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Kelvin–Helmholtz instability in solar Hα surges I.Zhelyazkova ∗ aFacultyofPhysics,SofiaUniversity,1164Sofia,Bulgaria T.V.Zaqarashvilib,c,R.Chandrad,A.K.Srivastavae,T.Mishonova bSpaceResearchInstitute,AustrianAcademyofSciences,8042Graz,Austria cAbastumaniAstrophysicalObservatoryatIliaStateUniversity,0162Tbilisi,Georgia 5 dDepartmentofPhysics,DSBCampus,KumaunUniversity,Nainital263002,India 1 0 eDepartmentofPhysics,IndianInstituteofTechnology(BanarasHinduUniversity),Varanasi221005,India 2 n a J 5 Abstract ] R WestudytheevolutionaryconditionsforKelvin–Helmholtz(KH)instabilityinaHαsolarsurgeobservedinNOAA S AR8227on1998May30. Thejetwithspeedsintherangeof45–50kms 1, widthof7Mm, andelectronnumber − . densityof3.83 1010 cm 3 isassumedtobeconfinedinatwistedmagneticfluxtubeembeddedinamagneticfield h − × p of 7 G. The temperature of the plasma flow is of the order of 105 K while that of its environment is taken to be - 2 106 K. The electron number density of surrounding magnetized plasma has a typical for the TR/lower corona o × regionvalue of 2 109 cm 3. Under these conditions, the Alfveń speed inside the jet is equalto 78.3 kms 1. We r × − − t modelthesurgeasamovingmagneticfluxtubefortwomagneticfieldconfigurations: (i)atwistedtubesurrounded s a byplasmawithhomogeneousbackgroundmagneticfield, and(ii)a twistedtubewhichenvironmentisplasmawith [ alsotwistedmagneticfield. Themagneticfieldtwistingivenregionischaracterizedbytheratioofazimuthaltothe 1 axial magnetic field componentsevaluated at the flux tube radius. The numericalstudies of appropriatedispersion v relationsofMHDmodessupportedbytheplasmaflowinbothmagneticfieldconfigurationsshowthatunstableagainst 7 Kelvin–HelmholtzinstabilitycanonlybetheMHDwaveswithhighnegativemodenumbersandtheinstabilityoccurs 6 atsub-Alfveńiccriticalflowvelocitiesintherangeof25–50kms 1. 8 − 0 Keywords: Solarsurges,Magnetohydrodynamicwaves,Kelvin–Helmholtzinstability,Numericalmethods 0 . 1 0 1. Introduction emission in He II 304 Å solar images (Bohlinetal., 5 1975;Georgakilasetal.,1999),obtainedbytheslitless 1 Surgesarephenomenainwhichdarkdensemassare XUV spectrograph on a Skylab mission and with the : v ejectedinthesolaratmospherefromchromosphericinto Extreme Ultraviolet Imaging Telescope on board the i coronal heights. Usually, they appear as straight or X Solar and Heliospheric Observatory (SOHO), respec- slightly curved ejective structures, and they often re- tively. With developing observationalinstruments and ar cur (Roy, 1973a,b; Sˇvestka, 1976; Tandberg-Hanssen, spacecrafts, chromospheric ejections were detected by 1977;Foukal, 1990). Atfirst, theywerestudiedin Hα usingdatafromthe50cmSwedishVacuumSolarTele- byNewton(1942)andbyMcMathandMohler(1948). scopeandtheTransitionRegionandCoronalExplorer Theyhaveatypicalsizeof38000–220000km,atrans- (TRACE, Brooksetal., 2007), as well as from the Big versevelocityof 30–200kms 1, and a lifetime of 10– − Bear Solar Observatory (Chen,Jiang,andMa, 2008) 20min. Therotationalorhelicalmotionswere,onoc- and with the Solar Ultraviolet Measurements of Emit- casions,alsoobservedinsurgeactivity(Guetal.,1994; tedRadiation(SUMER) spectroscopicobservationson Canfieldetal., 1996). Surges were also observed as boardSOHO(Chen,Innes,andSolanki,2008). Based on observational studies, it is now accepted Correspondingauthor ∗ Emailaddress:[email protected](I.Zhelyazkova) that the driving mechanism of mass ejection in surges PreprintsubmittedtoAdvancesinSpaceResearch January12,2015 is magnetic reconnection at chromospheric heights. ingitsmaximumriseandfoundanaveragetemperature KurokawaandKawai (1993) found that Hα surgesoc- andanumberdensityof2 106Kand4.17 109cm 3, − × × curattheverybeginningphasesofmagnetic-fluxemer- respectively. Schmiederetal.