented here, we neglect the contribu- tions of magnetism to entropy (magnetic buoyancy) and consider adiabatic motions. We expect this to be valid for MRI in the NSSL, but studies of MRI in the deep convection zone at high latitudes would need to incorporate these neglected effects. We choose our particular magnetic field configuration rather than a pure dipole because the radial component r θ ⋅ = ( )cos( ) r 0 B e B van- ishes at r = r1. The poloidal field strength in the photosphere is about 1 G, but measurements suggest sub-surface field strengths of about 50–150 G (ref. 9). The near-surface field should exhibit a strong hori- zontal(asopposedtoradial)character.Magneticpumping35 bysurface granulation within the outer 1% of the solar envelope could account forfilteringtheoutwardradialfield,withsunspotcoresbeingpromi- nent exceptions. Wealsotestpuredipolesandfieldswithanapproximately5%dipole contribution, yielding similar results. Furthermore, we test that the poloidal field is stable to current-driven instabilities. Our chosen confined field also has the advantage that eθ ⋅ B0 is constant to within 8% over r0 < r < r1. However, a pure dipole varies by about 37% across the domain. The RMS field amplitude is ∣B∣RMS ≈ 2B0 = 180 G, about 25% larger than the maximum-reported inferred dipole equivalent9 . However,projectingourfieldontoadipoletemplategivesanapproxi- mately 70 G equivalent at the r = r1 equator. Overall, the sub-surface field is the least constrained input to our calculations, the details of which change over several cycles.
- 6. Article Model equations.Respectively,thelinearizedanelasticmomentum, mass-continuityandmagneticinductionequationsare ρ ϖ ν ρ ∇ ∇ ∇ ∇ (∂ + × + × + ) = ⋅ ( ) + × + × , (21) t 0 0 0 0 0 0 σ u ω u ω u j B J b ρ ∇ ∇ ⋅ ( ) = 0, (22) 0 u η ∇ ∇ ∂ − ∇ = × ( × + × ), (23) t 2 0 0 b b u b u B where the traceless strain rate ∇ ∇ ∇ ∇ ∇ ∇ = + ( ) − 2 3 ⋅ . (24) σ u u u I ⊤ Tofindeigenstates,∂t → γ + iω,whereγisthereal-valuedgrowthrate, andωisareal-valuedoscillationfrequency.Theinductionequation(23) automatically produces MRI solutions satisfying ∇ ⋅ b = 0. Giventhevelocityperturbation,u,thevorticityω = ∇ × u.Giventhe magneticfield(Gaussincgsunits),thecurrentdensityperturbations j = ∇ × b/4π.Atlinearorder,theBernoullifunctionϖ h = ⋅ + ′ 0 u u ,where h′ represents enthalpy perturbations26 . Thevelocityperturbationsareimpenetrable(ur = 0)andstress-free (σrθ = σrϕ = 0)atbothboundaries.Forthemagneticfield,weenforceper- fectconductingconditionsattheinnerboundary(br = ∂rbθ = ∂rbϕ = 0).At theouterboundary,wetestthreedifferentchoicesincommonusage, asdifferentmagneticboundaryconditionshavedifferentimplications for magnetic helicity fluxes through the domain, and these can affect global dynamo outcomes36 . Two choices with zero helicity flux are perfectlyconductingandvacuumconditions,andwefindonlymodest differences in the results. We also test a vertical field or open bound- ary(thatis,∂rbr = bθ = bϕ = 0),which,althoughnon-physical,explicitly allowsahelicityflux.Theseopensystemsalsohadessentiallythesame results as the other two for growth rates and properties of eigenfunc- tions.Weconductmostofourexperimentsusingperfectlyconducting boundary conditions, which we prefer on the same physical grounds as the background field. We set constant and kinematic viscous and magnetic diffusivity parametersν = η = 10−6 inunitswhereΩ0 = R⊙ = 1.ThemagneticPrandtl numberν/η = Pm = 1assumesequaltransportofvectorsbytheturbu- lentdiffusivities.AmoredetailedanalysisoftheshearReynoldsnum- bers