Mon.Not.R.Astron.Soc.000,1–??(2012) Printed1August2012 (MNLATEXstylefilev2.2) Magnetic diffusion driven shear instability of solar flux tubes B.P. Pandey and Mark Wardle 2 Department of Physics, Astronomy & Research Centre for Astronomy, Astrophysics & Astrophotonics, 1 Macquarie University,Sydney, NSW 2109, Australia 0 2 l 1August2012 u J 1 ABSTRACT 3 The dynamics of the partially ionised solar photosphere–chromosphere can be de- scribed by a set of equations that are structurally similar to the magnetohydrody- ] R namic equations, except now the magnetic field is no longer frozen in the fluid but slips through it due to non–idealmagnetohydrodynamic effects which manifests itself S as Ohm, ambipolar and Hall diffusion. Macroscopic motion of the gas is widespread . h throughoutthesolaratmosphereandshearingmotionscoupletothenon–idealeffects, p altogether destabilising low frequency fluctuations in the medium. The origin of such - non-ideal magnetohydrodynamicinstability lies in the collisionalcoupling of the neu- o r tral particles to the magnetized plasma in the presence of sheared background flows. t Expectedly the maximum growthrate and most unstable wavenumberdepend on the s a flow gradient and ambient diffusivities. [ The orientationof the magnetic field, velocity shear andperturbationwavevector play crucial role in assisting the instability. In the presence of only vertical field and 1 vertical wavevector, ambipolar and Ohm diffusion can be combined together as Ped- v 2 ersen diffusion and causes only damping; in this case only Hall drift in tandem with 8 shearflowdrivestheinstability.However,fornon-verticalfieldandobliquewavevector, 1 both ambipolar diffusion and Hall drift assist the instability. 7 Weinvestigatethestabilityofamagneticelementinthenetworkandinternetwork. . Theshearscaleisnotyetobservationallydeterminedandthus,assumingtypicalshear 7 0 flow gradient ∼ 0.1s−1 we show that the magnetic diffusion shear instability grows 2 over a minute. Thus, it is plausible that network–internetworkmagnetic elements are 1 subject to this fast growing, diffusive shear instability. This fast–growing instability : could play important role in driving low frequency turbulence in the photosphere- v chromosphere plasma. i X Key words: Sun: Photosphere, MHD, waves. r a 1 INTRODUCTION heat the coronal plasma to high temperatures. This simple physical picture of wave excitation, propagation and ensu- ing coronal heating is quite attractive since it ties the heat The solar atmosphere is host to a number of highly ener- transportinthesolarcoronaandheliospheretotheultimate getic events such as coronal mass ejections (CMEs), flares, sourceofenergy–theshearingphotosphericconvectivemo- prominences, coronal heating. The huge reservoir of energy tions. Therefore the investigation of wave propagation and inthesolaratmosphereisrelatedtothemagneticfieldorig- concomitant heating of coronal plasma in the framework of inating in the convection zone where shearing motion of MHD has been very popular topic in solar physics (Priest fieldlinefootpointscanstretchandtwisttheanchoredfield 1987; Parhi et al. 1997a,b, 1998; Hasan & Kalkofen 1999; (Parker 1979). A variety of magnetohydrodynamic (MHD) Goedbloed & Poedts 2004; Aschwanden 2009). waves are generated in the corona due to convective mo- tioninthephotosphere.Theconvectiveshearingmotionnot Most solar atmospheric heating, with possible excep- only producesAlfv´en, fast or,slow magnetoacoustic waves, tionofflares,takesplaceinthechromosphere.Thechromo- but, also, brings topologically disparate parts of magnetic sphereishighlystratifiedandextendsuptoaboutninepres- configuration closer together resulting in the formation of sure scale heights, i.e. about 2 Mm above the photospheric current sheets. All in all, a tiny fraction of convective en- surface. The lower chromosphere is threaded by strong ( ∼ ergy carried by the waves to higher altitude may suffice to kiloGauss)verticalfluxtubeslocatedinthenetworkregions c 2012RAS (cid:13) 2 B.P.Pandey and Mark Wardle wheretheyareobservedasbrightpoints.Thesetubes,which work assumes ω ω . In fact ion carries the inertia of i ci ∼ have a low filling factor (< 1%) near the footpoint in the thefluid.Therefore,ontheonehand,thetime–independent photosphere,expandtofillabout 15% ofthelowerchromo- conductivitytensorimpliesω ω ,ontheotherhand, e,i ce,ci ≪ sphere ( 1Mm, where CaII emission features are observed MHDmomentumequationrequiresω ω .Clearly,inves- i ci ∼ ∼ inHandKlines)beforefillingtheentireatmosphereforming tigation of the collisional effects by merely modifying the acanopyinthechromosphere.Thequitesolarinternetwork induction and energy equation in the MHD framework is region is also magnetised, with patches of kG field concen- highly unsatisfactory. tration (de Wijn et al. 2009; Almeida et al. 2010) and an Any realistic model of the solar atmosphere must re- order of magnitude smaller field everywhere else. Observa- flect two basic observational facts: (a) the magnetic field tion suggests that localised active regions that emit 80% distributiononthesolarsurfaceisnotcontinuousbutisor- ∼ of the coronal radiative loss near solar maximum contain ganisedinthenetworkandinternetworkelements.Whereas plasma of chromospheric origin (Aschwanden 2001). This network field (& kG) is predominantly vertical organised fact raises the possibility that the same mechanism that in the flux tubes (diameter . 100km) located in the in- transports mechanical energy from the convection zone to tergranularlanes,theinternetworkfield[ (fewG kG)]in ∈ − the chromosphere to sustain its heating rate also supplies theinteriorofsupergranulecellsismostlyhorizontal(Hasan the energy needed to heat the corona and accelerate the 2009; Lites et al. 2008) 1; (b) plasma in the photosphere– solar wind. chromosphere is weakly ionised [with fractional ionisation, The absence of a unified theoretical framework to ad- whichisaratiooftheelectronandneutralnumberdensities dress the heating problem of the chromosphere–corona is Xe =ne/nn 10−4, VAL81]. ∼ partiallyrelatedtotheoveruseofidealMHD,fromthedeep Both active and quite phases of solar atmosphere are interior of the sun to the corona. This approach is valid highly dynamic and consist of convectively driven vortices only for the fully ionised plasma in which field and fluid and flows of various spatial and temporal scales (Bonet are well coupled to each other, a far cry from the low tem- et al. 2008; Wedemeyer-B¨ohm & Voort 2009; Balmaceda perature photosphere-chromosphere where non–ideal MHD et al. 2010; Bonet et al. 2010). Most vortices are small effects such as Hall, ambipolar and Ohm diffusion domi- (. 