(1996)studiedthecondi- gence,andsuggestedthatsurgesareproducedbymag- tions for flares and surges in AR 2744 on 1980 Octo- netic reconnection between the emerging flux and the ber 21 and 22 using observationsfromthe SolarMax- pre-existing magnetic flux. Canfieldetal. (1996) re- imum Mission satellite and coordinated ground-based portedthatcircumstancesfavorabletomagneticrecon- observations, which together covered a wide tempera- nection are produced by moving satellite spots in a turerangefrom<104Kto>107K.Inparticular,thede- surge-productiveregion. Uddinetal. (2012) presented tectedsurgeonOctober22hadatotalemissionmeasure amulti-wavelengthstudyofrecurrentsurgesoriginated of 4.9 1044 cm 3 and durationof about2000 s. The − × due to the photospheric reconnections. Very recently, rough estimations of temperature and electron number Chandraetal.(2015)reportedamulti-wavelengthstudy densityyieldedT 104Kandn 1012cm 3. Similar e e − ∼ ∼ ofsolarjetson2010December11usingtheSolarDy- valuesforthetemperatureandelectronnumberdensity namics Observatory (SDO, DeanPesnelletal., 2012) ofHαsurgeswereobtainedearlierbyJainandSorathia data.Theyfoundanincreaseintheamplitudeofoscilla- (1987)fromobservationsofasurgeprominenceinAR tionsclosetotheirfootpointsoftheobservedjets,which 17212 on 1980 October 30 made at an interval of 5 s provides the evidence for the wave-induced reconnec- and10sintheHαlinecenter,throughaHallefilterof tion as a mechanism for jets triggering. Canfieldetal. 0.7 Å passband in conjunction with a 15 cm aperture (1996) also showed that a high-temperature X-ray jet solarspartelescope. and a cooluntwisted surge can coexist located side by As seen, the electron number density and the tem- sideatthesite(seeFig.10cintheirpaper).Shibataetal. perature of solar surges can vary in rather wide lim- (1992)andYokoyamaandShibata(1995)succeededto its, from 1012 cm 3 and 104 K for cool Hα surgesto − ∼ reproduce surge mass ejections from chromospheric 109cm 3and106Kforhigh-temperatureEUVsurges. − ∼ heightsbymagneticreconnectionbetweenanemerging Since each surge is a jet in a well-defined magnetic flux and a pre-existingmagnetic field. This numerical flux tube, it is naturally to expect that the magnetohy- resultwas later confirmedbyNindosetal. (1998) who drodynamic waves propagating along the magnetized studiedtheradiopropertiesof18X-raycoronaljetsas plasma flow can become unstable against the Kelvin– observedbytheYohkohSXT(Tsunetaetal., 1991)us- Helmholtz (KH) instability. It is well-known that KH ingNobeyamaRadioheliograph17GHzdata.Fromthe instabilitiesoccurwhentwofluidsofdifferentdensities SXT images, Nindosetal. (1998) computed the coro- ordifferentspeedsflowbyeachother.Inthesolaratmo- nal plasma parameters at the location of the surge. At sphere,whichismadeofaveryhotandpracticallyfully the time of maximum surge activity, they found elec- ionized plasma, the two flows come from an expanse trontemperatureT = 2.8 106 K andemission mea- of plasma erupting off the Sun’s surface as it passes e × sure EM = 5.0 1045 cm 3, as well as derived con- by plasma that is not erupting. The difference in flow − × straints on the ejecta electron number density, notably speedsanddensitiesacrossthisboundarysparksthein- n <6.1 1010cm 3. HαsurgesandassociatedsoftX- stabilitythatbuildsintothewaves. Whentheinstability e − × rayloopswere also studiedby Schmiederetal. (1994) reachesitsnonlinearstage,vorticesmightform,recon- who performed simultaneous observations of NOAA nectionmightbe initiated and plasma structuresmight AR 6850 on 1991 October 7, made with the MSDP detach. Now, after the launch of SDO satellite, due to spectrograph operating on the solar tower in Meudon its high spatial and temporal resolution, the KH insta- and with the Yohkoh SXT. By measuring the volume bilityofthecoronalmassejectionreconnectionoutflow emission measures of the two flaring loops (northern layerinthelowercorona,occurredon2010November andsouthernones)andthesurgeregion(mid-partofthe 3, has been imaged by Foullonetal. (2013). Very re- surge),Schmiederetal.