yields U L ν Re = Rm = / ≈ 1,500 0 0 , where U0 ≈ 5,000 cm s−1 is the maximum shear velocity jump across the NSSL and L0 ≈ 0.06R⊙ is the distancebetweenminimumandmaximumshearvelocity(seesection ‘NSSL energetics and turbulence parameterization’ below). We compute the following scalar-potential decompositions a posteriori, u e e u ρ ρ ψ ∇ ∇ = + 1 × ( ), (25) ϕ ϕ ϕ 0 0 b a ∇ ∇ = + × ( ), (26) ϕ ϕ ϕ ϕ b e e whereboththemagneticscalarpotential,aϕ,andthestreamfunction, ψ, vanish at both boundaries. We, furthermore, compute the current helicity correlation relative to global RMS values, = ⋅ . (27) RMS RMS b j b j H Thereisnoinitialhelicityinthebackgroundpoloidalmagneticfield, A r θ ∇ ∇ ∇ ∇ = × ( ( , ) ) ⋅ ( × ) = 0. ϕ 0 0 0 0 ⇒ B e B B Linear dynamical perturbations, b(r, θ), will locally align with the background field and current. However, because the eigenmodes arewave-like,thesecontributionsvanishexactlywhenaveragedover hemispheres. ∇ ∇ ∇ ∇ ⟨ ⋅ ( × )⟩ = ⟨ ⋅ ( × )⟩ = 0. 0 0 b B B b The only possible hemispheric contributions arise when considering quadratic mode interactions, ∇ ∇ ⟨ ⋅ ( × )⟩ ≠ 0. b b This order is the first for which we could expect a non-trivial signal. Finally,wealsosolvethesystemusingseveraldifferentmathemati- callyequivalentequationformulations(forexample,usingamagnetic vectorpotentialb = ∇ × a,ordividingthemomentumequationsbyρ0). Inallcases,wefindexcellentagreementintheconvergedsolutions.We preferthisformulationbecauseofsatisfactorynumericalconditioning as parameters become more extreme. Computational considerations.TheDedaluscode25 usesgeneralten- sorcalculusinspherical-polarcoordinatesusingspin-weightedspheri- calharmonicsin(θ, ϕ)(refs. 37,38).Forthefiniteradialshell,thecode usesaweightedgeneralizedChebyshevserieswithsparserepresenta- tions for differentiation, radial geometric factors and non-constant coefficients(forexample,ρ0(r)).Asthebackgroundmagneticfieldand thedifferentialrotationareaxisymmetricandtheycontainonlyafew low-orderseparabletermsinlatitudeandradius,thesetwo-dimensional non-constant coefficients have a low-order representation in a joint expansionofspin-vectorharmonicsandChebyshevpolynomials.The resultisatwo-dimensionalgeneralizednon-Hermitianeigenvalueprob- lem Ax = λBx, where x represents the full system spectral-space state vector.Thematrices,AandB,arespectral-coefficientrepresentations of the relevant linear differential and multiplication operators. Cases 1 and 2 use 384 latitudinal and 64 radial modes (equivalently spatial points).ThematricesAandBremainsparse,withrespectivefillfactors ofabout8 × 10−4 and4 × 10−5 . The eigenvalues and eigenmodes presented here are converged to better than 1% relative absolute error (comparing 256 and 384 latitu- dinal modes). We also use two simple heuristics for rejecting poorly convergedsolutions.First,becauseλ0 iscomplexvalued,theresulting iteratedsolutionsdonotautomaticallyrespectHermitian-conjugate symmetry,whichweoftenfindviolatedforspurioussolutions.Second, the overall physical system is reflection symmetric about the equa- tor, implying the solutions fall into symmetric and anti-symmetric classes. Preserving the desired parity is a useful diagnostic tool for rejecting solutions with mixtures of the two parities, which we check individuallyforeachfieldquantity.Thepreciseparametersanddetailed implementation scripts are available at GitHub (https://github.com/ geoffvasil/nssl_mri). Analyticandsemi-analyticestimates Local equatorial calculation.Ourpreliminaryestimatesofthemaxi- mum poloidal field strength involve solving a simplified equatorial modelofthefullperturbationequations,settingthediffusionparam- eters ν, η → 0. Using a Lagrangian displacement vector, ξ, in Eulerian coordinates ξ ξ ξ u u u ∇ ∇ ∇ ∇ = ∂ + ⋅ − ⋅ , (28) t 0 0 ξ b B ∇ ∇ = × ( × ). (29) 0 Inlocalcylindricalcoordinatesneartheequator(r, ϕ, z),weassume all perturbations are axis-symmetric and depend harmonically ∝e k z ωt i( − ) z . The cylindrical assumption simplifies the analytical