0.5Mm) with average size 241 25km and typi- ∼ ± nate (Pandey & Wardle 2006, 2008; Pandey et al. 2008; cal lifetime 3 5min. though large vortices 20Mm ∼ − ∼ Khomenko & Collados 2012; Sykora et al. 2012; Pandey & withlifetime&20min.havealsobeenobserved(Attieetal. Wardle2012a).Itiswellknown[Vernazzaetal.(1981);here- 2009). The bright pointsassociated with the vortex motion afterVAL81]thatthenumberofplasmaparticlesinthepar- intheintergranularlanemoveswithtypicalspeed.2km/s tially ionised solar atmosphere, particularly below 2.5 Mm (Wedemeyer-B¨ohm&Voort 2009).Themagneticfieldsap- issmallcomparedtotheneutralsandthus,theplasmafluid pear to play crucial role in mediating vortex motion in the is not frozen in the magnetic field but can slip through it photosphereand chromosphere (Steiner& Rezaei 2012). duetofrequentcollisionswiththeneutrals(Mestel&Spizer Rotationhasbeeninvokedinthepasttoexplainthesta- 1956). bilityof fluxtubes(Schu¨ssler 1984). Models ofspicules also The inclusion of neutral dynamics not only destroys useconceptofrotatingfluxtubes(Kudoh&Shibata1997). the economy and simplicity of single fluid MHD descrip- Thenumericalsimulationofsolarconvectiondisplaysturbu- tion of fully ionised plasma but very concept of well de- lent vortex flows at intergranular lanes (Zirker 1993; Stein fined flux tube may be difficult to define in the multi–fluid & Nordlund 1998). The vorticity generation near bound- framework. Furthermore, high frequency Alfv´enwave may aries of granules have also been seen in the numerical sim- not survive the collision dominated photosphere and chro- ulation of photosphere (Nordlund et al. 2009; Muthsam et mosphere (Goodman 2004; Leake et al. 2005; Vranjes et al. al. 2010). The formation of small-scale, intergranular vor- 2007;Arberetal.2007;Vranjesetal.2008;Goodman2011). tices suggests that vorticity is formed due to interaction of This could havebeen anticipated on theground that in the thephotosphericplasmawith ambientmagneticfieldinthe photosphere(.500km)andchromosphere(.2500km)the intergranular lanes (Moll et al. 2011; Shelyag et al. 2011). plasma number density is much smaller than the neutral The high resolution simulations of non-ideal MHD effects number density and thus, high frequency (with respect to shows that Hall effect generates out-of-plane velocity fields theion-neutralcollisionfrequency)MHDwavesareseverely withmaximumspeed 0.1km/satinterfacelayersbetween ∼ dampedbycollisionaldissipationofthewaveenergy.Away weakly magnetized light bridges and neighbouring strong outofthisdifficultyistoretaintheMHDmomentumequa- field umbral regions (Cheung & Cameron 2012). To sum- tion and include collisional effects by modifying the induc- marise, both observation and numerical simulation points tion and energy equations via conductivity tensor (Erdelyi tothepresenceof shear flowat variousspatial scales in the &James2004;Leake&Arber2006),anapproachoftenem- solar photosphere. ployedtostudythelowerionosphereoftheEarth.However, Thelargescaleshearflowactsasasourceoffreeenergy derivation of the time–independent conductivity tensor re- in the solar plasma that can easily destabilise waves. For quirestheneglectoftimederivativesintheelectronandion example, shear driven Kelvin-Helmholtz instability (KHI), momentum equations, i.e. d /dt ω ω , implying which converts shear flow energy into vortex kinetic en- e,i e,i ce,ci plasma dynamical response freque∼ncy, ω≪is much smaller ergy, is invoked to explain the instability of the flux tubes e,i thantheirrespectivegyro–frequencies,ω .Therefore,ne- (Kolesnikov et al. 2004; Soler at al. 2010). Although solar ce,ci glecting plasma inertia, a linear relationship between the electric field E and plasma current J can be easily derived E = σ J where σ is the time-independent conductivity 1 Internetwork field could be as well isotropic (Almeida and · tensor. However, equation of motion in the MHD frame- Gonz´alez2011), butseealsoSteiner&Rezaei (2012). c 2012RAS,MNRAS000,1–?? (cid:13) Magnetic diffusion driven shear instability of solar flux tubes 3 atmospheremaybesusceptibletoKHI(Karpenetal.1993; mumgrowthrateoftheinstabilityisderived.Theexpression Kolesnikov et al. 2004; Soler at al. 2010; Zaqarashvili at for maximum growth rate and critical wavelength is given al.2010;Ofman&Thompson2011;Foullonetal.2011),the in various limiting cases. In Sec.4, variouslimiting cases of presenceofmagneticfieldisnotconducivetothisinstability. the dispersion relation is discussed. In Sec. 5 application of For example, magnetic field along the flow suppresses KHI the result is discussed. Finally, in Sec. 6 brief summary of whereas transverse (to the flow) field has no effect on the theresult is given. instability (Chandrasekhar 1961). However, magnetic field not only quenches shear instabilities but can also facilitate it. For example, the most important instability in accre- 2 BASIC MODEL tion discs, the magnetorotaional instability is caused by an interplay between the Keplerian shear flow and magnetic Thephotosphere–chromosphereplasmaconsistsofelectrons, field(Balbus&Hawley1998).Thus,dependingonthepres- protons,singly ionized metallic ions,H,HeI,HeII,andHe ence of flow gradient, magnetic field, when well coupled to III. We shall ignore the distinction between hydrogen and the plasma can suppress as well as driveshear instabilities. metallic ionsand assumethat photosphere-chromosphereis Therefore, before dwelling upon the role of magnetic field made up of electrons, singly charged ions and neutral Hy- onthesheardriveninstabilities,itispertinenttoknowhow drogen.Althoughbasicsetofequationsforpartiallyionised well is magnetic field coupled to the surroundingmatter. plasma was formulated more than 50 years ago (Cowling 1957;Braginskii1965),weshallusethesinglefluidformula- The field drift through weakly ionised matter