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arXiv:1208.6154v1 [astro-ph.SR] 30 Aug 2012 11896 Gizon Laurent solar of convection models challenges ∗ ne sn epfcsn,qarn emty o range a for geometry, times, quadrant integration deep-focusing, of a using [9]. ences for ago challenge years a fifty remained discovery has its applica- convec Mm of since One 35 of theory study scale scale length the function. typical preferred been with Green’s This has a two- [8]; helioseismology to solar The time-distance similar of role tion surface. a informat solar plays fundamental contains it the waves function acoustic on correlation solar point of locations times rando distant travel the heli the between measure of Time-distance to correlations field [7]. wave spatio-temporal l flows uses are subsurface i seismology times hand, to travel other sensitive wave the early [6], On helioseismology oscillations. -distance of frequencies mode s to unique large-scale coverage this of spatial used helioseismology full convection. high-precision al. and out et resolution a carry Hanasoge by high seconds of camera. 45 combination pixel every Full-Su of captured million superposition are convection. 16 random images near-surface velocity the by Doppler by motions excited caused waves the surface Solar seismic measures solar NASA’s the HMI onboard on [5] (SDO). (HMI) Observatory the Imager Dynamics of Magnetic launch and the mic since possible result are helioseismology surprises table interior, solar the into open window helioseismology new Whenever con a were models. features Desp by solar predicted [4]. above dently dynamo the now solar of the which none of a attempts, , theories valuable is the in zone role as convection central known the the a plays of shear, of base rotational bulk the of At The zone latitudes. latitude. high e at the and at than ro radius faster differentially, solar of rotates of envelope function convective inference a the as was tation achievement spectacular opacities. other underestimated on based expectations previous pe ii ncnetv eoiisa amncdegree harmonic an al., at et velocities superposi- Hanasoge convective to incoherent on According the limit waves. to upper seismic due solar of noise tion stochastic by inated lblmd rqece eeue omauetedphof depth 1980 the 0 mid measure at the envelope to convective In solar used o the were study [3]. frequencies the oscillations helioseismology, global-mode internal to due solar five-minute been have interior solar hydrodynamic popular a by l predicted upper simulation. than an numerical smaller report far They al. conv is interior. large-scale et that solar of the Hanasoge in amplitude velocities PNAS, the tive of stri on derive issue constraints to observations empirical this or- helioseismic many In recent employ spans [2] density magnitude. acros the of model where ders zone to convection difficult entire notoriously the is convection turbulent C Science of Academy National the of Proceedings to Submitted G¨ottingen, Germany 37077 1, Friedrich-Hund-Platz v a-lnkIsiu f¨ur Max-Pl Max-Planck-Institut Sonnensystemforschung, φ,ℓ aaoee l 2 esrdes-ettae-iediffer- travel-time east-west measured [2] al. et Hanasoge global- the affect not do flows convective order, first To no- most the perhaps is [2] al. et Hanasoge of work The itrcly ra dacsi u nesadn fthe of understanding our in advances great Historically, ǫ < – otdtruhteotrot3%o h u 1.Solar [1]. trans- is the of energy 30% which outermost by the through mechanism ported the is onvection 11898 sg τ ℓ /C PNAS ℓ † ∗ nti expression, this In . n ao .Birch C. Aaron and uy2,2012 24, July T pt 6h.Tae ie r dom- are times Travel hr. 96 to up , . 