- 7. calculationswhileallowingatransferenceofrelevantquantitiesfrom themorecomprehensivesphericalmodel.Thatis,weassumeapurely poloidalbackgroundfieldwiththesameradialformasthefullspher- ical computations, B0 = Bz(r)ez. We use the same radial density and angularrotationprofiles,ignoringlatitudinaldependence.Theradial displacement, ξr, determines all other dynamical quantities, ξ ω Ω ω k v ξ = − 2i − , (30) ϕ z r 2 2 A 2 ξ k r ρ rρ ξ r = i d( ) d (31) z z r 0 0 ϖ v B B ξ ω k r ρ rρ ξ r = ′ + d( ) d , (32) z z r z r A 2 2 2 0 0 wherev r B r ρ r ( ) = ( )/ 4π ( ) z A 0 . The radial momentum equation gives a second-ordertwo-pointboundary-valueproblemforξr(r).Theresult- ing real-valued differential equation depends on ω2 ; the instability transitionsdirectlyfromoscillationstoexponentialgrowthwhenω = 0. Weeliminatetermscontainingξ r ′( ) r withtheLiouvilletransformation Ψ r r B r ξ r ( ) = ( ) ( ) z r . The system for the critical magnetic field reduces to a Schrödinger-type equation, Ψ r k Ψ r V r Ψ r − ″( ) + ( ) + ( ) ( ) = 0, (33) z 2 with boundary conditions Ψ r r Ψ r r ( = ) = ( = ) = 0 (34) 0 1 and potential, V r v Ω r rρ B r rρ B r r = d d + d d 1 d d + 3 4 . (35) z z A 2 2 0 0 2 Upper bound. The maximum background field strength occurs in thelimitkz → 0.WithfixedfunctionalformsforΩ(r),ρ0(r),wesuppose B r B r r r r ( ) = 1 + 4( / ) 5( / ) , (36) z 1 1 5 1 3 withB1 = Bz(r1)settingandoverallamplitudeand B 1/ 1 2 servingasagen- eralized eigenvalue parameter. We solve the resulting system with DedalususingbothChebyshevandLegendreseriesfor64,128and256 spectral modes, all yielding the same result, B1 = 1,070 G. The results are also insensitive to detailed changes in the functional form of the background profile. Growth rate. We use a simplified formula for the MRI exponen- tial growth, proportional to eγt , in a regime not extremely far above onset22 .Thatis, γ α ω ΩS ω α ω Ω ≈ (2 − (1 + )) + 4 , (37) 2 2 A 2 A 2 2 A 2 2 where α = 2H/L ≈ 0.2–0.3 is the mode aspect ratio with latitudinal wavelength,L ≈ 20°–30°R⊙,andNSSLdepthH ≈ 0.05R⊙.Themaintext defines all other parameters. In the NSSL, S ≈ Ω. Therefore, γ/Ω ≈ 0.1, whenα ≈ 0.3andωA/Ω ≈ 1;andγ/Ω ≈ 0.01,whenα ≈ 0.2andωA/Ω ≈ 0.1. Saturation amplitude.Weusenon-dissipativequasi-lineartheory22 to estimate the amplitude of the overall saturation. In a finite-thickness domain, the MRI saturates by transporting mean magnetic flux and angularmomentumradially.Bothquantitiesare(approximately)glob- allyconserved;however,theinstabilityshiftsthemagneticfluxinward andangularmomentumoutward,sothepotentialfromequation(35) ispositiveeverywhereinthedomain. Given the cylindrical radius, r, the local angular momentum and magnetic flux density L ρ ru M ρ ra = , = . (38) ϕ ϕ 0 0 The respective local flux transport L L r b ∇ ∇ ∇ ∇ ∂ + ⋅ ( ) = ⋅ ( ), (39) t ϕ u b u M M ∇ ∇ ∂ + ⋅ ( ) = 0. (40) t For quadratic-order feedback, ρ r δu r b b ρ u u r b b ρ u u ∂ ( ) = ∂ ( ( − )) + ∂ ( ( − )), (41) t ϕ r ϕ r ϕ r z ϕ z ϕ z 0 2 2 0 2 0 ρ r δa r ρ a