in the tiongivenbyPandey&Wardle(2008)forwhichthespatial presenceofshearflowcanassistwavestogrow.Forexample, and temporal scales of partially ionised plasma has been in protoplanetary discs, both Hall and ambipolar diffusion clearly elucidated. assists magnetorotaional instability (Wardle 1999; Balbus Inthelowfrequencylimit,thecollisionaldynamicsand &Terquem2001;Kunz&Balbus2004;Desch2004;Wardle fractional ionisation can be incorporated in the basic set & Salmeron 2011; Pandey & Wardle 2012b). Clearly, drift of equations without having to deal with the complexity of of the magnetic field in a weakly ionised diffusive medium three–fluid equations. Thus the continuity equation of the provides new pathways through which shear energy can be bulk plasma fluid is channelled to the waves. However, the role of diffusion in destabilising partially ionised medium in the presence of ∂ρ + (ρv)=0. (1) shearflow isnot uniquetoweakly ionised discsbutis quite ∂t ∇· generic (Kunz 2008; Pandey & Wardle 2012a). The crucial Here ρ = ρ + ρ is the bulk mass density and ρ = i n i,n ingredients required to excite this diffusion–shear instabil- m n is the ion and neutral mass densities with i,n i,n ity is: (a) the presence of a shear flow, and, (b) favourable m ,n as the ion and neutral mass and number den- i,n i,n magnetic field topology. The ensuing instability is overtly sitiesrespectively;v=(ρ v +ρ v )/ρisthebulkvelocity, i i n n similartoKHI,andnotsurprisingly,thegrowthrateispro- and, v and v are bulk velocities of the ion and neutral i n portional tothesheargradient.However,unlikeKHIwhich fluidsrespectively. The momentum equation is is hydrodynamic in nature, this is magnetohydrodynamic instability. dv J×B ρ = P + , (2) A detailed investigation of diffusive shear instability in dt −∇ c thecontextofsolaratmosphereiscarriedoutinthepresent where J = en (v v ) is the current density, B is the e i e − work which although similar to our previous work [Pandey magnetic field and P = P +P +P is the total pressure. e i n & Wardle (2012a), hereafter PW12a] has following notable The induction equation is differences. (i) Unlike PW12a, where only vertical field and ∂B 4πη 4πη transverse fluctuation (vertical wavevector) is assumed, in = × (v×B) O J H J×b ∂t ∇ − c − c thepresentcasefieldtopologyismoregeneralandwavevec- (cid:20) tor is oblique; (ii) In PW12a back reaction of the fluid on +4πηA (J×b)×b , (3) themagneticfieldwascompletely ignored.Byretainingthe c (cid:21) back reaction of the fluid on the field we show that shear where b=B/B is the unit vector along themagnetic field, driven diffusive instability do not have any cut-off wave- and Ohm (η ), ambipolar (η ) and Hall (η ) diffusivities O A H length;(iii) Forvertical fieldandtransversefluctuationswe are showed in PW12a that only Hall diffusion assists theinsta- bilityandambipolarandOhm,whichcombinesasPedersen η = c2 ,η = D2B2 ,η = cB . (4) diffusiononlycausesdampingofwaves.Inthepresentwork O 4πσ A 4πρiνin H 4πene for general field topology and oblique wavevector, we shall Here see that both ambipolar and Hall diffusion can assist the cen ω ω instability; (iv) The general stability criterion for magnetic σ= e ce + ci (5) B ν ν diffusiondominatedplasmainthepresenceofshearflowsis (cid:20) e i (cid:21) presentedinthiswork.Therefore,thispaperismoregeneral is the parallel conductivity,ω =eB/m c is the particle′s cj j in scope and application than PW12a. cyclotron frequency where e,B,m ,c denotes the charge, j Thepaperisorganisedinthefollowingfashion.Theba- magneticfield,massandspeedoflightrespectivelyandD= sicsetofequationsanddispersionrelationisgiveninSec.2. ρ /ρistheratioofneutralandbulkdensities.Forelectrons n Insubsection2.1wegivelinearisedequationsintermsofdif- ν =ν andforionsν ν .Althoughν ,ν ,ν and,ν e en i in ee ei ii ie ≡ fusiontensorbeforedescribinggeneraldispersionrelation.In can become comparable to ν (see Table. 1), it is the en Sec. 3 the general stability criterion is described and maxi- neutral-plasma collision that gives rise to ambipolar and c 2012RAS,MNRAS000,1–?? (cid:13) 4 B.P.Pandey and Mark Wardle Halldiffusioninthemedium.2Theelectron-ioncollisioncon- tributesto Ohm diffusion. 12 (a)B = 1.2 kG η Defining theplasma Hall parameter β as A j )10 βj = ωνicnj , (6) η10( 8 ηH g η and the Hall frequency o 6 O l ωH = ρρi ωci, (7) 40 0.5 1 1.5 2 2.5 Height [Mm] thediffsivitiesinEq.(4)canbewritteninthecompactform 11 (Pandey & Wardle 2008) (b)B = 120 G η ηH =(cid:18)ωvAH2 (cid:19) ,ηA=D (cid:18)νvnA2i(cid:19) ,and,ηO =βe−1ηH, (8) ηg10( ) 79 ηηAHO o 5 where νni =ρiνin/ρn and vA=B/√4πρ. l TheinductionEq.(3)canbeexplicitlywritteninterms 3 0 0.5 1 1.5 2.0 2.5 of fluid and field velocities as (Wardle & Salmeron 2011) Height [Mm] ∂B 4πη = × (v+v )×B O J . (9) ∂t ∇ B − c k Figure1.TheOhm(ηO),Hall(ηH)andambipolar(ηA)diffusion (cid:20) (cid:21) profiles in the photosphere-chromosphere are shown for (a) B= where thefield drift velocity vB is 1.2kGand(b)B=120G. ( ×B) ×b ( ×B) v =η ∇ ⊥ η ∇ ⊥ , (10) B P B − H B densities D, and, various collision frequencies are given in and Table 1. The collision cross-section is not known for the ηP =ηO+ηA, (11) metallic ions-neutral collision, so we have employed the Messy-Mohr analytic formula for the cross-section (see ap- is the Pedersen diffusivity. The parallel and perpendicular pendix).Theresultingplasma-neutralcollision frequencyin componentsofcurrentintheprecedingequationrefertothe the present work is an order of magnitude smaller than orientation with respect to the background magnetic field. previously used values by Pandey et al. (2008) where a In the fluid frame, the magnetic drift velocity v contains B largerplasma-neutralcollisioncross-sectionwasassumedfor effectoffielddiffusion.Asweshallsee,theinductionEq.(9) ion H2 collision. intermsofmagneticdriftvelocityisparticularlyconvenient − We shall utilise the above magnetic field profile, along for linearization which we will do shortly. with the collision frequencies given in Table 1 to compute WenotefromEq.(4)thatOhmdiffusionisindependent diffusivities which are given in Table 2. In Fig 1 diffusivi- of the magnetic field. Whereas Hall has linear dependence ties are plotted against height which shows that Hall diffu- onthefield,i.e.