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Noname manuscript No. (will be inserted by the editor) Solar surface and atmospheric dynamics The Photosphere Mart´ınez Pillet, V. 3 1 0 Received: date/Accepted: date 2 n Abstract Various aspects of the magnetism of the quiet sun are reviewed. The a J suggestion that a small scale dynamo acting at granular scales generates what 9 we call the quiet sun fields is studied in some detail. Although dynamo action 2 has been proved numerically, it is argued that current simulations are still far from achieving the complexity that might be present on the Sun. We based this ] statement not so much on the low magnetic Reynolds numbers used in the simu- R lationsbut, aboveall,in the smallnessof the kineticReynolds numbers employed S by them. It is argued that the low magnetic Prandtl number at the solar surface . h mayposeunexpected problemsfortheidentificationof theobserved internetwork p fields with dynamo action at granular scales. Some form of turbulent dynamo at - bigger (and deeper) scales is favored. The comparison between the internetwork o r fields observed by Hinode and the magnetism inferred from Hanle measurements t areconvergingtowardsasimilardescription.Theyarebothdescribedasrandomly s a oriented,largelytransversefields in theseveral hecto-Gaussrange.These similar- [ ities are ever making more natural to assume that they are the same. However, 1 andbecauseofthelargevoidsofmagneticfluxobservedinthespatialdistribution v of the internetwork fields, it is argued that they are not likely to be generated 3 by dynamo action in the intergranular lanes. It is concluded that if a dynamo 3 is acting at granular scales, the end product might have not been observed yet 9 at current spatial resolutions and sensitivities with the Zeeman effect. Thus an 6 effort to increase these resolutions and polarimetric sensitivities must be made. . 1 New ground- and space-based telescopes are needed. The opportunity offered by 0 the Solar Orbiter mission to observe the Quiet Sun dynamics at the poles is seen 3 as oneof the mostimportanttestsfor confirmingtheexistence,or otherwise,of a 1 granularlydriven surface dynamo. : v i Keywords Quiet Sun magnetism · turbulent dynamo X r a InstitutodeAstrof´ısicadeCanarias 38200, LaLaguna, Tenerife,Spain Tel.:+34-922605237 Fax:+34-922605210 E-mail:[email protected] 2 Mart´ınezPillet,V. 1 Introduction 1 A consensus about the existence of a small-scale dynamo (SSD ) operating at the solar photosphere is being consolidated in today’s solar physics (see, e.g., Vo¨gler and Schu¨ssler2007;Abbett2007;PietarilaGraham et al.2010;Stein2012, for a review). High resolution magnetograms from ground and (mostly) space- based telescopesobservedin theinternetworkareoftenused toindicatethatsuch a surface dynamo exist (e.g. Danilovicet al. 2010a; Lites 2011). On theoretical grounds, simulations of various kinds have been used to suggest how universal the various ingredientsof such a dynamo seems to be (Moll et al. 2011). They all indicatethat turbulentshear stresses actingon the inertialrange act as the main mechanismabletoefficientlyconvertkineticintomagneticenergy.Theconclusions from the various simulationsof turbulence in a conducting fluid seems to be that it would have been a lot harder to explain the absence of a surface SSD than its presence. With an emphasis on the observationalside, we review in this work the currentstatusofthisconsensusandtrytopinpointwhichaspectsaremoresolidly established and which are less settled. Section 2 will be the only one that concentrates on simulations and the theoretical aspects related to the problem of the existence of SSDs on the Sun. It will addressthemostcontroversialargumentquestioningtheexistenceofsuchamech- anism,namelythefactthatthesolarconvectivezonehasamagneticPrandtlnum- ber thatisordersof magnitudesmallerthanone,whilesimulationsworknearthe ∼1regimemostofthetime.LowmagneticPrandtlnumbersareknowntoseverely discourage dynamo action since the early simulations of convectively driven tur- bulentdynamos(Nordlund et al.1992;Schekochihin et