Numerical Study of X-Ray Jets in Coronal Hole M

Numerical Study of X-Ray Jets in Coronal Hole M. Yamada, E. Belova, J. Latham*,

Princeton Plasma Physics Laboratory, Princeton University Theory Seminar February 2, 2021

1 2/5/21 *J. Latham et al: Published in Phys. Plasmas 2021. X-Ray Jets – Anemone Configuration

Fillipov, et al. Sol Phys, 254 259269 (2009).

2/5/21 https://www.britannica.com Shibata et al Astrophys. J. 431, L51–53 (1994) 2 Plasma jets with X ray emissions are observed when an inner dome configuration is tilted with respect to outer open field lines

Wyper, et al. Nature, 544 452-455 (2017) Spheromak tilt is proposed as an eruption mechanism

A model was proposed in 2019 for “spheromak formation and tilt”

2/5/21 4 Basic Concept A storage and release mechanism for solar flare eruption • We consider a dome-shaped magnetic configuration: spheromak • How does a spheromak which emerges slowly from the sun’s surface suddenly become unstable against an MHD instability? • The effects of line-tying on an elongated spheromak stability are presented • We conclude that magnetic reconnection occurs near the top of a half- sphere configuration to generate plasma jets with X-rays

5 2/5/21 Spheromak tilt: Line-Tying Improves Stability

Spheromak tilt: Oblate v. Prolate Line-tying should help stabilize tilt mode

Oblate

Prolate Tilt unstable Modeling of spheromak line-tied to the sun surface Calculation by HYM code (E. Belova) 2/5/21 6 Summary • We have verified that a spheromak can exist in a stable form with line-tied state to a single conducting surface • Marginal amount of line-tying will stabilize spheromak (more line-tying needed for elongated spheromak) • Magnetic reconnection on bottom of spheromak is necessary to detach flux for spheromak formation and for the tilt. • In the final stage, magnetic reconnection occurs at the top of inner dome and ejects a plasma jet to the open field.

2/5/21 7 Theory R-R seminar, PPPL, February 2, 2021

Numerical Study of Plasma Jets Formation in the Coronal Hole

E. Belova, M. Yamada (PPPL), J. Latham* (University of Michigan)

*Latham et al, Physics of Plasmas 28, 012901 (2021); https://doi.org/10.1063/5.0025136 111210-2 Singh et al. Phys. Plasmas 18, 111210 (2011)

FIG. 1. Various chromospheric anemone jets observed by solar optical tele- scope in Ca IIH. Note that the footpoint of jet has bright, inverse Y-shape or anemone shape structure (Ref. 2).

corona but also in the photosphere and chromosphere which are penumbral microjets,11 and fibrils. Since the discovery of fully collisional and partially ionized7,8 (cf. Table II). chromospheric anemone jets,1 several reconnection features have been observed in those jets and these features are summarized below. II. DYNAMICAL PROCESSES AND RECONNECTION FEATURES OBSERVED IN CHROMOSPHERIC ANEMONE JETS A. Apparent velocity and energy estimates The Ca IIH filtergram images from SOT/Hinode The apparent velocity of chromospheric anemone jets is 1 revealed that the chromosphere is very dynamic and filled typically 5 20 km sÀ . The Alfveń speed in the low 9,10 À 1 with numerous jet-like structures such as spicules, chromosphere is estimated to be VA 10 kmsÀ (B/100 G) 15 3 0.5 Anemone jets are explained by magnetic(n/10 cmÀ reconnection)À , so the apparent velocity of chromospheric anemone jet is close to the local Alfveń speed. Since the • X-ray anemonecharacteristic jet is a coronal velocity jet, ejected of reconnection from jet is close to the local an “anemone-Alfvetypeń” active speed, region. the apparent velocity of jet is consistent with • Both observationsthe reconnectionand numerical model. simulations An estimate of total stored energy indicate that the(thermal eruptionsmagnetic) leading to at the the footpoint suggests that a generation of Xsmall-amount-ray jetsþ results of from energy, magnetic lying in the nanoflare regime reconnection in(10 the23 corona.1024 erg) is released due to reconnection1 (cf. Table • Magnetic structureI). ItÀ iscan interesting be stable to to explore a long time the relationship between ubiq- before the eruption. Mechanism leading to eruption/reconnectionuitous jets is and unknown. coronal heating, however, at present the num- 1 • Several possibleberscenarios of these jets had isbeen too smallstudied to explain in the coronal heating. numerical simulations: [Singh, Phys. Plasmas 2011] - Flux emergenceB. Reconnection and reconnection, generated Alfveń waves Coronal jets parameters: L~104-105km, - Plasma footprintAlfve motion.ń waves can be generated by magnetic V~ VA~ 300 km/s, life-time ~10-100 tA reconnection.8,12–14 In case of anemone jet, the Alfveń wave 2 (more exactly kink wave) propagating along the jet should be observed because the jet is believed to be formed along the pre-existing magnetic field. Nishizuka et al.15 observed a giant chromospheric anemone jet from SOT/Hinode and reported propagating Alfveń wave along the jet. The velocity of Alfveń 1 wave 200 km sÀ , transverse velocity amplitude 5 15 km 1 À sÀ and wave period of 200 s is observed and this is in ac- FIG. 2. Schematic diagram of Anemone Jets in (a) corona and (b) chromo- cordance with the simulation of chromospheric anemone jet sphere in two-dimensions (2D). The three-dimensional (3D) evolution of chromospheric anemone jet (c) before and (d) after the reconnection is which also shows generation and propagation of Alfveń waves shown. along the cool jet. No. 2] Solar Coronal X-Ray Jets 365 Flux emergence/reconnectionNo. 2] Solar Coronal X-Ra scenarioy Jets 369 Table 3. Models for the oblique-coronal-field case. Temperature Coronal Magnetic Field Time = &4.0 Model Grid

