Kelvin–Helmholtz instability in solar cool surges

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Advances in Space Research xxx (2015) xxx–xxx

Kelvin–Helmholtz instability in solar cool surges

I. Zhelyazkov a,⇑, T.V. Zaqarashvili b,c, R. Chandra d, A.K. Srivastava e, T. Mishonov a

a Faculty of Physics, Sofia University, 1164 Sofia, Bulgariab Space Research Institute, Austrian Academy of Sciences, 8042 Graz, Austria

c Abastumani Astrophysical Observatory at Ilia State University, 0162 Tbilisi, Georgiad Department of Physics, DSB Campus, Kumaun University, Nainital 263 002, India

e Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi 221 005, India

Received 22 December 2014; received in revised form 27 April 2015; accepted 1 May 2015

Abstract

We study the conditions for onset of Kelvin–Helmholtz (KH) instability in a cool solar surge observed in NOAA AR 8227 on 1998May 30. The jet with speeds in the range of 45–50 km s�1, width of 7 Mm, and electron number density of 3:83� 1010 cm�3 is assumed tobe confined in a twisted magnetic flux tube embedded in a magnetic field of 7 G. The temperature of the plasma flow is of the order of105 K while that of its environment is taken to be 2� 106 K. The electron number density of surrounding magnetized plasma has a typ-ical value for the TR/lower corona region of 2� 109 cm�3. Under these conditions, the Alfven speed inside the jet is equal to 78.3 km s�1.We model the surge as a moving magnetic flux tube for two magnetic field configurations: (i) a twisted tube surrounded by plasma withhomogeneous background magnetic field, and (ii) a twisted tube which environment is plasma with also twisted magnetic field. The mag-netic field twist in given region is characterized by the ratio of azimuthal to the axial magnetic field components evaluated at the flux tuberadius. The numerical studies of appropriate dispersion relations of MHD modes supported by the plasma flow in both magnetic fieldconfigurations show that the Kelvin–Helmholtz instability can only occur for MHD waves propagating in axial direction, but with highnegative azimuthal mode numbers, and the instability occurs at sub-Alfvenic critical flow velocities in the range of 24–60 km s�1.� 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Solar surges; Magnetohydrodynamic waves; Kelvin–Helmholtz instability; Numerical methods

1. Introduction

Surges are phenomena in which dark dense mass areejected in the solar atmosphere from chromospheric intocoronal heights. Usually, they appear as straight or slightlycurved ejective structures, and they often recur (Roy,1973a,b; Svestka, 1976; Tandberg-Hanssen, 1977; Foukal,1990). At first, they were studied in Ha by Newton (1942)and by McMath and Mohler (1948). They have a typicalsize of 38,000–220,000 km, a transverse velocity of30–200 km s�1, and a lifetime of 10–20 min. The rotational

http://dx.doi.org/10.1016/j.asr.2015.05.003

0273-1177/� 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (I. Zhelyazkov).

Please cite this article in press as: Zhelyazkov, I., et al. Kelvin–Helmhodx.doi.org/10.1016/j.asr.2015.05.003

or helical motions were, on occasions, also observed in surgeactivity (Gu et al., 1994; Canfield et al., 1996). Surges werealso observed as emission in He II 304 A solar images(Bohlin et al., 1975; Georgakilas et al., 1999), obtained bythe slitless XUV spectrograph on a Skylab mission and withthe Extreme Ultraviolet Imaging Telescope on board theSolar and Heliospheric Observatory (SOHO; Domingoet al. (1995)), respectively. With developing observationalinstruments and spacecrafts, chromospheric ejections weredetected by using data from the 50 cm Swedish VacuumSolar Telescope and the Transition Region and Coronal

Explorer (TRACE) (Brooks et al., 2007), as well as fromthe Big Bear Solar Observatory (Chen et al., 2008) and withthe Solar Ultraviolet Measurements of Emitted Radiation

ltz instability in solar cool surges. Adv. Space Res. (2015), http://


mailto:[email protected]




2 I. Zhelyazkov et al. / Advances in Space Research xxx (2015) xxx–xxx

(SUMER; Wilhelm et al. (1995)) spectroscopic observationson board the SOHO (Chen et al., 2008).

Based on observational studies, it is now accepted thatthe driving mechanism of mass ejection in surges is mag-netic reconnection at chromospheric heights. Kurokawaand Kawai (1993) found that cool surges occur at the verybeginning phases of magnetic-flux emergence, and sug-gested that surges are produced by magnetic reconnectionbetween the emerging flux and the pre-existing magneticflux. Canfield et al. (1996) reported that circumstancesfavorable to magnetic reconnection are produced by mov-ing satellite spots in a surge-productive region. Uddinet al. (2012) presented a multi-wavelength study of recurrentsurges originated due to the photospheric reconnections.Very recently, Chandra et al. (2015) reported amulti-wavelength study of solar jets on 2010 December 11using the Solar Dynamics Observatory (SDO; Pesnell et al.(2012)) data. They found an increase in the amplitude ofoscillations close to their footpoints of the observed jets,which provides the evidence for the wave-induced reconnec-tion as a mechanism for jets triggering. Canfield et al. (1996)also showed that a high-temperature X-ray jet and a cooluntwisted surge can coexist located side by side at the site(see Fig. 10c in their paper). Kamio et al. (2010) using theEUV Imaging Spectrometer (EIS) on Hinode (Kosugiet al., 2007) and SUMER on SOHO reported an X-ray jetassociated with a cool macrospicule/surge eruption in He

