Tests of Dynamical Flux Emergence as a Mechanism for CME Initiation
aa r X i v : . [ a s t r o - ph . S R ] A ug Tests of Dynamical Flux Emergence as a Mechanism for CME Initiation
James E. Leake
U.S. Naval Research Lab 4555 Overlook Ave., SW Washington, DC 20375.College of Science, George Mason University, 4400 University Drive, Fairfax, Virginia [email protected]
Mark G. Linton
U.S. Naval Research Lab 4555 Overlook Ave., SW Washington, DC [email protected] andSpiro K. Antiochos
NASA Goddard Space Flight Center, Heliophysics Division, NASA/GSFC, Greenbelt, MD [email protected]
ABSTRACT
Current coronal mass ejection (CME) models set their lower boundary to be in the lower corona.They do not calculate accurately the transfer of free magnetic energy from the convection zone tothe magnetically dominated corona because they model the effects of flux emergence using kine-matic boundary conditions or simply assume the appearance of flux at these heights. We test theimportance of including dynamical flux emergence in CME modeling by simulating, in 2.5D, theemergence of sub-surface flux tubes into different coronal magnetic field configurations. We inves-tigate how much free magnetic energy, in the form of shear magnetic field, is transported from theconvection zone to the corona, and whether dynamical flux emergence can drive CMEs. We findthat multiple coronal flux ropes can be formed during flux emergence, and although they carry someshear field into the corona, the majority of shear field is confined to the lower atmosphere. Lessthan 10% of the magnetic energy in the corona is in the shear field, and this, combined with the factthat the coronal flux ropes bring up significant dense material, means that they do not erupt. Ourresults have significant implications for all CME models which rely on the transfer of free magneticenergy from the lower atmosphere into the corona but which do not explicitly model this transfer.Such studies of flux emergence and CMEs are timely, as we have new capabilities to observe thiswith Hinode and SDO, and therefore to test the models against observations.
Subject headings:
CMEs, Flux Emergence, MHD
1. INTRODUCTION1.1. CME modeling
Coronal mass ejections (CMEs) and eruptive flares are the most energetic manifestations of solar ac-tivity, with a typical CME accelerating 10 g of coronal plasma to speeds sometimes exceeding 1000 km/s(Gosling et al. 1976; Hildner 1977). It is now generally accepted that these giant eruptions of solar plasma andfield are due to the explosive release of magnetic energy stored in the corona prior to the event. Furthermore,the free energy for CMEs/flares is believed to be stored in the strongly sheared flux of filament channels (see,e.g, reviews by Forbes (2000); Klimchuk (2001); Linton & Moldwin (2009)). This flux is sheared in that that thefield lines are mainly parallel to the photospheric neutral line, rather than perpendicular as would be expectedfor a potential field. All CMEs/flares are associated with filament channels and these are the only locationsin the corona where the magnetic field exhibits strong non-potentiality. The basic picture of a CME is that, 2 –prior to the event, there is a force balance in the corona between the upward magnetic pressure of the shearedfilament channel field and the downward tension of overlying unsheared field. As the shear builds up or theoverlying field evolves, this force balance eventually disrupts producing an explosive outward expansion of thefilament channel flux and some of the overlying flux.It is evident from this discussion that the two central issues in understanding CMEs/eruptive flares arethe process by which magnetic shear forms in filament channels, and the mechanism that disrupts the forcebalance. These two issues have been at the heart of CME and flare research for decades and are the featuresthat distinguish the various CME models. The first issue of the shear formation process is especially critical,because the disruption mechanism is almost certain to depend on the topology of the filament channel field,which must be determined by the formation process. For example, some models invoke kink-like instabilities,which requires the formation of magnetic twist as well as shear in the filament channel field. We emphasize,however, that all the models and all the observations agree that the bulk of the magnetic free energy is in theform of shear. Note also that the issue of shear formation has broad implication for a number of important solarphysics problems, in particular the origin of prominences and filaments.There are only two general processes that can produce the observed shear in filament channels: either thefield emerges from below in a non-potential, sheared state, or the photospheric motions shear the field after itemerges into the corona. Of course, both processes must be present to some extent in the real corona. Thelatter process is likely to dominate in non-active region filament channels, such as those associated with high-latitude quiescent filaments, because these form well after the flux has emerged. On the other hand, in activeregions, especially in the strong complex regions that are the source of fast CMEs, the filament channels formwith the active region and, hence, the shear must emerge with the flux. Given that the shear formation is thefundamental driver of the eruption, it is clear that flux emergence must be explicitly included in any model forfast CMEs in order for the model to be physically rigorous.One such model, which generates fast CMEs from sheared filaments, but which should be modified toinclude the creation of these sheared filaments via flux emergence, is the magnetic breakout model (Antiochos1998; Antiochos et al. 1999). This model exploits the finite resistivity in the solar atmosphere to disrupt the pre-CME force balance by generating reconnection external to the filament channel. This reconnection reconfiguresthe magnetic field and allows the eruption of the highly sheared structure. The breakout model thus requiresa multipolar flux distribution with a separatrix where reconnection can occur. Numerical simulations haveinvestigated this model in both 2.5D, where the domain is axisymmetric and two dimensional but where allthree components of vector variables evolve (Lynch et al. 2004; MacNeice et al. 2004; DeVore & Antiochos 2005)and 3D, where the domain is three dimensional (Lynch et al. 2008; DeVore & Antiochos 2008). The basic modeluses a current-free quadrupole magnetic field configuration, consisting of a central active region dipole embeddedin an anti-parallel global dipole field, with a null point separating the two systems. This is shown in Figure1(a), taken from Lynch et al. (2004). Figure 2(a) shows, for this experiment, the vertical force balance in thecentral arcade along the latitude = 0 ◦ symmetry line for the initial equilibrium. The gravitational ( ρ g ) andgas pressure ( −∇ P ) forces balance, as do the magnetic tension ( B . ∇ B / π ) and magnetic pressure (- ∇ B / π ).Shear flows are applied at the lower boundary, within the central arcade, to generate magnetic field parallel tothe active region neutral line (in our 2.5D simulations the shear field will be the magnetic field in the ignorabledirection). Figure 2(b) shows the vertical forces after a short period of shearing. The presence of shear fieldexerts an upward magnetic pressure on the overlying field and stretches the original null point into a currentsheet, as shown in Figures 1(b) and (c). Reconnection occurs when the current sheet collapses below grid scaleand overlying field is removed by this reconnection, which allows further expansion of the central arcade. Theoutward expansion of the sheared arcade occurs exponentially as more and more overlying field is reconnected.Flare-like reconnection underneath the sheared dipole creates a disconnected flux rope which can then escape,as seen in Figures 1(d)-(f), at speeds comparable to observations of a fast CME.There are numerous other CME models which either assume the appearance of sheared magnetic flux at thelow corona, or use driving velocities to create sheared structures from coronal equilibria. Chen & Shibata (2000)and Dubey et al. (2006) simulated a 2.5D CME model where a pre-existing coronal flux rope, constrained byoverlying field, is allowed to escape when kinematically imposed emerging flux causes reconnection in a filament 3 –channel beneath the flux rope. They also found that CME eruptions occur when kinematic flux emergenceis used to drive reconnection outside the filament channel. The flux cancellation model (Amari et al. 2000,2003, 2007) is another model based on reconnection, but this model uses both boundary imposed shearing andthe cancellation of magnetic flux at the neutral line of filament sites to initiate a CME. Fan & Gibson (2007)developed a CME model where the lower corona is driven by the kinematic emergence of a sheared flux ropeinto a pre-existing coronal field. Here, loss of equilibrium via the kink or torus instability leads to an eruption,with the strength of the overlying field being an important constraint on the eruption. The breakout model hasalso been extended to cover different magnetic field configurations, and also to include a solar wind, in both2.5D and 3D simulations (van der Holst et al. 2007; Zuccarello et al. 2008; Soenen et al. 2009). Other CMEmodels exist which do not directly rely on flux emergence, such as those based on the loss of equilibrium of acoronal flux rope (Titov & D´emoulin 1999; Roussev et al. 2003; Kliem & T¨or¨ok 2006), and simulations of therise of a pre-formed coronal flux rope into the solar wind by Zhang & Wu (2009). Although not relying on fluxemergence directly, these models do assume that the flux ropes have already been formed in the corona andreached an equilibrium with the background coronal field, a state which is most likely caused by the emergenceof magnetic field from beneath the surface.All these models, constrained by the need to extend the simulation domain to at least a few solar radii,do not model the lower solar atmosphere. The lower boundary of these simulations has a typical density of3 × − g/cm and a typical plasma- β < . β = pB / π . (1)Based on the VAL model of the solar atmosphere (Vernazza et al. 1981) the photosphere, in contrast, has adensity of around 3 × − g/cm and β >
1. These CME initiation models simply assume the appearance offlux at the low density, low β corona and do not self-consistently calculate a process for the flux emergence.These simulations do not, therefore, address the critical question of whether and how newly emerging, shearedmagnetic flux can rise from its origins in the high β convection zone to the low corona where it is required todrive CME models. This is an important question to address now, as Hinode and SDO scientists are currentlymaking detailed observations of flux emergence with vector magnetograms, which will allow the comparison ofobservations and state of the art CME models. The current state of the theory of the formation of active regions suggests that dynamo actions in the solarinterior create twisted, buoyant flux tubes which rise through the convection zone, intersect the photosphereand create the observed sunspots and bipolar active regions (Parker 1955; Zirin 1970; Schuessler 1979; Parker1988; Zwaan 1987; Emonet & Moreno-Insertis 1998; Fisher et al. 2000). The subsequent expansion of theseflux tubes into the corona has been a subject of vigorous research for the past 30 years. The most likelydriver of flux emergence into the lower corona is an ideal MHD instability driven by magnetic buoyancy (themagnetic buoyancy instability), a particular mode of which was first suggested by Shibata et al. (1989a,b) in2D simulations using a crude model atmosphere and magnetic field configuration. They found that the rise ofmagnetic flux into the corona due to this instability matched observed rise velocities of magnetic flux.The process of flux emergence has been studied more recently in 3D simulations with a more detailedmodel atmosphere that includes a convection zone, isothermal photosphere/chromosphere, transition region, andisothermal corona, with a buoyant twisted flux tube embedded in the convection zone (Fan 2001; Manchester et al.2004; Archontis et al. 2004; Fan 2009). The reader is directed to a comprehensive review of this subject byArchontis (2008). These simulations show that as the flux tube intersects the photosphere, bipolar active re-gions are formed which exhibit shear flows along their neutral line. As the tube continues to emerge into thecorona, sigmoid-like structures are formed consistent with observations, with magnetic field being created par-allel to the neutral line (shear field). This suggests that flux emergence may be able to provide magnetic shearfield sufficient to drive CMEs, but it is unclear how this magnetic shear emerges into the corona, as the axis of 4 –the emerging flux tubes remains rooted near the surface.Manchester et al. (2004) showed that shearing motions associated with the emergence are able to createa sheared coronal flux rope which rises out of the simulation domain, although the authors do not associatethis with an eruption because the rise speed is limited by the amount of available ’axial flux’ to drive aneruption. Note that in this paper we use the term flux tube to refer to the sub-surface tubes which we initiateour simulations with, and flux rope to refer to the twisted structures formed in the corona. 