A Cancellation Nanoflare Model for Solar Chromospheric and Coronal Heating II. 2D Theory and Simulations
DDraft version January 10, 2019
Typeset using L A TEX twocolumn style in AASTeX62
A Cancellation Nanoflare Model for Solar Chromospheric and Coronal Heating II. 2D Theory and Simulations
P. Syntelis, E.R. Priest, and L.P. Chitta St Andrews University, Mathematics Institute, St Andrews KY16 9SS, UK Max Planck Institute for Solar System Research, G¨ottingen, Germany (Received; Revised; Accepted)
ABSTRACTRecent observations at high spatial resolution have shown that magnetic flux cancellation occurson the solar surface much more frequently than previously thought, and so this led Priest et al.(2018) to propose magnetic reconnection driven by photospheric flux cancellation as a mechanism forchromospheric and coronal heating. In particular, they estimated analytically the amount of energyreleased as heat and the height of the energy release during flux cancellation. In the present work,we take the next step in the theory by setting up a two-dimensional resistive MHD simulation of twocancelling polarities in the presence of a horizontal external field and a stratified atmosphere in orderto check and improve upon the analytical estimates. Computational evaluation of the energy releaseduring reconnection is found to be in good qualitative agreement with the analytical estimates. Inaddition, we go further and undertake an initial study of the atmospheric response to reconnection. Wefind that, during the cancellation, either hot ejections or cool ones or a combination of both hot andcool ejections can be formed, depending on the height of the reconnection location. The hot structurescan have the density and temperature of coronal loops, while the cooler structures are suggestive ofsurges and large spicules.
Keywords:
Sun: coronal heating – Sun: magnetic reconnection – Sun: activity Sun: Magnetic fields–Magnetohydrodynamics (MHD) –methods: numerical INTRODUCTIONThe emergence of new magnetic flux from below thephotosphere (Harvey & Martin 1973) and its reconnec-tion with the overlying magnetic field has long beenrecognised as being one way of heating part of the solarcorona, namely, X-ray bright points (Golub et al. 1974),and of heating small flares (Heyvaerts et al. 1977). Ithas also been proposed as a possible source of coronalX-ray jets (Shibata et al. 1992), for which there hasbeen a host of observational papers (e.g., Moore et al.2010; Shimojo & Shibata 2000) and numerical exper-iments (e.g., Yokoyama & Shibata 1996; Archontis &Hood 2010; Moreno-Insertis & Galsgaard 2013; Synteliset al. 2015). Indeed, it is now appreciated that reconnec-tion can produce a mixture of hot and cold structuresand that their origin can be highly subtle and complex
Corresponding author: P. [email protected] (e.g. Hansteen et al. 2017; N´obrega-Siverio et al. 2017,2018).The cancellation of photospheric magnetic flux is an-other common process (Martin et al. 1985) which hasbeen proposed as a mechanism for heating X-ray brightpoints (Priest et al. 1994; Parnell & Priest 1995), inwhich magnetic reconnection is driven in the overly-ing atmosphere during the approach of opposite-polaritymagnetic fragments before they actually cancel. Indeed,we shall include this pre-cancellation phase in our use ofthe words “flux cancellation”. Photospheric flux cancel-lation has been shown to be associated with both hotand cool jets and also with many different examples ofsmall-scale energy release, such as Ellerman bombs, UVbursts and IRIS bombs (e.g. Watanabe et al. 2011; Vis-sers et al. 2013; Peter et al. 2014; Kim et al. 2015; Visserset al. 2015; Rutten et al. 2015; Rezaei & Beck 2015; Nel-son et al. 2016; Tian et al. 2016; Reid et al. 2016; Rutten2016; Nelson et al. 2017; Toriumi et al. 2017; Hong et al.2017; Libbrecht et al. 2017; van der Voort et al. 2017). a r X i v : . [ a s t r o - ph . S R ] J a n A new achievement is the remarkable observationsfrom the Sunrise balloon mission (Solanki et al. 2010,2017), which have revealed images of the photosphericmagnetic field at a spatial resolution of 0.15 arcsec,which is six times better than the Helioseismic Imager(HMI) on the Solar Dynamics Observatory (SDO). Inparticular, they show that magnetic flux is emergingand cancelling at a rate of 1100 Mx cm − day − (Smithaet al. 2017), which is an order of magnitude higher thanpreviously realised. Furthermore, whereas, at the spa-tial resolution of HMI, coronal loops have their foot-points located in regions of uniform polarity, at Sun-rise resolution the footpoints are surprisingly revealedto have mixed polarity that is cancelling at a rate of10 Mx s − (Chitta et al. 2017b). Other examplesof flux cancellation producing coronal loop brighteninghave been presented by Tiwari et al. (2014), Huang et al.(2018), and Chitta et al. (2018)These observations led Priest et al. (2018) to pro-pose reconnection driven by photospheric flux cancella-tion as a mechanism for heating the chromosphere andcorona. They set up an analytical model for the ap-proach and cancellation of two opposite-polarity mag-netic fragments of flux ± F in the photosphere in thepresence of an overlying uniform horizontal magneticfield B , and found that the evolution of the systemdepends on the value of a key parameter, called the in-teraction distance , which, for three-dimensional sources,may be written as d D = (cid:18) FπB (cid:19) / . (1)Suppose the magnetic flux sources are separated bya distance 2 d . Then, when d > d D , the sources arenot connected magnetically, but when d = d D a nullpoint (or in 3D a separator ) forms in the photosphere.As the sources approach closer, such that d < d D , re-connection is driven and the reconnection location risesin the atmosphere to a maximum height proportional to d D . Thereafter, the reconnection location moves backtowards the solar surface, which it reaches when the twosources come into contact and cancel ( d = 0). Thus, themaximum reconnection height can be located in the pho-tosphere or chromosphere if d D is small enough or inthe corona if it is large enough.As well as calculate the way the reconnection heightdepends on flux ( F ) and overlying field strength ( B )through d D , Priest et al. (2018) made estimates for theenergy release, and found that, for reasonable values ofthe parameters, the heating rate is sufficient to heat thechromosphere and corona. In the present paper, we develop the model furtherby setting up a two-dimensional computational exper-iment for flux cancellation that has the same featuresas our analytical model, namely, two approaching fluxsources in the presence of an overlying horizontal mag-netic field, so that we can test the predictions of theanalytical model. However, we add an extra feature,namely, a simple stratified atmosphere in order to un-derstand some of the effects of stratification.Section 2 presents some more details of the theory ofreconnection in two dimensions, including Sweet-Parkerreconnection, fast reconnection, and energy conversion.Then, Section 3 presents our computational model andcompares it with the analytical theory, before a sum-mary discussion is given in the final Section. THEORY FOR ENERGY RELEASE DRIVEN BYPHOTOSPHERIC FLUX CANCELLATION IN 2DHere we make some theoretical estimates of the en-ergy release by steady-state magnetic reconnection intwo dimensions, developing the basic theory from Priest(2014) in new ways. We will start by briefly describingslow Sweet-Parker reconnection and fast reconnection,and then discussing reconnection driven by magneticflux cancellation.2.1.