(1994)concludedthatat10:24– cently,thatKHinstabilityobservationwasmodeledby 10:30UTthetemperaturewas(3–4) 106Kandthevol- Zhelyazkovetal.(2014b). Aconcisebutverygoodex- × umeemissionmeasurewas1047cm 3.Assumingavol- plorationofKHinstabilitiesinthesolaratmospherein − umeof(3–10) 1027cm3,theyderivedanelectronnum- viewoftheirinterpretationfromobservationsthereader × berdensityn =(3–6) 109cm 3.Kayshapetal.(2013) can find in TaroyanandRuderman (2012). The aim e − × haveobservedasolarsurgeinNOAAAR11271using of this study is to see whether MHD waves traveling theSDOdataon2011August25,possiblytriggeredby along the surge jet can become unstable within its ve- chromosphericactivity.Theyalsomeasuredthetemper- locityrangeof20–200kms 1. InthefollowingSect.2, − atureanddensitydistributionoftheobservedsurgedur- wewillbuildupsimplifiedmodelsforthesurge. Next 2 Sect.3dealswiththederivationoftheMHDwavedis- persion relations, while in Sect. 4 we will numerically analyze the dependenceof the linear/thresholdKH instability on relevant physical parameters of the surge and its environment. The last Sect. 5 summarizes our results. 2. Surge models, basic parameters, and governing equations We explore one of the four Hα surges observed by Brooksetal. (2007) in the solar active region NOAA AR 8227 (N26 , E09 ) on 1998 May 30 from 7:50 to ◦ ◦ 16:50UT.Theelectronnumberdensityderivedfromthe emission measure analysis of the TRACE Fe IX 171 Å imagesis equalto n = 3.83 1010 cm 3 (the label‘i’ i × − Figure1:EquilibriummagneticfieldgeometriesofaHαsolarsurge. stands for interior), and the jet velocity is of the order of 45–50 kms 1. Estimated chromospheric mag- − netic field is B = 25 G and from the conserva- foot ertheless,inthefollowingweconsidertheincompress- tion of magnetic flux between the reconnection region ibleplasmainsideandoutsidethesurge. andthephotosphereonefindsthatthecoronalmagnetic fieldis7–10G.Brooksetal.(2007)claimthattheirob- We model the surge as a vertically moving with a servationsconfirm the emergingflux regionsmodel of velocity v cylindrical flux tube with radius a = ∆ℓ/2 0 KurokawaandKawai (1993) and Shibataetal. (1994); (see Fig. 1), where ∆ℓ = 7 Mm is the surge width. moreoverthe clearevidenceofspatiotemporalcorrela- Our frame of reference is attached to the TR/coronal tionsbetweenchromospheric(CaIIK,Hα)andcoronal plasma that implies that v0 is the relative jet veloc- brightenings (TRACE Fe IX 171 Å) indicates that the ity with respect to its environment. We must men- chromosphere is heated up to coronal temperatures at tion that because the density contrast, η, is relatively the surge footpoint by the energy release during mag- high, in such a case, like in spicules, the occurrence neticfieldreconnection,asshowninthenumericalsim- of a KH instability, for instance of kink (m = 1) ulationofYokoyamaandShibata(1996). waves,becomespossibleatgenerallyhighAlfveńMach It is clear that the main body of the aforementioned numbers (the Alfveń Mach number is defined as the surgeispositionedattheTR/lowercoronaregion.Thus, ratio of jet velocity to Alfveń speed inside the jet, we can take the electron number density of the surge MA = v0/vAi)andcorrespondinglyathighcriticalflow environment to be equal to n = 2 109 cm 3 (the velocities being far beyond the speeds accessible for e − × label ‘e’ stands for exterior), and its temperature rea- surges/spicules in the solar atmosphere (Zhelyazkov, sonably can be T = 2 106 K. Concerningthe surge 2012; ZhelyazkovandZaqarashvili, 2012). This cir- e × temperature, we suppose it to be equal to T = 105 K. cumstance implies that the only possible way for i Then with a backgroundmagnetic field B = 7 G and emerging a KH instability in surges is the excitation e a density contrast η = ρ /ρ = 0.052 (we assume of higher MHD harmonics that can become unsta- e i