3D simulations(Fan 2009) have shown that without a pre-existing coronal field, the rise of coronal flux ropes created duringflux emergence is constrained by the expanding emerging field of the original flux tube, or envelope field. In asimilar manner, the work of Archontis & Hood (2008) and MacTaggart & Hood (2009a) showed that flux ropesformed from multiple flux tube emergence are ultimately trapped by the envelope fields. For an eruption tooccur, the overlying tension of the envelope field must be removed.Archontis & T¨or¨ok (2008) added a pre-existing horizontal coronal field to their flux emergence simulations.They found that a coronal flux rope could be formed by inflows and horizontal shearing motions which drivereconnection within the expanding volume of the flux tube. They also found that reconnection between theenvelope field and the pre-existing field reduced the overlying tension of the envelope field and allowed theflux rope to erupt. MacTaggart & Hood (2009b) found a similar result, but used a toroidal sub-surface fluxtube which was already arched towards the surface, which allowed the flux tube axis to emerge bodily intothe atmosphere. Archontis & Hood (2010) showed that the likelihood of a coronal flux rope escaping whenreconnection occurs between the ambient coronal field and the envelope field depends strongly on the strengthof the ambient field.Flux emergence simulations such as these occur over timescales of 1000 s, much quicker than timescalesof CME models, and do not extend high enough in the corona to allow a full comparison with CME modelsand observations. The development of coupled models which span the timescales and spatial scales of both fluxemergence and CME eruption is very important to fully understand the initiation of these explosive events.In this work we investigate whether magnetic flux emergence, in a 2.5D cartesian geometry, can provideenough magnetic shear energy to drive CME initiation, with focus on the initiation mechanism of the magneticbreakout model. Figure 2(c) shows the energy in the shear magnetic field (normalized to the energy in the totalfield) in a typical breakout model simulation as shearing is applied at the surface. Also shown is the total kineticenergy in the system. As the inner arcade is sheared at the β = 0 . β = 0 . β corona. This issue is key to understanding how to couple dynamical flux emergence simulations andcurrent CME simulations. We will investigate this using simulations of the evolution of sub-surface flux tubeswith a variety of tube profiles as they emerge into both field-free corona and corona with pre-existing fields. Wewill also investigate whether flux emergence can drive CME initiation in these configurations.This paper is structured as follows: § § §
2. MODEL DESCRIPTION2.1. Numerical Method
We model our system of a magnetized plasma with a polytropic equation of state (polytropic index γ = 5 / LareXd (Arber et al. 2001). The equations solved are presented below in Lagrangian form, using Gaussian units:
DρDt = − ρ ∇ . v (2) D v Dt = − ρ ∇ P + 1 ρ j ∧ B c + g + 1 ρ ∇ . S (3) D B Dt = ( B . ∇ ) v − B ( ∇ . v ) − c ∇ ∧ ( η j ) (4) DǫDt = − Pρ ∇ . v + ηj + ς ij S ij − ǫ − ǫ ( ρ ) τ (5)The gas density, pressure, and internal specific energy density are denoted by ρ , P , and ǫ respectively andare defined at the center of each numerical cell. The magnetic field, denoted by B , is defined at cell faces, j = c ∇ ∧ B / π is the current density, and c is the speed of light. The velocity, v , is defined at cell vertices.This staggered grid preserves ∇ . B during the simulation. The gravitational acceleration is denoted by g , ν isthe viscosity, set to 2 . × g . cm − s − , η is the resistivity, and S is the stress tenor which has components S ij = ν ( ς ij − δ ij ∇ . v ), with ς ij = ( ∂v i ∂x j + ∂v j ∂x i ) . The four equation are closed with a simple equation of staterepresenting an ideal gas: P = ρRT , where R is the gas constant.Equation 5, minus the last term, ( ǫ − ǫ ( ρ )) /τ , is similar to that used in the majority of flux emergencesimulations to date (Archontis 2008). When flux tubes emerge and expand into the corona, the associated coolingdue to the pressure terms can lead to unrealistic temperatures in the corona, as shown by Leake & Arber (2006).Leake & Arber (2006) also showed that by including a relaxation term, such as the last term in Equation 5,this unrealistic cooling could be avoided. This approach is a simple way to model the effects of terms that weare currently unable to formulate numerical equations for (such as chromospheric/coronal heating), and termssuch as thermal conduction and radiative losses which are beyond the scope of this particular numerical code.The term ( ǫ − ǫ ( ρ )) /τ relaxes the specific internal energy density back to its initial equilibrium values ǫ ( ρ ) ona timescale τ which scales with density: τ ( y ) = 23 (cid:18) ρ ( y ) ρ ( y = 0) (cid:19) − . s . (6)The equations are solved in 2.5D: the simulation box is 2D, with x and y being independent variables and z being ignorable, but all three components of the vector variables are evolved. The simulation box extendsfrom -3 Mm to 90 Mm in the vertical direction ( y ) and from -45 Mm to 45 Mm in the horizontal direction ( x ).This is much larger than most flux emergence simulations typically cover (Manchester et al. 2004; Fan 2009),but is designed to allow expansion of the flux tubes into coronal field as far as possible. The numerical grid isstretched in both x and y to provide better resolution in areas of interest. The grid spacing, δ , is smallest at thesurface, where δ = 0 . δ = 0 .
15 Mm.In the majority of previous flux emergence simulations either ideal MHD is used ( η = 0), or η is chosento be a constant such that the diffusion due to this explicit resistivity is greater than the numerical diffusionin the code. Theoretical evidence suggests that when the electron fluid flow speed, v e , exceeds the phase speedof the ion-acoustic model, c ia , then ion-acoustic turbulence has a strong effect on current sheet development(Bychenkov et al. 1988). In this case an effective formula for anomalous resistivity would be η = η max (cid:18) , v e c ia − (cid:19) (7) 6 –which we can rewrite, using v e ∼ | j | /ne , as η = η max (cid:18) , | j | j crit − (cid:19) (8)where j crit = nec ia . We choose η = 1 . × − s, and j crit = 2 . × statampere . cm − , and this anomalousdiffusion exceeds the numerical diffusion in the code. This approach has been used in MHD simulations of kinkinstabilities of coronal loops (Arber et al. 1999; Gerrard et al. 2001). The initial stratification is a simple 1D model of the temperature profile of the Sun which includes the upper3 Mm of the convection zone, a photosphere/chromosphere, transition zone, and corona. The temperature profileis given by T ( y ) = ( T ph − | g | R ( m +1) y, y ≤ T ph + ( T cor − T ph )2 h tanh (cid:16) y − y tr w tr (cid:17) + 1 i , y > y ≤
0) is a linear polytrope which is marginally unstable to convection, with m = γ − being the adiabatic index for a polytrope. The temperature in the photosphere and chromosphere isassumed to be constant, T ph = 5700 K, as is the temperature in the corona, T cor = 8 . × K. The heightof the transition region is y tr = 3 .
