Kink instability of triangular jets in the solar atmosphere
T. V. Zaqarashvili, S. Lomineishvili, P. Leitner, A. Hanslmeier, P. Gömöry, M. Roth
aa r X i v : . [ a s t r o - ph . S R ] F e b Astronomy & Astrophysicsmanuscript no. ms © ESO 2021February 22, 2021
Kink instability of triangular jets in the solar atmosphere
T. V. Zaqarashvili , , , S. Lomineishvili , , P. Leitner , A. Hanslmeier , P. Gömöry , and M. Roth Institute of physics, University of Graz Universitätsplatz 5, 8010 Graz, Austriae-mail: [email protected] Astronomical Institute, Slovak Academy of Sciences, 05960 Tatranská Lomnica, Slovakia Leibniz-Institut für Sonnenphysik (KIS), D-79104 Freiburg, Germany Ilia State University, Cholokashvili ave. 3 /
5, Tbilisi, Georgia Abastumani Astrophysical Observatory, Mount Kanobili, GeorgiaReceived ; accepted
ABSTRACT
Context.
It is known that hydrodynamic triangular jets (i. e. the jet with maximal velocity at its axis, which linearly decreases at bothsides) are unstable to antisymmetric kink perturbations. The inclusion of magnetic field may lead to the stabilisation of the jets. Jetsand complex magnetic fields are ubiquitous in the solar atmosphere, which suggests the possibility of the kink instability in certaincases.
Aims.
The aim of the paper is to study the kink instability of triangular jets sandwiched between magnetic tubes / slabs and its possibleconnection to observed properties of the jets in the solar atmosphere. Methods.
A dispersion equation governing the kink perturbations is obtained through matching of analytical solutions at the jetboundaries. The equation is solved analytically and numerically for di ff erent parameters of jets and surrounding plasma. The analyticalsolution is accompanied by a numerical simulation of fully nonlinear MHD equations for a particular situation of solar type II spicules. Results.
MHD triangular jets are unstable to the dynamic kink instability depending on the Alfvén Mach number (the ratio of flow toAlfvén speeds) and the ratio of internal and external densities. When the jet has the same density as the surrounding plasma, then onlysuper Alfvénic flows are unstable. However, denser jets are unstable also in sub Alfvénic regime. Jets with an angle to the ambientmagnetic field have much lower thresholds of instability than field-aligned flows. Growth times of the kink instability are estimatedas 6-15 min for type I spicules and 5-60 s for type II spicules matching with their observed life times. Numerical simulation of fullnonlinear equations shows that the transverse kink pulse locally destroys the jet in less than a minute in the conditions of type IIspicules.
Conclusions.
Dynamic kink instability may lead to full breakdown of MHD flows and consequently to observed disappearance ofspicules in the solar atmosphere.
Key words.
Sun: atmosphere – Sun: oscillations – Physical data and processes: Instabilities
1. Introduction
Flows and jets are essential building features of the solar at-mosphere. Many di ff erent types of jets are frequently observedin the solar chromosphere / corona like spicules / mottles (Beckers1968; De Pontieu et al. 2007; Rouppe van der Voort et al. 2009;Tsiropoula et al. 2012; Moore et al. 2011; Sterling et al. 2020),macrospicules (Pike & Mason 1998), X-ray jets (Shibata et al.1992; Savcheva et al. 2007; Moore et al. 2013; Sterling et al.2015, 2016, 2019), EUV jets (Chae et al. 1999; Zhang & Ji2014), chromospheric anemone jets (Shibata et al. 2007;Nishizuka et al. 2011) amongst others. Coronal X-ray jets can bedriven by magnetic reconnection after emergence of new bipolarmagnetic flux (Yokoyama & Shibata 1995; Moore et al. 2011) ormini-filaments (Sterling et al. 2015, 2016; Raouafi et al. 2016),while rebound shock waves may lead to classical spicules andmacrospicules (Hollweg 1982; Murawski & Zaqarashvili 2010;Murawski et al. 2011).Hydrodynamic flows are generally unstable (Chandrasekhar1961; Drazin & Reid 1981), which may lead to the energy dis-sipation and to the consecutive heating of solar atmosphericplasma. Di ff erent flow profiles lead to the di ff erent types of in-stabilities. The simplest is the basic flow of two fluids in par- allel infinite streams of di ff erent velocities, which is subject toKelvin-Helmholtz instability (Helmholtz 1868; Kelvin 1871).Kelvin-Helmhotz instability has been intensively observed inthe solar atmosphere at boundaries of rising coronal massejections (Ofman & Thompson 2011; Foullon et al. 2011, 2013;Möstl et al. 2013), in solar prominences (Berger et al. 2010;Ryutova et al. 2010) and in jets (Zhelyazkov et al. 2015, 2018;Li et al. 2019). Various types of flows with smooth transverseprofiles are also unstable in certain conditions (Drazin & Reid1981). Another interesting process is connected to the trans-verse displacement of jet axis, which becomes unstable to thedynamic kink instability due to the centripetal force acting onflows in a curved path (Zaqarashvili 2020). The instability maylead to the observed linear transverse motions of spicule axes(De Pontieu et al. 2012; Kuridze et al. 2015) in certain condi-tions.Flows with smoothed transversed profiles are more di ffi cultto be studied. The simplest case is the flow with a linear trans-verse profile, which can be analytically studied in various situ-ations (Drazin 2002). A jet having maximal velocity at the axiswhich linearly tends to zero at boundaries is a simple model butallows relevant conclusions about jets in the solar atmosphere. Article number, page 1 of 9 & Aproofs: manuscript no. ms