al.2004a,2005).Recently, progresshasbeenachieved,however,thatindicatesthatanSSDisindeedpossible inthelowmagneticPrandtlregime(Iskakov et al.2007;Schekochihin et al.2007; Brandenburg2011).But thesituationis notconclusiveand thepapersaddressing this issue often resort to the fact that a mixed polarity field is observed at the solar surfaceas the firmest indicationthat such a mechanism should exists. How- ever, and in the absence of a clear proof that this observed (internetwork) fields originatefromanSSD–andsuchaproofisnotavailableyet–theonlyprogressto settle this issue will come from further work in the simulationfront. The observational arguments that have been put forward to favor the presence of a solar surface dynamo are discussed later in Section 3. There are basically two such arguments. First, the evidence from the observed Hanle effect in lines such as the SrI 4607 ˚A line and the careful modeling of these signals indicate the existence of a tangled field with a mean strength of < B >∼ 130 G (TrujilloBueno et al. 2004) some few hundred kilometers above the solar surface (see also TrujilloBueno et al. 2006, and references therein). This number was originally derived under some model assumptions that made it uncertain to within a factor two. However, a recent study (Shchukina and Trujillo Bueno 2011) of the predicted Hanle signals from the MHD simulations described in PietarilaGraham et al. (2009b) has eliminated some of this model dependency and confirmed such large mean field strengths. As a mixed polarity tangled field at unresolved scales leaves basically no trace in the Zeeman profiles, these fields 1 Small scale here refers to generation of magnetic fields at scales smaller than the energy injectionone,thegranulation. Solarsurfaceandatmosphericdynamics 3 havealwaysbeen a primecandidateto be considered as originatedfrom a surface SSD.Ifthetanglingoccursatscalesnearorabovepresentresolutions,somesigna- turescanbedetected,though.Itisunclearifthesefieldshavebeenobservedusing theZeemaneffect(butseeLites et al.2009;Bellot Rubio and Orozco Suárez2012, andSection3).ThesecondobservationalargumentinfavorofanSSDcomesfrom theHinodespectropolarimeter(SP)instrument(Tsuneta et al.2008;Kosugi et al. 2007) and its unprecedented characterization of the internetwork fields using the Zeemaneffect(Lites et al.2008;Ishikawa and Tsuneta2011;Orozco Suárez and Bellot Rubio 2012). While transverse fields were known to exist in the quiet sun, as originally foundbytheAdvanced StokesPolarimeter(ASP;Lites et al.1996),itwastotally unexpected thatthesefieldshavea predominanttransversecharacter.Thistrans- verse nature seems to fit in a natural way with an origin related to a turbulent dynamo as shown by recent simulations(see Schu¨ssler and Vögler 2008). Apossibleoutcomegiventhisstateofaffairscouldbeasfollows.Theexistence ofanSSDactingatgranularscalesatthesolarsurfacecaneventuallybeconfirmed from a set of improved SSDs simulations(along the lines described in Section 2). In them, a continuous distributionof fields is obtained that is able to explain the Hanledepolarizationlevels from those fields created at the smaller scales and the largely horizontal internetwork fields from those at larger scales. The separation between these two sets of fields does not have to be sharp and a range of spatial scales can contribute to both the Zeeman and Hanle results (or, perhaps, that the two observed processes are due to fields exactly at the same scales as it can beinferredfromtherecentresultsofBellot Rubio and Orozco Suárez2012).Note that, such a field distribution would solely depend on the existence of the always present turbulent convective motions near the surface and, thus, should be inde- pendentoflatitudeandofactivitycyclephase.Whilethisconclusionseemsrather plausible given the current evidence, the aim of this work is to address some of the known problems that might prevent such an outcome. In particular, Section 4 describes some observations recently obtained with the IMaX/SUNRISE mag- netograph(Mart´ınez Pillet et al.2011;Solanki et al.2010)thatshowregionsthat display very little magnetic activity, either measured as