thatplasmadensitiesinbothmediaarehomogeneous), ble at sub-Alfveńic flow velocities in twisted tubes we havethe followingcharacteristicsoundandAlfveń (Zaqarashvilietal.,2010). Here,weconsidertwopos- speed inside the surge and in the surrounding magne- sible magnetic field geometries: (i) a moving twisted tizedplasma: c = 37kms 1,v (cid:27) 78.3kms 1 (more magnetic flux tube embedded in untwisted magnetic si − Ai − exactly78.314kms−1,whichvaluedeterminesthemag- field Be (the left tube in Fig. 1), and (ii) a moving neticfieldinsidethejettobeequaltoB =7.04G),and twisted magnetic flux tube surroundedby plasma with i c (cid:27) 196kms 1, v = 341kms 1. Thus, theplasma twisted magnetic field lines (the right tube in Fig. 1). se − Ae − betas of the two media are respectively β = 0.27 and In our cylindrical coordinate system (r,ϕ,z) the mag- i β = 0.28. Incompressible plasma is a good approxi- neticfieldhasthefollowingform: B= 0,B (r),B(r) , e ϕ z mationtostudytheKHinstability,thoughtheobserved andtheflowprofileinsidethetubeisv(cid:16) = (0,0,v ). I(cid:17)n 0 0 valuesarenotfavorablefortheincompressibility. Nev- general, v can be a functionof r, but we consider the 0 3 simplesthomogeneouscase. Theunperturbedmagnetic where A and B are constant. In the simpler case (left iz field B and the pressure p satisfy the pressure balance flux tube in Fig. 1) the magnetic field outside the tube equation is homogeneous, B = (0,0,B ), and the solution to e e Eq.(2) inside thetube boundedatthe tubeaxisis(see d p+ B2ϕ+B2z = B2ϕ, (1) Zaqarashvilietal.,2014) dr  2µ  −µr whereµisthemagneticpermeability. ptot(r6a)=αiIm(κir), (7) As the unperturbed parameters depend on the r co- whereI isthemodifiedBesselfunctionofordermand m ordinateonly,theperturbationscanbeFourieranalyzed α isaconstant.Transversedisplacementcanbewritten i withexp[ i(ωt mϕ kzz)].Theequationsgoverningthe usingEq.(5)as − − − incompressible plasma dynamics are (Goossensetal., 1992) d2ptot + C3 d rD dptot ξir = αri ρi (cid:16)ΩΩ22−−ωω2A2Aii(cid:17)2κ−ir4Im′A(2κωir2A)i/µ dr2 "rDdr C3!# dr  (cid:16) (cid:17) 2mAωAiIm(κir)/√µρi +"rCD3 ddr rCC31!+ D12 (cid:16)C2C3−C12(cid:17)#ptot =0. (2) −ρi Ω2−ω2Ai 2−4A2ω2Ai/µ, (8) where where the atten(cid:16)uation co(cid:17)efficient κ and Alfveń fre- i D=ρ Ω2 ω2 , C = 2mBϕ mB +k B , quencyωAi aregivenby (cid:16) − A(cid:17) 1 − µr2 (cid:18)r ϕ z z(cid:19) κ =k 1 4A2ω2 /µρ Ω2 ω2 2 1/2, (9) C = m2 +k2 , i z(cid:20) − Ai i(cid:16) − Ai(cid:17) (cid:21) 2 − r2 z! mA+k B ω = z iz, (10) C =D2+D2Bϕ d Bϕ 4B2ϕρω2, Ai √µρi 3 µ dr r !− µr2 A and prime sign means a differentiation by the Bessel k B 1 m functionargument. ωA = √·µρ = √µρ(cid:18)r Bϕ+kzBz(cid:19) (3) ThesolutiontoEq.(2)outsidethefluxtubebounded atinfinityis isthelocalAlfveńfrequency, p (r>a)=α K (k r), (11) Ω=ω k v (4) tot e m z 0 − · whereK isthemodifiedBesselfunctionofordermand istheDoppler-shiftedfrequency,and p istotal(ther- m tot α isaconstant.Transversedisplacementcanbewritten mal+magnetic)pressureperturbation.Radialdisplace- e as mentξ isexpressedthroughthetotalpressureperturba- r α k rK (k r) tionas ξer = e z m′ z , (12) D dp C r ρ ω2 ω2 ξ = tot + 1p . (5) e − Ae r C dr C tot (cid:16) (cid:17) 3 3 where,asbefore,theprimesignmeansadifferentiation The solution to this equation dependson the magnetic by the Bessel function argument, and the local Alfveń fieldanddensityprofile. Toobtainthe dispersionrela- frequencyis tion of MHD modes, we find the solutions to Eqs. (2) and(5)insideandoutsidethetubeandmergethesolu- ω = kzBe =k v . (13) tionsthroughboundaryconditions. Ae √µρe z Ae Here, vAe = Be/√µρe is the Alfveń speed in the tube 3. Wavedispersionrelations environment. The boundaryconditionswhich merge the solutions Beforeobtainingthewavedispersionrelationsforthe insideandoutsidethetwistedmagneticfluxtubearethe two magnetic configurations, we first specify the twist continuity of the radial component of the Lagrangian