This triangular jet is unstable to the antisymmetric or kink in-stability for long wavelength perturbations in hydrodynamics(Drazin 2002). As the solar atmosphere is generally perceived bythe magnetic field, the stability of triangular jets must be studiedin magnetohydrodynamic (MHD) approximation.A magnetic field aligned with the axis of a flow generallystabilises the sub Alfvénic flows (Chandrasekhar 1961), whilethe transverse magnetic field seems to have no e ff ect on the in-stabilities (Sen 1963; Ferrari et al. 1981; Cohn 1983). Therefore,the magnetic field strength (namely Alfvén Mach number i.e. theratio of flow to Alfvén speeds) and topology are crucial for thethreshold of flow instability. Flows with angles to the magneticfield, e. g. axially moving twisted magnetic flux tubes, can bealways unstable (Zaqarashvili et al. 2010, 2014). The magneticfiled in the solar atmosphere is highly inhomogeneous and hasa complex topology. Therefore, the field may suppress the flowinstability in some places leading to the formation of relativelystable flows. However, in some places flows may become unsta-ble leading to the heating and turbulence of plasma.Here we study the stability of triangular jets in the presenceof a magnetic field with application to the solar atmosphere.We derive the analytical dispersion equations for antisymmetric(kink) modes of a triangular jets imbedded in external magneticfield. The dispersion equations have complex solutions in certainconditions, which indicate to the instability of the jets. We alsoperformed numerical simulations, which fairly confirmed the an-alytical thresholds and growth rates. Finally, the results wereapplied to type I and type II spicules in the solar atmosphere,which showed interesting coincidence of instability propertiesand spicule dynamics.
2. Main equations
For stability analysis of inhomogeneous jets we use simplest in-compressible approximation. Compressibility in general a ff ectsthe flow stability (Sen, 1964, Gill, 1965), but the basic propertiesof the instability are still seen in the incompressible limit. There-fore, we consider single fluid incompressible linearized magne-tohydrodynamic (MHD) equations: ρ " ∂∂ t + V ·∇ u + ρ ( u ·∇ ) V = −∇ p t + π ( B ·∇ ) b , (1) " ∂∂ t + V ·∇ b = ( B ·∇ ) u + ( b ·∇ ) V , (2) ∇· u = , ∇· b = , (3)where ρ , B and V are unperturbed density, magnetic field andflow, while b , u and p t are perturbations in the magnetic field, ve-locity and total (hydrodynamic plus magnetic) pressure. The un-perturbed magnetic field is assumed to be homogeneous. Gravitye ff ects are neglected at this stage.We consider a Cartesian coordinate system ( x , y , z ) and aplasma jet, which has a slab structure along the x axis withthe half width of d and flows in ( y , z ) plane. The velocity ofthe jet is homogeneous with regards y and z , but can be ei-ther homogeneous or inhomogeneous across the slab. The un-perturbed magnetic field is directed along the z axis. The solu-tions of Eqs. (1-3) can be searched in terms of normal modes Ψ ( x ) exp i ( k y y + k z z − ω t ). Then the continuity of Lagrangiandisplacement and total pressure at the slab boundaries gives the dispersion relation for the normal modes with generally com-plex ω . Note, that the incompressible limit neglects the incom-ing and outgoing waves from the slab (i.e. leaky modes are ab-sent), therefore the complex frequency means real instability ofthe normal modes.The magnetic field of quiet Sun regions is concentrated inthin tubes at the photospheric level. However, the tubes rapidlyexpand upwards in the chromosphere and may merge at someheights (Fig. 1). The plasma, which forms spicules at higherheights, may flow in three di ff erent channels: inside the tubes,between the neighbouring tubes with the same polarity and be-tween the tubes with opposite polarities (shown by arrows onFig. 1). There is no firm observational evidence in which ofthese channels plasma flows. The di ff erent channels may sup-port the formation of spicules with di ff erent stability properties.The direction and speed of flows strongly depend on the forma-tion mechanism of di ff erent jets, which is beyond the scope ofpresent paper. Here we assume that the jets are already formedby some mechanism and study their stability for di ff erent pa-rameters. The stability of hydromagnetic flows depends on thetransverse profile of the flow and the direction of the flow withrespect to the magnetic field. Before we go to triangular jets, webriefly review the stability of homogeneous jets.