II 304 A above the limb. Shibata et al. (1992) andYokoyama and Shibata (1995) succeeded to reproducesurge mass ejections from chromospheric heights by mag-netic reconnection between an emerging flux and apre-existing magnetic field. This numerical result was laterconfirmed by Nindos et al. (1998) who studied the radioproperties of 18 X-ray coronal jets as observed by theYohkoh SXT (Tsuneta et al., 1991) using NobeyamaRadioheliograph 17 GHz data. From the SXT images,Nindos et al. (1998) computed the coronal plasma parame-ters at the location of the surge. At the time of maximumsurge activity, they found electron temperature

T e ¼ 2:8� 106 K and emission measure

EM ¼ 5:0� 1045 cm�3, as well as derived constraints onthe ejecta electron number density, notably

ne < 6:1� 1010 cm�3. A close correlation among coolsurges, magnetic flux cancelations, and UV brighteningsfound at the edge of an emerging flux region was reportedby Yoshimura et al. (2003). The authors showed that thesurge activities in Ha and the brightenings in TRACE

1600 A images correlate well in both time and space withthe cancellation of magnetic fluxes around an emerging fluxregion. These facts are consistent with the magnetic recon-nection model. The released energy through magnetic

reconnection, which is estimated to be 1028 erg, is suffi-ciently large to produce surge activities.

Cool solar surges and associated soft X-ray loops werealso studied by Schmieder et al. (1994) who performedsimultaneous observations of NOAA AR 6850 on 1991


October 7, made with the MSDP spectrograph operatingon the solar tower in Meudon and with the Yohkoh SXT.By measuring the volume emission measures of the twoflaring loops (northern and southern ones) and the surgeregion (mid-part of the surge), Schmieder et al. (1994) con-cluded that at 10:24–10:30 UT the temperature was

ð3–4Þ � 106 K and the volume emission measure was

1047 cm�3. Assuming a volume of ð3–10Þ � 1027 cm3, they

derived an electron number density ne ¼ ð3–6Þ � 109 cm�3.Kayshap et al. (2013) have observed a solar surge inNOAA AR 11271 using the SDO data on 2011 August25, possibly triggered by chromospheric activity. They alsomeasured the temperature and density distribution of theobserved surge during its maximum rise and found an aver-

age temperature and a number density of 2� 106 K and

4:17� 109 cm�3, respectively. Schmieder et al. (1996) stud-ied the conditions for flares and surges in AR 2744 on 1980October 21 and 22 using observations from the Solar

Maximum Mission satellite and coordinatedground-based observations, which together covered a wide

temperature range from <104 K to >107 K. In particular,the detected surge on October 22 had a total emission mea-

sure of 4:9� 1044 cm�3 and duration of about 2000 s. Therough estimations of temperature and electron number

density yielded T e � 104 K and ne � 1012 cm�3, respec-tively. Similar values for the temperature and electronnumber density of cool surges were obtained earlier byJain and Sorathia (1987) from observations of a surgeprominence in AR 17212 on 1980 October 30 made at aninterval of 5 s and 10 s in the Ha line center, through aHalle filter of 0.7 A passband in conjunction with a15 cm aperture solar spar telescope. Successive chromo-spheric surges ejected from a filament in active regionNOAA AR 10407 on 2003 July 14, as observed by theTHEMIS telescope in the center of the Ha line duringthe time interval 9:20–9:40 UT, were reported byZuccarello et al. (2011). An eruption of a miniature Ha fil-ament that took the form of a surge was also detected byYang et al. (2012) on 2001 March 27 in NOAA AR 9401by high-resolution Ha observations from the Big BearSolar Observatory. Observations with the 1.6 m apertureNew Solar Telescope (NST) of the Big Bear SolarObservatory made next steps in our understanding of thesolar surface structure. Recently, Vargas-Domınguezet al. (2014) presented a multi-wavelength analysis ofobservations, acquired with the NST, the SDO, theInterface Region Imaging Spectrograph (IRIS; DePontieu et al. (2014)) and the Hinode mission, of a transientemerging flux event of 2013 August 7 in the vicinity ofactive region NOAA AR 11810, resulting in generationof a surge/jet and compact heating at coronal-height loca-tions. The maximum thickness of the detected surge wasabout �2:2 Mm, and the longitudinal expansion velocitywas estimated to be of about 30 km s�1 for the first ejectionand �10 km s�1 for the following retraction-ejectionperiods.




I. Zhelyazkov et al. / Advances in Space Research xxx (2015) xxx–xxx 3

As seen, the electron number density and the tempera-ture of solar surges can vary in rather wide limits, from

�1012 cm�3 and 104 K for cool surges to �109 cm�3 and

106 K for high-temperature EUV surges. Since each surgeis a jet in a well-defined magnetic flux tube, it is naturallyto expect that the magnetohydrodynamic waves propagat-ing along the magnetized plasma flow can become unstableagainst the Kelvin–Helmholtz (KH) instability. It iswell-known that KH instabilities occur when two fluidsof different densities or different speeds flow by each other.In the solar atmosphere, which is made of a very hot andpractically fully ionized plasma, the two flows come froman expanse of plasma erupting off the Sun’s surface as itpasses by plasma that is not erupting. The difference in flowspeeds and densities across this boundary sparks the insta-bility that builds into the waves. When the instabilityreaches its nonlinear stage, vortices might form, reconnec-tion might be initiated and plasma structures might detach.Note, however, that except the existence of a longitudinalvelocity shear, leading to KH instability, we can also havetransverse velocity shear initiating this instability. Ofmanet al. (1994), in numerically studying the coronal heatingby the resonant absorption of Alfven waves in an inhomo-geneous slab, have shown that a shear Alfven wave can trig-ger a KH instability and its onset depends on the phase shiftbetween azimuthal velocity and magnetic field perturbations.Transverse oscillations in flowing prominence threads andfine strand-like structures in the solar corona can lead toKH instabilities that deform the cross-sectional area of loops(Terradas et al., 2008; Antolin et al., 2014).