η B,thedependenceofambipolardiffu- sion on the magnHeti∝c field is quadratic, i.e. η B2. Thus, sion dominates Ohm and ambipolar diffusion in the photo- A ∝ sphere and lower chromosphere in the intense field regions as the magnetic field strength in a flux tubedecreases with [Fig 1(a)]. However in the weak field ( 100G at h = 0) increasing altitude, the drop in the Hall diffusivity will not ∼ regions Ohmand Hallare comparable below 0.3Mm before be as severe as in ambipolar diffusion. The altitude depen- Hall becomes dominant diffusion in theentire photosphere- dence of diffusivities can be easily quantified if variation of chromosphere [Fig 1(b)]. the magnetic field with altitude is known. Thus, in order It has been suggested in the past that Pedersen diffu- to compute diffusivities at various altitude, we shall choose sion (which is the sum of Ohmic and ambipolar diffusions) powerlawvariationofthemagneticfieldwithneutralnum- could play an important role in the chromospheric heating ber densityn n (Goodman 2004). We find that in the sunspots, active re- n 0.3 gions,poresandintergranularregionswherefieldisintense, B=B0 nn0 , (12) HallandambipolarwilldominateOhmdiffusion.Clearlyre- (cid:18) (cid:19) alisticmodellingofvariousenergeticprocessessuchaschro- where n0 is the number density of neutrals at the surface mospheric heating, CMEs etc. must include these diffusive (h=0) and B0 =1.2kG is the typical value of the field at processes. It would appear that since Hall diffusion is non– intergranularboundaries.Suchafieldprofilecapturesessen- dissipativeinnature,itmaynothaveanyroleintheenergy tial height dependenceof observed field in fluxtubes(Mar- extractionfromtheconvectivemotionofthelargelyneutral tinezetal.1997).Wenotethatabovescalingofthemagnetic medium. However, as we have shown recently (Pandey & field differssomewhat from themass density scaling (Leake Wardle 2012a), Hall diffusion in the presence of shear flow et al. 2005) although we have retained the same power law destabilises low frequency fluctuations which in turn may index 0.3. leadtotheturbulentcascadeofconvectiveenergytosmaller Thevalueofneutralnumberdensityandfractionalion- scaleswheredissipationcanconvertittoheat.Therefore,it isation (Model C, VAL81), the ratio of neutral and bulk isquiteplausiblethattheHalleffectmayplayanimportant role in heating thechromospheric plasma. 2 Toleadingordercollisionsbetweenlikeplasmaparticlesνee,νii How much diffusion is too much in the solar atmo- donotcausediffusion(Longmire&Rosenbluth1956). sphere? For example, if magnetic diffusion dominates fluid c 2012RAS,MNRAS000,1–?? (cid:13) Magnetic diffusion driven shear instability of solar flux tubes 5 Table 1. The neutral number density nn, the fractional ionisation Xe ne/nn [Model C, VAL81], the ≡ ratioofneutraltobulkmassdensitiesD=ρn/ρ,theion-neutral,electron-neutral,andelectron–ioncollision frequenciesaregiveninthetable.WehaveassumedB=B0(nn/n0)0.3 withB0=1.2kGandmi=30mp andmn=2.3mp wheremp=1.67 10−24 gistheprotonmass. × h(km) nn(cm−3) Xe D νin(Hz) νen(Hz) νei(Hz) 0 1.2 1017 5.5 10−3 1 108 7.4 109 1.3 109 250 2.3·1016 ·10−4 1 107 1.3·109 1.2·108 515 2.1·1015 1.2 10−4 1 106 ·109 1.3·107 1065 1.7·1013 ·10−2 0.91 104 106 3.1·106 1515 ·1012 6 10−2 0.53 8 102 6.6 104 2.0 ·106 2050 7.7 1010 5·10−1 0.12 · 60 5.3·103 9.6·105 2298 3.2·109 1· 100 0.07 6.3 ·103 ·105 2543 ·109 1.2·100 0.05 6.3 5.3 102 3.2 102 · · · Table 2. The values of Ohm, ηO, ambipolar, ηA and Hall, ηH diffusivities at different altitude are given in the table. The value of variouscollisionfrequencies havebeentakenfromTable1. h(km) ηO(cm2/s) ηA(cm2/s) ηH(cm2/s) h(km) ηO(cm2/s) ηA(cm2/s) ηH(cm2/s) 0 3.6 107 3.6 105 9.3 107 1180 2.1 106 1.2 1010 4.5 109 100 1.2·108 3.1·106 4.8·108 1380 5.5·105 1.1·1010 3.1·109 250 1.5·108 2.9·107 1.3·109 1605 2.1·105 1.1·1010 2.7·109 515 1.2·108 5.2·108 7.1·109 1925 7.3·104 7.2·109 2.6·109 555 1.2·108 9.4·108 9.2·109 2016 5.1· 104 4.4· 109 2.3·109 755 3.6·107 5.3·109 9.8·109 2104 3.5 ·104 2 1·09 2.1·109 855 1.5·107 4.6·109 6.3·109 2255 3.6·104 2.·7 109 3 1·09 980 5.9·106 7.6·109 4.8·109 2543 2.4·105 4 1·010 2.·1 1010 · · · · · · convectionintheinductionEq.(3),fieldwillbepoorlycou- than the Ohm diffusion (Sykora et al. 2012). Clearly, both pled to the plasma (Wardle 2007). Thus, we shall compare ambipolarand Halldiffusiontermsareimportant inthein- advectionterm ×(v×B) vB/Lwithvariousdiffusion duction equation. ∇ ∼ termintheinductionEq.(3)bydefiningmagneticReynolds numbersRm,Am and Hm as vL vL vL 2.1 Dispersion relation Rm= ,Am= &Hm= . (13) η η η O A H The magnetic element in the photosphere-chromosphere NotethatbothAmandHmdependsonhowwelltheplasma which can bemodelled as cylindrical tubesorplaner sheets is coupled to the magnetic field since both ambipolar and are highly dynamic accompanied by numerous flows with Hall diffusion depends on the magnetic field. Further ηA = differentspatial/ temporalscales. Recentnumericalsimula- βiηH and ηH =βeηO. Thus tions of the umbral magneto convection suggests that the dynamicalscaleoverwhichHalldiffusioncangeneratemag- Am=Rm/(β β ) , Hm=Rm/β . (14) i e e netic and velocity fields is much faster [ 10–20 km spatial ∼ Thedependenceof AmandHmon theplasmaHallparam- scaleand 300stemporalscale(Cheung&Cameron2012)] ∼ eter β is not surprising since the origin of both ambipolar thanthespatialandtemporalscaleofatypicalfluxtube(& j and Hall diffusion is inherently linked to the magnetisation few hundred km and few days). The simulation results ∼ ofthemedium.Thisalsoexplainsinherentlydifferentnature are easily scalable to 2km with temporal scale 2s. We ∼ oftheOhmandambipolardiffusion:whereasOhmdiffusion note that at present the best achievable resolution is 90km acts isotropically, ambipolar diffusion owing to its depen- (Bonet et al. 2008). dence on the magnetic field is anisotropic. As we shall see, Since the spatial scale over which the flow and field in the presence of shear flow, the anisotropic nature of am- generation occurs, is much smaller than the typical tube bipolar diffusion is at the centreof wavedestabilization. diameter, we shall approximate the cylindrical tube by a TheheightdependenceofRm,HmandAmforbothkG planer sheet and work in the Cartesian coordinate system and 0.1 kG fields were discussed in PW12a. It was shown where x,y,z represents radial, azimuthal and vertical di- that when v v (where v is the Alfv´enspeed), Rm 1 rections locally. We shall assume an initial homogeneous A A suggesting th∼at the Ohm diffusion is unimportant in c≫om- state with azimuthal shear flow v = v0′xy. The magnetic parison with theadvectionterm.However,HmandAmare fieldintheintergranular lanesatthenetworkboundariesis three to four orders of magnitude smaller than Rm. The clumped into elements or flux tubes that are generally ver- recent2Dnumericalsimulationofpartiallyionizedsolarat- tical (Martinezet al. 1997; Hasan 2009) buthighly inclined mosphere also suggests that in the weak field (. 