75 Mm, and its width is w tr = 0 .
75 Mm. The density and gas pressure arespecified by initially assuming hydrostatic equilibrium. The resulting stratification is shown in Figure 3(a).
We insert a cylindrical magnetic flux tube into the model convection zone at a height of y t = − . B z = B e − r /a (10)where r = p x + ( y − y t ) is the radial distance from the center of the tube, B = 7800 G is the axial fieldstrength at the center ( r = 0), and a = 0 . B θ in the cylindrical coordinate system, B x and B y in cartesian). A minimumamount of twist is required to play the role of surface tension and keep the flux tube coherent as it rises in theconvection zone (Emonet & Moreno-Insertis 1998). We denote the twist by Θ, defined asΘ = B θ rB z . (11)The three twist profiles we use are given byTube 1 : Θ ( r ) = c (12)Tube 2 : Θ ( r ) = c (cid:16) − e − r /c (cid:17) (13)Tube 3 : Θ ( r ) = ce − r /c (14)where c = 1 /a , c = 0 .
15 Mm and c = 1 . B . ∇ ) B π = (cid:0) − B θ /r (cid:1) ˆ e r π . (15)The magnetic field, twist (Θ), and tension profiles for the three tubes are shown in Figure 4. Tubes 1 and 3 havesimilar twist and tension profiles. Tube 2 has lower tension near its axis due to the lower magnitude of B θ closeto the center of the tube. From their 3D simulations, Murray & Hood (2008) concluded that the twist profilewas not as important a factor in the emergence of the flux tubes as the axial field strength, which affected therate of emergence but not the extent of the emergence. The purpose of choosing these three tubes in our studyis to cover a range of flux tube behavior whilst keeping the potential amount of shear energy provided by fluxemergence optimal.The plasma β at the center of the flux tubes is approximately 10, and the flux tubes are all non–force-free. To initiate a buoyant flux tube we perturb the background gas pressure by an amount p such that( ∇ p ) r = ( j ∧ B ) r , which can be integrated to give p ( r ) = − B ( r )8 π + Z ∞ r B θ (4 πr ) dr (16)so that the tube is in radial force balance. Assuming that the flux tube is in thermal equilibrium with itssurroundings makes the tube less dense that the surrounding plasma and initiates its buoyant rise to thesurface.We will investigate the emergence of these tubes first into a field-free corona and then into differentquadrupole and dipole coronal field configurations to test how efficient flux emergence is at providing mag-netic shear energy to the corona. After investigating the emergence of flux tubes into a field free corona, we investigate the interactionof coronal field and emerging flux within the context of magnetic breakout. To do this we first impose aquadrupole coronal field above the magnetic flux tube. We construct the quadrupole as in Karpen et al. (1996)by first defining the magnetic vector potential along a source surface at y = − . M m (which is outside thecomputational domain): A z ( x, − . q d ( x h − ( x/x a ) i , ≤ | x | ≤ x a , | x | > x a (17)with x a = 21 Mm being the horizontal extent of vertical qaudrupole field at this height. The vector potentialin the interior of the domain is then determined by integration of Laplace’s equation with Equation 17 as thelower boundary condition and A z = 0 at y = ∞ . The quadrupole consists of an inner dipole and an overlyingdipole of opposite orientation, separated by a null point at y ∼ x = 0 line, is a certain fractionof the horizontal flux contained between the center of the sub-surface flux tube and the null point. We choose q d to be [11.7, 5.85, 3.9] × Gcm, which give quadrupole surface strengths at x = x a / β at certain heights above the surface. The curves in Figure 3(b) shows the β profilesfor the configurations of a sub-surface flux tube plus three different strength quadrupoles. These profiles aregenerally consistent with β profile models of active regions developed by Gary & Alexander (1999) and Gary(2001) which take into account Soft X-ray Telescope limb observations. While emerging a flux tube into a quadrupole is the first step to coupling flux emergence and the breakoutmodel, we also investigate a more self-consistent configuration, with a flux tube emerging into a backgrounddipole field. By using a dipole with opposite orientation to the upper half of the flux tube, reconnection betweenthese two systems creates a quadrupole structure self-consistently.The dipole field is represented by the vector potential A = A z e z where A z ( x, y ) = d y − y d r , (18)with r = p x + ( y − y d ) being the distance from the source. We chose y d to be -15 Mm so that the fluxtube is far from the source of the dipole. To cover various dipole strengths we pick a range of d = [18, 9,4.5] × Gcm which gives a magnetic field strength at x = 0, y = 0 of [105, 53, 26] G respectively. In a similarfashion to the quadrupole field, we choose these values so that the horizontal flux contained between the axisof the flux tube and the separatrix which separates the flux tube and the dipole is a certain fraction of thehorizontal flux above this separatrix. With decreasing dipole field strength, these factors are [1, 0.5, 0.25]. The β profile for the three dipole choices overlying a sub-surface flux tube are shown in Figure 3(b). These choicesof dipole strength allow for a range in the β profiles, but are still consistent with the models of β in the solaratmosphere developed by Gary & Alexander (1999) and Gary (2001).