Now, after the launch of SDO satellite, due to itshigh spatial and temporal resolution, the KH instabilityof the coronal mass ejection reconnection outflow layerin the lower corona was for the first time detected byOfman and Thompson (2011) and Foullon et al.(2011). The KH instability occurred in the CME on2010 November 3, has been imaged by Foullon et al.(2013). From observations, the authors obtained thatthe instability growth rate, cKH, was equal to 0.033 s�1

with a wavelength kKH ¼ 18:5 Mm. Very recently, thatKH instability was modeled by Zhelyazkov et al.(2015). It is worth pointing out that there exists somecorrelation between the CMEs and the oscillations andwaves in surges. SOHO/SUMER observations of promi-nence oscillation before eruption of CME as reported byChen et al. (2008), gave ground to consider thelong-time prominence oscillations as a precursor forthe CME. The observations indicated that a siphon flowwas moving from the proximity of the prominence to asite at a projected distance of 27000, which was followedby repetitive cool surges and continual prominence oscil-lations. The oscillation lasted 4 h before the prominenceerupted as a blob-like CME, subsequently subject to aKH instability. A concise but very good exploration ofKH instabilities in the solar atmosphere in view of theirinterpretation from observations the reader can find in


Taroyan and Ruderman (2011). The aim of this studyis to establish whether MHD waves traveling along thesurge jet can become unstable within its velocity rangeof 20–200 km s�1. In the following Section 2, we willbuild up simplified models for the surge. NextSection 3 deals with the derivation of the MHD wavedispersion relations, while in Section 4 we will numeri-cally analyze the dependence of the linear/thresholdKH instability on relevant physical parameters of thesurge and its environment. The final, Section 5, summa-rizes the results derived in this paper.

2. Surge models, basic parameters, and governing equations

We explore one of the four surges observed by Brookset al. (2007) in the solar active region NOAA AR 8227(N26�, E09�) on 1998 May 30 from 7:50 to 16:50 UT.The electron number density derived from the emissionmeasure analysis of the TRACE Fe IX 171 A images is

equal to ni ¼ 3:83� 1010 cm�3 (the label ‘i’ stands for inte-

rior), and the jet velocity is of the order of 45–50 km s�1.Estimated chromospheric magnetic field is Bfoot ¼ 25 Gand from the conservation of magnetic flux between thereconnection region and the photosphere one finds thatthe coronal magnetic field is 7–10 G. Brooks et al. (2007)claim that their observations confirm the emerging fluxregions model of Kurokawa and Kawai (1993) andShibata et al. (1994); moreover the clear evidence of spa-tiotemporal correlations between chromospheric (Ca II K,Ha) and coronal brightenings (TRACE Fe IX 171 A) indi-cates that the chromosphere is heated up to coronal tem-peratures at the surge footpoint by the energy releaseduring magnetic field reconnection, as shown in the numer-ical simulation of Yokoyama and Shibata (1996).

It is clear that the main body of the aforementionedsurge is positioned at the TR/lower corona region. Thus,we can take the electron number density of the surge envi-

ronment to be equal to ne ¼ 2� 109 cm�3 (the label ‘e’stands for exterior), and its temperature reasonably can

be T e ¼ 2� 106 K. Concerning the surge temperature, we

suppose it to be equal to T i ¼ 105 K. Then with a back-ground magnetic field Be ¼ 7 G and a density contrastg ¼ qe=qi ¼ 0:052 (we assume that plasma densities inboth media are homogeneous), we have the following char-acteristic sound and Alfven speed inside the surge and inthe surrounding magnetized plasma: csi ¼ 37 km s�1,vAi ffi 78:3 km s�1 (more exactly 78.314 km s�1, whichvalue determines the magnetic field inside the jet to beequal to Bi ¼ 7:04 G), and cse ffi 196 km s�1, vAe ¼341 km s�1, respectively. Thus, the plasma betas of thetwo media are correspondingly bi ¼ 0:27 and be ¼ 0:28.Incompressible plasma is a good approximation to studythe KH instability, though the observed values are notfavorable for the incompressibility. Nevertheless, in thefollowing we consider the incompressible plasma insideand outside the surge.





We model the surge as a vertically moving with a veloc-ity v0 cylindrical flux tube with radius a ¼ D‘=2 (see Fig. 1),where D‘ ¼ 7 Mm is the surge width. Our frame of refer-ence is attached to the TR/coronal plasma that implies thatv0 is the relative jet velocity with respect to its environment.We must mention that because the density contrast, g, isrelatively high, in such a case, like in spicules, the occur-rence of a KH instability, for instance of kink (m ¼ 1)waves, becomes possible at generally high Alfven Machnumbers (the Alfven Mach number is defined as the ratioof jet velocity to Alfven speed inside the jet, MA ¼ v0=vAi)and respectively high critical flow velocities being farbeyond the speeds accessible for surges/spicules in the solaratmosphere (Zhelyazkov, 2012; Zhelyazkov andZaqarashvili, 2012). We note that this requirement appliesfor KH instability occurring in the longitudinal direction(subject to longitudinal shear flow) and it is less strictand far more plausible for that occurring in the transversedirection (subject to transverse shear flow). Therefore, inour case of longitudinal shear flow the only possible wayfor emerging a KH instability in surges is the excitationof higher MHD harmonics that can become unstable atsub-Alfvenic flow velocities in twisted tubes (Zaqarashviliet al., 2010). Here, we consider two possible magnetic fieldgeometries: (i) a moving twisted magnetic flux tube embed-ded in untwisted magnetic field Be (the left tube in Fig. 1)(ii) a moving twisted magnetic flux tube surrounded byplasma with twisted magnetic field lines (the right tube inFig. 1). In our cylindrical coordinate system (r;u; z) themagnetic field has the following form: B ¼ ð0;BuðrÞ;BzðrÞÞ, and the flow profile inside the tube isv0 ¼ ð0; 0; v0Þ. In general, v0 can be a function of r, butwe consider the simplest homogeneous case. The unper-turbed magnetic field B and the pressure p satisfy thepressure balance equation

d

drp þ

B2u þ B2

z

2l

!¼ �

B2u

lr; ð1Þ

Fig. 1. Equilibrium magnetic field geometries of a cool solar surge.