100 G) fields have also been reported in the literature (Stenflo et regions in the chromosphere, the Hall diffusion is two or- al. 1987; Solanki at al. 1987). The internetwork magnetic ders of magnitude larger than the Ohm diffusion whereas elements have predominantly horizontal field (Hasan 2009; ambipolar diffusion is four to six order of magnitude larger Steiner&Rezaei2012).Therefore,weshallassumeuniform c 2012RAS,MNRAS000,1–?? (cid:13) 6 B.P.Pandey and Mark Wardle backgroundfieldthathavebothazimuthalaswellasvertical Wehaveusedfollowingnormalisationintheaboveequations component, i.e. B=(0,By,Bz). σ kv η v′ The focus of the present investigation is low frequency σ˜ = , k˜= A , η˜= | 0|. (25) v′ v′ v2 behaviour of the medium, and thus, we shall work in the | 0| | 0| A Boussinesq approximation limit which is valid if the mo- Eq. (22) reduces to known dispersion relation (Kunz 2008) tion in the medium is very slow (Spiegel & Veronis 1960). when D=1 and η =0. When D=0, ambipolar diffusion O Thus linearising Eqs. (1) and (2) and assuming an ax- expectedly,dropsout of thedispersion relation. isymmetricperturbationsoftheformexp(ik x+σt),with · k=(k ,0,k ), in Boussinesq approximation, we get x z 3 NON-IDEAL MHD INSTABILITIES k δv=0. (15) · A general stability criterion in the magnetic diffusion dom- δp i inated plasma can be derived from the dispersion relation, σδv = ik + [(k B) δB (B δB) k ] , x − x ρ 4πρ · x− · x Eq.(22) byrecasting it in the following simple form σδvy+v0′ δvx = 4πiρ(k·B) δBy, ak˜4+bk˜2+c=0, (26) δp i where thecoefficients a,b and c are σδv = ik + [(k B) δB (B δB)k ] . (16) z − z ρ 4πρ · z− · z a=β γµ2, b=δ γσ˜2, c=σ˜4, (27) − − Eliminating the pressure perturbation in favour of ve- with locity and making use of Eq. (15), from preceding equation we get for the(x,y) components β= µ2η˜H2 +η˜Pη˜T σ˜2+(η˜P+η˜T) µ2σ˜+µ4, δvˆ= ikσv2Aµ (cid:18) −σv0′ σ0 (cid:19) δBˆ . (17) (cid:0) δ(cid:1)=(cid:2)(η˜Pγ+=η˜Tα)(σ˜g+η˜A2+µ2s(cid:3)η˜σH˜2),. (28) Here δvˆ = δv/v and δBˆ = δB/B, v′ = dv(x)/dx, µ = The dispersion relation, Eq. (26) can also berecast as A 0 k˜·b ≡kˆzbz, k˜=k/k and kˆz =kz/k. Since βk˜4+δk˜2+σ˜4=γk˜2 σ˜2+µ2k˜2 . (29) (cid:16) (cid:17) δvB =ik ηH µδBˆ ×b b δBˆ k˜ b Thusin theneighbourhohod of σ˜ ik˜ 1 where − · × ∼ ≪ h n+ηP µδB(cid:16) b δ(cid:17)Bˆ k˜ o, (18) β µ4, δ 2µ2σ˜2, (30) − · ∼ ∼ n (cid:16) (cid:17) oi Eq.(29) becomes the linearised induction equation (x,y components) can be written as µ2k˜2+σ˜2 =γ2σ˜2, (31) σ 0 k2v2 µ2 σ 0 (cid:20)(cid:18) −v0′ σ (cid:19)+ σA2 (cid:18) −v0′ σ (cid:19) , which can bσ˜e w2ritten as +k2 ηxx ηxy δBˆ =0, (19) γ−µ2 = k˜ . (32) (cid:18) ηyx ηyy (cid:19)(cid:21) (cid:18) (cid:19) From Eq. (32) we see that for positive σ right hand side is where η is thediffusivity tensor with following components positive. This implies that γ µ2 >0. Therefore, we arrive − η =η +b2η , η =sη +gη , at thegeneral stability criterion which states that if xx O z A xy H A ηyx =(gηA−sηH)/kˆz2, ηyy =ηO+ 1−kˆx2b2z ηA, (20) α (gη˜A+sη˜H)>µ2, (33) (cid:16) (cid:17) the waves are unstable in the medium. The Ohm diffusion Here do not appear in the above expression which is not sur- g= kˆxkˆzbybz, (21) prising considering that the above criterion pertains to the − longwavelengthfluctuations.Fordefiniteness,insubsequent andhelicity s=µkˆ b kˆ2.Wenotethatforpurelyverti- z ≡ z z analysis, we shall assume α=1. calfieldthesignofhelicitysisdeterminedbytheprojection The diffusion-shear instability is caused by a competi- of the vertical magnetic field on thevorticity,∇×v0. tionbetweenthefluidadvectionandfielddriftintheplasma. Following dispersion relation can be derived from (19) Thiscan beseen from they and xcomponentsof Eqs.(17) σ˜4+k˜2 (η˜P+η˜T) σ˜3+C2σ˜2+C1σ˜+C0=0, (22) and (18) respectively, which suggest that when σ = 0, the advection of fluidin thex direction, δvˆ isequal andoppo- x where sitetothemagneticfielddrift velocityδvˆ .Bycombining Bx C2 =k˜4 η˜Pη˜T+µ2η˜H2 +k˜2 2µ2 α (gη˜A+sη˜H) , y component of Eqs. (17) and x component of Eq. (18)] we − get (cid:0) (cid:1) (cid:2) C1 =k˜4µ2 (η˜P+η˜T(cid:3)) , C0 =k˜4µ2 µ2−α (gη˜A+sη˜H) , (23) δvˆx+δvˆBx =ik˜µ −γα+µ2µ2 δBˆy+ kˆ12 η˜AδBˆx . (34) α= v′/ v′ 1 and (cid:2) (cid:3) (cid:20) z (cid:21) − 0 | 0|≡± HerewehaveneglectedOhmdiffusion.Bysetting γ+µ2 = ηT =ηO+µ2ηA. (24) 0 near marginal state, in the vicinity of η˜A −1, we get ≪ c 2012RAS,MNRAS000,1–?? (cid:13) Magnetic diffusion driven shear instability of solar flux tubes 7 δvˆ +δvˆ 0. This provides a simple physical explana- medium is threaded only by the vertical field and wave is x Bx ≈ tion on how the magnetic diffusion helps the shear flow in propagatingalongthefield,i.e.k=(0,0,k ),j =(j ,j ,0), z x y destabilising the waves. The outward slippage of the field both Ohm and ambipolar diffusion cause damping of the in the x–direction due to magnetic diffusion weakens the waves since (j b)= 0 and E′ J =η J2. However, when P · · magnetictension force. Asaresult magnetic restoringforce µ=1,lastterminEq.(38)canhelpfluctuationsgrowsince 6 in the wave is dominated by the inertia force, resulting in theincreased inwarddriftofthefluidelements.Thisishow E′·J = 4c2π ηO+ 1−(j·b)2 ηA J2 magnetic diffusion assist the waves to grow in the presence of shear flow. +4c2π ηA(cid:8)(j·b)(cid:2)[J·(j×(J(cid:3) ×b(cid:9)))] . (39) Thestabilitycriterion,Eq.