hot layer 7. 400 x 480 25 3?r/4 ^ (corona) 8. 200 x 240 25 7T 9. 200 x 240 25 157I-/16 10. 200 x 240 25 7TT/8 cool layer Downloaded from https://academic.oup.com/pasj/article-abstract/48/2/353/2948719 by guest on 12 July 2020 K /photosphere/ \ 11. 200 x 240 25 3TT/4 \ chromosphere' 12. 200 x 240 25 5TT/8 L-convection layer 13. 200 x 240 25 TT/2 200 x 240 25 3TT/8 Downloaded from https://academic.oup.com/pasj/article-abstract/48/2/353/2948719 by guest on 12 July 2020 0 / N Xmax 14. 15. 200 x 240 25 TT/4 Initially Perturbed Region 16. 200 x 240 25 TT/8 Magnetic Flux Sheet 0 20 40 60 80 0 20 40 60 80 0 20 40 60 80

Fig. 12. Simulation results for the case where the inclination angle is 6cor = TT/2, i.e. vertical for the oblique-coronal field Fig. 8. Schematic picture of the initial conditions in the case (model 13). The displayed quantities are the temperature T distribution (gray-scale plot), the magnetic field (solid Yokoyma and Shibata [PASJ 1996] lines), and the velocity field (arrows). The intensity levels are shown by a gray-scale color bar. The spacing between the oblique-coronal-field case. The angle 6COT is the in- created.field Figur lines ise 0.2 15 0of thshowe magnetis thc fluxe .one-dimensiona The scale of the velocitly vectordistribus is indicate- d by the arrow at the top-right corner (2Dclinatio simulationsn of the initia ofl corona coronall field. jets) The other no- tion at thof eace hcollisio panel for na velocitsitey. oAf V detaile= 5.0. d analysis of the jump tation is the same as in figure 2a. conditions in the colliding flow shows that this pressure • Flux emergence is due to magneticjum buoyancyp can Currenbe identifie instabilityt densitd wity h (Parkerthe fast-mode instability) MHD shock [Parker 1966]. (figure 9c)) r . / s[Ou y ryiyyyyT analysis. shows that the jump condi- (coarse) mes• h Fluxwas non-unifor emergencem in thande ^-directio reconnectionn with tion cans foleadr fas tto an and slow-mod anemonee MH shapeD shock ats ar footpointe satisfied of the x-ray jet. Az = 0.15 (0.3) for z < 35 and with increasing spacing within errors of a few tens of percent. This error may be from Az =• 0.15Issues: (0.3) to continuous Az = 0.375 (0.75 reconnection;) for z > 35. It 2Ddue pictureto the unsteadines - does snot and producecomplicated circular geometry patternof the in 3D? is uniform in the ^-direction with Ax = 0.2 (0.4). The shocks caused by its curvature, non-uniform flow distri- models examined for the oblique-field case are summa- bution and so on.] At the shock, the jet is compressed, 3 rized in table 3. and the flow changes direction to along the field. The flow is accelerated again by the enhanced gas pressure behind 4.2. Results the shock. This final ejection (called hot jet hereafter) is Figure 9 shows the simulation results of a typical model observed as an X-ray jet. The velocity of the hot jet is -1 (model 7) for the oblique coronal-field case. The initial V « 4 « 0.57 VA.cor (~ 48 km s ). The temperature is T « 40 « 1.6 Tcor ~ T /ß , where ß « 0.75 (see coronal field of this model is inclined (0cor = 37r/4), and COT COT COT the x-component of the field is anti-parallel to the flux discussion in subsection 3.2), and the density contrast sheet in the convection layer. between the jet and ambient plasma is small. The initial evolution of the emerging flux is almost In addition to the upward ejection of the hot jet, this the same as that in the horizontal-field case (see subsec- simulation also reveals the whip-like motion of a cool tion 3.2). The only difference is the formation site of the dense plasma (see figure 9b). The cool plasma involved