3. RESULTS3.1. Initial Evolution in the Convection Zone and Lower Atmosphere: Effect of Twist Profiles
The evolution of the flux tubes in the convection zone is similar to previous simulations of rising flux tubes(Emonet & Moreno-Insertis 1998; Magara 2001; Leake & Arber 2006). The initial buoyancy causes the flux tubeto rise to the surface, during which time the cross-section increases and flux conservation decreases the axial, orshear, field strength. As the flux tube meets the convectively stable photosphere its rise is halted and horizontalexpansion spreads the flux tube out to form a contact layer with the plasma above. The density above thislayer is higher than the density below due to the concentration of magnetic field and the layer is unstable to aRayleigh-Taylor like instability known as the magnetic buoyancy instability (Newcomb 1961; Athay & Thomas1961; Acheson 1979). The stability of the contact layer involves a competition between the destabilizing gradientin the magnetic field and the stabilizing sub-adiabatic temperature gradient (Gilman 1970). The stabilizing termis dependent on the local β , and as the center of the tube rises up to this contact layer, the local β falls andthe instability allows the upper portion of the flux tube to expand into the atmosphere, as in Archontis et al.(2004).As can be seen in Figure 4(c), the choice of twist profile affects the amount of tension in the tubes. The 9 –increasing twist tube (Tube 2) has relatively less tension near its axis than the constant twist tube (Tube 1)and the decreasing twist tube (Tube 3), and also a higher buoyancy at its center. This difference distinguishesthe evolution of the increasing twist tube from the other two, and from now on we refer to the low tensioncase to mean Tube 2 and the high tension case to mean Tubes 1 and 3. In Figure 6 we can see how the choiceof twist profile affects the initial expansion into the field-free corona. The tubes act nearly identically as theyrise buoyantly in the convection zone and also in the lower atmosphere, as the magnetic buoyancy instabilitydevelops and the outer fieldlines extend into the corona. However, the two cases then differ in how the centerof the tube reacts. In the high tension cases, the lower fieldlines near the center of the tube remain near thesurface, shown in Figure 6(d), and the shear field is concentrated at the center, creating a single neutral line. Inthe low tension case, the decreased tension allows the center of the flux tube to rise, forming a crescent shape,shown in Figure 6(h). During this evolution, the mass in the center of the tube drains to the lowest possiblelocation which is at either end of this crescent shape. The coupling of shear field and density in this 2.5Dsimulation means that the shear field also concentrates in these regions. Note that this emerging structure nowhas three neutral lines at the photosphere. The center of the high tension flux tube remains at the surface for the duration of the simulation. Wefollow the continued evolution of the low tension flux tube in Figure 7, which shows the shear field and thelog of density, along with magnetic fieldlines. The deformation of the center of the tube into a crescent shapeforms a current sheet as field of opposite sign is brought together on either side of this crescent. This enhancedcurrent triggers the anomalous resistivity and reconnection sets in. As can be seen from Figures 7(b) and (e),the reconnection sites are at either side of the apex of the current sheet arch. Plasmoids are formed whichare expelled along the current sheet. The half width of the current sheet, l , is approximately 0.15 Mm, anda typical distance between the plasmoids is 5 Mm. Thus the ratio of half current sheet width to the lengthbetween islands, λ , is approximately l/λ = 0 .
03. Based on Priest & Forbes (2000), the wavelength range for thetearing mode is 12 π (cid:18) Dlv A (cid:19) / < lλ < / (2 π ) ∼ .
16 (19)where D is the magnetic diffusivity ( D = ηc / π ) and v A is the Alfv´en speed. Using values from our simulationsgives the lower limit as 0.01. Hence the current sheet dimensions, l/λ = 0 .
03, are compatible with a tearingmode instability. Later in time, plasmoids are formed closer to the apex of the current sheet and coalesce,creating an isolated flux rope at the center. Both shear field and mass are trapped in this flux rope, but themajority of the shear field concentrates at either end of the current sheet, in the low-lying dips in the magneticfield.To determine what happens to this coronal flux rope which is formed from the original center of the lowtension flux tube, we can examine the forces in the flux rope. Figure 8 shows the four vertical forces associatedwith gravity, magnetic tension, magnetic pressure, and gas pressure along the central, x = 0, line at twodifferent times. The forces are shown for the initial sub-surface flux tubes in Figures 8(a) and (b), where theinitial hydrostatic equilibrium can be seen in the large scale trends for ρ g and −∇ P , and the magnetic forcescan be seen over the range -2.5 Mm to -1.5 Mm. The outward directed magnetic pressure force, due primarilyto the shear field, dominates over the inward directed tension force of the twist field. The gas pressure forceis directed inwards to balance this excess magnetic pressure force. Figures 8(c) and (d) show the forces in thecorona after the flux tube has emerged. For the low tension case, the forces in the newly formed coronal fluxrope, the center of which is at y = 7 Mm, can be seen in Figure 8(d). The presence of mass can be detected inthe small dip in the gravitational force. Here, in contrast to the configuration of the initial flux tube, the inwarddirected tension of the twist field dominates over the outward directed magnetic pressure force. Thus the gaspressure force is directed outward here, to balance the excess tension force. This shows that, in contrast to thebreakout model, shear field does not create a strong unbalanced upward/outward pressure force, and so there isno impetus which can drive a CME eruption. This shows the dramatic effect the dense lower solar atmosphere 10 –can have on the role of shear in driving CMEs.As far as we are aware, the mechanism shown here for the creation of a coronal flux rope from an emergingflux tube whose cross section distorts into a crescent shape has not been seen before. We expect that suchcoronal flux ropes will form from 3D flux tubes which emerge with a large radius of