Slow Sweet-Parker Reconnection
Consider first a simple Sweet-Parker current sheet ofgiven length L , depth L s and width l situated betweenoppositely directed magnetic fields B i and − B i (Fig.1a). If plasma and magnetic field is brought in fromboth sides at a speed v i , then a balance between in-wards advection and outwards diffusion of magnetic fluximplies v i = ηl . (2)Furthermore, if the plasma has uniform density ρ i , bal-ancing the rates of inflow and outflow of mass gives Lv i = lv Ai , (3)where v Ai = B i / √ µρ i is the outflow speed from thecurrent sheet, namely, the Alfv´en speed based on theinflow magnetic field.Eliminating l between Equation (2) and (3) producesan expression for the dimensionless inflow speed orAlfv´en Mach number ( M Ai = v i /v Ai ), i.e., the recon-nection rate , of M Ai = 1 R / mi , (4)where R mi = Lv Ai /η is the external magnetic Reynoldsnumber based on the global external length-scale ( L )and Alfv´en speed ( v Ai ). For a given current-sheet length Sweet-Parker
Fast Reconnection(a) (b)
Figure 1.
The nomenclature for energy release in a simplereconnection region consisting of either (a) a slow Sweet-Parker current sheet or (b) a fast reconnection region with asmall sheet and four slow-mode shock waves. ( L ) and external magnetic field ( B i ), Equation (4) thusprovides the Sweet-Parker rate.Half of the magnetic energy that comes into the re-connection region from both sides is converted into heatand the other half into kinetic energy (which can lateritself dissipate viscously or through shock waves). Therate of inflow of magnetic energy from one side throughan area of LL s is just the Poynting influx ( EH i LL s = EB i LL s /µ ), where the magnitude of the electric field is E = v i B i , and so the rate of conversion to heat of mag-netic energy coming in from both sides of the currentsheet is dWdt = v i B i µ LL s = 1 R / me v Ai B i µ LL s , (5)after substituting for v i from Equation (4). The phrase“two-dimensional” can refer to a situation in which thevariables are situated in three dimensions but they de-pend on only two of them, such as x and y , but it canalso refer to variables that exist only in two dimensions,in which case the above expression would need to be di-vided by L s . In what follows it should be clear which ofthe two definitions is being inferred.2.2. Fast Reconnection
Next, suppose the inflow speed is faster than v Ai /R / me ,while the inflow magnetic field ( B i ) and area L s are thesame as before (Fig. 1b). Then, three possibilities have been studied. Firstly, according to fast steady-statereconnection theory (either Petschek (1964) or Almost-Uniform (Priest & Forbes 1986)), the reconnection re-gion possesses a complex internal structure consistingof a central small Sweet-Parker current sheet togetherwith four slow-mode shock waves propagating from theirends and standing in the flow. As the speed increases,the central sheet diminishes in size, while the lengthand inclination of the shock waves increases. Most ofthe energy conversion then takes place at the shockwaves, with of the inflowing magnetic energy beingconverted to heat (rather than the that is found inSweet-Parker reconnection) and the remainder going tokinetic energy.Secondly, fast collisionless reconnection is helped bythe Hall effect, when the resistive diffusion region is re-placed by an ion diffusion region and a smaller electrondiffusion region. In this case, a similar fast maximumrate of reconnection as in Petschek’s mechanism results(Shay & Drake 1998; Birn et al. 2001; Huba 2003; Huba& Rudakov 2004; Shay et al. 2007; Birn & Priest 2007).Thirdly, when the central sheet is long enough, it goesunstable to secondary tearing mode instability and aregime of impulsive bursty reconnection results, first de-scribed by Priest (1986); Lee & Fu (1986); Biskamp(1986); Forbes & Priest (1987) and later studied byLoureiro et al. (2007, 2012, 2013); Bhattacharjee et al.(2009). Reconnection is then fast but time-dependentand impulsive, although the mean rate is likely to besimilar to the previous cases.For each of the three above scenarios, Equation (2) nolonger holds, but the same mass conservation relationholds as before for the reconnection region as a whole,namely, Lv i = lv Ai , (6)where L now refers to the length of the whole reconnec-tion region (including shock waves and central currentsheet) rather than the length of just the central sheet,and variables with subscript i refer to values at the in-flow to that whole region. Equation (6) determines theoverall width ( l ) of the complex reconnection region fora given L , v i and v Ai . The conversion rate of inflowingenergy from both sides of the current sheet then becomes dWdt = 45 v i B i µ LL s , (7)where v i possesses any value up to a maximum of typi-cally 0.01–0.1 of the Alfv´en speed ( v Ai ).2.3. Energy Conversion during Photospheric FluxCancellation in 2D