3. Homogeneous jets
Homogeneous jets are studied elsewhere, therefore we brieflydescribe the main properties of its instability. The main conclu-sion about the stability is that the flow-aligned magnetic field sta-bilises the sub Alfvénic flows (Chandrasekhar 1961; Sen 1964;Ferrari et al. 1980; Hardee & Norman 1988; Singh & Talwar1994). However, the perpendicular magnetic field does not af-fect the instability, therefore one always can find the exponen-tially growing unstable modes in a jet that has a perpendicularcomponent to the field.The dispersion relation of antisymmetric perturbations for amagnetised plane jet of half-width d can be written as " + ρ ρ e tanh( kd ) ω − ρ ρ e ( k · V ) tanh( kd ) ω + ρ ρ e tanh( kd )( k · V ) − ρ ρ e tanh( kd )( k · V A ) − ( k · V Ae ) = , (4)where ρ ( ρ e ) and V A ( V Ae ) are the density and Alfvén speedinside (outside) the jet, k is the wave vector in yz -plane. Thisequation can be obtained from the general dispersion relation ofSingh & Talwar (1994) as an incompressible limit. If the jet is di-rected along the unperturbed magnetic field (i.e. V k V A ), wherethe Alfvén speed is assumed to have the same strength and direc-tion inside and outside the slab for simplicity, then phase speedis always real for any sub Alfvénic flows ( V < V A ). However, ifthe jet has an angle with the magnetic field, then the harmonicsperpendicular to the magnetic field i.e. with k · V A = Article number, page 2 of 9. V. Zaqarashvili et al.: Kink instability of triangular jets in the solar atmosphere
Magnetic tube Flow a) Magnetic tubeFlow b) Magnetic tube Flow c) Fig. 1.
Possible channels of plasma flows in the solar atmosphere: a) between neighbouring magnetic flux tubes of the same polarity; b) inside aflux tube; c) between neighbouring flux tubes of opposite polarity.
Magnetic field
Jet xz d Fig. 2.
Sketch of considered analytical set up, which corresponds to thesituation of Figure 1c. A nonmagnetic triangular jet is located betweenthe magnetic fields of opposite polarities. inclined at the tube boundaries, and thus make an angle with re-spect to upward flows. Therefore, the flows between the tubesmay have the component perpendicular to the magnetic field andthen they can be unstable due to the the dynamic kink instability.