1slrrdu,dee than deeper radius, solar 71 o.109 vol. ǫ sg ∗ ≪ nkSrß ,311Ktebr-idu emn,and Germany, Katlenburg-Lindau, 37191 2, anck-Straße stesignal-to- the is 1 o 30 no. fteUie ttso America of States United the of s quator ngent imit tion olar An- ion: ℓ eis- ec- in- ite . fi- m o- is n n s s s - f iue5o upeetr aeil stersl fnumeric of result the is material) supplementary of 5 Figure C dac oprdt h ale praho aaoee al. et Hanasoge of approach where earlier [11] the to compared advance ℓ cnitn ih[3 ntelmto large of limit the approximat in is value [13] expectation with the (consistent and field velocity the of os ai ftetae ie dmntdb h contributi the by and (dominated supergranulation) times travel from the of ratio noise eetmtdfo the from estimated be rvltms ic h aineo ueniebhvslk 1 of like measurement behaves noise The pure of variance [10]. the since times, travel essshrclhroi degree harmonic spherical versus h bevtoa pe ii rmSOHIhloesooyat helioseismology SDO/HMI from limit upper observational ro hw hoeia oe ii ae ngoa dynami global on over based equipartition limit mode lower suming theoretical a shows arrow rnlsuigSOHIitniyiae 1] oieteex the velociti Notice surface from [18]. images is intensity curve SDO/HMI red using The granules [2]. averaging day one u osprrnlto.Rslsfo npht fnumeric of snapshots at from sliced Results vection supergranulation. to due lecresostesetu rmasagr[6 near-surfa [16] size stagger of a simula from simulation spherical-shell spectrum ASH the shows an curve from blue spectrum the shows curve i.1. Fig. eevdfrPbiainFootnotes Publication for Reserved v ℓ ny(oiae ysohsi os) h quantity The noise). stochastic by (dominated only φ 2 i sdaoet ovr rvltmsit eoiis(e [2], (see velocities into times travel convert to above used / = 2 oprsno iei nryspectra energy kinetic of Comparison P ℓ ǫ ≥ sg 0 96 0 . 98 E a sue.Teclbainconstant calibration The assumed. was 1 = × φ

3 − 2 † R E

φ (k m s ) f¨ur Georg-August-Universit¨atInstitut Astrophysik, G¨ ( ⊙ 96 10 10 10 10 10 10 10 ℓ ) −2 −1 0 1 2 3 −3 /r r ie ytebu n ih lecre.Teblue The curves. blue light and blue the by given are 10 × where , 0 Seismology 20 ASH T < ℓ ℓ h lc uv bv h ryae hw the shows area gray the above curve black The . www.pnas.org/cgi/doi/10.1073/pnas.1208875109 dpnec ftevrac fthe of variance the of -dependence Mm 10 1 ǫ 750 v τ 3 sg φ ℓ h oiotlbakln n associated and line black horizontal The . ℓ ). sterstae iea scale at time travel rms the is stelniuia pord)component (prograde) longitudinal the is edefined We . yHnsg ta.[]i an is [2] al. et Hanasoge by 10 ℓ 2 tracking Granulation E stagger 10 φ 3 flniuia oa velocities solar longitudinal of E db oiotlaverage horizontal a by ed 10 φ erdaiecompressible radiative ce smaue ytracking by measured es espwrat power cess 4 sagmns[4,as- [14], arguments cs in[2 n h light the and [12] tion lsmltoso con- of simulations al tradius at radius 0 . 96 r ǫ uhthat such ℓ sg R ottingen, ∼ ⊙ may 120 /T and on al forward modelling and leads to an upper limit on the velocity lation. The kinetic energy spectra of the two simulations at at the target depth. r = 0.98R⊙ roughly agree around ℓ ∼ 150. This suggests The interpretation of travel-time measurements is a topic that ASH is capturing some of the general dynamics there, of current research, especially regarding the effect of time- despite using simplified and missing the driving by dependent turbulent velocities. Hanasoge et al. assumed strong cooling at the solar surface. frozen convection to obtain the calibration constant Cℓ, which SDO/HMI provides at least two other means than helio- is only an approximation when the lifetime of convection is seismology to observe surface flows. The first method is direct less than T . This may lead to an underestimation of the so- Doppler measurements of the line-of-sight component of veloc- lar velocities. Also the vertical correlations of the convective ity [17]. The second method is based on tracking the motions velocities were ignored: Eddies of vertical sizes less than the of granules or other small features and provides both com- first Fresnel zone cannot be detected. These two points may ponents of the horizontal velocities. In the Figure, we plot potentially affect the inferred upper limit on the convective the granulation-tracking result from Roudier et al. [18], who velocities. Furthermore, experience with other experiments employed SDO/HMI intensity images. Notice that the ASH suggests that systematic effects, e.g. center to limb