residual signals in time- averageddeepmagnetogramsorasevidencedbyalackoffluxemergenceepisodes in the form of small-scaleloops (as discovered by Mart´ınez Gonza´lez et al. 2012). It is unclear how these voids are compatible with a granularly driven SSD. That the situation is far from clear has been corroborated recently by the study of (Stenflo 2012, based on SDO/HMI magnetograms)who proposes the existence of a basal flux of order 3 G that is suggested to be an upper limit to the efficiency of an SSD at the solar surface. Accordingto this result all the internetworkfields observed with Hinode/SP will not be generated through such a mechanism and only the Hanledepolarizing fields could be originatedthrough it (if at all). In spite of this somewhat confusing situation, it is important to stress that our understandingabout the natureand the propertiesof the quiet sun fields has improved enormously in recent years. But it is clear that a number of important questionsremainsonthetheoretical/modellingsideandontheobservationalfront. Section 5 finishes this work proposing a way forward to further advance in this understandingofthequietsun magnetism.Notsurprisingly,wepromotean effort toincreasethepolarimetricsensitivityandthespatialandtemporalresolutionsof both,theHanleand theZeemanobservations.Studyingthestatisticsofthequiet sun fields at various latitudes will also provecrucial. 4 Mart´ınezPillet,V. 2 Small scale dynamo action at low Pm. Implications for the solar case Theseminalreferencethattriggeredthepresentdebateon theexistenceof acon- vectively driven turbulent dynamo at the solar surface was the work of Cattaneo (1999), although the debate is older (see, e.g., Petrovay and Szakaly 1993; Lin 1995). It is important to point out that this work mentioned, both, the granular andsupergranularscalesaspossiblecontributorstosuchnon-helicaldynamo.The simulations used closed upper and lower boundaries with vertical fields in both of them. The Reynolds and magnetic Reynolds numbers could be clearly defined for this simulationthanksto the fixed computationalgridused to solvethe MHD equations. They were Re=ul/ν =200 and Rem =ul/η =1000, with l the characteristic length of the energy injecting convective cells, u the velocity of these cells and ν and η the molecular viscosity and the magnetic diffusivity, respec- tively. These numbers are large enough to ensure the development of turbulence. But it is important to note that Re was five times smaller than Rem (magnetic Prandtl number of Pm = Rem/Re = ν/η = 5 > 1). Under these circumstances, themagneticfield sees a smoothmean flow efficientlyactingon it.Thenumerical simulationresultedindynamoactionsaturatingat20%ofthekineticenergyflow. Thecrucialingredientwasthechaoticnatureofthedrivingflows.Figure2ofthis paper already showed that, at the surface, the strong fields were localized in the downflow lanes, while the cell interiors showed no (vertical) field signature. The situation was different in deeper layers where fluctuating fields were filling basically the whole volume. At the time of the publication, the dominant transverse nature of the internetwork was not known (see Section 3) and this aspect was not analyzed. For this reason, the profiles synthesized by Sánchez Almeida et al. (2003) using these simulations concentrated on the study of the asymmetries in- ducedintheStokesVprofiles(circularpolarization)anditscomparisonwiththose observed in the internetwork. A shortage of asymmetries indicated that the simulations still did not achieveas much complexityas present in the Sun. However, usingthissynthesis,andafterincludingeffectssuchastelescopediffraction,itwas predictedthatwhenimprovingthespatialresolutionfrom1 arcsecto0.15arcsec, one should detect four times more Stokes V polarizationsignals. A numberofassumptionsmadein thissimulation(suchas theBoussinesqap- proximation)havebeen relaxed in morerecent works.The morerealisticones (in terms of their proximity to the physical conditions on the Sun) are those made with the MURaM code (Vo¨glerand Schu¨ssler 2007; Schu¨ssler and Vo¨gler 2008; PietarilaGraham et al. 2010). In particular, they have addressed the important questionoftheroleplayedbytheclosedboundaryconditionsassumedinCattaneo (1999).Stein et al.