curvature along their axis.A large radius of curvature implies that the draining of mass along the axis will not exceed the draining ofmass perpendicular to the axis seen in these 2.5D simulations, and so the behavior will be similar. However,3D simulations of similar flux tubes by Murray & Hood (2008) do not show this behavior. We find that westill see the formation of our coronal flux ropes when we use the same parameters ( y t =-0.17 Mm, B =3900 G, a =0.425 Mm, c = 3 Mm − and c =0.17 Mm) as one of the low tension tubes used in Murray & Hood (2008).3D simulations are required to investigate this issue further.Figure 9(a) shows the height of the centers of the three flux tubes as they emerge into the field-free corona.The center (which is initially the axis) is defined at the point where B x | x =0 changes sign. The centers of thehigh tension tubes stay at the photosphere, while the center of the low tension tube rises to a height of about8Mm as the tube is deformed into a crescent shaped current sheet and the resulting coronal flux rope rises. Thecoronal flux rope does not escape, most likely due to the presence of mass accumulation in the rope and due tothe lack of magnetic shear.Figures 9(b) and (c) show the normalized magnetic shear energy above two different heights as a functionof time for all three tubes. This normalized energy is given by E shear = R y =90 Mmy = y R x =45 Mmx = − Mm B z dxdy R y =90 Mmy = y R x =45 Mmx = − Mm | B | dxdy (20)The first height, y = 1 . β = 0 . y = 5 Mm above the surface and is at a location just below where the isolated flux rope is formed from thecenter of the low tension tube.As the low tension case develops ( t =1700 s to t =2200 s), the crescent shaped current sheet is formed andmass and shear field drain to the ends, below 1.2 Mm. Thus the shear energy above 1.2 Mm drops as shown bythe green/dashed curve in Figure 9(b). The coalescence of plasmoids, which leads to a concentration of shearflux, then creates an increase in E shear at t =2300 s. The expulsion of these plasmoids along the current sheet,down below 1.2 Mm, creates the sharp decrease seen shortly after at t =2400 s. As the flux rope formed fromthis current sheet then rises into the corona, carrying shear field with it, the shear energy above 5 Mm rises, ascan be seen in the green/dashed plot in Figure 9(c). The gradual rise in E shear for the high tension cases above1.2 Mm after t =2800 s (red/dot-dashed and blue/solid plots in Figure 9(b)) is caused by reconnection at nearlyvertical current sheets which form at the edges of the active region. This reconnection creates plasmoids whichare able to transfer a small amount of shear above 1.2 Mm but which do not rise to the 5 Mm level.The magnetic shear energy above 1.2 Mm for all three tubes does not exceed 10% of the total magneticenergy above 1.2 Mm. For the emergence of a 2.5D flux tube into a field-free corona, the shear field is generallyconfined to the high β lower atmosphere. The shear field which does rise in the coronal flux rope is not strongenough to create magnetic pressure which will counterbalance the magnetic tension and gravitational force in therope and hence drive a further rise or an eruption. In the next section we look at the expansion into a quadrupolefield, to see if the small amount of shear field which does emerge into the corona can drive reconnection at thenull point of the quadrupole, thus reconfiguring the field to allow an eruption as in the breakout model. We present results of the emergence of the low tension flux tube into a pre-existing coronal quadrupolemagnetic field. The quadrupole configuration can be destabilized by the presence of magnetic shear, as in the 11 –breakout model, but the effects of the lower atmosphere have not yet been investigated. We know from theprevious section that the emergence of magnetic flux can bring shear into the corona, but for a field-free corona,the coronal flux ropes which then form are not sheared enough to erupt. In this section we emerge the lowtension flux tube into a quadrupole of varying strength, and examine the interaction between emerging field andcoronal field. We also look at the effect of the quadrupole strength on the overall emergence. We test whetherthe quadrupole configuration can be destabilized by flux emergence-supplied magnetic shear energy. The initialconfiguration is shown in Figure 5 for the medium strength quadrupole ( B | x = x a / ,y =0 = 98 G).The initial rise of the flux tube is similar to the field-free corona case up to about 1000 s. The presence ofquadrupole field below the surface has a negligible effect on the rise of the tube, and its initial expansion into theatmosphere. Figure 10 shows the evolution of the low tension flux tube and the medium strength quadrupoleat 6 different times (the coupling of B z and ρ in these 2.5D simulations allows us to only show the density todemonstrate the structure of the coronal flux ropes). The field of the inner dipole of the quadrupole is alignedwith the twist field of the upper part of the tube which expands into the corona. Thus the inner flux systemof the quadrupole field is pushed upwards towards the null by the emergence, as shown in Figures 10(a) and(b). The expanding shell of the tube causes a crescent shaped current sheet to be formed as the original null isstretched over this shell. Reconnection sets in on either side of the apex of the current sheet. The current sheetcontinues to rise as the expanding shell of the emerging flux tube pushes it up as shown in Figures 10(d)-(f),while the reconnection creates a coronal flux rope at a height of 18 Mm.In Figure 10 we can also see the deformation of the emerging flux tube’s center, and the creation of a fluxrope by reconnection at the current sheet formed at about 5 Mm, just as in the field-free corona simulations.Thus there are two coronal flux ropes formed, as can be seen in Figure 10(f). The upper rope is created bythe reconnection at the separatrix between the two systems of the quadrupole, driven by the expansion of theflux tube into the atmosphere. The lower rope is formed by reconnection at the current sheet formed from thecenter of the emerging flux tube. Both of these flux ropes carry shear field and mass.Figure 11 shows the vertical forces in these two coronal flux ropes at later times of t = 2392 s and t = 2760 s. Both ropes carry significant amounts of mass and in both ropes the magnetic tension acts inwards,counteracting any outward force due to gas or