where l is the magnetic permeability.As the unperturbed parameters depend on the r coordi-

nate only, the perturbations can be Fourier analyzed withexp½�iðxt � mu� kzzÞ�. The equations governing theincompressible plasma dynamics are Goossens et al. (1992)

d2ptot

dr2þ C3

rDd

drrDC3

� �� dptot

dr

þ C3

rDd

drrC1

C3

� �þ 1

D2C2C3 � C2

1

� �� ptot ¼ 0: ð2Þ

where

D ¼ q X2 � x2A

� �; C1 ¼ �

2mBu

lr2

mr

Bu þ kzBz

� ;

C2 ¼ �m2

r2þ k2

z

� �;

C3 ¼ D2 þ D2Bu

ld

drBu

r

� ��

4B2u

lr2qx2

A;

xA ¼k � Bffiffiffiffiffiffi

lqp ¼ 1ffiffiffiffiffiffi

lqp

mr

Bu þ kzBz

� ð3Þ

is the local Alfven frequency,

X ¼ x� k � v0 ð4Þ

is the Doppler-shifted frequency, and ptot is total (ther-mal + magnetic) pressure perturbation. Radial displace-ment nr is expressed through the total pressureperturbation as

nr ¼DC3

dptot

drþ C1

C3

ptot: ð5Þ

The solution to this equation depends on the magnetic fieldand density profile. To obtain the dispersion relation ofMHD modes, we find the solutions to Eqs. (2) and (5)inside and outside the tube and merge the solutionsthrough boundary conditions.

3. Wave dispersion relations

Prior to obtaining the wave dispersion relations for thetwo magnetic configurations, we first specify the twist ofthe magnetic flux tube to be a uniform one, i.e.,

Bi ¼ ð0;Ar;BizÞ; ð6Þ

where A and Biz are constant. We specify the twist of themagnetic field, e, by the ratio Biu=Biz, where Biu is evaluatedat the inner surface of the tube, that is, e ¼ Aa=Biz. In thefollowing, we only consider e 1, therefore the configura-tion is stable for kink and sausage instabilities. In the sim-pler case (left flux tube in Fig. 1) the magnetic field outsidethe tube is homogeneous, Be ¼ ð0; 0;BeÞ, and the solutionto Eq. (2) inside the tube bounded at the tube axis is (seeZaqarashvili et al. (2014))

ptotðr 6 aÞ ¼ aiImðjirÞ; ð7Þ





where Im is the modified Bessel function of order m and ai isa constant. Transverse displacement can be written usingEq. (5) as

nir ¼ai

rX2 � x2

Ai

� �jirI 0mðjirÞ

qi X2 � x2Ai

� �2 � 4A2x2Ai=l

(

�2mAxAiImðjirÞ=

ffiffiffiffiffiffiffilqi

p

qi X2 � x2Ai

� �2 � 4A2x2Ai=l

); ð8Þ

where the attenuation coefficient ji and Alfven frequencyxAi are given by

ji ¼ kz 1� 4A2x2Ai=lqi X2 � x2

Ai

� �2h i1=2

; ð9Þ

xAi ¼mAþ kzBizffiffiffiffiffiffiffi

lqi

p ð10Þ

and prime sign means a differentiation by the Bessel func-tion argument.

The solution to Eq. (2) outside the flux tube bounded atinfinity is

ptotðr > aÞ ¼ aeKmðkzrÞ; ð11Þwhere Km is the modified Bessel function of order m and ae

is a constant. Transverse displacement can be written as

ner ¼ae

rkzrK 0mðkzrÞ

qe x2 � x2Ae

� � ; ð12Þ

where, as before, the prime sign means a differentiation by theBessel function argument, and the local Alfven frequency is

xAe ¼kzBeffiffiffiffiffiffiffilqe

p ¼ kzvAe: ð13Þ

Here, vAe ¼ Be=ffiffiffiffiffiffiffilqe

pis the Alfven speed in the tube

environment.The boundary conditions which merge the solutions

inside and outside the twisted magnetic flux tube are thecontinuity of the radial component of the Lagrangiandisplacement

nirjr¼a ¼ nerjr¼a ð14Þand the total pressure perturbation (Bennett et al., 1999)

ptot i �B2

iu

lanir

��r¼a

¼ ptot ejr¼a; ð15Þ

where total pressure perturbations ptot i and ptot e are givenby Eqs. (7) and (11), respectively. Applying these boundarycondition, after some algebra we finally derive the disper-sion relation of the normal MHD modes propagatingalong a twisted magnetic flux tube with axial mass flowv0 surrounded by plasma embedded in a homogeneousmagnetic field

X2 � x2Ai

� �F mðjiaÞ � 2mAxAi=


p

X2 � x2Ai

� �2 � 4A2x2Ai=lqi

¼ P mðkzaÞqe

qix2 � x2

Ae

� �þ A2P mðkzaÞ=lqi

; ð16Þ


where, recall, X ¼ x� k � v0 is the Doppler-shifted wavefrequency in the moving flux tube, and

F mðjiaÞ ¼jiaI 0mðjiaÞ

ImðjiaÞand P mðkzaÞ ¼

kzaK 0mðkzaÞKmðkzaÞ

:

A derivation of Eq. (16) starting from the basic equationsof ideal magnetohydrodynamics the reader can see inZhelyazkov and Zaqarashvili (2012).