ofthemagneticfluxtubetobeauniformone,thatis, displacement B =(0,Ar,B ), (6) ξ =ξ (14) i iz irr=a err=a | | 4 andthetotalpressureperturbation(Bennettetal.,1999) and 4B2 ω2 ptoti− Bµ2iaϕξir(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r=a = ptote|r=a, (15) Asolutionκtoe2 =Eqk.z2(119)−bµoρuend(cid:16)ωed2a−etϕωin2Afien(cid:17)2itay2is. (21) where total pressure pertu(cid:12)rbations p and p are toti tote given by Eqs. (7) and (11). Applying these boundary a2 condition,aftersomealgebrawe finallyderivethedis- ptot(r>a)=αer2Kν(κer), (22) persionrelationofthenormalMHDmodespropagating alongatwistedmagneticfluxtubewithaxialmassflow whereν= √4+n2andα isaconstant. e v0 surroundedbyplasma embeddedin a homogeneous Transversaldisplacementcanbewrittenas magneticfield r ω2 ω2 κ rK (κ r) Ω2−ω2Ai Fm(κia)−2mAωAi/√µρi ξer =αe (cid:16) − Ae(cid:17)2e ν′ e (cid:16) Ω2 (cid:17)ω2 2 4A2ω2 /µρ a2ρe ω2−ω2Ae −4B2eϕω2/µ − Ai − Ai i (cid:16) (cid:17) = ρρ(cid:16)ei (cid:16)ω2−ω(cid:17)2APe(cid:17)m+(kzAa2)Pm(kza)/µρi, (16) −αear a2ρe(cid:16)ω22−a(cid:16)ωω2A2e−(cid:17)2ω−2A4e(cid:17)B2eϕω2/µ wwahveeref,rreeqmueenmcbyeirn,Ωthe=mωo−vinkg·flv0uxistuthbee,Daonpdpler-shifted + a2ρe ω2m2−Beωϕω2AeAe2/−√4µBρe2eϕω2/µKν(κer). (23) F (κa)= κiaIm′(κia) and P (k a)= kzaKm′(kza). By applyi(cid:16)ng bound(cid:17)ary conditions(14) and (15), m i I (κa) m z K (k a) m i m z where the transversal displacements are given by A derivation of Eq. (16) starting from the basic equa- Eqs.(12)and(23),andtotalpressureperturbationsptoti tionsofidealmagnetohydrodynamicsthereadercansee and ptote, accordingly, by Eqs. (7) and (22), we ob- inZhelyazkovandZaqarashvili,2012. tain the dispersionrelation of the normalMHD modes In the case when the outside magnetic field is also propagatingalongatwistedmagneticfluxtubewithax- twisted (the rightflux tubein Fig. 1), we considerthat ial mass flow v0 surrounded by plasma embedded in magnetic field, B , has the form (Zaqarashvilietal., twistedmagneticfield e 2014) B = 0,B a,B a 2 , (17) Ω2−ω2Ai Fm(κia)−2mAωAi/√µρi e eϕr ez(cid:18)r(cid:19) ! (cid:16) Ω2 (cid:17)ω2 2 4A2ω2 /µρ − Ai − Ai i andthedensityispresentedasρ = ρ (a/r)4,sothatthe (cid:16) (cid:17) e Alfveńfrequency = a2(cid:16)ω2−ω2Ae(cid:17)Qν(κer)−G , (24) L H a2 ω2 ω2 Q (κ r) G mBeϕ+kzaBez − − Ae ν e − ω = (18) h (cid:16) (cid:17) i Ae √µρea where itsiocnotnosttahnet,gowvheircnhinagllEowq.s(u2s).tTohfientdotaanlparneaslsyutirceaplesrotulur-- Qν(κer)= κeKaK(ν′κ(κae)a), L=a2ρe ω2−ω2Ae 2−4B2eµϕω2, bation outside the tube is governedby the Bessel-type ν e (cid:16) (cid:17) equation d2ptot + 5dptot n2 +κ2 p =0, (19) H = B2eϕ A2, G =2a2 ω2 ω2 + 2maBeϕωAe. dr2 r dr − r2 e! tot µa2 − µ (cid:16) − Ae(cid:17) √µρe where Note thatthe left-handsidesof dispersionequations (16) and (24) are identical (this is not surprising), but 4m2B2 8mB ω n2 =m2 eϕ + eϕ Ae , (20) theright-handsidesarecompletelydifferentduetothe −µρea2 ω2−ω2Ae √µρea ω2−ω2Ae verydifferentmagneticfieldenvironments. (cid:16) (cid:17) (cid:16) (cid:17) 5 4. Numericalcalculationsandresults 0.3 ε = 0.1 m = –3 MA = 0.615 The main goal of our study is to determine under 0.25 ε = 0.2 which conditions the MHD waves propagating along MA = 0.59 0.2 thejetcanbecomeunstable. Toconductthisinvestiga- )Ai v tion, it is necessaryto assume thatthe wave frequency /ph 0.15 ω is a complex quantity, that is, ω → ω + iγ, where m(v γ is the instability growth rate, while the longitudinal I 0.1 wavenumberk isarealvariableinthewavedispersion z ε = 0.4 relation. Since the occurrence of the expected KH in- 0.05 MA = 0.575 stability is determinedprimarilyby the jetvelocity, by 0 searchingforacriticalorthresholdvalueofit, wewill 0.5 1 1.5 2 k a graduallychangeits magnitudefromzero to that criti- z cal value and beyond. Thus, we have to solve the dis- 0.65 persionrelationsincomplexvariablesobtainingthereal 0.64 ε = 0.025 m = –3 MA = 0.64 andimaginarypartsofthewavefrequency,orasiscom- 0.63 monlyaccepted,ofthewavephasevelocityv =ω/k , absetfwuneecntiothnessoufrkgzeaatnvdariitosuesnvviarlounemsoefntt,hvev.elopchitysheazr /v)hAi 00..6612 εM =A 0=. 