neutral point. In the present case, it is situated slightly in this motion is originally from the chromosphere, whic= h Fig. 13. Simulation results for the case where the inclination angle is 0COr ^74 for the oblique-coronal-field case I to the right of the exact top of the expanding loop. Other is carrieThd e uremaninp witg hnotatio then expandinis the same as gin loops figure 5. , and is ejected by physical processes concerning the current sheet, such as the sling-shot effect due to reconnection, which produces the formation and coalescence of magnetic islands and a whip-like motion. The velocity of this whip-like motion 1 the ejection of the island, are the same (figure 9). Af- is V = 4.0 - 7.0 (« 48 - 84 km s" ). The final configu- ter these processes, a clear 'X' shape structure of slow- ration of the cool plasma is a vertical collimated feature, mode MHD shocks is formed at the reconnection site, which may be observed as a cool jet. The cool jet and the as shown in the current-density plot of figure 9c. These hot jet are ejected side by side with each other (figure 9; slow-mode shocks are a characteristic feature of the fast t = 105), which may explain the simultaneous observa- reconnection model proposed by Petschek (1964). In fig- tion of ©an AstronomicaX-ray jet (hol Societt jet)y anof dJapa an nH a• Providesurge (cood byl jetthe) NASA Astrophysics Data System ure 10 we show the one-dimensional distribution across (Shibata et al. 1992b; Canfield et al. 1995). Note that the slow-mode shocks. Due to the tension force of mag- it has long been thought that the cool Ha surges could netic field lines, a pair of reconnection jets are ejected not be explained by magnetic reconnection, because re- from the neutral point. The velocity of the reconnection connection would heat any cool plasma to X-ray tem- jet is V « 7 A¿ VA,cor (~ 84 km s-1). One of these pair of peratures; hence, surges require a separate mechanism jets goes up and collides with the magnetic field lines at to accelerate the plasma without heating it at the same the side. At the collision site a high-pressure plasma is time. Our simulations now provide a possible physical

© Astronomical Society of Japan • Provided by the NASA Astrophysics Data System The Astrophysical Journal, 880:62 (13pp), 2019 July 20 The Astrophysical Journal, 880:62 (13pp), 2019 July 20 Meyer et al. Meyer et al.

Foot-print circular motion mechanism

• Simulations start with ‘flux emergence’ –like 2D configuration. • Due to circular motion at the base, magnetic field lines twist leading to kinking and the reconnection. • This mechanism does produce magnetic configuration similar to the dome-shaped Meyer et al. ApJ 2019 (0.6 and 4 full rotations at the base) (anemone) configuration, but several full rotations are needed to drive the reconnection – not very realistic.

Figure 3. Closed (magenta) and open (blue) magnetic field linesFigure viewed 3. inClosed the xz-plane(magenta at y)and=9 open Mm,( forblue(a))magneticPAD09, (fibeld) PAD10, lines viewed and (c in) PAD09R. the xz-plane For at(ay)and=9( Mm,b), for (a) PAD09, (b) PAD10, and (c) PAD09R. For (a) and (b), plots are shown at t=30 time steps (left, 0.6 rotations), t=70 (plotscenter, are 1.4 shown rotations at t—=after30 time onset steps of kink(left,), 0.6 and rotationst=200),(right,t=70 4( rotationscenter, 1.4). For rotations(c), PAD09R,—after onset plots of are kink), and t=200 (right, 4 rotations). For (c), PAD09R, plots are shown at t=0 time steps (left), t=70 (center), and t=200 (rightshown). Note at t that=0 PAD09 time steps and( PAD09Rleft), t= are70 ( identicalcenter), and at tt==0200 and (tright=30). Notetime steps, that PAD09 so plot and(a) left PAD09R and (c) are identical at t=0 and t=30 time steps, so plot (a) left and (c) left apply to both simulations. An animation of the xz-plane at lefty= apply9 Mm to for both the simulations.(a) PAD09, An(b) animationPAD10, and of( thec) PAD09Rxz-plane at simulationsy=9 Mm from for the time(a step) PAD09, 0 to 200(b) isPAD10, and (c) PAD09R simulations from time step 0 to 200 is provided online. provided online. (An animation of this figure is available.) (An animation of this figure is available.) function of time. Initially, the null steadily increasesfunction in height of as time. Initially,result from the including null steadily an explicit, increases non-ideal in height diffusion as result effect from(such including an explicit, non-ideal diffusion effect (such the jet dome expands. Once the kink occurs around thet= jet50 dome time expands.as in Once Pariat the et kink al. 2009 occurs). This around alsot means=50 time that the magnetofric-as in Pariat et al. 2009). This also means that the magnetofric- steps, the symmetry of the system is broken and thesteps, jet domethe symmetrytional of themodel system cannot is broken accurately and simulate the jet dome the dynamictional and model rapid cannot accurately simulate the dynamic and rapid begins to tip over. This also allows for changes in connectivitybegins to tip over.energy This also release allows of for an changes eruptive in connectivity event or instability;energy it can, release of an eruptive event or instability; it can, or “reconnection” to occur, similar to Pariat et al.or(2009“reconnection). In ”however,to occur, follow similar the to buildup Pariat ofetenergy al. (2009 within). In the simulationhowever, up follow the buildup of energy within the simulation up Rachmeler et al. (2010), the axis also kinks, but theirRachmeler FLUX et al. to(2010 the), point the axis at which also kinks, the instability but their forms. FLUX Figure to4(b the)(black point at which the instability forms. Figure 4(b)(black code models an ideal evolution, thus it does notcode allow models for anline ideal) shows evolution, the free thus magnetic it does energy not allow as a function for ofline time) shows for the the free magnetic energy as a function of time for the reconnection and there is no eruption. Since our codereconnection follows an and therePAD09 is no simulation. eruption. Since We do our see code a follows drop in an the freePAD09 magnetic simulation. We do see a drop in the free magnetic ideal evolution, and cannot accurately simulate reconnection,ideal evolution, we andenergy cannot after accurately the kink simulate instability reconnection, occurs, with we the initialenergy peak after in the kink instability occurs, with the initial peak in refer instead to “changing connectivity” when discussingrefer instead our to “freechanging magnetic connectivity energy occurring” when discussing around t= our70. Thisfree release magnetic of energy occurring around t=70. This release of magnetofrictional simulations. This changing connectivitymagnetofrictional is a energy simulations. takes longer This changing to onset than connectivity in the simulation is a ofenergy Pariat et takes al. longer to onset than in the simulation of Pariat et al. result of numerical diffusion, which is unavoidable inresult any offinite numerical(2009 diffusion,) due which to the is quasi-static unavoidable nature in any offinite the magnetofrictional(2009) due to the quasi-static nature of the magnetofrictional difference numerical model, but is much less than whatdifference would numericalmethod. model, but is much less than what would method.