u r ρ a u ∂ ( ) = − ∂ ( ) − ∂ ( ). (42) t ϕ r ϕ r z ϕ z 0 2 2 0 2 0 For linear meridional perturbations, u ψ u rρ ψ rρ = − ∂ , = ∂ ( ) , (43) r z z r 0 0 b a b ra r = − ∂ , = ∂ ( ) . (44) r z ϕ z r ϕ For the angular components, u Ω S ψ B ρ b ∂ = ∂ (2 − ) + 4π , (45) t ϕ z z ϕ 0 a B ψ ∂ = ∂ ( ), (46) t ϕ z z b B u S a ∂ = ∂ ( + ). (47) t ϕ z z ϕ ϕ Usingthelinearbalances,wetimeintegratetoobtainthelatitudinal- mean rotational and magnetic feedback, L δΩ r ρ r ρ = 1 ∂ ( ), (48) r 3 0 2 0 δA r ρ r ρ = 1 ∂ ( Φ). (49) r 2 0 2 0 where angle brackets represent z averages and L B a u Ω S a B = 2 ⟨ ⟩ − (2 − ) ⟨ ⟩ 2 , (50) z ϕ ϕ ϕ z 2 2 a B Φ = ⟨ ⟩ 2 . (51) ϕ z 2 The dynamic shear and magnetic corrections, δS r δΩ δB r rδA = − ∂ , = 1 ∂ ( ). (52) r z r
- 8. Article We derive an overall amplitude estimate by considering the functional ∫ V Ψ Ψ r ∇ ∇ = ( + )d , (53) 2 2 F whichresultsfromintegratingequation(33)withrespecttoΨ*(r).The saturation condition is F F δ = − . (54) The left-hand side includes all linear-order perturbations in the potential, δV, and wavefunction, δΨ, where δV r v ΩδΩ r δB B r v Ω r rρ B r rρ δB r δB B rρ B r rρ B r = 2 d( ) d − 2 d d + d d 1 d d − d d 1 d d , (55) z z z z z z z z A 2 A 2 2 0 0 0 0 δΨ δB B Ψ = . (56) z z Allreferenceandperturbationquantitiesderivefromthefullsphere numerical eigenmode calculation. We translate to cylindrical coor- dinates by approximating z averages with latitudinal θ averages. The sphericaleigenmodeslocalizeneartheequator,andtheNSSLthickness isonlyabout5%ofthesolarradius,justifyingthecylindricalapproxi- mation in the amplitude estimate. Empirically, the first δV term dominates the overall feedback cal- culation,owingtotheshearcorrections δΩ r H ∝d /d ~ 1/ r 2 .Isolatingthe shear effect produces the simple phenomenological formula in equation (3). NSSLenergeticsandturbulenceparameterization We estimate that the order-of-magnitude energetics of the NSSL are consistentwiththeamplitudesoftorsionaloscillations.Thetorsional oscillations comprise |Ω′| ≈ 1 nHz rotational perturbation, relative to the Ω0 ≈ 466 nHz equatorial frame rotation rate. However, the NSSL contains ΔΩ ≈ 11 nHz mean rotational shear estimated from the func- tionalforminequations(10)–(13).Intermsofvelocities,theshearinthe NSSL has a peak contrast of roughly U0 ≈ 5 × 103 cm s−1 across a length scaleL0 ≈ 0.06R⊙.Therelativeamplitudesofthetorsionaloscillations to the NSSL background, |Ω′|/ΔΩ, are thus about 10%. Meanwhile, the radial and latitudinal global differential rotations have amplitudes of the order of about 100 nHz. The relative energies are approximately thesquaresofthese,implyingthattheΔKEofthetorsionaloscillations is about 0.01% to the differential rotation and about 1% to the NSSL. TheseroughestimatesshowthattheNSSLandthedifferentialrotation can provide ample energy reservoirs for driving an MRI dynamo, and theamplitudeofthetorsionaloscillationsisconsistentwithnonlinear responses seen in classical convection-zone dynamos17 . Vigoroushydrodynamicconvectiveturbulenceprobablyestablishes thedifferentialrotationoftheNSSL.Thelargereservoirofshearinthe solarinteriorplaystheanaloguepartofgravityandKeplerianshearin accretiondisks.Thedetailsofsolarconvectionareneitherwellunder- stood nor well constrained by observations. There are indications, however, that the maintenance of the NSSL is separate from the solar cycle because neither the global differential rotation nor the NSSL shows substantial changes during the solar cycle other than in the torsional