10.615 0 p Before starting the numerical job, we have to nor- e(v 0.6 R malize all variables and to specified the input parame- 0.59 ε = 0.2 ters. Thewavephasevelocity,v ,andtheotherspeeds 0.58 MA = 0.59 ph are normalized to the Alfveń speed inside the jet, vAi, 0.57 ε = 0.4 MA = 0.575 which is calculated on using the axial magnetic fields 0.56 B . The wavelength, λ = 2π/k , is normalized to the 0.5 1 1.5 2 iz z k a z tuberadius,a,thatisequivalenttointroducingadimen- sionless wavenumberkza. For normalizingthe Alfveń Figure 2: (Toppanel) Growth rates ofthe unstable m = 3MHD − frequency in the environment, ω , except the density modepropagatingonincompressiblejetsinfourdifferenttwistedin- Ae ternal magnetic fields (with ε = 0.052, 0.1, 0.2, and 0.4) at η = contrastηandthetuberadiusa,wehavetoadditionally 0.052, b = 1, and corresponding critical Alfveń Mach numbers. specifytheratiooftheaxialmagneticfieldcomponents Forkza = 1.8thewavelength oftheunstablem = 3harmonic is inbothmedia,b = Bez/Biz. Foroursurgeanditsenvi- λKH=12.2Mm,andthewavegrowthrateisγKH=0−.008s−1.(Bot- ronmentthatratioisequalto0.994373andbecausewe tompanel)Marginaldispersioncurvesoftheunstablem= 3MHD − modeforthecriticalAlfveńMachnumbersasfunctionsofthemag- considerthe twoplasmasasincompressiblemedia, we neticfieldtwistparameterε. Atkza =1.8thecriticaljetvelocityis shalltakeb = 1. Inthedimensionlessanalysistheflow vcr=45kms 1. 0 − speed,v ,willbepresentedbytheAlfveńMachnumber 0 M =v /v . A 0 Ai We first begin with the numerical solving Eq. (16). uponthe magneticfield twist parameterε (see Fig. 2). Aswealreadysaid,thekink(m = 1)modemightbein ThecriticaljetspeedsforemergenceofaKHinstability principal exited, but owing to the relatively high den- accordinglyare 50.1kms 1, 48.2kms 1, 46.2kms 1, − − − sity contrast, η = 0.052, calculations show that this and 45 kms 1. The two narrow windows correspond- − mode can become unstable at Alfveń Mach numbers ingtoε = 0.025andε = 0.1arepracticallyinapplica- largerthan 6.36, which meansa critical jet velocity of ble to our surge–environmentconfiguration: the wave- 498kms 1,whichisinaccessibleforsolarsurges. The length, λ = π∆ℓ/k a, of unstable m = 3 harmonics − z − excitationsofunstableMHDwaveswithhigherpositive becomescomparabletothesurge’sheight;forinstance, modenumbersm = 2orm = 3alsorequiresveryhigh fork a = 0.4(themiddleofthesecondinstabilitywin- z critical plasma flow speeds beyond the upper limit of dow)thewavelengthoftheunstablem = 3harmonic − 200kms 1 forsurges. Adistinctivedecreaseofthein- atε = 0.1isλ = 55Mm. Actuallyonlyintheforth − KH stabilitycriticalAlfveńMachnumber/jetspeedonecan instability window (at ε = 0.4) one can have real un- achieveatthepropagationofMHDwaveswithnegative stable m = 3 MHD mode: for instance, at k a = 1.8 z − modenumbers. NumericalcalculationsofEq.(16)for thewavelengthisλ =12.2Mm,andthecorrespond- KH the m = 3 MHD mode give in four instability win- ing dimensionless wave phase velocity growth rate is − dowsonthek a-axiswhosepositionandwidthdepends equal to 0.19245 which implies a wave growth rate z 6 0.35 0.06 0.3 m = –2 εεei == 00..301 0.05 m = –2 εεei == 00..301 0.25 m = –3 v)Ai 0.2 m = –3 v)Ai 0.04 m = –4 /ph m = –4 /ph 0.03 m(v 0.15 m(v I I 0.02 0.1 0.05 0.01 0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 1.2 1.4 1.6 1.8 2 k a k a z z 0.65 0.6 0.6 εεei == 00..301 0.5 m = –2 εεei == 00..031 m = –4 m = –2 MA = 0.587 MA = 0.587 MA = 0.472 )Ai 0.55 )Ai 0.4 v v /h /h p p e(v 0.5 m = –3 MA = 0.524 e(v 0.3 R R m = –3 0.45 m = –4 MA = 0.472 0.2 MA = 0.524 0.4 0.1 0.6 0.8 1 1.2 1.4 1.6 1.8 1.2 1.4 1.6 1.8 2 k a k a z z Figure3:(Toppanel)Growthratesoftheunstablem= 2,m= 3, Figure4:(Toppanel)Growthratesoftheunstablem= 2,m= 3, − − − − andm= 4MHDmodespropagatingonincompressibletwistedjets andm= 4MHDmodespropagatingonincompressibletwistedjets − − with εi = 0.3in a twisted external magnetic field with εe = 0.01 with εi = 0.3in atwisted external magnetic field with εe = 0.01 at η = 0.052, b = 1, and critical Alfveń Mach numbers equal to atη = 0.052,b = 1,andcorrespondingcriticalAlfveńMachnum- 0.587,0524,and0.472. (Bottompanel)Marginaldispersioncurves bers.Forkza=2thewavelengthoftheunstablem= 4harmonicis oftheunstablem = −2,m = −3,andm = −4MHDmodesforthe λKH=11Mm,andthewavegrowthrateisγKH=0.00−1s−1.