(33)inthedimensionalform Therefore, ambipolar diffusion plays dualrole in apartially becomes ionized medium: whereas, in one direction it can cause dis- −v0′ (gηA+sηH)>µ2v2A, (35) sipation like Ohm, in the other direction the dissipation is ′ considerably smaller. The directional dissipation is the hall whichsuggeststhatwhen v >0,thelinearcombinationof 0 − mark of ambipolar diffusion which causes it to assist the ambipolarandHalldiffusivitiesmultipliedbysuitabletopo- instability (Desch 2004). logical factors g and s must exceed square of the oblique From stability criterion Eq. (35) it is clear that the Alfv´enspeed. Above equation provides a simple stability Hall–Ohmstablemedium( v′ η 6b v2)mayormaynot criterion of diffusive medium. For example, low frequency − 0 H z A beambipolarunstablewhereasHall–Ohmunstablemedium fluctuationsin theHall–Ohm dominated photosphere-lower ( v′ η >b v2)canalwaysbecomeambipolarunstablefor chromosphere (ηA =0) are unstableif − 0 H z A non–vertical field and oblique wavevector since g > 0 can −v0′ηH >bzvA2 , (36) be easily satisfied. The ambipolar diffusion not only drives shear flow instability when g > 0, but also enlarges the or, in terms of Hall frequencies parameter space over which Hall can destabilise the waves. ′ v0 >bzωH ωHZ. (37) ThereforeHalldiffusioncandriveshearflowinstabilitywhen − ≡ HerewehaveusedηH =vA2/ωH (PW12a)andbzωH =ωHZ η > bzvA2 g η . (40) istheHallfrequencydefinedintermofverticalfield.Clearly H v′ − s A if the Hall frequency is less than the shear frequency v′, − 0 0 − Since b ,b [0,1] and maximum g =0.25, above inequal- Hall diffusion can driveshear flow instability. z y ∈ When v′ η /v2 = b , since η > 0, ambipolar dif- ity implies that for non-zero ηA, Hall diffusion drives shear 0 H A − z A instabilityatmuchlargernegativevaluethanwhenthefield fusion can drive shear flow instability provided g > 0. has only vertical component. For purely vertical field or, transverse fluctuations (vertical The stability criterion can also be recast as wavevector)wheng=0,ambipolardiffusioncanonlycause the damping of waves. Therefore, the question of ambipo- kˆ b 1 v2 1 llainrkdeidffutosiotnheasasmisbtiinengtthfieeldshgeaeormfloewtryinasntadbiwliatvyeisobinlihqeureennetlsys − kˆxz byz!> ηA (cid:18)−Av0′ − bz ηH(cid:19) . (41) which together is encapsulated in the topological factor g. Assuming positive left hand side in the preceding equation The important role of g in the ambipolar diffusion driven weseethatabovecriterioniseasilysatisfiedfornon-zeroηA shear flow instability was discovered by Desch (2004). when kˆz 0. Therefore, waves propagating almost along x → How does ambipolar diffusion help drive shear flow in- directioni.e.whenthefluctuationsarealmostmagnetosonic stability? In order to see this, we first note that when the are always unstable. wavevector is perfectly aligned to the ambient magnetic The maximum growth rate of the instability can be field,i.e.µ kˆ b =1,η η inEqs.(22)–(24),Ohmand foundbysetting thediscriminant b2 4ac=0 inEq.(26). z z T P ≡ ≡ − ambipolardiffusioncombinestogetherasPedersendiffusion This yield and their effect on the wave propagation is identical–they gη +sη both cause wave damping. Only when µ = 1, this diffusive σ˜0 = A H . (42) 6 (η +η )+2 µ2η2 +η η degeneracy is lifted and Ohm and ambipolar diffusion can P T H P T no longer be combined together as single diffusion. After Weseehthatthemaximpumgrowthrateidependsonbothgη A removal of degeneracy whereas Ohm diffusion still causes and sη and for comparable η and η [as is the case in H A H isotropic damping of the waves, damping by the ambipolar large part of the solar atmosphere; see Fig. (1)], the bigger diffusion becomes anisotropic. Therefore, the non–vertical contributiontothegrowthratecomesfromtheHalldiffusion magnetic field and oblique wavevector (which is crucial in since the maximum value of parameter g is 0.25 whereas removing this degeneracy) plays important role in the am- maximum value of thehelicity s is one. bipolar diffusion driven shear instability. This can also be In theabsence of Halland Ohm,themaximum growth seen from the genelarlised induction Eq. (3) if we rewrite rate, Eq. (42) becomes theelectric field as ′ g v E′ = 4c2π ηO+ 1−(j·b)2 ηA J+ 4c2πηHJ×b σ0 = (1+| µ0|)2 , (43) (cid:8) (cid:2) +4π η(cid:3) (j(cid:9)b) [j×(J×b)] , (38) which suggests that when the field drift is solely dueto the c2 A · ambipolar diffusion the maximum growth rate is indepen- where the electric field is written in the neutral frame, j = dent of the diffusivity as well as the strength of the back- J/J . It is clear from the preceding equation that when ground magnetic field. However, the signature of the mag- | | c 2012RAS,MNRAS000,1–?? (cid:13) 8 B.P.Pandey and Mark Wardle 0.6 (a) B = 1.2 kG 0.4 v’|0 σ / |00.2 00 0.5 1 1.5 2 2.5 Height [Mm] 0.5 (b) B = 120 G 0.4 v’|0 σ / |00.2 0 Figure 3.Abovecartoondepictsgrowthrateagainstwavenum- 0 0.5 1 1.5 2 2.5 ber. Height [Mm] kFbˆzzig==ur11e/(√b2o.2lTdahnliednmeb)za,x=kˆizm0=.u1mkˆ(xdgar=sohwbeztdh=lrian−teeb),yfEo=rq.(1a(/4)√2B)2i=(sdso1ht.o2tewkdnGlifnaonre)dkˆaz(nb=d) gη˜A+sη˜H >2+ µ2η˜ηP˜H2++η˜Tη˜Pη˜T , (46) B=120G. maximum growthprate occurs at finite k0 [see Fig. (3)]. In the opposite limit k0 becomes imaginary implying that maximum growth rate occurs at infinity. For parameters of network-internetwork magnetic elements, with shear gradi- neticfieldinEq.(43)appearsthroughthetopologicalfactors g and µ. ent v0′ ∼ 10−2s−1 and vA ∼ 105 kms−1, k0 is imaginary [corresponding totwo similar curvesin Fig. (3)]. When the field drift in the plasma is solely due to the Hall diffusion, the maximum growth rate Eq. (42) becomes ′ σ0 = |v20|kˆz, (44) 4 RLIEMLIATTINIOGNCASES OF THE DISPERSION whichimpliesthatthewaveswithkˆz =1aremostunstable. Themagneticfielddriftinthepartiallyionizedplasmaopens A comparison with Eq. (43) shows that in purely ambipo- upnewpathwaysthroughwhichthefreeshearenergyofthe lar or, purely Hall case, growth rate is independent of the fluid can betransferred to thewaves (PW12a). To see this, ambient diffusivity. we first note that since σ˜ 1, three terms in Eq. (19) are We see from Fig. 2(a) that for purely vertical field and 1,k˜2,k˜2η and thus, one∼of the following three scenarios vertical wavevector (kˆz = bz = 1, g = 0, bold lines) the in- m∼ay prevail in a diffusive medium. stability grows at maximum rate v0′ /2 in thephotosphere- A. Ideal MHD: In this limit k˜2 1 k˜2η˜, i.e. η˜ 1 lower chromosphere in the strong| fi|eld region. Recall that and last term in Eq. (19) can be n∼eglec≫ted. However≪, as inthisintervalHallis thedominantdiffusion mechanismin has been shown in PW12a, this limit is applicable to the the network and internetwork regions (Fig. 1) and dissipa- longwavelength fluctuations&103km andtherefore, isnot tionduetotheOhmandambipolardiffusion(whichcanbe relevant to thepresent analysis. combined together as Pedersen diffusion for this topology) B. Cyclotron limit: In this case k˜2η˜ k˜2 1. This is small. is the low frequency limit and first term∼in Eq≫. (19) can When the field is weak [Fig. 2(a)] the instability be neglected. The low frequency, short wavelength dressed grows at close to maximum rate in the entire photosphere- ion-cyclotronwavewithfrequencyω =µω isthenormal C H chromosphere except very close to the surface (. 