5 5 The Astrophysical Journal, 875:10 (17pp), 2019 April 10 Nayak et al.

Can anemone magneticThe Astrophysical Journal, 875:10structure(17pp), 2019 April 10 be a spheromak?Nayak et al.

• Confined plasma can relax into Taylor-state spheromak with ∇×B = �B – is that relevant for astrophysics/solar physics? • Unconfined spheromak will expand indefinitely in the absence of external field. • In the ambient magnetic field, spheromak will be unstable to tilt instability. It will tilt (on few tA time scale) and reconnect. - Tilt instability occurs because the magnetic moment of a spheromak is anti-aligned with an external magnetic field. • Tilt instability can be reduced by line-tying [Finn, Reiman, PF1982; Hooper, Phys. Plasma 2009]. Can it be stabilized

by line-tying alone? [Nayak et al. ApJ 2019] (3D data constrained Figure 8. Left column shows close ups of the magnetic nulls (in pink) with corresponding MFLs and the MFLs for the QSL (in descending order). The right column is thesimulatio correspondingn top-down) view where the bottom boundary is superimposed with ln Q contours having lnQ {} 3, 6 . The footpoints tracing the Q-contours, which have large values, signify a sharp change in MFL connectivity. A region with QSL is characterized by its large Q-value. The presence of highly twisted MFLs—in yellow at the left panel, white at the right panel—which may constitute a ﬂux rope,5 is important.