(Bottom criticalAlfveńMachnumbersatthemagneticfieldstwistparameters panel)Marginaldispersioncurvesoftheunstablem = 2,m = 3, − − εarie=co0r.r3esapnodndεieng=ly0.e0q1u.alTthoe4c6riktimcasl−s1u,r4g1ekvmelosc−i1t,ieasndof3t7heksmesm−o1.des athnedmmag=ne−ti4cMfieHldDs tmwoisdtepsafroarmtehteercsriεtiica=l A0.l3fvaeńndMεaech=n0u.m01b.erTshaet critical surgevelocity ofthem = 4harmonicatkza = 2isequal to25.2kms 1. Bycontrasttothei−nstabilitywindowsplottedinthe − γ = 0.008 s 1. If we shift to the left at k a = 1.5 bottompanelofFig.3,inthisfamilyofinstabilitywindowsthenor- KH − z malizedphasevelocitiesarenotconstant. (withmaximalnormalizedwavephasevelocitygrowth rate),thewavelengthisλ = 14.7Mm,andthecorre- KH spondingwavegrowthrateisalittlebithigher,namely γ = 0.0085s 1. Thenumericalstudyof thefluting- dealswithunstablevortices. KH − like harmonic m = 2 yields similar results; the only A much more interesting picture one obtains when − differenceisthatthefourthinstabilitywindowlocksat studythemorecomplicatedcaseofatwistedmagnetic k a = 1.6. AcceptableKH instabilitywavelengthsand fluxtubesurroundedbyplasmaembeddedinatwisted z growth rates one can get again for ε = 0.4. The criti- background magnetic field. Our choice for the twist caljet velocitiesforthe m = 2 mode, however,are a characteristics of the two magnetic fields (internaland − littlebithigher—theyareintherangeof55–60kms 1. externalones)areε = 0.3andε = 0.01,correspond- − i e Notethatnormalizedwavephasevelocityongivendis- ingly. Numerical solving Eq. (24) for the three mode persion curve in the bottom panels of Figs. 2 and 3 is numbersm = 2, m = 3,andm = 4givesforeach − − − equaltoitslabelM . Therefore,theunstableperturba- mode number two instability windows: one of them A tions are frozen in the flow and consequently they are with relatively high maximal growth rate, and a sec- vorticesratherthan waves. This observationis consis- ond window, next to the former, with one order lower tent with the KH instability in the hydrodynamicsthat maximalgrowthrate. Thesetwofamiliesofinstability 7 windows, for clarity, are presented in separate figures, velocityof45kms 1 andwavelengthλ = 12.2Mm − KH Figs.3and4. AsseenfromFig.3, arealKH instabil- with a linear growthrate γ = 0.008s 1 as the mag- KH − ityonecanobservemostlyforthem = 4MHDmode netic field twist parameter ε = 0.4. We note, that ex- − andpartlyforthem = 3harmonic. Thewavegrowth ploringtheKHinstability(Zhelyazkovetal.,2014a)in − ratesofunstablemodesareofthesameorderlikethose a high-temperature solar surge, like that observed by illustratedinFig.2,i.e.,fewinversemilliseconds. The Kayshapetal. (2013), for the same mode at the same critical flow velocities for the m = 2, m = 3, and position on the k a-axis and the same value of ε, one z − − m = 4modesarecorrespondinglyequalto46kms 1, obtains a wave growth rate of 0.033 s 1 being exactly − − − 41kms 1, and37 kms 1. Thesecondfamilyofinsta- equal to the growth rate of the imaged KH instabil- − − bility windows, shown in Fig. 4, has two distinct pe- ity in a coronal mass ejecta in the lower corona by culiarities: first, the instability windows are shifted to Foullonetal.