In the case when the outside magnetic field is alsotwisted (the right flux tube in Fig. 1), we consider that mag-netic field, Be, has the form Zaqarashvili et al. (2014)

Be ¼ 0;Beuar;Bez

ar

� 2� �

ð17Þ

and the density is presented as q ¼ qeða=rÞ4, so that theAlfven frequency

xAe ¼mBeu þ kzaBezffiffiffiffiffiffiffi

lqe

pa

ð18Þ

is constant, which allows us to find an analytical solutionto the governing Eq. (2). The total pressure perturbationoutside the tube is governed by the Bessel-type equation

d2ptot

dr2þ 5

rdptot

dr� n2

r2þ j2

e

� �ptot ¼ 0; ð19Þ

where

n2 ¼ m2 �4m2B2

eu

lqea2 x2 � x2Ae

� �þ 8mBeuxAeffiffiffiffiffiffiffilqe

pa x2 � x2

Ae

� � ð20Þ

and

j2e ¼ k2

z 1�4B2

eux2

lqe x2 � x2Ae

� �2a2

!: ð21Þ

A solution to Eq. (19) bounded at infinity is

ptotðr > aÞ ¼ ae

a2

r2KmðjerÞ; ð22Þ

where m ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi4þ n2p

and ae is a constant.Transversal displacement can be written as

ner ¼ ae

r x2 � x2Ae

� �jerK 0mðjerÞ

a2qe x2 � x2Ae

� �2 � 4B2eux2=l

� ae

ra

2a x2 � x2Ae

� �a2qe x2 � x2

Ae

� �2 � 4B2eux2=l

(

þ2mBeuxAe=

ffiffiffiffiffiffiffilqe

p

a2qe x2 � x2Ae

� �2 � 4B2eux2=l

)KmðjerÞ: ð23Þ

By applying boundary conditions (14) and (15), where thetransversal displacements are given by Eqs. (12) and (23),and total pressure perturbations ptot i and ptot e, accordingly,by Eqs. (7) and (22), we obtain the dispersion relation ofthe normal MHD modes propagating along a twisted mag-netic flux tube with axial mass flow v0 surrounded byplasma embedded in twisted magnetic field




0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Im(v

ph/v

Ai)

kza

m = –2ε = 0.1

MA = 0.741

ε = 0.2MA = 0.719

ε = 0.4MA = 0.698

0.68

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Re(

v ph/

v Ai)

kza

m = –2ε = 0.025MA = 0.762

ε = 0.1MA = 0.741

ε = 0.2MA = 0.719

ε = 0.4 MA = 0.698

Fig. 2. (Top panel) Growth rates of the unstable m ¼ �2 MHD modepropagating on incompressible jets in four different twisted internalmagnetic fields (with e ¼ 0:025; 0:1; 0:2, and 0.4) at g ¼ 0:052; b ¼ 1,and corresponding critical MA numbers. For kza ¼ 1:5 the wavelength ofthe unstable m ¼ �2 harmonic is kKH ¼ 14:7 Mm, and the mode growthrate is cKH ¼ 0:0055 s�1. (Bottom panel) Marginal dispersion curves of theunstable m ¼ �2 MHD mode for the critical MA numbers as functions ofthe magnetic field twist parameter e. At kza ¼ 1:5 the critical jet velocity isvcr

0 ¼ 50:8 km s�1.


X2 � x2Ai

� �F mðjiaÞ � 2mAxAi=


p

X2 � x2Ai

� �2 � 4A2x2Ai=lqi

¼a2 x2 � x2

Ae

� �QmðjeaÞ � G

L� H a2 x2 � x2Ae

� �QmðjeaÞ � G

� ; ð24Þ

where

QmðjeaÞ ¼jeaK 0mðjeaÞ

KmðjeaÞ; L ¼ a2qe x2 � x2

Ae

� �2 �4B2

eux2

l;

H ¼B2

eu

la2� A2

l; G ¼ 2a2 x2 � x2

Ae

� �þ 2maBeuxAeffiffiffiffiffiffiffi

lqe

p :

Note that the left-hand sides of dispersion Eqs. (16) and(24) are identical (this is not surprising), but the right-handsides are completely different due to the very different mag-netic field environments.

4. Numerical calculations and results

The main goal of our study is to determine under whichconditions the MHD waves propagating along the movingflux tube can become unstable. To conduct this investiga-tion, it is necessary to assume that the wave frequency xis a complex quantity, that is, x! xþ ic, where c is theinstability growth rate, while the longitudinal wave numberkz is a real variable in the wave dispersion relation. Theoccurrence of the expected KH instability is determinedprimarily by the jet velocity and in searching for a criticalor threshold value of it, we will gradually change velocitymagnitude from zero to that critical value (and beyond).Thus, we have to solve dispersion relations in complex vari-ables, obtaining the real and imaginary parts of the wavefrequency, or as is commonly accepted, of the wave phasevelocity vph ¼ x=kz, as functions of kz at various values ofthe velocity shear between the surge and its environment,v0.

Before starting the numerical task, we have to normalizeall variables and to specify the input parameters. The wavephase velocity, vph, and the other speeds are normalized tothe Alfven speed inside the jet, vAi, which is calculatedusing the axial magnetic fields Biz. The wavelength,k ¼ 2p=kz, is normalized to the tube radius, a, that is equiv-alent to introducing a dimensionless wavenumber kza. Fornormalizing the Alfven frequency in the environment, xAe,except the density contrast g and the tube radius a, we haveto additionally specify the ratio of the axial magnetic fieldcomponents in both media, b ¼ Bez=Biz. For our surge andits environment that ratio is equal to 0.994373 and we shallsimply take b ¼ 1. In the dimensionless analysis the flowspeed, v0, will be presented by the Alfven Mach numberMA ¼ v0=vAi.