The constants B0 and L0 are selected as the average magnetic uniform ambient state, satisfies an elliptic boundary value field strength and length-scale of the vector magnetogram. problem—generated by imposing the discretized incompres- sibility constraint (Equation 11(b)) on the discrete integral Furthermore, vBa 0 40 is the Alfvén speed and ρ0 is the form of the momentum equation (Equation 11(a));see constant mass density. The τa and τν are, respectively, Alfvén transit time (τ = L /v ) and viscous diffusion timescale Bhattacharyya et al. (2010) and the references therein. An a 0 a identical procedure with an ad hocauxiliarypotentialaddedto ( L 2 ). The kinematic viscosity is denoted by ν. The 0 the induction equation (Equation 11(c)) is employed to keep ratio τa/τν is an effective viscosity of the system which, along B solenoidal, see Ghizaru et al. (2010) and Smolarkiewicz & fl fl Figure 8. Left column shows close ups of the magnetic nulls (in pink) with correspondingwith theMFLs other and the forces, MFLs forin theuences QSL (in the descending magneto orderuid). The evolution. right column is Charbonneau (2013) for details. The following summarizes the corresponding top-down view where the bottom boundary is superimposed with ln EquationsQ contours having11(aln)–Q(d) {}are 3, 6 . solved The footpoints by using tracing the the Q well-contours, estab- which only important features of the EULAG-MHD whereas details have large values, signify a sharp change in MFL connectivity. A region with QSLlished is characterized magnetohydrodynamic by its large Q-value. The numerical presence of highly model twisted EULAG- MFLs—in yellow at the left panel, white at the right panel—which may constitute a flux rope, is important. are in Smolarkiewicz & Charbonneau (2013) and its MHD (Smolarkiewicz & Charbonneau 2013),whichisan bibliography. Central to the model is the spatio-temporally extension of the hydrodynamic model EULAG predominantly second-order accurate nonoscillatory forward-in-time multi- The constants B0 and L0 are selected as the average magnetic useduniform in atmospheric ambient andstate, climate satisfies research an elliptic(Prusa boundary et al. 2008 value). dimensional positive definite advection transport algorithm, field strength and length-scale of the vector magnetogram. Theproblem pressure—generated perturbation by imposing(p),aboutathermodynamically the discretized incompres- MPDATA (Smolarkiewicz 2006).TheMPDATAisprovento sibility constraint (Equation 11(b)) on the discrete integral Furthermore, vBa 0 40 is the Alfvén speed and ρ0 is the form of the momentum equation (Equation 11(a));see8 constant mass density. The τa and τν are, respectively, Alfvén transit time (τ = L /v ) and viscous diffusion timescale Bhattacharyya et al. (2010) and the references therein. An a 0 a identical procedure with an ad hocauxiliarypotentialaddedto ( L 2 ). The kinematic viscosity is denoted by ν. The 0 the induction equation (Equation 11(c)) is employed to keep ratio τa/τν is an effective viscosity of the system which, along B solenoidal, see Ghizaru et al. (2010) and Smolarkiewicz & fl fl with the other forces, in uences the magneto uid evolution. Charbonneau (2013) for details. The following summarizes Equations 11(a)–(d) are solved by using the well estab- only important features of the EULAG-MHD whereas details lished magnetohydrodynamic numerical model EULAG- are in Smolarkiewicz & Charbonneau (2013) and its MHD (Smolarkiewicz & Charbonneau 2013),whichisan bibliography. Central to the model is the spatio-temporally extension of the hydrodynamic model EULAG predominantly second-order accurate nonoscillatory forward-in-time multi- used in atmospheric and climate research (Prusa et al. 2008). dimensional positive definite advection transport algorithm, The pressure perturbation (p),aboutathermodynamically MPDATA (Smolarkiewicz 2006).TheMPDATAisprovento

8 Tilt stabilization by line-tying

Initial configuration • 3D MHD simulations used to study the n=1 tilt � mode for a spheromak line-tied to a conducting surface. • Initial conditions: low � (~0.02) GS solution with z � RB�~ � L/2 • Stability depends mostly on two parameters: R - elongation � = L/R (changed by changing �),

r - fraction of line-tied internal flux � = �tied/�0. • For stabilization conducting surface needs to be conducting surface placed very close to spheromak crossing the separatrix. • Stability thresholds are found using linearized simulations using HYM code.