0.2Mm). mode of thesystem (Pandey & Wardle 2006, 2008). FWigh.en2(akˆ)z–(=b)]kˆzthe=gbrozw=th−rabtye=is 1sm/√al2ler[dtohttaendtchuervperseviin- get η˜Assuηmvin′g/v|v20′| ∼100.−012s−110a−n4d vA1∼a5nd×t1h0u5skimt ws−as1 iwne- ous case. This is because in this case the instability is not ferred≡(PW|102|a)Ath∼at the c−yclotron≪limit is not valid in the onlyduetotheHalldiffusionbutisalsoassistedbytheam- photosphere–chromosphere. However, at an increased shear bipolar diffusion through directional dissipation of waves. frequency v′ . 1s−1 we get η˜ η v′ /v2 . 1 and this Therefore,thegrowthrateisalwayssmallerwhenbothHall limit becom|e0s|important.Sincere≡cent|n0u|meAricalsimulation and ambipolar diffusion are present in the medium. With with 10 20km resolution (Cheung & Cameron 2012) can the decreasing vertical field [bz = 0.1, dash-dot curves in beeasily−scaledto2kmwhichfortypicalv0 2kms−1gives Fig. 2(a)–(b)] the maximum growth rate decreases further v′ 1s−1,weconcludethattheproperan∼alysisofthecy- 0 owing to smaller g and s. |clo|tr∼on limit will provide important insight to the ongoing The most unstable wavenumberis numerical simulations of the photosphere–chromosphere. σ˜2 Taking k , and thus setting a(σ) = 0 in Eq. (26) k˜02 = 0 . (45) or, neglecting →firs∞t matrix in Eq. (19), we get following dis- µ2η˜H2 +η˜Pη˜T σ˜0−µ2 persion relation It is c(cid:16)lepar from precedin(cid:17)g equation that when µ2η˜H2 +η˜Pη˜T σ˜2+µ2 (η˜P+η˜T) σ˜ =µ2 γ µ2 . (47) − (cid:0) (cid:1) (cid:0) (cid:1) c 2012RAS,MNRAS000,1–?? (cid:13) Magnetic diffusion driven shear instability of solar flux tubes 9 Positiveσ requiresγ >µ2 sincecoefficientsonthelefthand When ambipolar diffusion is negligible, Hall diffusion side of preceding equation are positive. Thus the stability can as well driveshear instability provided criterion in the cyclotron limit is identical to general sta- v′ sη >k˜2 η˜2 +µ2η˜2 . (56) bility criterion, Eq. (33). This is not surprising since above − 0 H O H dispersion relation is short wavelength limit of the general Above criterion(cid:0)is similar t(cid:1)o Eq. (24) of PW12a. Note that case. in PW12a, s = 1 as both the field and the wavevector is The growth rate of theinstability becomes vertical whereas here we are dealing with the general field σ˜ = 2 (η˜Pη˜Tµ+2 µ2η˜H2) −(η˜P+η˜T)+√∆ , (48) ntoapteodTlohgreeygi(imnsse6=tawb1i)ill.litgyroiwn athtethHeamll-aaxmimbuipmolarartediffusion domi- h i where ∆=(η˜P−η˜T)2+4η˜H2 γ−µ2 +4 γη˜µP2η˜T . (49) σ˜0 =γ (η˜P+(η˜η˜PT−)−η˜T2)p2−η˜P4η˜µT2+η˜H2µ2η˜H2 , (57) (cid:18) (cid:19) (cid:0) (cid:1) which for purely Hall (ηA = ηO = 0) case reduces to Above equation acquire particularly simple form in purely Eq. (44). In the absence of Hall diffusion the maximum Hall or ambipolar limits. For example in the Hall diffusion growth rate becomes dominated regime, from Eq. (48) we get gη˜A σ= v′ s µ2 vA2 1/2 v , (50) σ˜0 = √η˜P+√η˜T 2 . (58) (cid:20)− 0 ηH − ηH2 (cid:21) A When(cid:0)η˜O =0 abov(cid:1)e equation reduces to Eq. (43). which in µ<1 limit becomes Most unstable wavelength in the highly diffusive limit σ v′ sω 1/2 . (51) becomes ≈ − 0 H 2σ˜2 k˜2 = 0 . (59) Thus(cid:2)thegrowt(cid:3)h rate of thedressed ion-cyclotron wave ap- 0 [α (gη˜A+sη˜H) (η˜P+η˜T) σ˜0] proximatelyequalsthegeometricmeanoftheshearandHall − For purely Hall case and (α=1) preceding acquires partic- frequenciesandattainsmaximumvalueonlyforpositivehe- ularly simple form licity s=1. In purely ambipolar diffusion dominated case setting 1 η =η =0, we get k˜0 = . (60) O H r2bzη˜H σ= vA2 1+µ2 + (1 µ2)2 4gv0′ ηA . (52) In purely ambipolar regime k˜0 becomes, 2ηA "− s − − vA2 # 1 g (cid:0) (cid:1) k˜0 = , (61) We see from Eq. (52) that when g =0, ambipolar diffusion (1+µ) µη˜A r causes onlydamping.Asnotedin theprevioussection, it is fromwhereitisclearthatthemostunstablewavenumberis only when topological factor g is non-zero that anisotropic nonzero only when the topological parameter g is nonzero. ambipolar diffusion can driveshear flow instability. This is not surprising given that very existence of the am- C. Highly diffusive limit : In this limit k˜2η˜ 1 bipolardiffusiondrivenshearinstabilitydependsonthefield k˜2, i.e. η˜ 1 and k˜2 1. Only the first a∼nd la≫st geometry and obliqueness of thewave vector. ≫ ≪ term in Eq. (19) are retained. For typical values of η [Ta- ble (2)], this limit gives λ . 6km which fits within a pressure scale height. Therefore, as noted in our previous 5 DISCUSSION work (PW12a), highly diffusive limit is applicable to the photosphere-chromosphere. Thesolar photosphereisthreadedbykGfieldconcentrated The dispersion relation in this limit becomes in the vertical flux tubes (radius 100 200km) at in- ∼ − tergranular boundaries (Hasan 2009). Although less fre- (η˜P+η˜T) σ˜k˜2+σ˜2 =k˜2 γ k˜2 η˜Pη˜T+µ2η˜H2 , (53) quently (. 