(2003)pointedoutthat,inthesolarconvectivezone,fieldsare submerged efficientlyto the bottomof the convectivezone by strong and concentrated downflows showing little recirculation near the surface. This recirculation wasartificiallyenhancedinthesimulationsofCattaneo(1999)bytheuseofclosed boundary conditions. Stein et al. (2003) concluded that diverging upflows sweep thefluidintodownflows,oftenvortical,wherestretchingandtwistingbecomesef- fective(andbalancedbydiffusion)butallthesefieldsareratherrapidlysubmerged down into the bulk of the convectivezone. The energy added to the flux that vis- its the surface was a very small fraction of the global budget of magnetic energy and the effect cannot be considered a local dynamo. This criticism has, however, been superseded by the MUraM simulations which used an open boundary and Solarsurfaceandatmosphericdynamics 5 allowfora non-zeropointingflux atthebottomboundary.Thewayin whichthis boundary condition was implemented in the simulations of Vögler and Schu¨ssler (2007) was by imposing an artificially increased magnetic diffusivity there. This diffusivityensuredthathorizontalfieldsmovingdownwardinthesimulationleave the box unimpeded while, at the same time, prevented horizontal flux from en- teringthedomain.Allflux leavingthebottomboundaryin thesesimulationswas created by dynamo action inside the box. Vo¨gler and Schu¨ssler (2007) concluded thatwhilethedownwardpumpingoffluxoutsideofthedomaindoesindeedreduce thegrowthrateofthedynamo,itdoesnotshutitdown.Theypredictthataslong as a sufficiently high Rem is used (above the critical magnetic Reynolds number, ReC), dynamo action will be observed in the simulations. Specifically, they see m exponential growth of the magnetic energy for Rem of 2600, while they find a decrease for Rem below 1300. Now, here we should caution that the exact values of Rem and Re (and thus of Pm) are not as easily defined as in Cattaneo (1999) simulations. The need to resolve shocks with artificial viscosity schemes or the implementationof the bottom boundary diffusivity necessarily implies that these numbersaremoredifficulttoascertain.AnestimateoftheeffectiveReynoldsand PrandtlnumbersfortheMURaMsimulationswasgivenbyPietarilaGraham et al. (2010)usingvariousmomentsofthevelocityandmagneticspectra(orTaylormi- croscales). The various dynamo runs available from this code turn out to have Rem ∈ [2100,8300] (with the latter value using a grid resolution of 4 km) and Pm ∈ [0.8,2]. This set of simulations all displayed an SSD generating magnetic fields inside their volume. The conclusion was that current MURaM simulations (withPm ∼1) willshow dynamoaction as long as Rem >ReCm ∼2000.Themain mechanism identified for this generation was the stretching and twisting of field linesbyfluidmotionsintheinertialrangeofthespectrumofvelocityfluctuations. Inparticular,itwasconcludedthatdynamoactionisconcentratedintheturbulent downflows as field line stretching against magnetic tension is very efficient there (e.g. Stein 2012, for a review). In order to clarify the nature of the observed dynamoaction,Moll et al.(2011)haveanalyzedtheunderlyingphysicalmechanism under various physical conditions and assumptions. They concluded that for the casesstudied(incompressibleMHD,Boussinesqconvectionandcompressiblesolar convection), the field is amplified by similar inertial range shear stresses that are independent of the conditions at the injection scale. The inclusion of compress- ibilityeffects or the asymmetry between upflows and downflowsgenerated by the strong stratification did not influence the result. They, thus, termed the dynamo mechanism as universal. OneconcernremainsabouttheexistenceofapossibleSSDontheSun,though. The problem has been known since the early studies of dynamo action in conducting fluids. It was originallyformulated by Batchelor (1950) who studied how turbulent motions stretch the field lines and amplify the magnetic energy as long asthisprocessremainsunimpededbyohmicdiffusion.Fieldlinestretchingispro- duced