Magnetic configuration
Consider sources of positive and negative photosphericmagnetic flux ( ± F ) situated at points B ( d,
0) and A( − d,
0) on the x -axis in a region of uniform magneticfield B ˆx , and suppose they approach one another atspeeds ± v .The resulting magnetic field (in two dimensions) abovethe photosphere ( y >
0) is given by B = Fπ ˆr r − Fπ ˆr r + B ˆx , (8)where r = ( x − d ) ˆx + y ˆy , r = ( x + d ) ˆx + y ˆy , are the vector distances from the two sources to a pointP( x, y ).It is natural in Equation (8) to non-dimensionalise themagnetic field with respect to B and distances with re-spect to the 2D version of the interaction distance (Long-cope 1998), namely, d = 2 FπB , (9)and so define¯ B x = B x B , ¯ d = dd , ¯ y = yd . Then the magnetic field on the y -axis becomes¯ B x = (cid:18) − ¯ d ¯ y + ¯ d + 1 (cid:19) . (10)Consider what happens as the two sources approacheach other. The evolution of the topology is similarto what happens in three dimensions, as described indetail in Sec. 2.1 of Priest et al. (2018). When the twosources are far away ( d > d ), they are not connectedmagnetically and two first-order null points lie on the x -axis between the sources (Fig. 2a). When d = d ,a high-order null point appears at the origin (Fig. 2b).As the sources approach one another ( d < d ), the nullpoint rises above the photosphere (Fig. 2c). The locationof the null point at y = y N is given by¯ y N = (cid:112) ¯ d − ¯ d , and it rises along the y -axis to a maximum of y = d when d = d . Then, it falls, reaching the origin when d = 0, as shown in Fig. 3a. When d = 0, the flux of thetwo sources has completely cancelled.2.3.2. Inflowing Plasma Speed ( v i ) and Magnetic Field( B i ) at the Reconnection Site To analyze the energy release during flux cancellation,the natural parameters, for each value of the source sep-aration (2 d ), are the magnetic diffusivity ( η ), the criti-cal source half-separation distance ( d ), the flux sourcespeed ( v = ˙ d = dd/dt ) and the overlying field strength( B ). We now proceed to calculate the inflow speed( v i ) and magnetic field ( B i ) to the current sheet andthe sheet length ( L ) as functions of these parametersin the cases of slow reconnection and fast reconnection.The magnetic configuration driven by flux cancellationis shown in Fig. 2d.First we consider B i . The components of the potentialmagnetic field sufficiently close to a 2D X-point can bewritten as (Priest 2014) B y + iB x = kz, where k is a constant and z = x + iy is the complexvariable. Suppose that the configuration with a recon-necting current sheet of length L is represented by B y + iB x = k ( z + L ) / , (11)such that the sheet is a cut in the complex plane between z = ± iL . Then, putting z = 0+ implies that B i = kL, which is the required expression for B i when the x -component of the field in the potential state near thenull has the form B x = ky . The value of k is calculatedas follows. The horizontal field B x near y = y N maybe obtained by putting y = y N (1 + (cid:15) ), where (cid:15) (cid:28) B x = 2 (cid:15) (1 − ¯ d ) , or B x = 2 (cid:114) d − dd (cid:18) y − y N d (cid:19) B . (12)This determines the value of k , and so our required ex-pression becomes B i B = (cid:114) d d − Ld . (13)Next, consider v i . This may be calculated from therate of change ( ˙ ψ ≡ dψ/dt ) of magnetic flux, since v i B i = ˙ ψ, (14)or, in dimensionless form v i v A = ˙ ψv A B B B i , (15) Figure 2.
Magnetic topology during reconnection driven by photospheric flux cancellation when (a) d > d , (b) d = d and (c) d < d , where d is half the separation distance of the two flux sources and d is the flux interaction distance. (d) The notationused for the reconnection region. where v A a hybrid Alfv´en speed based on the mag-netic field B and the density of the inflowing material,namely, v A = B √ µρ i . (16)In turn, ˙ ψ may be calculated from the reconnected flux( ψ ), as estimated from the magnetic flux below the nullpoint, namely, ψ = (cid:90) y N B − F dπ ( y + d ) dy = 2 Fπ (cid:90) ¯ y N − ¯ d (¯ y + ¯ d ) d ¯ y = 2 Fπ (cid:32)(cid:112) ¯ d − ¯ d − tan − (cid:112) − ¯ d √ ¯ d (cid:33) . (17)It can be seen from Fig. 3b that, as expected, the recon-nected flux vanishes when d = d and increases mono-tonically to a value of F as the separation (2 d ) betweenthe sources approaches zero.Then, differentiating Equation (17) with respect totime determines ˙ ψ in terms of ˙ d = v , and Equation(15) becomes v i = v d L . (18)2.3.3.