40%), field of similar strength have also been − from wherewe see thatfohr positiv(cid:0)e σ left hand si(cid:1)dieis posi- observed in thequiet solar internetwork region (Almeidaet al. 2010). Outside kG patches, internetwork field is much tive.Thusright hand side mustbepositive. Thuswe arrive weaker ( 100G). The flux tube is often modelled as non- at following general stability criterion ∼ rotatingcylindricaltubeswiththeplasmaandthemagnetic v0′ (gηA+sηH)>k˜2 η˜Pη˜T+µ2η˜H2 . (54) pressurebalanceprovidingrequiredstability.Inthepresent − work we have approximated flux tubes by a planer sheet In theabsence of Hall d(cid:0)iffusion, above(cid:1)criterion becomes where x and y coordinate locally correspond to the radial v0′ gηA >k˜2η˜Pη˜T. (55) and azimuthal directions locally. − Although the chromosphere has limited extension As right hand side in the above equation is positive, the in comparison to the corona, its net radiative loss stability criterion implies that in thepresence of favourable 107ergcm−3s−1 is10timesmorethanthecorona. Furthe∼r, ′ shearflowgradient( v >0)ambipolardiffusionwill drive exceptfortheflares,most solar atmospheric heatingoccurs 0 − the instability only if g > 0. Positive topological factor g inthechromosphere(Aschwanden2001;Goodman2001).A guaranteesthattheambipolardiffusionwillassisttheshear strong correlation between core emission of calcium K and flow in destabilising thewaves. Hresonancelinesandquitesun magneticfield(Schrijveret c 2012RAS,MNRAS000,1–?? (cid:13) 10 B.P.Pandey and Mark Wardle al. 1989) suggest that the origin of chromospheric heating B = 1.2 kG dwc(∼hioffrrou1kms0ic7ooaensnprghdpserrcriovmivcei−ndh2eessaha−tei1vna)irgaibssinlimensctmaaegbencicrlehiuttaiyccniaipislnmrionnpgtaoortseueedrdxieep.nilnaTtsihntreheteqhmuepairegrexendsceeetntsiocst σ / |v’|000.02.55 (a) kkkzzz /// kkk === 100,.. 94B,, zBB /zz B// BB= ==1 00..99 excite this instability namely shear flow and magnetic field 0 is always present in the network–internetwork region. The 0 0.5 1 1.5 2 2.5 Height [Mm] attractivefeatureofthisfastgrowingdiffusiveshearinstabil- ityisthatallwavelengthsoffluctuationsarelikelytobeex- 1 (b) citedasthereisnocut-offwavelength.Thus,asweseefrom H Fbyigt.h(2is),innsettawboilriktyfi.eIlndtbheeloiwnt1erMnemtwisorlikkeellyemtoenbtesd,etshtiasbiinlisstead- λ / 00.5 bilitymayoperateintheentirephotosphere–chromosphere. 0 Only uncertainty involved is the lack of information 0 0.5 1 1.5 2.0 2.5 Height [Mm] about the scale of shear flow gradient which can not be re- solved by current observations. However, small whirlpools B = 120 G with size similar to terrestrial hurricanes [. 0.5Mm with 0.5 (a) t(sy2ca0pl0iec8a)vl]oslriuftegictgeiemssteatt∼hseu5ppmreerisnger.nacnoenuolaftrhsuejcushnocfllatoirwonssus.rLwfaoicntehgBltayospntiienctgaleltalirfagele-. σ / |v’|000.25 time 1 2h with enhanced CaII emission have also been observ∼ed−(Attie et al. 2009). The swirl motion in the chro- 00 0.5 1 1.5 2 2.5 Height [Mm] mosphere has been recently detected by Wedemeyer-B¨ohm &preVseonocretof(fl20ow09g).raCdlieeanrtlsyi,notbhseeprvhaottioosnpshseurigcg-ecshtroumboiqspuhiteoruics 0.02.52 (b) kkzz // kk == 10,. 9B, zB /z B/ B= =1 0.9 H kz / k = 0.4, Bz / B = 0.9 plasma. Past numerical simulations also indicated thepres- λ / 0 0.1 ence of vortex flows in intergranular lanes (Zirker 1993; Stein & Nordlund 1998). The typical vorticity of a vor- tex is 6 10−3 s−1 which corresponds to rotation period 00 0.5 1 1.5 2 2.5 Height [Mm] ∼ × 35 minutes (Bonet et al. 2010). Thus it would appear ∼ thatHallinstabilitydoesnothavetimetodevelopsincethe Figure 4. The growth rate and most unstable wavelength is growth rate(v0′ /2=3 10−3 s−1) isvery small. However, shownfor1.2kG[Fig.4(a)]and120G[Fig.4(b)]fields.Following | | × above vorticity value is limited by the upper limit in the parameters have been used in the above figure: kˆz = 1,bz = 1 vorticity resolution [ 4 10−2 s−1, Bonet et al. (2010)]. (bold line), kˆz = bz = 0.9 (dashed line) and kˆz = 0.4,bz = 0.9 ∼ × The numerical simulation gives much higher vorticity value (dotted line). ( 0.1 0.2 s−1) in the photosphere-lower chromosphere ∼ − (Fig.31,Stein&Nordlund (1998)).Thegrowthratecorre- sponding to v′ =0.2 s−1 is one minute. 0 For alm|ost|magnetosonic waves (kˆ 0), maximum mumratewhereaswithdecreasingkˆ or,b thegrowthrate z z z → growthrateoftheHalldiffusiondrivenshearinstabilitymay diminishes. becomesquitesmall. Themaximum growth ratein theam- In Fig. 4(b) we plot most unstable wavelength against bipolardiffusiondominatedmiddleandupperchromosphere heightforsameparametersasinFig.4(a).Thewavelengthis will be only one fifth of the Hall diffusion dominated case normalized against scale height calculated self–consistently for the maximum g = 0.25 and µ = 0.5. Therefore, vortex usingmodelC,VAL81.ForbothkGandweakerfieldλ0 fits motions with typical lifetime & 15min. will be susceptible well within a scale height and thustheinstability will grow to the ambipolar diffusion driven shear instability. Since, at maximum rate in the entire photosphere–chromosphere. vortex motions of various spatial and temporal scales are However, it is only in the photosphere and lower chromo- observed, it is likely that non–ideal MHD effects will play sphere (. 1Mm) where the instability will grow at maxi- important role in exciting low frequency turbulence in the mum rate for a kG field. In the middle and upper chromo- medium. sphere the growth rate tapers off and becomes one eighth The non-ideal MHD description of the photosphere- ofshearfrequency.Therefore,inthestrongfieldregion,dif- chromosphere provides several pathways through which fusive instability will be efficient in destabilizing magnetic shear energy can be channelled to the waves by magnetic element in the photosphereand lower chromosphere. When field.Forexampleinexcessivelydiffusivelimitwhenδv/v the field is weak ( 100G), the instability can operate in ≪ ∼ δB/B, diffusion in tandem with the shear flow can desta- theentire photosphere-chromosphereat maximum rate. bilisethenetwork-internetworkfield.Forpurelyverticalfield How does nonlinear saturated state of diffusive insta- and vertical wavevector this limit has been discussed in de- bility will look like? The answer to this can be given only tailinPW12a.InordertocomparewithPW12a, webriefly by numerical simulations. However, if the nonlinear results describe the effect of the field topology and wave orienta- oftheprotoplanetarydiscsandstarformingregionsareany tion in the highly diffusive limit. Comparing Figs. 2 and 4 guidethen this instability should be quiteefficient in excit- we conclude that the maximum growth rate is similar in ingthelowfrequencyturbulenceandheatingoftheplasma. both cases. When b = kˆ = 1, instability grows at maxi- In fact interplay between the vortex flow and magnetic dif- z z c 2012RAS,MNRAS000,1–?? (cid:13)