by fluid motions in the inertial range whose dissipation scale is set by the viscosityofthefluid.Highviscosityν couplesthefieldlinestotheflowandallows ittobendthemefficiently.Largemagneticdiffusivityη decouplestheplasma(and theflows)fromthefield linesand preventsthebending.Thus,itwasalwaysclear thatthe efficiency of theSSD was goingto be controlledby theinterplayof these twoeffects as measured by theratioPm =ν/η. Clearly,both conditionsRem ≫1 (tofavorfieldlinestretching)andPm ≫1(tocouplefluidmotionsandfieldlines) 6 Mart´ınezPillet,V. Fig. 1 The Reynolds number, magnetic Reynolds number and (inverse) Prandtl number in the solarconvective zone according tothe mixing-lengthmodel of Spruit(1974). TheX near thesurfacemarksthetipicalvaluesachievedofbothReandRem inthesimulations. boost local dynamo action. Based on an analogy between vorticity and magnetic fields Batchelor(1950)even concludedthatforPm <1 noSSD waspossible.This conclusion was later criticized by a number of authors as the analogy cannot include the different initial and boundary conditions seen by these two fields (see, e.g.Boldyrev and Cattaneo2004).However,simulationsintheearly90’s(see,e.g., Nordlund et al. 1992), including compressibilityand strong stratification,already resulted in dynamo action only for Pm ≥ 1 with efficient shutting down of the dynamo for Pm < 1. Other simulations encountering the same problem are discussed in Boldyrev and Cattaneo (2004) and in Schekochihin et al. (2005). Thus, thequestionoftheexistenceofanSSDatlowPrandtlnumbershasreceivedsome attentionin recent years. Before we briefly describe the results from the numerical studies about the existence of an SSD at low Prandtl numbers, it is important to remember what are the actualnumbers that occur in the solar convectivezone and get an idea of howfarorhowclosearewefromsimulatingtheseconditions.Tothisend,Figure1 −1 shows Re, Rem and Pm as computed in themixing-lengthbased model of Spruit (1974). The velocities at the injection scale u are obtained as part of the model and the characteristic length is assumed here to be l = z/2, with z the depth 5 inside the convective zone. The magnetic Reynolds number changes from 10 at 9 thephotosphereto10 atthebottomoftheconvectivezone,whereastheReynolds 12 −7 −3 number stays constant at a level of 10 . This makes Pm ∈[10 ,10 ], with the smallest value reached at the photosphere. Thus, everywhere in the Sun, we have Solarsurfaceandatmosphericdynamics 7 Re ≫ Rem ≫ 1 and Pm ≪ 1 which is exactly the regime where the existence of an SSD becomes problematic. It is generally believed that for a sufficiently high Rem there will always be an SSD at work. But the above mentioned simulations prompted a deeper study about the existence and nature of an SSD under solar conditions.Asalreadymentioned,thesimulationsbyCattaneo(1999)hadPm =5 and those from the MURaM code alwaysmove close to the Pm ∼1 case (there is onewithPm =0.8thatis discussed below).NotethattheregimewhereRem and Re are similar is actually very favorable for the numerical codes as similar grid sizes resolvethe dissipativescales of both, magnetic and velocityfields. WhatisthephysicalargumentbehindthisdifficultytogenerateanSSDwhen Pm ≪ 1?. Under such conditions the viscous (lν) and resistive (lη) scales follow lη/lν ∼Pm−3/4 ≫1(Schekochihin et al.2004a, 2005)and theresistivescalelη falls in the middle of the inertial range. Turbulent eddies of scales l > lη do the nec- essary field line bending and twisting for dynamo action at a rate of u /l (with l ul the typical flow velocity at this scale). If Pm > 1 only these eddies occur and the field lines always see a spatially smooth flow. However, if Pm < 1 one has now eddies below the resistive scale l < lη. These eddies act on the field as a turbulent diffusion with diffusivityu l and destroy magneticenergy. It is the pre- l dominanceofthislastprocesswhatcanmakedynamoactionimpossibleasitwas seen in the previously mentioned simulations. Using an incompressible spectral 2 MHD code and the PENCIL code , Schekochihin et al. (2005) studied what are the possible asymptotic limits when Re ≫ Rem and the corresponding values of the ReC for the existence of a dynamo. The above described effect always trans- m lates into a sharp increase in ReCm as Pm → 0 (see also PietarilaGraham et al. 2009a), but does not prevent the existence of a dynamo in this regime. The two asymptotic limits are, first, as Re → ∞, ReC → const, so that dynamo action is m possible for higher Rem and, second, ReCm → ∞ with ReCm/Re → PCm =const, in which case no dynamo is possible (turbulent diffusion efficiently dissipates magnetic energy at small scales). Which exactly of the two asymptoticlimitsprevails has been under much debate in recent years. While Schekochihin et al. (2004a) andSchekochihin et al.(2005)(seetheirFigure2)favortheexistenceofaPC (no m dynamo)fromsimulationsofincompressiblemagnetoconvectionreachingvaluesof Pm assmallas0.15,Boldyrev and Cattaneo(2004)providedanalyticalarguments favoringthe existence of a ReC. m Whilethedebatein themidlastdecadedidnotlookpromisingforconfirming the existence of an SSD at the solar surface, the situation has changed recently (even ifnot completelysettled).Schekochihin et al.(2007) (seealsoIskakov et al. 2007)performedsimulationsofincompressibleMHDturbulencereachingvaluesof Pm =0.1 and with a sufficiently high Rem that indicated a plateau region where a ReC is observed (see their Figure 1b). Admittedly, the number of such simula- m tions proving the existence of this plateau is very small but the authors consider it enough numerical certainty. The results from Brandenburg (2011) using the PENCIL code (that includes compressible effects) and low values of Pm resulted indynamoactionbeingactivatedaswell.Pietarila Graham et al.(2010)alsofind growth of magnetic energy in the one case they analyzed with Pm = 0.8. Thus, themostadvancedexistingnumericalsimulationsofsmall-scaledynamoactionin 2 Seehttp://www.nordita.dk/software/pencil-code. 8 Mart´ınezPillet,V. turbulent MHD currently favor the occurrence of such a process in the low Pm regime. However,a number of caveats remain: – First, and most importantly, ReCm increases with decreasing Pm. The exact factor depends on the specificities of the simulations. Boldyrev and Cattaneo (2004) suggest a factor 7 increase in ReCm when shifting from the Pm ≫ 1 to the Pm ≪ 1 case. They use an analytical model of isotropic and homogeneous turbulence that includes the extra roughness of the velocity field for Pm < 1. Schekochihin et al. (2007) find from incompressible forced turbulence a factor 3 increase. As in the MURaM simulations one has ReC ∼2000 m (PietarilaGraham et al. 2010), this means that as soon as we move into the low-Pm regime, we need magnetic Reynolds numbers above at least 6000 to be able to trigger dynamo action. These magnetic Reynolds numbers are not currently achieved by this code. In particular, none of the runs used by Danilovicet al. (2010a) would be able to actually sustain dynamo action. On top of that, the saturated field strength is known to decrease with decreasing Pm, thus the expected field strengths willbe smallerthan thosecomputed for Pm ∼ 1. The exact amount of this reduction is still a very controversial issue (seeSchekochihin et al.2004a,2007;Brandenburg2011)anditsmagnitudefor the solar case unknown. But the net effect will be a reduction in the fields as compared to those computed for Pm >1. – Tocompletethedemonstrationoftheexistenceofadynamodrivenbyfluidmo- tionsintheinertialrangeatlowPmvalues,agrowthrateofthedynamoscaling withRe1/2 mustbeobtainedfromthesimulations.NeitherSchekochihin et al. m (2007) nor PietarilaGraham et al. (2010) have reached that (see Figure 3 of thelatterwork).Simulationswithanincreasedresolutionareneededtofinally settle this issue. Schekochihin et al. (2007) concludes that, as long as this is not achieved, the mechanism that sustains the growth of the magnetic field fluctuationsin the low-Pm regime will remain basically unknown. – Schekochihin et al.(2007)andIskakov et al.