Energy Release
The rate of inflow of magnetic energy from one sideof a current sheet of length L , at speed v i , with fieldstrength B i and density ρ i is the Poynting flux throughthat surface. In 2D the surface of the current sheet willbe a line of length L , so the Poynting influx is EH i L = EB i L/µ . Since the electric field is E = v i B i , and themagnetic energy inflow occurs from both sides of thecurrent sheet, the Poynting influx from both sides willbe S i = 2 v i B i µ L (19)This has units of energy/time/length, since we assumehere a purely 2D configuration with no depth in the thirddimension. To derive the energy release, the length ofthe current sheet and the conversion rate to heat has tobe estimated. Both will depend on the type of recon-nection (Sweet-Parker or fast). For a configuration withdepth L S in the third dimension this would be multi-plied by L S .2.3.4. Slow Sweet-Parker Reconnection
Here we calculate the energy release for Sweet-Parkerreconnection. After eliminating l between the Sweet-Parker relations (Equations (2) and (3)) we find thatthe current sheet length is L = ηv Ai v i , (20) (a) y N / d (b) −1.2−1.0−0.8−0.6−0.4−0.2−0.0 ψ / F Figure 3. (a) The height of the null point ( y N ) given byEquation (11) as a function of the distance ( d ) of the sourcesfrom the origin (as shown in Fig. 2c). (b) The magnetic flux( ψ ) below the null given by Equation (17). which can be nondimensionalised in terms of d to give Ld = 1 R m B i B v A v i , where R m = d v A /η is the magnetic Reynolds numberbased on d and v A . We then substitute for B i /B fromEquation (13) and v i from Equation (18) to give L SP d = R m v v A (cid:112) d /d − . (21)where the subscript SP denotes the Sweet-Parker cur-rent sheet length. Finally, by substituting in Equation(19) the values of v i (Equation (18)), B i (Equation (13))and L = L SP (Equation (21)), the rate of Poynting in-flux becomes S i SP = 2 v B µ d (cid:112) d /d − R m M A , (22)in terms of the Alfv´en Mach number ( M A = v /v A )based on the flux source speed v . Since half of the mag-netic energy is converted to heat during Sweet-Parker reconnection, the energy release rate will be: dW SP dt = v B µ d (cid:112) d /d − R m M A , (23)which has units of energy/time/length for our 2D theory.2.4. Fast Reconnection
We derive now the energy release for fast reconnec-tion driven by flux cancellation. During fast reconnec-tion, the length of the current sheet is much smallerthan the Sweet-Parker one. L is determined by assum-ing the inflow speed v i = αv Ai . By writing v i = αv Ai = αv A B i /B and then using Equation (13) and (18), L becomes L d = v αv A (cid:112) d /d − . (24)Then, after substituting for v i , B i and L into Equation(19), we find the rate of energy inflow for fast reconnec-tion as S i = 2 v B µ d (cid:112) d /d − M A α . (25)During fast reconnection, of the magnetic energy isconverted to heat and to kinetic energy. Therefore,the rates of kinetic energy release and energy release asheat become dKdt = 1 . v B µ d (cid:112) d /d − M A α (26)and dWdt = 0 . v B µ d (cid:112) d /d − M A α . (27) NUMERICAL COMPUTATIONS3.1.
Numerical Setup
To perform the computations, we numerically solvethe 2D MHD equations in Cartesian geometry using theLare3D code (v3.2) of Arber et al. (2001). The equationsin dimensionless form are: ∂ρ∂t + ∇ · ( ρ v ) = 0 , (28) ∂ ( ρ v ) ∂t = −∇ · ( ρ vv ) + ( ∇ × B ) × B − ∇ P + ρ g , (29) ∂ ( ρ(cid:15) ) ∂t = −∇ · ( ρ(cid:15) v ) − P ∇ · v + Q j + Q v + Q c , (30) ∂ B ∂t = ∇ × ( v × B ) − ∇ × ( η ∇ × B ) , (31) (cid:15) = P ( γ − ρ , (32) P = ρk B Tµ m , (33) T ( K ) −18 −16 −14 −12 −10 −8 −6 −4 ρ ( g c m − ) Figure 4.
The atmospheric temperature (solid black) anddensity (solid blue). The dashed lines show the temperatureand density of the 1D C7 model of Avrett & Loeser (2008). where ρ , v , B and P are density, velocity vector, mag-netic field vector and gas pressure. Gravity is g =274 m s − . We assume a perfect gas with specificheat of γ = 5 /
3. Viscous heating ( Q v ) and Joule dis-sipation ( Q j ) are included. Heat conduction ( Q c ) istreated using super-time stepping (Meyer et al. 2012),similarly to Johnston et al. (2017). The reduced massis µ m = m f m p , where m p is the mass of proton and m f = 1 . k B is the Boltzmann constant.The normalization is based on the photospheric valuesof density ρ u = 1 . × − g cm − , length H u = 180 kmand magnetic field strength B u = 300 G. From thesewe obtain temperature T u = 6234 K, pressure P u =7 . × erg cm − , velocity v u = 2 . − and time t u = 86 . x ∈ [ − ,
30] Mm in the horizontal direction and y ∈ [0 ,
30] Mm in the vertical direction, on a 2048 × y = 0. To mimicthe steep temperature increase from the photosphere tothe corona, we assume an hyperbolic tangent profile forthe atmospheric temperature T ( y ) = T ph + T cor − T ph (cid:18) tanh y − y cor w tr + 1 (cid:19) , (34)where T ph = 6109 K, T cor = 0 .
61 MK, y cor = 2 .
12 Mmand w tr = 0 .
18 Mm. These parameters create anisothermal photospheric-chromospheric layer at 0 Mm ≤ y < .
96 Mm, a transition region at 1 .
96 Mm ≤ y < . . ≤ y <
30 Mm.To derive the atmospheric density, we assume the at-mosphere is in hydrostatic equilibrium. We do so bynumerically solving the hydrostatic equation dP/dy = − gy , assuming a photospheric density of ρ ph = 1 . × − g cm − . The atmospheric temperature (solid black) and density (solid blue) are shown in Fig. 4. Forcomparison, we plot with dashed lines the temperatureand density for the 1D model atmosphere (model C7) ofAvrett & Loeser (2008).We adopt an anomalous resistivity η = η , | j | < j crit η + η | j | /j crit . | j | > j crit , (35)where η = 10 − , η = 10 − and j crit = 10 − . Theresistivity can be anomalous away from the boundaries( x ∈ [ − ,
28] Mm and y ∈ [2 ,
28] Mm). Elsewhere,it is uniform with η = 10 − . Anomalous resistivity(Yokoyama & Shibata 1994) has been previously chosento drive fast reconnection. Other methods (e.g. hyper-diffusion Nordlund & Stein 1990; van Ballegooijen &Cranmer 2008; Mart´ınez-Sykora et al. 2011) can alsobe used to initiate a fast reconnection by permittingenhanced resistivity in current sheets.The initial magnetic field is the sum of two magneticsources and a horizontal field: B = Fπ ˆr r − Fπ ˆr r − B ˆx , (36)where ˆr = ( x + d s )ˆ x + ( y − y )ˆ y, (37) ˆr = ( x − d s )ˆ x + ( y − y )ˆ y (38)are the position vectors of the left and right sources,respectively, d s = 1 . x = 0, and y = − .