4. Triangular jets
Now we consider a jet, which has transverse inhomogeneity invelocity, and study how this transverse structure a ff ects the in-stability properties of the jet. For simplicity, we assume simplestlinear profile of the flow and consider a triangular jet, whichhas maximum velocity at the slab axis and linearly decreases to-wards slab boundaries. The stability of the jet is well studied innonmagnetic fluids. It was shown that the jet is generally unsta-ble for long wavelength antisymmetric normal modes, while itis generally stable for the symmetric modes (Drazin 2002). It isof our interest to study how an external magnetic filed influencesthe stability properties of the jet. Analytical solution of MHD tri-angular jets is complicated, therefore we consider the situation when a nonmagnetic jet is sandwiched between two magneticenvironments (Fig. 2). Especially interesting situation may arisewhen a jet flows between the two tubes with opposite polarities(Figure 1c), where a neutral sheet is formed. The magnetic fieldbecomes negligible between the tubes and the nonmagnetic jetis a good approximation.Therefore, for the triangular jet we consider | B z | = const , V = , for | x | > d , (5) B z = , V = V − α | x | d ! , for | x | < d , (6)where the flow, V , has y and z components. The flow velocity ismaximal at the slab axis and linearly decreases towards bound-aries. The parameter 0 ≤ α ≤ α = α = α = ρ , in general, is di ff erent from thesurrounding plasma i.e ρ , ρ e ( ρ e is the density in externalmedium).Then Eqs. (1-3) lead to the equations ρ ( k · V − ω ) " ∂ ∂ x − k u x = k z B z π " ∂ ∂ x − k b x , (7)( k · V − ω ) b x = k z B z u x , (8)where k = q k y + k z .When ω , k · V and ω , k z V A , then the plasma dynamicsinside and outside of the slab is governed by the equation " ∂ ∂ x − k u x = . (9)The solution of this equation is a combination of exponentialfunctions and it depends on boundary conditions on the slab cen-ter, boundaries and infinity.We require that the solution vanishes at infinity outside theslab, therefore the resulted expression is u x = Ae − k ( | x |− d ) , for | x | > d . (10)The solutions inside the slab can be antisymmetric (sinuous) orsymmetric (varicose) with regards to the slab center. Article number, page 3 of 9 & Aproofs: manuscript no. ms k z d R e k z d -0.5-0.2500.250.5 I m Fig. 3.
Solutions of antisymmetric (kink) mode (Eq. 15) for nonmag-netic case ( B z =
0) when α = k y , V y = k z d > . k z d → k z V z . For the antisymmetric kink mode the solution inside the jetis (Drazin 2002) u x = B cosh kx cosh kd + D sinh k | x | sinh kd , for | x | < d . (11)These solutions must satisfy the continuity of Lagrangiantransverse velocity and total pressure at the slab boundaries,which give the equations: u x k · V − ω = const , at x = ± d , (12) ρ k · V − ω − k z v A k · V − ω ∂ u x ∂ x − ρ k · V ′ − k z v A ( k · V − ω ) u x == const , at x = ± d . (13)Additionally, the third equation is obtained from the pressurecontinuity condition at the slab center: ρ [ k · V − ω ] ∂ u x ∂ x − ρ k · V ′ u x = const , at x = . (14)Eqs. (10-11) and (12-14) give the following dispersion rela-tion for the antisymmetric mode " + ρ ρ e tanh kd ω + k · V " α tanh kdkd − ρ ρ e (3 − α ) tanh kd − ω + ( k · V ) " α kd ρ ρ e − tanh kdkd ! + ρ ρ e (1 − α )(3 − α ) tanh kd − k z V A ( k · V ) ω + ρ ρ e (1 − α )( k · V ) " α k d tanh kd − α kd − (1 − α ) tanh kd + k · V k z V A " − α tanh kdkd = . (15)This is a general dispersion relation, which can be analysedin di ff erent context. If the magnetic field is negligible ( V A ≈ XZ plane (i. e. k y , V y =
0) and Eq. (15) istransformed (for α =
1) into the dispersion equation of non-magnetised triangular jet (see Drazin 2002, Equation 8.43) " + ρ ρ e tanh k z d ω + k z V z " tanh k z dk z d − ρ ρ e tanh k z d − ω + k z V z k z d ρ ρ e − tanh k z dk z d ! = . (16)This equation has complex solutions in the interval 0 < k y d < . < k z d < . k z d > . V y , k y , k z = k z and V z arereplaced by k y and V y . Hence, the flow is always unstable forlong wavelength perturbations even in the presence of magneticfield. This confirms the previous results that the normal-to-flowmagnetic field does not a ff ect the Kelvin-Helmholtz instability(Chandrasekhar 1961; Sen 1964; Zaqarashvili et al. 2010, 2014). α = (homogeneousjet) limit Next we consider the limit of α =
0, which actually means a ho-mogeneous jet. Then the Eq. (15) is transformed into the equa-tion " + ρ ρ e tanh( kd ) ω − ρ ρ e ( k · V ) tanh( kd ) ω + ρ ρ e tanh( kd )( k · V ) − k z V A ! ( ω − k · V ) = , (17)When ω , k · V , then we recover Eq. (4) (with V A = V =
0) (seeEdwin & Roberts 1982, Equation 12). The antisymmetric modesare unstable when( k · V ) k z V A > ρ e + ρ tanh( kd ) ρ tanh( kd ) . (18)When k y = ρ e = ρ , then this inequality is replaced by V z / V A > p / (1 − e − kd ), which actually means that the flowis stabilised for all wavelength perturbations when V A / V z > √ . ≈ . V A / V z < √ .