(2007)concludedthatinthePm ≪ 1 regime, the magnetic energy spectra is fundamentally different from that foundinthePm ≫1.Thespatialdistributionofthegrowingmagneticfieldsis qualitativelydifferenttoo(seeFigure2inSchekochihin et al.2007).Thisindi- cates that the use of simulationsin the Pm ∼1 range to compute the ensuing Stokes profiles and its comparisonwith those observed may not be justified. – TheoftensimulatedcasewithPm ∼1hasthesamespectralenergyproperties and field distribution as the Pm ≫ 1 case (Schekochihin et al. 2004b). This is probably why the case with Pm =0.8 simulated by PietarilaGraham et al. (2010) was so similarto the those in the range of Pm ∈[1,2]. Letusfinalizethissectionbystressingthatifthesituationlooksconfusing,itis becausethishasindeedbeenthecaseinthistopicforsometime(seeIskakov et al. 2007; Schekochihin et al. 2007, who speak about a frustrating outcome). One argument commonly given to promote the existence of an SSD is the observation of a mixed polarityfield in the internetworkregions of the Sun (and many of the abovementionedworksusethisargumentonewayoranother).Figure2 of,both, Cattaneo (1999) and Vo¨gler and Schu¨ssler (2007) clearly indicate that this is a reasonable argument. The point we want to stress here is that the same applies to Figure 10 from Stein and Nordlund (2006), which does not include an SSD. PietarilaGraham et al. (2009a) estimate for this simulation a Rem ∼ 600, which Solarsurfaceandatmosphericdynamics 9 isknowntobetoosmalltodevelopdynamoaction.However,theirmixedpolarity distributionlocated in the interganularlines looks as ‘solar’as in the other cases. In the work of Stein and Nordlund (2006) emphasis is made on divergingupflows bringing flux to the surface, expulsion to the intergranular lanes and sweeping of field lines into strong downflows that carry the flux into deeper layers. These simulations extent typically further down than those that concentrate in SSD generation and also include larger scales such as those associated with the meso- granulation.AsshowninFig.1, thedeeper wemoveintotheSun,thelargerRem andPm (althoughstillsmallerthanone).Thus,avalidquestionisifitisnotmore natural to ask if a solar SSD exists at meso- and supergranular scales and, if so, whether they dominate over that that might exists at granular ones. This point will be further discussed in Section 4. 3 Observed signatures of small scale dynamo action at the solar surface We now turn to the observational aspect of the discussion and ask the question: HaveweseenthefieldsproducedbyapossibleSSDoperatingatthesolarsurface? As it will become evident, there has been, as before, solid observational progress and areas with much confusion. Basically, two candidates exist that are often considered as by-products of an SSD, the internetwork fields and the, so-called, hidden fields that generate the Hanle depolarizationsignatures. 3.1 Zeeman signals There is no question that Hinode/SP data has produced a major quantitative andqualitativejumpforwardinourunderstandingoftheinternetworkfields.The publication by Lites et al. (2008) of a gigantic slit scanned map with consistent −3 10 polarimetric sensitivity and homogeneous spatial resolution of 0.3 arcsec changed our view of the quiet sun magnetism. In this map, a myriad of patches withpredominantlinearpolarizationsignatureswasdiscovered.TheASPalready found the existence of episodic burst of largely transverse fields (the Horizontal InternetworkFeatures,HIF, Lites et al.1996) but they werethoughttobe rather sporadic.Theonlypreviousindicationoftheirexistenceandglobalcharactercame from the SOLIS instrument as found by Harvey et al. (2007). But no prediction fromthesimulationsorestimateoftheirmagnitudewasavailable.Theexistenceof this ubiquitous horizontalfield has now received full confirmationfrom the SUN- RISE/IMaX data(Danilovicet al.2010b) who could makethefirststudy of their evolution(seetheanimationin Solanki et al.2010)andestablishasolidstatistics of their lifetimes.Both instruments,Hinode/SP and SUNRISE/IMaX coincidein locatingthese HIF at the borders of the upflowinggranules for a largefraction of their evolution. Lites et al. (2008) emphasized that the linear polarization signatures were not co-spatialwith line-of-sightfields that were more frequentlyfound in the intergranularlanes. BeforethepublicationofLites et al.