6 Physics of Plasmas ARTICLE scitation.org/journal/php

to the plasma gun.22 In these simulations, the spheromak shape and stability properties were determined by the shape of the flux conserver, which had a small aspect ratio Lc=Rc 1:6 (corresponding to a tilt stable regime) and did not include the external field. Previous laboratory experiments of line-tied spheromaks25 studied the spheromak formation and expansion into a large vacuum chamber. Since the ambient field was not applied, these spheromaks were tilt stable, in contrast to the present study. For a solar case, the spheromak configuration of interest does not have up–down symmetry—there is only one line-tying boundary, the solar surface. In addition, this boundary cuts through the separatrix, creating a dome-like structure in the corona, while tying-up a part of the spheromak’s internal flux. In the simulations, this was modeled by placing the bottom of the simulation box very close to the spheromak, crossing the separatrix, and moving all other boundaries further away until the converged results were obtained. Two parameters were changed in a set of linearized 3D MHD simulations: (1) the elongation of the spheromak and (2) the amount of poloidal flux within the FIG. 3. The critical line-tying fraction v vs the elongation of the spheromak. Error spheromak separatrix that was line-tied to the solar surface. bars are the margin between zero-crossing-pointÃ and the adjacent data point’s v Figure 2 shows the calculated linear growth rate of the n 1 tilt value in Fig. 2. Curves of v e (blue) show approximate relation between elonga- ¼ tion and line-tying for a flux-emergenceð Þ scenario. mode against the line-tying fraction v (defined as wtied =w0, where wtied is the poloidal magnetic flux of the spheromak, which pierces the conducting boundary, and w0 is the total poloidal flux of the spheromak, simulations demonstrate that the spheromak configuration partially as in Fig. 1). The growth rate c is normalized by the Alfven time embedded into the solar surface can remain stable with respect to the sA R=vA,whereR is the separatrix radius and vA is the Alfven tilt as long as its elongation remains relatively small, and the line-tying speed. For each elongation of the spheromak (e 1:43; 1:69; 1:96; is sufficiently strong, and it will be destabilized if either its elongation ¼ 2:75), the growth rate of the n 1 mode was calculated for an ensem- is increased or the fraction of the line-tied flux is reduced below the ¼ ble of line-tying fractions ranging from v 0 to v 1, where a case of threshold. ¼ ¼ v 1 corresponds to a boundary crossing the spheromak in the mid- Since a magnetic configuration on the surface of the sun is not ¼ dle (through the magnetic axis). enclosed in a conducting box, the size of the simulation region was For each elongation, there was a particular amount of tied flux at increased to ensure that the results of our simulations converge. The which the spheromak transitioned from unstable to stable; this stabil- results of this test are summarized in Table I where the relative simula- ity threshold, v ,inFig. 3,is,inFig. 2, the point where the growth rate tion box size is represented by Rc=R (Rc is the radius of the simulation LargerÃ elongations require stronger line-tying for stability curve hits csA 0. The stability threshold v increases with the elon- boundary and R is the radius of the separatrix). The ratio of simulation ¼ Ã gation of the spheromak, ranging from 0.14 (e 1:4) to v 0:78 z axis length to separatrix length L roughly followed Rc=R.The ¼ Ã ¼ (for the largest elongation considered e 2:75). Thus, the linearized ¼ TABLE I. Convergence of threshold for expanding boundary. e L=R spheromak ¼ elongation; Rc radius of conducting boundary; R radius of spheromak; ¼ ¼ zc distance from the center of spheromak to upper axial boundary; L axial length¼ of• spheromak;Simulation and v critical box line-tying size fraction was for stability. increased¼ until the Ã ¼ converged results were obtained. e Rc=R 2zc=L v Ã 1.43• Spheromak 1.18 can be 1.37stable to tilt if 0.0 line-tying is 1.43sufficiently 1.50 strong. 2.09 0.13 1.43 2.23 3.11 0.14 1.69• For spheromak 1.18with 1.12large elongation 0.18 more line- 1.68 1.49 1.77 0.15 1.69tying is 2.23required for 2.65 stability: 0.15 1.96 1.21 1.78 0.25 1.96- for 1.61� = L/R ≲2, � 1.65*≈ 0.15-0.2, 0.23 1.92 2.40 2.50 0.21 2.75- for 1.06 � = 2.75, �*≈ 1.13 0.8. 0.52 2.75• Results 1.73are consistent 1.26with previous 0.74 studies of FIG. 2.GrowthCurves of therate growthof ratethe ofn=1 the n tilt1 tiltmode mode vsvs theline line-tying-tying fraction 2.75 2.59 1.88 0.78 v w =w , for different elongations of spheromak¼ e L=R. parameter,tied 0 �, for spheromaks with 4 different spheromak stability [Finn PF 1981; Bondeson PF 1981; elongations [Latham et al. PoP 2021]. Finn PF1982; Hooper, PoP 2009].

Phys. Plasmas 28, 012901 (2021); doi: 10.1063/5.0025136 28, 012901-3 Published under license by AIP Publishing