magnetic pressure. Figure 11(a) shows that both the magnetictension and magnetic pressure forces are directed inwards in the upper flux rope, thus a strong outward pressureforce is required for force balance. In addition, the downward gravity force of the trapped mass is as strongas any of the pressure or tension forces. These other forces all have an upward directed asymmetry to helpbalance this large gravity force, but nonetheless, as can be seen in Figure 11(c), they cannot and the flux ropestarts to sink. In contrast, the gravity force is relatively small in the lower flux rope, as shown in 11(b), and themagnetic pressure force is directed outwards. Yet this outward magnetic pressure force is well balanced by theinward magnetic tension. Thus, while the ropes differ in the way the magnetic and gas pressures act, the keyfact is that the amount of shear field in them is not enough to create an unbalanced outward acting magneticpressure force to cause any further expansion.The fraction of the magnetic energy contained in the shear field above the heights of 1.2 Mm and 5 Mm forthe three choices of quadrupole and the field-free corona case are shown in Figure 12. In general, increasing thestrength of the quadrupole decreases this fraction as the total magnetic energy in the system increases. Howeverthis also increases the amount of reconnection at the separatrix of the quadrupole as the configuration containsmore flux above and below the null.We have shown that the emergence of a flux tube into a quadrupole field does cause reconnection at theseparatrix between the two flux systems in the quadrupole. However, the nature of the reconnection meansthat it does not remove significant amounts of horizontal field from above and below the original null point,as in the single X-point reconnection seen in the breakout model. The reconnection at the null is driven hereby expansion of the outer part of the emerging flux tube, and the current sheet formed prefers reconnectionoff-center, creating a coronal flux rope which does not have enough magnetic pressure to escape. This flux ropeeventually falls back down, and joins with the flux rope which was created from the center of the emerging lowtension flux tube. The magnetic energy in the shear field above the β = 0 . In this section we present results from simulations of the emergence of the three different flux tubes into aparticular dipole field to highlight the importance of the choice of tube profile. We choose the dipole field with B | x =0 ,y =0 = 52 G. This choice of dipole best highlights the differences between the high and low tension cases.We will investigate the choice of dipole strength in § x = 0 line for the three tubes. Initially the paths of the centersand separatrices are the same for all three tubes as they expand and push the dipole field upwards. After t = 1500 s the low tension tube undergoes reconnection at its center, forming the lower of the two flux ropes.Reconnection at the separatrix between the emerging tube and the dipole field in all three simulations createsthe coronal rope higher in the corona (solid lines). These coronal flux ropes eventually fall back down throughthe transition region after t = 4000 s.The amount of normalized magnetic shear energy supplied to the two heights of 1.2 Mm and 5 Mm forall three tubes emerging into the dipole can be seen in Figures 14(b) and 14(c). The normalized shear energyabove 1.2 Mm increases after 2000 s for the experiment where the low tension tube emerges into the dipole,shown as the green/dashed curves on Figure 14(b). This is caused by the removal of some horizontal flux byreconnection in the current sheets. While this is similar to the reconnection which leads to an eruption in thebreakout model, this reconnection does not remove enough horizontal field to raise the importance of the shearfield in the coronal flux ropes formed by this reconnection. As a result the shear magnetic energy is still lessthan 10% of the total magnetic energy above the β = 0 . By varying the dipole strength we can control the location and amount of reconnection at the currentsheet formed between the flux tube and the dipole, and through this affect the amount of shear energy (relativeto total magnetic energy) which is transferred to the low β corona. In this section we present results of theemergence of the low tension flux tube into a range of dipoles. These dipoles have horizontal fluxes which arefactors of the horizontal flux that is initially in the top half of the emerging tube of 1, 0.5 and 0.25. We alsocompare these three cases to the field-free corona case.The strength of the dipole affects the expansion of the outer part of the flux tube, as the stronger the dipoleis the more tension it has to resist the expansion of the upper part of the flux tube into the corona. The strongerthe dipole, the quicker (and lower down in the corona) the two systems are forced together by the emergenceand so the quicker the reconnection occurs between them and the quicker the coronal flux rope is formed. Thiscan be seen in Figure 15 which shows snapshots at t=2530 s for the low tension flux tube emerging into the3 different dipoles (of decreasing strength from left to right) and a field-free corona (far right panel). At thistime, the stronger dipole case has already formed a coronal flux rope, while the weaker dipole case has only juststarted reconnection between the emerging flux tube and the dipole.Figure 16(a) shows the heights of the flux tube centers and the flux tube/dipole separatrices for the differentchoices in dipole strength. The separatrices all reach similar heights, but do so at different times, due to thediffering tension in the dipoles which resists the pushing up of the dipole by the expanding flux tube.Figures 16(b) and 16(c) show the normalized shear energy above 1.2 Mm and 5 Mm. In general, themagnetic shear energy due to flux emergence is confined too low in the atmosphere to have a strong enougheffect on the magnetic field in the corona. The shear energy is still always below 10% of the total energy abovethe low β corona. Any shear field that is transported to the corona by coronal flux ropes is too small to affectthe twist field at these heights. The coronal flux ropes formed are not sheared enough and contain significantamounts of dense material. In order to drive a CME by flux emergence in these simulations, we need a methodfor transferring more shear field to coronal heights while transferring less mass.