We first begin with the numerical solving Eq. (16) for thefluting-like m ¼ �2 mode and one obtains four instabilitywindows on the kza-axis whose position and width dependsupon the magnetic field twist parameter e (see Fig. 2). Thecritical jet speeds for emergence of a KH instability


accordingly are 59.6, 57.7, 55.2, and 50.8 km s�1. Thesevelocities are higher than the speeds registered by Brookset al. (2007), but they (the velocities) are generally accessi-ble for cool surges. The three narrow windows correspond-ing to e ¼ 0:025; e ¼ 0:1, and e ¼ 0:2 are practicallyinapplicable to our surge–environment configuration: thewavelength, k ¼ pDL=kza, of unstable m ¼ �2 harmonicsbecomes comparable to the surge’s height; for instance,for kza ¼ 0:3 (the middle of the second instability window)the wavelength of the unstable m ¼ �2 harmonic at e ¼ 0:1is kKH ¼ 73 Mm. Actually only in the forth instability win-dow (at e ¼ 0:4) one can have an observable unstablem ¼ �2 MHD mode: for instance, at kza ¼ 1:5 the wave-length is kKH ¼ 14:7 Mm, and the corresponding dimen-sionless mode phase velocity growth rate is equal to0.17638 which implies a mode growth ratecKH ¼ 0:0055 s�1.




0

0.05

0.1

0.15

0.2

0.25

0.5 1 1.5 2 2.5

Im(v

ph/v

Ai)

kza

m = –4ε = 0.1MA = 0.573

ε = 0.2MA = 0.542

ε = 0.4MA = 0.511

0.58

0.6

0.62

0.64

/vAi

)

m = –4

ε = 0.025MA = 0.604

ε = 0.1MA = 0.573


Next two figures, Figs. 3 and 4, show the results for them ¼ �3 and m ¼ �4 MHD modes. The first observation,comparing Figs. 2–4, is that with increasing the modulusof the MHD harmonic, the range of wave propagation(for a set of four magnetic filed twist parameters equal to0.025, 0.1, 0.2, and 0.4) becomes wider. On the other hand,the instability windows do not recover all dimensionlesswavenumber’s values—there appears gaps between thewindows. That is well-pronounced for the m ¼ �4 MHDmode. Those gaps in Figs. 3 and 4 can be filled in by per-forming the calculation for some other values of the mag-netic field twist parameter, e, say for e ¼ 0:3. The secondobservation is that with increasing the modulus of theMHD mode the maximal values of the normalized modephase velocity growth rate decrease. Common in all casesis the circumstance that one can really have unstableMHD harmonics (with reasonable wavelengths) only inthe forth window for e ¼ 0:4. Especially for the m ¼ �4MHD mode one could also expect an unstable wavelengthof kKH ¼ 14:7 Mm at e ¼ 0:3.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.5 1 1.5 2

Im(v

ph/v

Ai)

kza

m = –3ε = 0.1MA = 0.615

ε = 0.2MA = 0.59

ε = 0.4MA = 0.575

0.56

0.57

0.58

0.59

0.6

0.61

0.62

0.63

0.64

0.65

0.5 1 1.5 2

Re(

v ph/

v Ai)

kza

m = –3ε = 0.025MA = 0.64

ε = 0.1MA = 0.615

ε = 0.2MA = 0.59

ε = 0.4 MA = 0.575

Fig. 3. (Top panel) Similar as Fig. 2 but for the case m ¼ �3. Forkza ¼ 1:8 the wavelength of the unstable m ¼ �3 harmonic iskKH ¼ 12:2 Mm, and the mode growth rate is cKH ¼ 0:007 s�1. (Bottom

panel) Marginal dispersion curves of the unstable m ¼ �3 MHD mode forthe critical MA numbers as functions of the magnetic field twist parametere. At kza ¼ 1:8 the critical jet velocity is vcr

0 ¼ 41:8 km s�1.

0.48

0.5

0.52

0.54

0.56

0.5 1 1.5 2 2.5

Re(

v ph

kza

ε = 0.2MA = 0.542

ε = 0.4 MA = 0.511

Fig. 4. (Top panel) Similar as Fig. 2 but for the case m ¼ �4. Forkza ¼ 2:3 the wavelength of the unstable m ¼ �4 harmonic iskKH ffi 9:6 Mm, and the mode growth rate is cKH ¼ 0:0055 s�1. (Bottom

panel) Marginal dispersion curves of the unstable m ¼ �4 MHD mode forthe critical MA numbers as functions of the magnetic field twist parametere. At kza ¼ 2:3 the critical jet velocity is vcr

0 ¼ 37:2 km s�1.