(2013). Ontheotherhand,theaforemen- theright-handsideofthek a-axis,i.e.,thepropagation tioned γ = 0.008 s 1 is of the same order as the z KH − rangeofunstableMHDmodesisextended,andsecond, growthrateof0.003s 1ofvortex-shapedfeaturesalong − the marginal dispersion curves are not constant—the theinterfacebetweenanerupting(dimming)regionand normalized wave phase velocities gradually decrease thesurroundingcoronaimagedbytheSDO/AIAasre- with increasing the dimensionless wavenumber. If we portedby Ofman&Thompson(2011). Allthese com- fixthenormalizedwavenumbertobek a=2,thewave- parisons allows us to believe that the KH instability z lengthoftheunstablem = 4modeisλ = 11Mm, might be imaged in surges, too. The second magnetic KH − and at Im(v /v ) = 0.02456 the wave growth rate field configuration (the right flux tube in Fig. 1), re- ph Ai is γ = 0.001 s 1, much lower than the observable veals some new aspects of the KH instability, notably KH − growth rates in the first family of instability windows. for a fixed pair of magnetic fields twist parameters, in Note also, that the critical jet velocity for emerging a our case equal to ε = 0.3 and ε = 0.01, for given i e KH instability now is remarkably lower—its normal- high-harmonic mode one appears two instability win- izedvalueis 0.322thatimpliesvcr = 25.2kms 1, i.e., dowsonthek a-axis,nexttoeachother.Thus,therange 0 − z the half of the surge speed evaluated by Brooksetal. ofKHinstabilityisextended.True,inthesecondinsta- (2007). Thus, one can conclude that high-harmonic bilitywindowsthegrowthratesaremuchlowerthanin MHD modes can become unstable against the KH in- thestandardinstabilitywindows,butneverthelessthese stability for accessible sub-Alfveńicvelocities—thisis weak/slow instabilities can occur. Moreover, the criti- morepronouncedinthecasewhenbothmagneticfields calflow velocityforemergingKH instability mightbe aretwisted. relatively lower. For example, the m = 4 harmonic − with wavelengthof 11 Mm can becomeunstableat jet speed of only (cid:27)25 kms 1 and its growth rate is equal 5. Discussionandconclusion − to0.001s 1. SolarHαsurgesaregenerallysmall-scale − In this paper, we have studied the condition under eruptiveeventsand their contributionsto solar coronal which MHD modes traveling on a Hα solar surge can heatingduetotriggeredbytheKHinstabilitywavetur- become unstable against the Kelvin–Helmholtz insta- bulenceismodest.Therearesomecases,forinstanceas bility. Our model for the surge is a vertically mov- surgesaredetectedinUVandEUVspectrallines,when ingtwistedmagneticcylindricalfluxtubethatmightbe the developingKH instabilitycan bringfortha notice- surroundedbyplasmaembeddedinhomogeneousmag- ablecontributiontothesolarcoronaenergybudget. netic field or by magnetizedplasma with twisted magneticfield.Ineachcasethetwistofgivenmagneticfield Acknowledgments The works of I.Zh., R.C., (internalorexternalone)ischaracterizedbytheratioof A.K.S.,andT.M.weresupportedbytheBulgarianSci- azimuthalmagneticfieldcomponentattheinnersurface ence Fund and the Departmentof Science & Technol- of the tube to its longitudinalcomponent. Among the ogy, Government of India Fund under Indo-Bulgarian MHD modes which propagate along the moving tube bilateral project CSTC/INDIA 01/7, /Int/Bulgaria/P- only the high harmonics with negative mode numbers 2/12.TheworkofT.Z.wassupportedbyFP7-PEOPLE- can become unstable at accessible critical flow veloci- 2010-IRSES-269299 project- SOLSPANET, by Shota ties. One can observepractically a developingKH in- Rustaveli National Science Foundation grant DI/14/6- stabilityifonlytheparameterεcharacterizingthemag- 310/12 and by the Austrian Fonds zur Fo¨rderung der neticfieldtwistislargeenough.Forinstance,atthefirst wissenschaftlichen Forschung under project P26181- magneticfieldconfiguration(theleftfluxtubeinFig.1) N27. The authors are indebted to Dr. Snezhana Yor- them = 3harmonicbecomesunstableatcriticalflow danovafordrawingonefigure. − 8 References Roy,J.-R.,1973a.Themagneticpropertiesofsolarsurges.SolarPhys. 28(1),95–114. 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