oscillations. StrongdynamicalturbulenceintheouterlayersoftheSunisanuncer- taintyofourMRIdynamoframework,butscaleseparationgiveshope for progress. From our linear instability calculations, the solar MRI operates relatively close to the onset and happens predominantly on large scales. If the fast turbulence of the outer layers of the Sun acts mainly as an enhanced dissipation, then the solar MRI should survive relativelyunaffected.Treatingscale-separateddynamicsinthisfashion hasgoodprecedent:large-scalebaroclinicinstabilityintheatmosphere ofEarthgracefullyploughsthroughthevigorousmoisttropospheric convection(thunderstorms).Scale-separateddynamicsareparticularly relevant because the MRI represents a type of essentially nonlinear dynamo,whichcyclicallyreconfiguresanexistingmagneticfieldusing kineticenergyasacatalyst.Frompreviouswork,itisclearthatthedeep solarconvectionzonecanproduceglobal-scalefields,butthesefields generally have properties very different from the observed fields17 . EssentiallynonlinearMHDdynamoshaveanaloguesinpipeturbulence, and, similar to those systems, the self-sustaining process leads to an attractor in which the dynamo settles into a cyclic state independent of its beginnings. A full nonlinear treatment of turbulence in the NSSL-MRI setting awaitsfuturework.Hereweadoptasimplifiedturbulencemodelusing enhanced dissipation. To model the effects of turbulence, we assume thattheviscousandmagneticdiffusivitiesareenhancedsuchthatthe turbulentmagneticPrandtlnumberPm = 1(withnoprincipleofturbu- lencesuggestingotherwise).ThemomentumandmagneticReynolds numbersareRe = Rm ≈ 1.5 × 103 .Thesevaluesarevastlymoredissipative than the microphysical properties of solar plasma (that is, Re ~ 1012 ), andthemicrophysicalPm ≪ 1,implyingthatRm ≪ Re.Thestudiescon- ducted here find relative independence in the MRI on the choices of Rewithinamodestrange.Bycontrast,otherinstabilities(forexample, convection) depend strongly on Re. We compute sample simulations downtoRe ≈ 50withqualitativelysimilarresults,althoughtheymatch the observed patterns less well and require somewhat stronger back- ground fields. Our adopted value of Re ≈ 1,500 strikes a good balance for an extremely under-constrained process. Our turbulent param- eterizations also produce falsifiable predictions: our proposed MRI dynamomechanismwouldfaceseverechallengesiffuturehelioseismic studiesoftheSunsuggestthattheturbulentdissipationismuchlarger than expected (for example, if the effective Re ≪ 1). However, it is dif- ficult to imagine how any nonlinear dynamics would happen in this scenario. Dataavailability TheraweigenfunctionandeigenvaluedatausedtogenerateFig.2can befoundalongwiththeanalysisscriptsatGitHub(https://github.com/ geoffvasil/nssl_mri)39 . Codeavailability WeusetheDedaluscodeandadditionalanalysistoolswritteninPython, as noted and referenced in the Methods. Beyond the main Dedalus installation, all scripts are available at GitHub (https://github.com/ geoffvasil/nssl_mri)39 . 30. Brown, B. P., Vasil, G. M. & Zweibel, E. G. Energy conservation and gravity waves in sound-proof treatments of stellar interiors. Part I. Anelastic approximations. Astrophys. J. 756, 109 (2012). 31. Vasil, G. M., Lecoanet, D., Brown, B. P., Wood, T. S. & Zweibel, E. G. Energy conservation and gravity waves in sound-proof treatments of stellar interiors: II. Lagrangian constrained analysis. Astrophys. J. 773, 169 (2013). 32. Anders, E. H. The photometric variability of massive stars due to gravity waves excited by core convection. Nat. Astron. 