7 TiltPhysics stability of Plasmas – elongation vs lineARTICLE -tyingscitation.org/journal/php

to the plasma gun.22 In these simulations, the spheromak shape and • Spheromakstability can propertiesremain were stable determined due by the to shape line of the tying flux conserver, to which had a small aspect ratio Lc=Rc 1:6 (corresponding to a tilt the conductingstable regime) surface and did (solar not include surface) the external until field. Previous the labora- increase oftory spheromak experiments of line-tied elongation spheromaks 25and/orstudied the spheromak formation and expansion into a large vacuum chamber. Since the reduction ofambient the field fraction was not applied, of its these line spheromaks-tied flux were tiltwill stable, in contrast to the present study. make it unstableFor a solarto the case, thetilt spheromak instability configuration. of interest does not have up–down symmetry—there is only one line-tying boundary, the • Elongationsolarcan surface.increase In addition, due thisto boundaryflux cutsemergence, through the separatrix, but that wouldcreatingchange a dome-like structureline-tied in thefraction. corona, while tying-up a part of the spheromak’s internal flux. In the simulations, this was modeled by • placing the bottom of the simulation� box very� close to the spheromak, Approximatecrossing relation the separatrix, between and moving alland other boundariesas further away spheromak untilgrows the converged through results line were-tied obtained. flux Two injection parameters were can be obtained:changed in � a( set�) of = linearized 1 – a/ 3D� MHD(a=const). simulations: (1) the elongation of the spheromak and (2) the amount of poloidal flux within the FIG. 3. The critical line-tying fraction v vs the elongation of the spheromak. Error spheromak separatrix that was line-tied to the solar surface. barsLine are the-tying margin stability between zero-crossing-point threshold,Ã and �*, the vs adjacent elongation data point’s v • Another possibilityFigure 2 forshowscrossing the calculatedthe linearstability growth rate of the n 1 tilt value in Fig. 2. Curves of v�e �(blue) show approximate� relation between elonga- ¼ tion(red). and line-tying Relation for a flux-emergence ð (Þ ) = 1 scenario. – a/ is shown in blue. mode against the line-tying fraction v (defined as wtied =w0, where wtied threshold isisthe the poloidal reduction magnetic fluxof ofline the spheromak,-tying which(through pierces the con- [Latham et al. PoP 2021] reconnectionducting on boundary, the bottom). and w0 is the total poloidal flux of the spheromak, simulations demonstrate that the spheromak configuration partially as in Fig. 1). The growth rate c is normalized by the Alfven time embedded into the solar surface can remain stable with respect to the sA R=vA,whereR is the separatrix radius and vA is the Alfven tilt as long as its elongation remains relatively small, and the line-tying speed. For each elongation of the spheromak (e 1:43; 1:69; 1:96; is sufficiently strong, and it will be destabilized if either its elongation ¼ 2:75), the growth rate of the n 1 mode was calculated for an ensem- is increased or the fraction of the line-tied flux is reduced below the ¼ 8 ble of line-tying fractions ranging from v 0 to v 1, where a case of threshold. ¼ ¼ v 1 corresponds to a boundary crossing the spheromak in the mid- Since a magnetic configuration on the surface of the sun is not ¼ dle (through the magnetic axis). enclosed in a conducting box, the size of the simulation region was For each elongation, there was a particular amount of tied flux at increased to ensure that the results of our simulations converge. The which the spheromak transitioned from unstable to stable; this stabil- results of this test are summarized in Table I where the relative simula- ity threshold, v ,inFig. 3,is,inFig. 2, the point where the growth rate tion box size is represented by Rc=R (Rc is the radius of the simulation Ã curve hits csA 0. The stability threshold v increases with the elon- boundary and R is the radius of the separatrix). The ratio of simulation ¼ Ã gation of the spheromak, ranging from 0.14 (e 1:4) to v 0:78 z axis length to separatrix length L roughly followed Rc=R.The ¼ Ã ¼ (for the largest elongation considered e 2:75). Thus, the linearized ¼ TABLE I. Convergence of threshold for expanding boundary. e L=R spheromak ¼ elongation; Rc radius of conducting boundary; R radius of spheromak; ¼ ¼ zc distance from the center of spheromak to upper axial boundary; L axial length¼ of spheromak; and v critical line-tying fraction for stability. ¼ Ã ¼

e Rc=R 2zc=L v Ã 1.43 1.18 1.37 0.0 1.43 1.50 2.09 0.13 1.43 2.23 3.11 0.14 1.69 1.18 1.12 0.18 1.68 1.49 1.77 0.15 1.69 2.23 2.65 0.15 1.96 1.21 1.78 0.25 1.96 1.61 1.65 0.23 1.92 2.40 2.50 0.21 2.75 1.06 1.13 0.52 2.75 1.73 1.26 0.74 FIG. 2. Curves of the growth rate of the n 1 tilt mode vs the line-tying fraction 2.75 2.59 1.88 0.78 v w =w , for different elongations of spheromak¼ e L=R. tied 0

Phys. Plasmas 28, 012901 (2021); doi: 10.1063/5.0025136 28, 012901-3 Published under license by AIP Publishing Physics of Plasmas ARTICLE scitation.org/journal/php

simulations were performed for four different values of elongation, where the elongation was changed by changing the spheromak toroi- r dal field profile taken in the form: RB/ w . For approximately the same value of the spheromak elongation, the critical line-tying fraction v was found to converge as the box size increased. This convergence indicatesÃ that the simulation results are valid for sufficiently large simulation box, approaching a half-infinite domain. IV. NONLINEAR 3D SIMULATIONS 3D nonlinear simulations have been performed showing the evolution of the tilt instability in the case of an unstable spheromak. The high resolution simulations (grid size is nz nr n/ 240 190 64) include the nonlinear interaction of allÂ the toroidalÂ ¼ har- monics.Â Â Simulations were performed for spheromaks of the elongation e 1:67 and the line-tied fraction v 11%. (For this elongation, the ¼ ¼ stability threshold v is 15%, so this case is linearly unstable.) The nonlinear simulation wasÃ allowed to run until the spheromak was tilted at a relatively large angle, when the spheromak plasma and current came into contact with the conducting surface. The simulation results that show the earlier stages of tilting are shown in Figs. 4 and 5(b). Figure 4 shows the contour plot of the toroi- CHAPTER 2. SIMULATION RESULTS dal component26 of the plasma current, with blue/red color indicating CHAPTER 2. SIMULATION RESULTS the direction26 in/out of the page. The contour plot is drawn in the plane Nonlinear evolutionof the tilt. Black arrowsof representthe flow velocity,tilt with leads a maximum value to top reconnection CHAPTER 2. SIMULATION RESULTS 26

B, Jϕ V, Jϕ • Nonlinear simulations of tilt unstable spheromak show formation of current sheet at the