5, then long wavelengthperturbations are stable, while the short wavelength perturba-tions are unstable. The critical wavenumber can be estimated as k z d = ln (cid:16) − V A / V z (cid:17) − / . α = limit Next we consider the limit when the jet velocity vanishes at theslab boundaries ( x = ± d ) i.e. α =
1. In this case, Eq. (15) leadsto the equation (see also Zaqarashvili (2011)) " + ρ ρ e tanh kd ω + k · V " tanh kdkd − ρ ρ e tanh kd − ω + Article number, page 4 of 9. V. Zaqarashvili et al.: Kink instability of triangular jets in the solar atmosphere k z d I m Fig. 4.
Growth rate (imaginary part of frequency) of antisymmetric(kink) mode vs normalised wavenumber k z d calculated from Eq. (15)when α = k y , V y = ff erent ratios of inverse AlfvénMach number, M − A . Green, blue, cyan and red solid lines correspondto M − A = α =
0, which is already zerofor M − A = ρ /ρ e = k z V z . k z d I m Fig. 5.
Same as on Fig. 4, but for the density ratio of ρ /ρ e =
10. Green,blue, cyan, and red solid lines correspond to M − A = k z V z . ( k · V ) kd ρ ρ e − tanh kdkd ! − k z V A ( k · V ) ω + k · V k z V A " − tanh kdkd = . (19)This equation is transformed into Eq. (16) for non-magnetised case.Fig. 4 shows the growth rates of unstable modes againstnormalised wavenumber calculated from the dispersion relation(15) for α = ff erent values of Alfvén Machnumber, M A = V z / V A (the ratio of the flow speed at the slab k z d I m Fig. 6.
Growth rate of antisymmetric (kink) mode vs wavenumber formodes with two di ff erent propagation angles, k y / k z , when flow is di-rected with 45 degree to the magnetic field, V y / V z =
1. Green dashedline corresponds to M − A = k y / k z =
1, while green, blue, andred solid lines correspond to M − A =
1, 5, and 10 respectively, when k y / k z = center and the external Alfvén speed), for ρ /ρ e =
1. It is seenthat the flow is unstable only for finite interval of the wave num-bers. Therefore, only the harmonics with certain wave lengthsare unstable. On the other hand, homogeneous flows ( α = ffi ciently strong magnetic field suppresses theinstability when M − A > V z < V A are stable. But homogeneous jetscan be stabilised even for slightly super Alfvénic regime, thoughthey have relatively stronger growth rates.Fig. 5 displays the growth rates for di ff erent values of AlfvénMach number for the denser jet, ρ /ρ e =
10. It shows that thedenser jet needs stronger external Alfvén speed to stabilise thekink instability (in this case Alfvén speed needs to be 2.5 higherthan the flow speed at the slab center). It is clear from physicalpoint of view that the denser jet has stronger kinetic energy andtherefore the stronger magnetic energy it required for stabilisa-tion.If the flow is directed with some angle to the magnetic field,i.e. V y ,
0, then the instability is not completely suppressed bythe magnetic field as it was mentioned in the subsection 4.1. Fig.6 shows the growth rates of unstable modes against wavenum-ber for V y = V z , i.e. the flow is directed with 45 degree to themagnetic field. In this case, the modes with the propagation an-gle of 45 degree to the magnetic field, k y / k z = ffi ciently strong magnetic field. But themodes with the propagation angle close to 90 degree to the mag-netic field ( k y / k z =
10, solid lines) are unstable even for very highAlfvén speed of V A / V z =
5. Discussion
It was shown by Drazin (2002) that hydrodynamic 2D triangularjets are unstable to antisymmetric kink modes, while they are sta-ble to symmetric sausage modes. The kink modes are unstable incertain interval of wavelengths, while short and long wavelengthperturbations are stable (see Fig. 3). Here we studied the influ-ence of magnetic field on the kink instability of triangular jets
Article number, page 5 of 9 & Aproofs: manuscript no. ms k z d R e k z d I m
535 540 545 550 555 560 565 570 575
Wavelength, km P e r i od , s
535 540 545 550 555 560 565 570 575
Wavelength, km G r o w t h t i m e , s Fig. 7.