36 Mm is the depth of thesources below the photosphere (the sources are outsidethe numerical domain). The flux of each source is F =2 . × Mx cm − . The polarities produced at thephotosphere have a maximum field strength of 2.2 kGand a size of about 1 Mm (defined as the length where | B y | >
100 G) (Fig. 5a). The flux of each polarity is F m = 2 . × Mx cm − . The horizontal field has astrength of B = 45 G.The boundary conditions on the upper boundary are v = 0 and zero gradients for B , ρ , (cid:15) . The photosphericboundary conditions are zero gradients for ρ , (cid:15) . Themagnetic field at the photospheric boundary changesaccording to the driver. To drive the cancellation, wemove the sources with a velocity of v ( t ) = v max (cid:18) tanh t − t w + 1 (cid:19) , (39)where v max = 1 km s − , t = 10 . w = 1 . d ( t ) = −20 −10 0 10 20−2−1012 (a) −20 −10 0 10 20x (Mm)−2−1012 B y ( k G ) (b) P o l a r i t y P o s i t i on ( Mm ) d(t) (y<0)d m (t) (y=0) Figure 5. (a) The variation with x of the vertical magneticfield ( B y ) at the photosphere. (b) Black line: the position ( d )of the sources as a function of time. Blue line: the position( d m ) of the photospheric polarities as a function of time. d s − x ( t ), where x ( t ) = v max w (cid:20) ln (cid:18) cosh t − t w (cid:19) − ln (cid:18) cosh t w (cid:19)(cid:21) (40)+ v max t. The simulation is driven by changing the magnetic fieldat the lower boundary ( y = 0 − (ghost cells)) usingEquation (36) and d ( t ). The half-separation ( d ( t )) of thesources (below the photosphere) as a function of time isplotted in Fig. 5b (black line). The blue lines show thepositions of the polarities at y = 0 (found by measuringthe location of maximum B y ). The latter reflects theresponse of the photosphere to the driver.For the parametric study of Sec. 3.4, we vary the mag-netic field strength ( B ) of the atmosphere in order tovary the height of the null point. The values of B andthe corresponding null height at t = 0 min are shown inTable 1.3.2. Comparison of Theory with Simulation
Table 1.
Initial Conditions for theSimulationsName B (G) y N (Mm)Case 1 45 7 . . . . . In this section, we discuss our reconnection experi-ment driven by flux cancellation and compare its resultswith the theory presented in Sec. 2. For this, we shallfocus on Case 1 of Table 1.3.2.1.
Brief Description of Simulation
The magnetic field at t = 0 is shown in Fig. 6a, witha null point at ( x, y ) = (0 , .
6) Mm. As the driver isswitched on, reconnection is driven at the null pointdue to the converging photospheric polarities. The en-ergy released by reconnection spreads above and belowthe null (and shows up as a “horizontal” heated re-gion and an underlying heated arcade in Fig. 6b). Theheated material is denser than the background atmo-sphere (Fig. 6c).At the photosphere, the positions of the polarities at y = 0 ( d m ( t ), blue line, Fig. 5b) do not keep followingthe driver after t = 37 min (black line). At this time,the magnitude of the photospheric field has decreasedto the point that β >
1. As a result, the driver cannotmove the overlying field anymore. The reconnection atthe null follows the response of the atmospheric field tothe driver and gradually stops.The interaction distance for this simulation is d =200 based on the sources and d m = 2 F m / ( πB ) = 173 . Comparison Methodology
In our simulation, we set the gradient of B to be zeroat the boundaries (besides the photosphere). The rea-son for this is as follows. After flux cancellation, if thepolarities are completely cancelled, the remaining atmo-spheric field ought to be a horizontal field with strength B . This cannot happen in a finite numerical domain,but only in a semi-infinite one. To achieve that in thesimulation domain, we use a zero gradient boundarycondition. This “straightens” the field lines, mimicingthe effect we require.In Sec. 2, the inflow speed is found by taking intoaccount the rate of change of flux and conservation of (a) −20 −10 0 10 20x (Mm)0510152025 y ( Mm ) T ( K ) (b) −20 −10 0 10 20x (Mm)0510152025 y ( Mm ) T ( K ) (c) −20 −10 0 10 20x (Mm)0510152025 y ( Mm ) −15−14−13−12−11−10−9−8−7 ρ ( g / c m ) Figure 6.