7, 1228–1234 (2023). 33. Christensen-Dalsgaard, J. et al. The current state of solar modeling. Science 272, 1286–1292 (1996). 34. Howe, R. Solar interior rotation and its variation. Living Rev. Sol. Phys. 6, 1 (2009). 35. Tobias, S. M., Brummell, N. H., Clune, T. L. & Toomre, J. Transport and storage of magnetic field by overshooting turbulent compressible convection. Astrophys. J. 549, 1183–1203 (2001). 36. Käpylä, P. J., Korpi, M. J. & Brandenburg, A. Open and closed boundaries in large-scale convective dynamos. Astron. Astrophys. 518, A22 (2010). 37. Vasil, G. M., Lecoanet, D., Burns, K. J., Oishi, J. S. & Brown, B. P. Tensor calculus in spherical coordinates using Jacobi polynomials. Part-I: mathematical analysis and derivations. J. Comput. Phys. X 3, 100013 (2019).
- 9. 38. Lecoanet, D., Vasil, G. M., Burns, K. J., Brown, B. P. & Oishi, J. S. Tensor calculus in spherical coordinates using Jacobi polynomials. Part-II: implementation and examples. J. Comput. Phys. X 3, 100012 (2019). 39. Vasil, G. et al. GitHub https://github.com/geoffvasil/nssl_mri (2024). Acknowledgements D.L., K.A., K.J.B., J.S.O. and B.P.B. are supported in part by the NASA HTMS grant 80NSSC20K1280. D.L., K.J.B., J.S.O. and B.P.B. are partly supported by the NASA OSTFL grant 80NSSC22K1738. N.B. acknowledges support from the NASA grant 80NSSC22M0162. Computations were conducted with support from the NASA High-End Computing Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center on the Pleiades supercomputer with allocation GID s2276. Our dear co-author, Keith Julien, passed away unexpectedly while this paper was in the press. We want to acknowledge the outstanding friend and colleague he was for so many years. He did much to make this work a reality and was thrilled to see it completed. He will be sorely missed. AuthorcontributionsG.M.V., D.L., J.S.O. and K.J. conceived and designed the experiments. G.M.V. performed analytical and semi-analytical estimates. K.J.B. and D.L. developed the multidimensional spherical eigenvalue solver used in the computations. K.A. and D.L. performed and verified the eigenvalue computations. K.J.B., G.M.V., J.S.O., D.L. and B.P.B. contributed to the numerical simulation and analysis tools. G.M.V., D.L., B.P.B. and J.S.O. wrote and, along with K.J.B., K.A., K.J. and N.B., revised the paper. All authors participated in discussions around the experimental methods, results and the Article. Competing interests The authors declare no competing interests. Additional information Correspondence and requests for materials should be addressed to Geoffrey M. Vasil. Peer review information Nature thanks Steven Balbus and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Reprints and permissions information is available at http://www.nature.com/reprints.
- 10. Article ExtendedDataFig.1|Fulldiagramofcomplex-valuedeigen-spectrum. Thetimedependencesends∂t → γ + iω,witheachreal/imaginarycomponent measuredintermsofΩ0 = 466 nHz.Themodesalongtheverticalaxisappear tolieonacontinuum,accumulatingatalowervalue.Theisolatemodesappear tobepointspectra.Theredcirclerepresentsthecase(i)“fastbranch”fromthe maintext.Thepurplecircle(withitscomplexconjugate)representthecase (ii)presented“slowbranch”.
- 11. ExtendedDataFig.2|Thecompletecollectionof“fastbranch”modes.ThegrowthratescorrespondtotheverticalaxisofExtendedDataFig.1.Eachcase containsnodiscernibleoscillations.Forcompleteness,weshow(boxedingrey)thete = 60 dayfast-branchcase(i)presentedinthemaintextFig.2(a).