Let us now give some numbers for critical velocities,unstable wavelengths and corresponding mode growthrates of the m ¼ �3 and m ¼ �4 MHD modes. For them ¼ �3 harmonic (see Fig. 3), the critical jet velocitiesfor emergence of a KH instability correspondingly are50.1, 47.9, 45.3, and 41.8 km s�1 as the lowest value isobtained in the forth instability window with e ¼ 0:4. Ifwe pick out a kza ¼ 1:8, that is, an unstablekKH ¼ 12:2 Mm, the corresponding dimensionless modephase velocity growth rate is equal to 0.19245 whichimplies a mode growth rate cKH ¼ 0:007 s�1. If we shiftto the left at kza ¼ 1:5 (with maximal normalized modephase velocity growth rate), the wavelength iskKH ¼ 14:7 Mm, and the corresponding mode growth rateis a little bit higher, namely cKH ¼ 0:008 s�1, but neverthe-less of the same order. With exciting the m ¼ �4 harmonic,one can have relatively shorter unstable wavelengths aswell as lower critical flow velocities—the latter are, indescending order, equal to 47.3, 44.6, 41.6, and37.2 km s�1, respectively—as seen in the range of deducedby Brooks et al. (2007) threshold speeds. For kza ¼ 2:3 (see




0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.6 0.8 1 1.2 1.4 1.6 1.8

Im(v

ph/v

Ai)

kza

εi = 0.3εe = 0.01

m = –2

m = –3

m = –4

0.4

0.45

0.5

0.55

0.6

0.65

0.6 0.8 1 1.2 1.4 1.6 1.8

Re(

v ph/

v Ai)

kza

εi = 0.3εe = 0.01

m = –2 MA = 0.587

m = –3 MA = 0.524

m = –4 MA = 0.472

Fig. 5. (Top panel) Growth rates of the unstable m ¼ �2; m ¼ �3, andm ¼ �4 MHD modes propagating on incompressible twisted jets withei ¼ 0:3 in a twisted external magnetic field with ee ¼ 0:01 atg ¼ 0:052; b ¼ 1, and critical MA numbers equal to 0:587; 0524, and0.472, respectively. (Bottom panel) Marginal dispersion curves of theunstable m ¼ �2; m ¼ �3, and m ¼ �4 MHD modes for the critical MA

numbers. The critical surge velocities of these modes are correspondinglyequal to 44 km s�1, 39.3 km s�1, and 35.4 km s�1.


Fig. 4), kKH ¼ 9:6 Mm and the corresponding Imðvph=vAiÞvalue is 0.115032 that yields cKH ¼ 0:0055 s�1. If we choosekza ffi 1:92—the value at which the dimensionless modephase velocity growth rate is maximal—the correspondingunstable wavelength is kKH ¼ 11:5 Mm, and forImðvph=vAiÞmax ¼ 0:204124 the mode growth rate iscKH ¼ 0:0081 s�1. This obviously is the biggest achievablemode growth rate at a reasonable KH instability wave-length. Note that for each unstable MHD harmonic thenormalized mode phase velocity on given dispersion curve(see the bottom panels in Figs. 2–4) is equal to its label MA.Therefore, the unstable perturbations are frozen in the flowand consequently they are vortices rather than waves. Thisobservation is consistent with the KH instability in thehydrodynamics that deals with unstable vortices.

A surge implies both an upward and a downwardmotion of plasma in an often ballistic fashion. These twostages therefore introduce two regimes, one with v0 > 0and one with v0 < 0 for the flow. If the waves are being


continuously generated somewhere in the chromosphereand thus keep propagating upward during the entire life-time of the surge, the Doppler shifted frequency of thewave, x� k � v0, will change drastically at the moment offlow velocity reversal. Computations show, however, thatthe only change is associated with a reverse of the wavephase velocity, ReðxÞ=kz—the instability growth rates,ImðxÞ=kz, are the same as in an upward plasma motion.The wave dispersion curves in a downward plasma floware the mirror images of the dispersion curves in upwardflowing plasma.

One obtains a much more interesting picture whenstudying the more complicated case of a twisted magneticflux tube surrounded by plasma embedded in a twistedbackground magnetic field. Our choice for the twist charac-teristics of the two magnetic fields (internal and externalones) are ei ¼ 0:3 and ee ¼ 0:01, respectively. Numericalsolving Eq. (24) for the three mode numbersm ¼ �2; m ¼ �3, and m ¼ �4 gives for each mode num-ber two instability windows: one of them with relativelyhigh maximal growth rate, and a second window, next tothe former, with one order lower maximal growth rate.These two families of instability windows, for clarity, arepresented in separate figures, Figs. 5 and 6, respectively.As seen from Fig. 5, one can observe a KH instabilitymostly for the m ¼ �4 MHD mode and partly for them ¼ �3 harmonic. The wave growth rates of unstablemodes are of the same order like those illustrated inFigs. 2–4, that is, few inverse milliseconds. The critical flowvelocities for the m ¼ �2; m ¼ �3, and m ¼ �4 modes arecorrespondingly equal to 44, 39.3, and 35.4 km s�1. Thesecond family of instability windows, shown in Fig. 6,has two distinct peculiarities: first, the instability windowsare shifted to the right-hand side of the kza-axis, that is,the propagation range of unstable MHD modes isextended, and second, the marginal dispersion curves arenot constant—the normalized wave phase velocities gradu-ally decrease with increasing wavenumber. If we fix thenormalized wavenumber to be kza ¼ 2, the wavelength ofthe unstable m ¼ �4 mode is kKH ¼ 11 Mm, and atImðvph=vAiÞ ¼ 0:02456 the mode growth rate iscKH ¼ 0:001 s�1, much lower than the growth rates in thefirst family of instability windows. Note also, that the crit-ical jet velocity for emerging a KH instability now isremarkably lower—its normalized value is 0.322036 thatimplies vcr

0 ¼ 24:2 km s�1, that is, the half of the surgespeed evaluated by Brooks et al. (2007). Thus, one can con-clude that high-harmonic MHD modes can become unsta-ble against the KH instability for accessible sub-Alfvenicvelocities—this is more pronounced in the case when bothmagnetic fields are twisted.