Case 1 simulation. (a) Temperature and mag-netic field lines at t = 0. (b) Temperature and (c) density at t = 40 min. flux. To compare the simulation with theory, we needto carefully calculate the fluxes inside the numerical do-main. The Appendix calculates how much flux shouldbe found inside and outside the finite numerical do-main, from which we deduce a flux correcting factor f (Equation A14, Fig. 13). This is used to multiply sev-eral quantities ( v i → f v i , L sp → f L sp , L → f L , dW/dt → f dW/dt ), since the simulation uses a finitedomain rather than a semi-infinite one (as on the Sun,for which f → d , v ), and(ii) the values measured at the photosphere (or else-where), which mimic an actual observation.We will refer to the latter quantities with a subscript m . Thus, the half-separation of the sources is d , whereasthe half-separation of the photospheric polarities is d m .To compare the theory with the simulation, we firstidentify the current sheet. It is located along the y -axis and is the vertical region of increased temperaturelocated above the apex of the arcade (Fig. 7a, orange linesegment). We identify the coordinates of its lowest ( S l )and highest ( S h ) point throughout the simulation. Theseparation of these two points is the measured length ofthe current sheet, L m (solid black line, Fig. 7e).We measure the inflowing magnetic field strength, ve-locity and density in the following manner. At bothsides of the current sheet, we identify the regions thatare parallel to the current sheet and at a distance of∆ x = 0 . B i m , v i m and ρ i m .The total inflow of Poynting flux ( S i m ) into the cur-rent sheet is measured by taking into account the Poynt-ing flux along both AC and CD.3.2.3. Current Sheet Length, Inflowing Magnetic Field andInflow Velocity
Fig. 7b (solid line) shows the inflow magnetic fieldstrength, B i m . The dashed line is the predicted B i us-ing Equation (13) with ( d, d , L m ). The dashed-dot lineis B i using Equation (13) with ( d m , d m , L m ). Compar-ing both approaches, we see that the theory is in goodagreement with the simulation. Indeed, the second ap-proach, where we take into account only the response ofthe simulation to the driver, is in better agreement.The simulation’s inflow velocity ( v i m ) is plotted inFig. 7c (solid line). The dashed line shows v i us-ing Equation (18) (times f ) with ( v , d , L m ). Thedashed-dot line is v i using Equation (18) (times f ) with( v m , d m , L m ). Again, both agree well with the simula-tion.Before estimating the length of the current sheet forfast reconnection (Equation (24)), we need to measuretwo quantities: α , which is the Alfv´en Mach number ofthe inflow and v A (Equation (16)). In Fig. 7d, we plotthe Alfv´en Mach number of the inflow. Between t =10 min and 40 min, when the cancellation occurs, it0 (a) −1 0 1x (Mm)4.04.55.05.56.06.57.0 y ( Mm ) T ( K ) S l S h AB DC (b) B i ( G ) B im simulationB i theory (d, d , L m )B i theory (d m , d , L m ) (c) v i ( k m / s ) v im simulationv i theory (v , d , L m )v i theory (v , d , L m ) (d) v i / v A (e) L ( Mm ) L m simulationL ( α m , d, d , v , v A0m )L ( α m , d m , d , v , v A0m ) (f) L ( Mm ) L m simulationL ( α m , d, d , v , v A0m )L ( α m , d m , d , v , v A0m )L SP Figure 7. (a) Temperature around the reconnection site at t = 30 min. Comparison between simulation and theory for (b)the inflow magnetic field, (c) the inflow velocity, (d) the Mach Alfv´en number of the inflow, (e) the length of the current sheet,(f) the length of the current sheet together with its value for Sweet-Parker reconnection. has an average value of α m = 0 .
05. This value of α is typical for fast reconnection (Priest 2014). For thehybrid Alfv´en speed we use v A m = B / √ µρ i m .The length of the simulation’s current sheet ( L m ) isplotted in Fig. 7e (solid line). The dashed line is L us-ing Equation (24) (times f ) with ( α m , d, d , v , v A m ).The dashed-dot line is L using Equation (24) (times f )with ( α m , d m , d m , v m , v A m ). Both approaches showthe theory to be in good agreement with the simulation.In Fig. 7f, we plot the quantities of panel e, andoverplot the length of the current sheet assuming Sweet-Parker reconnection (triple-dot dashed line, with L SP calculated from Equation (21) (times f ) using( d m , v m , v A m )). The predicted current sheet for anassumption of slow Sweet-Parker reconnection is longerthan the simulated one by an order of magnitude, andso we deduce that fast reconnection with an inflowAlfv´en Mach speed of 0.05 well describes the simulation.3.2.4. Energy Release
We study energy release only for fast reconnection andfirst focus on the Poynting flux inflow ( S i m ), which is1plotted in Fig. 8a (solid lines). The dashed curve is S i from Equation (25) (times f ) based on ( v , v A m , B , d , d ). The dashed-dot curve is S i based on ( v m , v A m , B , d m , d m ). Both agree well with the simulation. Theapproach of using only “measured” values is in excellentagreement with the simulation results.Next, we consider the conversion of Poynting flux tokinetic and thermal energy, for which we calculate theenergy integral terms: (cid:90) C µ E × B · d C = − (cid:90) A η j dA + (cid:90) A j · ( v × B ) dA − (cid:90) A ∂∂t (cid:18) B µ (cid:19) dA. (41)The curve C is ABCDA in Fig. 7a and the surface A isits area. The term (cid:82) C E × B · d C = (cid:82) A ∇ · S dA , where S is the Poynting vector, is the rate of energy inflowand can be compared with Equation (25). The term − (cid:82) A η j dA is the rate of energy converted to joule heat-ing during reconnection, which can be compared withEquation (27). The term (cid:82) A j · ( v × B ) dA is the rate ofenergy converted to kinetic energy and can be comparedwith Equation (26). The term − (cid:82) A [1 / (2 µ )] ∂B /∂t dA is negligible.We first check whether the energy conversion rates ofthe simulation agree with those of fast reconnection, i.e.whether of the total Poynting influx is converted tokinetic energy and is converted to Joule heating. Ifthe conversion rates are such, then we should find in thesimulation that35 (cid:90) A ∇ · S dA = (cid:90) A j · ( v × B ) dA. (42)We plot these terms in Fig. 8b, from which it can be seenthat the left (solid line) and right (dashed line) terms arein agreement. Furthermore, we examine if25 (cid:90) A ∇ · S dA = − (cid:90) A η j dA. (43)These terms are plotted in Fig. 8c, from which againthe left (solid line) and right (dashed line) terms are inagreement. So, indeed the energy release in the simula-tion agrees with the rates predicted by fast reconnection.We now compare the energy release from the simu-lation with the theoretical predictions. The kinetic en-ergy release rate is calculated from Equation (26) basedon ( v m , v A m , B , d m , d m ) and is plotted in Fig. 8b(dashed-dot line). This is in fact just the dashed-dotline of Fig. 8a multiplied by 0.6. Next, we calculate thetotal rate of conversion of energy to heat from Equa-tion (26) based on ( v m , v A m , B , d m , d m ) and plot (a) d S / d t ( x e r g / s / c m ) SimulationdS/dt (v , B , d , d)dS/dt (v , B , d , d m ) (b) d K / d t ( x e r g / s / c m ) Simulation: 0.6(div term)Simulation: j(vxB) termTheory: 0.6 dS/dt (c) d W / d t ( x e r g / s / c m ) Simulation: 0.4(div term)Simulation: η j termTheory: 0.4 dS/dt Figure 8.