(2008)mostofthediscussionontheinter- network flux concentrated on obtainingits mean unsigned flux (see Solanki 2009, for a review on this topic previous to the impact of the Hinode measurements). The idea was that an intricately complex field with mixed polarities observed 10 Mart´ınezPillet,V. in the best available Stokes V magnetograms was the outstanding description of the internetwork fields. Increased resolution (or sensitivity) will result into ever increasing amounts of longitudinal signals observed as there was less and less cancellation due to instrumental effects. Either because of lack of reliable measurements of the field inclination or because of a habit to focus studies of solar magnetism exclusively in longitudinal magnetograms, no mention to its possible transversecharacterwas traditionallymade. As the field strengths were expected to be near or below equipartitionvalues (less than 500 G, see Keller et al. 1994), these fields were not thought to be necessarily vertical either. Before the Hinode results, measurements of < |B | > (the spatially averaged unsigned longitudinal L flux in the internetwork) were routinely being made and its variation with the spatial resolution closely followed (Sánchez Almeida et al. 2003). In a way, Hin- ode/SP results have made this emphasis obsolete. We now know that these fields arelargelytransverseandoneshouldmainlycareabout<|B |>orsimplyabout T thespatiallyaveraged<|B|>.We shouldcautionherethatthesemagnitudesare obtained by observations of different Stokes parameters that have different sensi- tivitiestotherealmagneticfieldcomponentsontheSunandtothefractionofthe observed pixel that they occupy (the filling factor). Thus, the steps to compute < |B| > from the observed < |B | > and < |B | > are more problematic than L T what one might anticipate. Lites et al. (2008) estimated that the quiet sun map obtainedbyHinode/SPhad a<|Bapp|>of11G (orMx cm−2) anda <|Bapp|> L T of 55 G. These estimates were based on using integrals of the Stokes parameters thatwerecalibratedagainstmagneticfluxesbutwithnoaccountforthefractionof thepixeloccupiedby thefields.Thisiswhythey arenamed‘apparent’fluxes.An analysisperformedbyOrozco Suárez et al.(2007)ofthesamedata,but thistime using a Milne-Eddington (M-E) inversion code including a filling factor as a free parameter,resultedalsoinapredominantlytransversenatureoftheinternetwork, albeit with a smaller ratioof transverseto longitudinalapparent fluxes. It is no exaggeration to say that the ratio measured by Lites et al. (2008) <|Bapp|>/<|Bapp|>∼5 cameas a surpriseand was, thus, subjected to a deep T L scrutiny by the community. Several factors can create a systematic bias in this ratio. Spatially averaging noise affects a positively defined quantity such as B in a way different than T what it does to a signed quantity (B ). As the Zeeman effect has a sensitivity L different for each of these two components, the visibility of a given field strength is different depending on whether it is a field aligned with the LOS or perpen- dicular to it. In particular, fields close to the noise levels translate into different visibilitythresholds. Last but not least, there is the already mentioned difference in how the filling factor couples to the real field strengths and inclinations for eachoneofthetwocomponentsand dependingon thespecific methodofanalysis used. These effects and their impact into the factor 5 obtained by the first Hin- ode/SPmeasurementshavebeenstudiedbyvariousauthors(Asensio Ramos2009; Borrero and Kobel 2011; Stenflo 2011; Sánchez Almeida and Mart´ınez Gonza´lez 2011; Steiner and Rezaei 2012, see the latter for a review). It all translates into understanding how exactly noise influences the final result given the method of analysis one follows and the various thresholds for inclusion of a given pixel or not. Depending on the specific case, different values for < |Bapp| >, < |Bapp| >, T L or inverted parameters B, inclination, azimuth and filling factor are obtained. Figure 2 shows the central portion of the same magnetogram used in Lites et al.