5. Discussion and conclusion

In this paper, we have studied the condition under whichMHD modes traveling on a cool solar surge can become




0

0.01

0.02

0.03

0.04

0.05

0.06

1.2 1.4 1.6 1.8 2

Im(v

ph/v

Ai)

kza

εi = 0.3εe = 0.01m = –2

m = –3

m =–4

0.1

0.2

0.3

0.4

0.5

0.6

1.2 1.4 1.6 1.8 2

Re(

v ph/

v Ai)

kza

εi = 0.3εe = 0.01

m = –2MA = 0.587

m = –3MA = 0.524

m = –4MA = 0.472

Fig. 6. (Top panel) Similar as Fig. 5 but for the second family of instabilitywindows. For kza ¼ 2 the wavelength of the unstable m ¼ �4 harmonic iskKH ¼ 11 Mm, and the mode growth rate is cKH ¼ 0:001 s�1. (Bottom

panel) The critical surge velocity of the m ¼ �4 harmonic at kza ¼ 2 isequal to 24.2 km s�1. By contrast to the instability windows plotted in thebottom panel of Fig. 5, in this family of instability windows thenormalized phase velocities are not constant.


unstable against the Kelvin–Helmholtz instability. Ourmodel for the surge is a vertically moving twisted magneticcylindrical flux tube that might be surrounded by plasmaembedded in homogeneous magnetic field or by magne-tized plasma with twisted magnetic field. In each case thetwist of a given magnetic field (internal or external one)is characterized by the ratio of azimuthal magnetic fieldcomponent at the inner surface of the tube to its longitudi-nal component, e. In the simplest case, the tube becomesunstable to the kink instability when e > 2 (Zaqarashviliet al., 2014). The sausage (pinch) instability needs evenhigher values. In this paper, we only consider the weak azi-muthal magnetic field limit taking e 1, therefore the con-figuration is stable for the magnetic instabilities.

Among the MHD modes which propagate along themoving tube only the high harmonics with negative azi-muthal mode numbers can become unstable at accessiblecritical flow velocities. It is known that the perturbationswith wave vector nearly perpendicular to the magneticfield, that is, k � B 0, are the most unstable ones


(Pataraya and Zaqarashvili, 1995; Zaqarashvili et al.,2014). These modes are pure vortices in the incompressiblelimit, therefore they have the strongest growth rate due toKH instability. This condition yields opposite signs for azi-muthal mode (m) and axial (kz) wave numbers. Here weconsider positive kz, therefore the harmonics with the neg-ative m have maximal growth rates. If one takes negativekz, then the harmonics with positive m will have maximalgrowth rates. Negative kz means the harmonics propagat-ing opposite to the flow, hence a superposition of unstableharmonics with positive and negative m may lead to a lessvelocity of KH vortices compared to the flow speed. Thismay explain why the KH vortices travel with the half speedof CME in the case of Foullon et al. (2013).

One can observe practically a developing KH instabilityonly if the parameter e characterizing the magnetic fieldtwist is large enough. For instance, at the first magneticfield configuration (the left flux tube in Fig. 1) them ¼ �3 harmonic becomes unstable at critical flow velocityof 41.8 km s�1 and wavelength kKH ¼ 12:2 Mm with a lin-ear growth rate cKH ¼ 0:007 s�1 as the magnetic field twistparameter e ¼ 0:4. We note, that exploring the KH insta-bility (Zhelyazkov et al., 2015) in a high-temperature solarsurge at coronal plasma densities, like that observed byKayshap et al. (2013), for the same mode at the same posi-tion on the kza-axis and the same value of e, one obtains amode growth rate of 0.033 s�1 being exactly equal to thegrowth rate of the imaged KH instability in a coronal massejecta in the lower corona by Foullon et al. (2013). On theother hand, the aforementioned cKH ¼ 0:007 s�1 is of thesame order as the growth rate of 0.003 s�1 ofvortex-shaped features along the interface between anerupting (dimming) region and the surrounding coronaimaged by the SDO/AIA as reported by Ofman andThompson (2011). All these comparisons allows us tobelieve that the KH instability might be imaged in surges,too. The second magnetic field configuration (the right fluxtube in Fig. 1), reveals some new aspects of the KH insta-bility, notably for a fixed pair of magnetic fields twistparameters, in our case equal to ei ¼ 0:3 and ee ¼ 0:01,for a given high-harmonic mode we obtain two instabilitywindows on the kza-axis, next to each other. Thus, therange of KH instability is extended. True, in the secondinstability windows the growth rates are much lower thanin the standard instability windows, but nevertheless theseweak/slow instabilities can occur. Moreover, the criticalflow velocity for emerging KH instability might be rela-tively lower. For example, the m ¼ �4 harmonic withwavelength of 11 Mm can become unstable at jet speedof only ffi 24:2 km s�1 and its growth rate is equal to0.001 s�1. Solar cool surges are generally small-scale erup-tive events and their contributions to solar coronal heatingdue to triggered by the KH instability wave turbulence ismodest. There are some cases, for instance as surges aredetected in UV and EUV spectral lines, when the develop-ing KH instability can bring forth a noticeable contribu-tion to the solar corona energy budget.





Acknowledgments

The works of I.Zh., R.C., A.K.S., and T.M. were sup-ported by the Bulgarian Science Fund and theDepartment of Science & Technology, Government ofIndia Fund under Indo-Bulgarian bilateral projectCSTC/INDIA 01/7, /Int/Bulgaria/P-2/12. The workof T.Z. was supported by FP7-PEOPLE-2010-IRSES-269299 project- SOLSPANET, by Shota RustaveliNational Science Foundation Grant DI/14/6-310/12, bythe Austrian Fonds zur Forderung der wissenschaftlichenForschung under project P26181-N27, and by AustrianScientific-Technology Collaboration (WTZ) Grant IN10/2013. The authors would like to thank the twoanonymous reviewers for their helpful and constructivecomments that greatly contributed to improving the finalversion of the manuscript. We are also indebted toSnezhana Yordanova for drawing one figure.

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Kelvin–Helmholtz instability in solar cool surges

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