Comparison between simulation and theory for(a) the total inflow of Poynting flux, (b) the rate of energyconverted to kinetic energy and (c) the rate of energy con-verted to heat during the reconnection. it Fig. 8c (dashed-dot line). In both cases, the theo-retical predictions are in excellent agreement with thesimulation. 3.3.
Atmospheric Response
In this section we briefly discuss the atmosphericresponse to reconnection driven by flux cancellation.First, we study the time evolution of one individualcase. Then, we vary the height of the null at t = 0by changing the strength of the external horizontal fieldand studying five cases with different B . The valuesof B and the corresponding y N are shown in Table 1.In Fig. 9, we plot the y N values (vertical lines) and theinitial temperature stratification (solid line), in order tobetter visualize the initial location of the null point rela-2 T ( K ) Figure 9.
The vertical lines show the height of the nullat t = 0 min for all cases of Table 1, plotted against thebackground temperature stratification. tive to the corona, transition region, chromosphere andphotosphere.We focus on the time evolution of the temperature anddensity for case 2 (Fig. 10). The null point is initially lo-cated at the base of the corona. As reconnection starts,hot material is ejected along the post-reconnection fieldlines (panels a1, b1). A hot “loop” of 1.8 MK and den-sity of 2 × − g cm − is formed above the recon-nection site. Below the null, the top of the arcade isheated to 2.6 MK (panels a2, b2). As the polaritiesconverge the null height decreases, as predicted fromthe theory. When the null point reaches the base ofthe transition region and below, dense, cool plasma isejected along the reconnected field lines (Fig. 10, pan-els a3, b3). Due to the higher density of the region,the resulting heat released from the reconnection can-not raise the plasma temperature to millions of Kelvin.As the process continues, a cooler and denser ejection isformed, with temperature of 0 . − .
12 MK, and a den-sity of 2 × − − × − g cm − . It propagates withvelocity up to 105 km s − , extending from transitionregion to coronal heights (panels a4, b4).In Fig. 11 we plot the temperature (first column) anddensity (second column) for cases 2 − t = 40 min,while Case 1 is shown in Fig. 6b,c. An important qual-itative difference appears between the cases. When thenull point is initially located in the corona, both a hotand a cool plasma region develop above the null duringthe cancellation. When the initial null point is placedat progressively lower heights (top to bottom row), theamount of hot material decreases, while the cool mate-rial increases. Eventually, for a null point placed at thechromosphere (bottom row), the resulting post recon-nection plasma does not have a high-temperature com-ponent. In this case, the region above the null contains (i) a very cool component of photospheric or chromo-spheric material (around 6300 K) which is “slingshot-ted” upwards from the tension of the reconnected lineswith speed of 10-20 km s − and(ii) a cool plasma component which is heated by re-connection to around 0 . − .
03 MK.In Fig. 12 we plot the time evolution of the maximumvelocity of the hot (
T >
T < . − (105 km s − ),79 km s − (87 km s − ), 35 km s − (70 km s − ) and0 km s − (57 km s − ) respectively. Notice also that thehot and cool components are produced with a time de-lay, as shown previously in Fig. 10. The time differencebetween the acceleration of the hot and cold materialdecreases as the null point is situated lower. In cases 2and 3, the hot material appears first and the cold ma-terial after. For case 4, the hot and cold ejections arealmost co-temporal.3.4. Atmospheric Response DISCUSSIONIn Priest et al. (2018), we proposed magnetic recon-nection driven by photospheric flux cancellation as amechanism for energizing coronal loops and heating thechromosphere. We also derived analytical expressionsthat predict the energy release during reconnection. Inthe present work, we begin to numerically validate ourtheory by developing the theory in 2D and comparingit with computations of two converging polarities insidea stratified atmosphere containing a background hori-zontal field. As the polarities converge, reconnection isdriven at the null point.To compare the theory with simulations, we evaluatedseveral quantities from the simulations. For example, wecalculated the velocity of approach of the opposite po-larities in two ways. One was to use the values that cor-respond to the simulation’s driver and the other was tomeasure the response of the photosphere to that driver.We found excellent agreement between theory andsimulation, especially with the second approach. Theresponse to the driver is to initiate motions in the nu-merical domain which lead to reconnection. It is foundthat the theory agrees well with the system’s response tothe driver, which is encouraging, since it shows that thatour theory could be used to derive estimates of the en-ergy released during flux cancellation from solar obser-vations, as observations measure the photospheric andatmospheric response, without knowledge of the sub-photospheric conditions driving the cancellation. Weconclude, based on our 2D computational experiments,3 y ( Mm ) (a1) t=10.1 min0246810 y ( Mm ) (a2) t=16.6 min0246810 y ( Mm ) (a3) t=36.2 min0246810 y ( Mm ) −20 −10 0 10 20x [Mm](a4) t=39.9 min T ( K ) (b1) (b2) (b3) −20 −10 0 10 20x [Mm] (b4) −15.0−14.5−14.0−13.5−13.0−12.5−12.0−11.5−11.0−10.5−10.0−9.5−9.0−8.5−8.0−7.5−7.0 ρ ( g / c m ) Figure 10.