(a) (t=550.0) top, where magnetic field of (a) (t=550.0) tilted spheromak opposes the (a) (t=550.0) FIG. 5. Field lines of tilting spheromak. Field lines were chosen to help visualize thebackground separatrix. An internal flux magnetic surface near the separatrix field. is covered by the multi- color (mostly green) field line. In blue, curving around the spheromak and extending upwards are the field lines of the ambient magnetic field. Figures drawn with • VisItLeft program, half from of simulation the data. spheromak, (a) depicts the field lines which at t 0, while (b) depicts field lines at the same time as Fig. 4. ¼ tilts upwards, flows into the of v 0:039v , where the Alfven velocity is defined in terms of the current¼ A sheet and forces the (b) (t=700.0) magnetic field at the center of the spheromak and the background density.reconnection. The tilting motion is clockwise. The tilting of the spheromak (b) (t=700.0) is also apparent in Fig. 5, where the 3D magnetic field lines of the (b) (t=700.0) • spheromakTwo models are plottedof for tworesistivity time frames ofwere the same non-linear simulation as Fig. 4. An internal flux surface is covered4 by the multi- colorused: (mostly1) green) uniform field line. In with blue, curving S~10 around, and the spheromak and extending upwards are the field lines of the ambient magnetic FIG.Contour 4. Contour plotplots of plasmaof current,toroidal where thecurrent colors indicate and the vector current direc- field.2) anomalous resistivity model as tion in or out of the page. Black arrows represent flow velocity. Current sheet forma- When the spheromak tilts, the magnetic field lines from the top tionplots is visible of at(a) the magnetic top, close to thefield separatrix. and Simulation (b) velocity parameters at are ofin the [ tiltedYokoyama spheromak opposePASJ the1996; external Sato field lines, and resulting in an e 1:67; v 11%. The close-up on the current sheet by the separatrix shows appearance of the reconnection layer. The current sheet formation is that¼different the left half¼ of times. the spheromak, Tilting which tiltsmotion upwards, flowsis clockwise. into the current sheet Hayashi PF 1979]. (c) (t=800.0) and forces the reconnection between the spheromak’s magnetic field and the back- visible at the top of the spheromak close to the separatrix in Fig. 4.The (c) (t=800.0) ground field. It should be noted that the current sheet appears at the left hand side close-up on the current sheet shows that the left half of the spheromak,9 (the opposite side of tilting) of the major axis of the inner spheromak. which tilts upwards, flows into the current sheet. This tilting motion (c) (t=800.0)

Figure 2.10: Simulation of a line-tied spheromak tilting. Shown in color are the contour plotsFigure of 2.10: toroidalSimulation current. of Arrows a line-tied are spheromak magnetic field. tilting. Starting Shown in in color the first are the frame contour and Phys. Plasmas 28, 012901 (2021); doi: 10.1063/5.0025136 28, 012901-4 veryFigureplots visible of 2.10: toroidal inSimulation the current. last frame, of Arrows a line-tied current are sheetspheromak magnetic formation field. tilting. is Starting visibleShown closein in color the to first are the the separatrix.frame contour and Published under license by AIP Publishing Simulationplotsvery visible of toroidal parameters in the current. last are: frame, Arrows✏ =1 current.67, are sheet=6.5% magnetic formation (For field. this iselongation Starting visible close in✏ =1 the to.67, first the the separatrix.frame stability and thresholdverySimulation visible parameters inis the 8.3%, last so are: frame, this✏ is=1 current an.67, unstable sheet=6.5% spheromak.) formation (For this iselongation visible close✏ =1 to.67, the the separatrix. stability ⇤ Simulationthreshold parametersis 8.3%, so are: this✏ is=1 an.67, unstable=6.5% spheromak.) (For this elongation ✏ =1.67, the stability ⇤ threshold is 8.3%, so this is an unstable spheromak.) ⇤ Tilting motion pushes spheromak field vs external field

B, Jϕ V, Jϕ

Contour plots of toroidal current and vector plots of (a) magnetic field and (b) velocity. 3 Tilting motion is clockwise. Solar surface is modelled as a dense cold plasma with n= 10 ncorona.

10 Later stages of instability show outflows with v∼vA

Vector plots of plasma flow; lines are integrated (Bz, BR) streamlines. Tilting motion is clockwise. Contour plot of velocity normalized to VA show outflow velocities up to V ≈VA.

0.0 4.4E-01 8.8E-01 1.3E+0

11 Summary

• It has been demonstrated that a spheromak can exist in a stable form when line-tied to a single conducting surface (solar surface). • Marginal amount of line-tying will stabilize the tilt instability in an oblate spheromak (more line-tying needed for very elongated spheromak). • Stable spheromak can be destabilized due to 1) elongation, 2) flux emergence or 3) reduction of line-tying (flux detachment via reconnection at the bottom). • In the tilt unstable spheromak, magnetic reconnection occurs at the top of dome leading to ejection of a plasma jet.