effects, kinetic energies are above the granulation-tracking value for could also play a role at the m/s level. ℓ< 80. This is surprising since one would expect the convec- It is enlightening to consider the helioseismology results tive velocities to decrease in amplitude with increasing depth. in the context of existing models of convection. Hanasoge A striking feature in the granulation-tracking curve is the et al. [2] showed that in the range ℓ < 60 their helioseismol- excess kinetic energy at ℓ ∼ 120 due to supergranulation. ogy upper limit for longitudinal convective velocities at radius Current simulations of convection are not ideal for modelling 0.92R⊙ is orders of magnitude less than what is predicted by the supergranulation in detail; currently this scale is near an Anelastic Spherical Harmonic (ASH) hydrodynamic sim- the largest scale of the stagger code and the smallest scale ulation [12]. This is a main concern of the authors. For the of ASH. The helioseismology inferences from Hanasoge et al. outside commentator, there is no clear way to reconcile this stop short of the supergranular scale; the method of anal- severe disagreement. ysis was not optimized for that purpose. Overall, a better Here we supplement the comparison with additional the- observational coverage and theoretical understanding of the oretical, numerical, and observational constraints, which we intermediate spatial scales would help connect the local and have combined in the figure. We chose the kinetic energy global scales of convection. The next generation of convection density Eφ(ℓ) to characterize the strength of longitudinal con- models are expected to cover the supergranulation range. vective velocities [13], which is a partial but useful description Assuming that the helioseismology upper limit on convec- of convection. tive velocities from Hanasoge et al. [2] can be taken at face Miesch et al. [14] recently obtained an interesting the- value, this will force a rethinking of the large-scale dynamics oretical lower limit of 30 m s−1 for convective velocities at of the solar convective zone. One particular question is how to radius 0.95R⊙ for scales ℓ . 750. This calculation is based on model very highly turbulent regimes, e.g. by including deep the idea that the observed large-scale flows (differential rota- thermal plumes [19]. tion and meridional circulation) are maintained by convective Any viable theory of convection ought to explain convec- Reynolds stresses. Partitioning the kinetic energy evenly over tion in other . In this respect asteroseismology may play 3 −2 all modes ℓ < 750, we find Eφ > 0.4 km s , which is well an important role. The observed amplitudes of oscillation in above the helioseismology upper limits at the lowest ℓ values other Sun-like stars [20] contain information about the vigour (see figure). More work is needed to determine the ℓ depen- of surface convection in these stars, which in turn will place dence of this theoretical lower limit. constraints on stellar convection models. The ASH simulation is truncated at radius 0.98R⊙, above which additional physics is needed, for example compressibil- ACKNOWLEDGMENTS. We thank Mark Miesch (Anelastic Spherical Harmonic ity and radiative transfer. Convection in the near-surface code), Thierry Roudier (granulation tracking) and Robert Stein (stagger code) for layers has been modelled with great success (judging from making their data available. We have received support from Deutsches Zentrum fr Luft- und Raumfahrt (DLR) grant ”German Data Center for SDO”, European Re- comparisons with surface observations [15]) using fully com- search Council Starting Grant ”Seismic Imaging of the Solar Interior”, and Deutsche pressible radiative simulations in local Cartesian simulation Forschungsgemeinschaft SFB-963 grant ”Astrophysical Flow Instabilities and Turbu- boxes. For example a recent stagger simulation [16] covers lence.” r > 0.97R⊙, which overlaps in radius with the ASH simu-

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Footline Author PNAS July 24, 2012 vol. 109 no. 30 11897