Solutions of antisymmetric (kink) mode (Eq. 15) in the conditions of type I spicules: ρ /ρ e = d =
400 km, V =
30 km / s. Left panelsshow the real (upper) and the imaginary (lower) parts of frequencies, which are normalised by k z V z . Blue lines show the purely hydrodynamiccase i.e. V A =
0. Red lines show the case when the external Alfvén speed is V A =
200 km / s. Right panels show the periods (upper) and growth times(lower) of unstable harmonics vs wavelength in the interval of k z d = α = k y , V y = k z d R e k z d I m
200 400 600 800 1000 1200 1400
Wavelength, km P e r i od , s
200 400 600 800 1000 1200 1400
Wavelength, km G r o w t h t i m e , s Fig. 8.
Solutions of antisymmetric (kink) mode (Eq. 15) in the conditions of type II spicules: ρ /ρ e = d =
100 km, V =
100 km / s. Left panelsshow the real (upper) and the imaginary (lower) parts of frequencies, which are normalised by k z V z . Blue lines show the purely hydrodynamiccase i.e. V A =
0. Red lines show the case when the external Alfvén speed is V A =
200 km / s. Right panels show the periods and growth times ofunstable harmonics vs wavelength in the interval of k z d = α = k y , V y = in the connection to solar atmospheric physics. In order to ob-tain analytical dispersion equations, we considered a triangularjet sandwiched between magnetic field tubes (or slabs), whichallowed us to solve the linearised problem and led to analyticaldispersion equation (Eq. 15) using appropriate boundary condi-tions.The dispersion equation has solutions in the form of complexwave frequency, which means instability of corresponding wavemodes. The kink instability occurs in the certain interval of wavenumbers depending on flow speed at the axis, or more preciselyon the ratio of flow to Alfvén speeds (Alfvén Mach number).Decreasing of Alfvén Mach number leads to the shortening ofthe instability interval and the shifting of the interval to higherwave numbers (see Fig. 4). After certain Alfvén Mach numberthe instability is terminated, hence the magnetic field stabilisesthe kink instability. For the density ratio of ρ /ρ e =
1, the insta-bility is ceased when M − A > ρ /ρ e =
10, the instability isceased when M − A > . Article number, page 6 of 9. V. Zaqarashvili et al.: Kink instability of triangular jets in the solar atmosphere from the Lorentz force, hence the flow instability starts in subAlfvénic regimes.The antisymmetric kink instability discussed here can beused to model the instability of various types of jets in the so-lar atmosphere. Here we consider the stability of spicules as itis seen under obtained results. It must be mentioned, however,that the real conditions of the solar atmosphere (magnetic fieldstructure, jet density structure, etc.) are much more complicatedcomparing with simplified set up of this paper. Therefore, theresults show only general properties of jet instability, which cansignificantly vary for di ff erent types of jets and solar atmosphericconditions. Spicules are chromospheric plasma jets flowing upwards intothe lower corona, therefore they are almost 100 times denserand cooler than the corona. Spicules are known for long time asthey have been frequently observed at the solar limb (Beckers1968). Typical life time of the spicules is 5-15 min and up-ward velocity is ∼ / s. Note that the disk counter-parts of spicules are called mottles (Tsiropoula et al. 2012). Re-cent Hinode observations with high spatial and temporal res-olutions revealed spicules with di ff erent properties known astype II spicules (De Pontieu et al. 2007). The type II spiculeshave much shorter life time of 10-150 s and higher upwardvelocities of 50-150 km / s than the classical spicules (nowknown as type I spicules). The disk counterparts of type IIspicules are known as Rapid Blue / Rapid Red shifted excur-sions (RBEs / RREs) and were first observed by Swedish SolarTelescope (Rouppe van der Voort et al. 2009). Generation mech-anism for spicules is poorly understood. Classical spicules canbe easily formed due to rebound shocks after photosphericpulses (Hollweg 1982; Murawski & Zaqarashvili 2010), buttype II spicules are more di ffi cult to be excited. Recent mod-els of emerging bipoles resulting in magnetic reconnection(Moore et al. 2011; Sterling et al. 2015, 2020) or in release oftwist by ambipolar di ff usion (Martínez-Sykora et al. 2017) seemto capture essential features of type II spicules, though morework is required in this regards. Density and temperature aresimilar in both types of spicules, therefore the short life timeof type II spicules can be caused either due to their rapid dif-fusion or due to rapid heating, which may lead to their dis-appearance in chromospheric spectral lines. The appearanceof spicules in transition region spectral lines show that typeII spicules are rapidly heated (Pereira et al. 2014), though theheating