4. CONCLUSIONS
We have investigated the initiation of CMEs by dynamical flux emergence using 2.5D cartesian simulationsof the emergence of a range of sub-surface flux tubes into a range of coronal magnetic field configurations. Mostcurrent CME models do not include the lower dense atmosphere in their models, but the boundary conditionsfor the shear energy and dynamical flows they use are most likely results of flux emergence through this region.In a first attempt to couple the modeling of CME initiation and flux emergence, we have taken coronal fieldconfigurations which are based on the magnetic breakout model. However, our results are generally applicableto any CME model which assumes the appearance of highly sheared flux at the low β corona.The evolution of flux tubes into a field-free corona can be separated into low and high tension cases, as inMurray & Hood (2008), based on the relative amount of tension near the center of the flux tube. During theemergence of a low tension tube, an isolated flux rope was created by the deformation of the flux tube center intoa current sheet and the reconnection of fieldlines on either side of the apex of this current sheet. Since twistedflux ropes are essential for several CME models (Titov & D´emoulin 1999; Roussev et al. 2003; Kliem & T¨or¨ok2006; Fan & Gibson 2007) further investigations are warranted, especially in 3D, to take into account the effectof curvature along the tube’s axis. The comparison of 3D simulations with observations of neutral lines duringflux emergence will be able to verify if this behavior is occurring on the Sun.In an attempt to test whether the shear field brought into the corona by flux emergence could destabilizethe quadrupole configuration of the magnetic breakout model, we added a pre-existing quadrupole coronal fieldabove the emerging flux tubes. The expanding outer shell of the flux tube stretched the original null point of thequadrupole into a current sheet, and reconnection in this current sheet created another coronal flux rope, higher 14 –in the corona. Both types of flux ropes were created in similar fashions: by the formation of a crescent shapedcurrent sheet, with preferential reconnection occurring off-center, forming a coronal flux rope at the apex of thecrescent. However, one type of flux rope was created by the deformation of the center of an emerging flux tube,and one was created along the separatrix between the two flux systems of the quadrupole.The emergence of sub-surface flux tubes into a simpler dipole coronal field configuration was simulated,in an attempt to remove the twist field of the flux tube which first emerges into the corona. Reconnectionat the separatrix between emerging field and pre-existing dipole field yielded an isolated flux rope, as did thedeformation of the center of the emerging low tension flux tubeAlthough the coronal flux ropes that are formed in these simulations of flux emergence transport shearfield into the low β corona, the shear field is weak compared to the twist field and they are laden with photo-spheric/chromospheric material. Any outward directed forces from the shear field are insufficient to overcomethe inward directed tension force and the downward directed gravity force. These coronal flux ropes are thereforenot candidates for erupting structures, and eventually disappear or fall back into the lower atmosphere.We have covered a range of flux tube profiles and coronal field structures and strengths, and in all casesfound that the emergence of sub-surface flux tubes in these 2.5D simulations is an inefficient method for thetransfer of magnetic shear energy into the low β corona. The amount of magnetic shear energy supplied to thelow β corona was calculated, at heights of 1.2 Mm and 5 Mm. The lower of these two heights corresponds to thelowest point at which β = 0 . β < . We conclude that the simple observation of flux emergence and shear at the photosphere is not sufficient evidencethat a CME can be driven.
Further investigations must be carried out to help understand the conditions underwhich emerging shear can be transported up to the low corona, where it is needed.For 3D flux tubes with significant curvature along the tube axis, flows along fieldlines may be an importantphenomena in flux emergence simulations. These simulations should be repeated in 3D to see if this has anaffect on the formation of coronal flux ropes and their eventual evolution in the solar atmosphere.We have tested the robustness of these results by varying the resolution and introducing asymmetry intothe flux emergence and coronal fields, and find similar results regardless of the position of the flux tube relativeto the center of the quadrupole/dipole.In these simulations, and almost all previous flux emergence simulations, the high β lower atmosphereis assumed to be fully ionized, which is not the case for the solar chromosphere. The presence of neutralsin the nearly neutral region near the temperature minimum may have an important effect on these results.Leake & Arber (2006) and Arber et al. (2007) showed that this nearly neutral region can dramatically affectflux emergence. In particular, cross-field currents in emerging structures are destroyed by ion-neutral collisions,and the magnetic field lines are able to slip through the neutral gas during emergence. Thus the flux tubes liftup less mass from the chromosphere when the effects of partial ionization are taken into account. All this occurslower down in the solar atmosphere around the β = 1 region and will directly affect the transfer of both massand shear energy to the low β corona above. These effects, along with 3D effects, may provide a mechanism forthe transfer of magnetic shear energy into the corona without the transfer of mass, which may be key to drivinga CME with dynamical flux emergence. This topic will be investigated in future publications.J. E. Leake and M. G. Linton acknowledge support from NASA SR&T grant number NNH06AD58I, fromONR/NRL 6.1 basic research funds, and from the NRL-Hinode analysis program. Hinode is a Japanese missiondeveloped and launched by ISAS/JAXA, collaborating with NAOJ as domestic partner, and NASA (USA)and STFC (UK) as international partners. Scientific operation of the Hinode mission is conducted by theHinode science team organized at ISAS/JAXA. This team mainly consists of scientists from institutes in thepartner countries. Support for the post-launch operation is provided by JAXA and NAOJ, STFC, NASA, ESA(European Space Agency), and NSC (Norway). We are grateful to the Hinode team for all their efforts in 15 –the design, build and operation of the mission. The work by S. K. Antiochos was supported by the NASAHTP, TR&T, and SR&T Programs. The authors would like to thank C. R. DeVore for enlightening discussionconcerning the magnetic breakout model for CME initiation and the numerical modeling of CMEs. REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
18 –Fig. 1.— Meridional projections of magnetic field lines throughout the magnetic breakout eruption process atsix different times. Reprinted with permission from Lynch et al. (2004). Spatial units are solar radii. 19 –Fig. 2.— Panels (a) and (b): Vertical forces along the latitude=0 ◦ line at the center of the inner arcade of themagnetic quadrupole in the breakout model at t=0 s, panel (a), and t=30000 s, panel (b), after shearing ofthe arcade by imposed boundary flows. Panel (c): The normalized shear magnetic energy (solid line) and 2Dintegral of kinetic energy density (dashed line) in the system during this shearing period. 20 –Fig. 3.— Panel (a): the initial height profile of the model stratified atmosphere showing the temperature (solidline), gas pressure (dashed line), and magnetic pressure (dot-dashed line), up to 20 Mm. The magnetic pressureprofile is for a sub-surface magnetic flux tube and an overlying, anti-parallel dipole. Panel (b): The initialplasma β for six different simulations: Three simulations with background dipole field and three simulationswith background quadrupole field. The straight black dotted line shows a constant value of β = 0 .