mechanism is not yet completely clear. Ion-neutral col-lisions, Kelvin-Helmholtz instability, or both together mightlead to the rapid heating (Zaqarashvili et al. 2015; Kuridze et al.2016; Martínez-Sykora et al. 2017; Antolin et al. 2018), but it isnot yet fully established. Another possibility is that the spiculesare quickly destroyed by some instability process. Zaqarashvili(2020) suggested that the dynamic kink instability of homoge-neous jet may be responsible for the disappearance of type IIspicules at some height of expanding magnetic flux tubes. Herewe will examine the e ff ects of transverse inhomogeneity of flowon dynamic kink instability in both types of spicules separately. The diameter of type I spicules varies from 300 to 1100 km(Pasacho ff et al. 2009), therefore we take a mean value of 800 km, which leads to d =
400 km for the half width. Plasma flowmay reach to 20-30 km / s so we take V z =
30 km / s for the ve-locity at the jet axis (Beckers 1968; Pasacho ff et al. 2009). Wealso assume for the external Alfvén speed in low corona as 200km / s and for the density ratio of spicules and lower corona as100. Left panels of Fig. 7 shows the solutions of the dispersionrelation (15) in the parameters of type I spicules. It is seen thatthe jets is almost fully stabilised (red lines). Only negligible in-stability region around k z d = k z d < Type II spicules are generally thinner than type I spicules witha diameter of <
200 km (De Pontieu et al. 2007), therefore wetake d =
100 km for the half width. Plasma flow may reach to50-150 km / s so we take V z =
100 km / s for the velocity at the jetaxis. We assume the same external Alfvén speed and the den-sity ratio as for the case of the type I spicules i. e. Alfvén speedof 200 km / s and the density ratio of spicules and lower coronaequal to 100. Left panels of Fig. 8 show the solutions of the dis-persion relation (15) in these parameters. It is seen that the jetsbecomes unstable for the wavenumbers of k z d = k z d < Chromospheric plasma is partially ionised, therefore the colli-sion between ions and neutral atoms may have influence on thekink instability in spicules. Since our analysis concerns onlyfully ionised plasma, it is of importance to estimate the e ff ects ofion-neutral collisions. Kuridze et al. (2016) estimated the heat-ing time due to ion-neutral collision e ff ects as t heat ∼ βδ in D V A − ξ n ξ n , (20)where β = π p / B z is the plasma beta, D is the spatial scaleof perturbations, δ in is the ion-neutral collision frequency, ξ n isthe ratio of neutral to total particle density. Let us estimate theheating time for type II spicules, taking the spatial scale of un-stable harmonics as D =
400 km (see previous subsection). Wetake the following values for other parameters as δ in = Hz, β = . V A =
100 km / s and ξ n = , s, which isby the two orders of magnitude longer than the growth time of Article number, page 7 of 9 & Aproofs: manuscript no. ms kink instability. Therefore, ion-neutral collision e ff ects are neg-ligible at the initial stage of instability. However, when the kinkinstability is fully developed then the energy will be transferredto smaller scales, which will decrease the heating time. There-fore, ion-neutral collision e ff ect may have influence on plasmadynamics only on later stage of kink instability. This is not thescope of the present paper and will be studied in the nearest fu-ture. The analytical model considered here is obviously simplified.The solutions are linear, therefore we only model the initial lin-ear stage of the instability, while it is also of importance to seethe full development of the instability, i.e. what is happening tothe jets later on. Additionally, the absence of magnetic field in-side the jet in considered analytical model is not good approxi-mation for spicules. Therefore, numerical simulations should beinvoked to test the analytical results.In this section we present results of 2-D numerical simula-tions obtained with the code PLUTO, e.g. Mignone et al. (2007),accompanying our discussion on antisymmetric kink modes inSect. 4 in a uniform background magnetic field. The code is em-ployed to solve the ideal MHD equations in conjuction with thethermal ideal equation of state for closure. For the flux computa-tion an approximate Riemann solver, i.e. Harten, Lax, Van Leer(HLL) has been used. A more detailed and systematic numericalanalysis is under way and will be discussed in a separate paper.In di ff erence with the analytical solution (with no magneticfield inside the jet), here we consider the homogeneous magneticfield inside and outside the jet with the same strength of B z = d =
100 km andits speed at the axis to V z =
100 km / s. The inner temperature isset to 10 K. The mass density ρ was chosen such that a pressureequilibrium at the jet boundary is maintained, thus depending onthe external mass density of ρ e = × − g / cm and externaltemperature of T e = × K, which means ρ /ρ e =
40. We useda cartesian grid with cell sizes of ∆ x = ∆ z = /
50 of the jet’s half width, to easily capture the turbulentevolution of the plasma flow in the non-linear regime. The initialtemporal resolution was set to 5 × − s and let adjust accordingto a maximum CFL number of 0.2. Neumann zero-gradient out-flow boundary conditions have been applied on the right as wellas on top and on bottom sides while an open inflow condition isspecified on the left boundary.We show that the jet becomes unstable to kink modes, whensubjected to transverse antisymmetric perturbations as discussedin Sect. 4. The perturbation amplitude of the transverse veloc-ity was set to ∼
10% of the flow speed V z and the pertur-bation FWHM 2 √ σ of the initial Gaussian pulse g ( z ) = ( √ πσ ) − exp {− / · [( z − µ ) /σ ] } , centered around the location µ (where the initial pulse is seeded), is related to typical sizes ofgranulation cells. We scanned the dispersion relation, Fig. (3),for various wave numbers and indeed found a strong instabilityin the depicted regime. The simulation presented below is basedon an initial perturbation with k z d =
1, i.e. to the spatial scale of628 km.Fig. 9 shows the dynamics of the plasma density, longitudi-nal components of velocity and magnetic field in the jet and sur-roundings at di ff erent times. A significant displacement of the jetaxis is already seen 15 s following the initial perturbation (top-most panels). After 45 s (bottom line) the transverse displace- ment already shows nonlinear character, therefore the growthtime of perturbations can be estimated around 30 s, which iscomparable to the analytical growth time (Fig. 8). Around 45 safter the initial perturbation, the jet is almost destroyed locally.Hence, in less than 1 min after the initial excitation, the trans-verse velocity pulse transformed into a fully developed local in-stability. If one initially excites a harmonic wave instead of lo-cal pulse, then the instability will be developed along the wholewave train which will collapse the jet (numerical simulations willbe presented in a separate paper). This fairly agrees with the lifetime of type II spicules, therefore the spicules could be destroyedby the kink instability.The propagation speed of the excited kink pulse is estimatedas ≈
54 km / s as measured in the inertial frame. The instabilitykeeps growing until the jet is finally destroyed. The simulationsshow that the development of the instability critically depends onthe flow speed of the jet as expected: at a lower V z of 30 km / s,the initially generated kink wave forms only a narrow, low am-plitude pulse that is propagating with the flow without tearingdown the underlying jet.
6. Conclusions
The stability of triangular jets sandwiched between two mag-netic tubes / slabs in the solar atmosphere was studied. Disper-sion equation governing the antisymmetric kink perturbationswas obtained, which was solved analytically and numerically fordi ff erent cases of Alfvén Mach number, M A , and the ratio of in-ternal and external densities, ρ /ρ e . It has been shown that trian-gular jets are unstable to the dynamic kink instability dependingon Alfvén Mach number and the density ratio. For example, jetswith ρ /ρ e = M − A < ρ /ρ e =
10 are unstable also in sub Alfvénicregime ( M − A < . Acknowledgements.
The work was funded by the Austrian Science Fund (FWF,projects P30695-N27 and I 3955-N27) and by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) - Projektnummer 407727365. SLand PG acknowledge the support of the project VEGA 2 / /
20. Acknowledg-ment is also given to the developers of the code PLUTO we have been using forperforming the simulation presented in this paper. The authors thank the anony-mous referee for stimulating comments, which led to improve the paper.
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Article number, page 8 of 9. V. Zaqarashvili et al.: Kink instability of triangular jets in the solar atmosphere
Fig. 9.
MHD simulation of a magnetised triangular jet in conditions of type II spicules. The halfwidth of the jet is 100 km and the total lengthof the simulated jet is about 7 Mm. The initial transverse perturbation has an amplitude of ∼
10 km / s (corresponding to 10% of the flow speedat the jet centre) and a spatial scale of around 600 km. Shown are the mass density ρ (left panels), the flow velocity V z (middle panels) and the z -component of the magnetic field B z (right panels) at consecutive snapshots, each 15 s apart (upper panels correspond to the initial set up). Apronounced kink is developing around 30 s after initial perturbation turning into an instability after ≈
45 s. Corresponding movies can be foundonline.
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