Evolution of temperature (left column) and density (right column) for case 2. that the energy released during photospheric cancella-tion can be accurately estimated from a knowledge ofthe converging velocity, the separation and strengths ofthe converging fluxes, the strength of the backgroundmagnetic field, and the density and Alfv´en Mach num-ber of the material flowing into the current sheet.The promising results from these 2D simulations sug-gest that our analytical estimates can indeed be usedto predict energy release. Ideal observational candi-dates for such a comparison in future include many caseswhere photospheric flux cancellation is associated withsmall-scale energy release, such as Ellerman bombs, UVbursts and IRIS bombs or the energy injected into coro-nal loops due to flux cancellation at their feet.We have also presented an initial study of the at-mospheric response to reconnection (for more sophisti-cated and realistic simulations, see, e.g., Danilovic et al.(2017); Hansteen et al. (2017); N´obrega-Siverio et al.(2018)). The maximum height of the null point in 2D is d . As the polarities converge, the null point movesup to its maximum height and then down towards thephotosphere. The atmospheric response during photo-spheric cancellation is as follows. When the null pointis located initially at a coronal height, a hot “loop”(around 1 − − and a temperature of 0 . − . y ( Mm ) (a1) Case 20246810 y ( Mm ) (a2) Case 30246810 y ( Mm ) (a3) Case 40246810 y ( Mm ) −20 −10 0 10 20x [Mm](a4) Case 5 T ( K ) (b1) (b2) (b3) −20 −10 0 10 20x [Mm] (b4) −15.0−14.5−14.0−13.5−13.0−12.5−12.0−11.5−11.0−10.5−10.0−9.5−9.0−8.5−8.0−7.5−7.0 ρ ( g / c m ) Figure 11.
Temperature (left column) and density (right column) for cases 2-5 at t = 40 min. loop, whereas the cool structures have values that arereminiscent of surges or larger spicules.Note that, if only part of the photospheric flux cancels,the null point stops moving towards the photosphereat some intermediate height. Then, reconnection oc-curs only between the initial and final height of the null,which produces a shorter, less energetic burst of energyrelease.There is an important difference between 2D and 3Dsimulations. In 2D, the magnetic field of a source fallsoff with distance like 1 /r , whereas in 3D, the fall-off be-haves like 1 /r . As a result, the interaction distancein 2D ( d = 2 F/ ( πB )) is larger than its 3D value( d = (cid:112) F/ ( πB )) (see Priest et al. 2018), which pro-duces a higher location for the null point in 2D than in3D for a given polarity separation distance, photosphericflux and background field. Thus, in our 2D simulations,in order to place the null at a particular height in thestratified atmosphere, we adopt a stronger background field than would be needed in 3D. During the recon-nection, this higher background field produces a largerPoynting influx into the current sheet in 2D than in 3D.The result is that more energy is converted into heatand kinetic energy. Therefore, we leave a detailed dis-cussion of temperature and density distributions alongreconnected field lines for a future 3D experiment. In3D, the energy release may well accelerate the coolerplasma to form shorter structures than in 2D.In our model we have assumed a horizontal externalfield in order to be able to make a direct comparison withour analytical theory. An oblique external field wouldhave several extra effects. Firstly, it would enhanceplasma draining along field lines, changing the maxi-mum length and density of the heated plasma structures,an effect that would be stronger for the cooler ejections.Secondly, the null point would move sideways, as well asvertically. This could affect the width of the structuresand possibly produce “thread-like” ejections. Thirdly,5 (a) v ( k m / s ) Case 2
Hot ejectionCool ejection (b) v ( k m / s ) Case 3
Hot ejectionCool ejection (c) v ( k m / s ) Case 4
Hot ejectionCool ejection (d) v ( k m / s ) Case 5
Hot ejectionCool ejection
Figure 12.
Maximum velocity of the hot (
T >
T < . after reconnection, the flows above the null point willhave up and down components, instead of being mainlyhorizontal. The resulting magnetic “loop” would haveits footpoints rooted in the photosphere and the ejec-tion would be dominated by a single inclined upflow(together with a much shorter downflow), rather thanconsisting of two opposite directly horizontal flows (e.g.,Fig. 10). Jet-like structures have been observed at thefeet of coronal loops (Chitta et al. 2017a,b) which couldbe related to the upflows we expect in the oblique field.However, we do not expect the energy release to changedrastically. In Sec. 2, we derived the rate of heating byassuming it is half the total inflow of Poynting flux. Foran oblique field, the flux function ψ would be different,but, during flux cancellation, the same amount of fluxwill be cancelled, irrespective of the orientation of theexternal field. Consequently, the same Poynting influxinto the current sheet would be produced over the sametime scale. Thus, the energy release rate should not besignificantly different. We aim to check this numericallyin future. In this work, we have positively validated our ana-lytical theory using 2D numerical computations. Thissuggests that nanoflares driven by magnetic flux can-cellation can indeed be an important mechanism forheating the chromosphere and corona, as proposed inPriest et al. (2018), which is built upon recent observa-tional findings. In future, we aim to extend our modelin several ways to make it more realistic and to con-sider more cases. In particular, we shall study obliqueexternal fields in order to determine in more detail theways in which chromospheric and coronal loops may beheated by reconnection at their footpoints. We shallalso set up a fully three-dimensional computation in or-der to study the extent and implications of our theoryand to deduce in more detail the atmospheric responseto energy release.L.P.C. received funding from the European Union’sHorizon 2020 research and innovation programmeunder the Marie Sk(cid:32)lodowska-Curie grant agreementNo. 707837. This research has made use of NASA’s As-trophysics Data System. The authors are most gratefulfor invaluable discussions with Hardi Peter, Clare Par-nell and Alan Hood.6 APPENDIX A. FLUX CORRECTION FACTORAt x = 0, the flux contained between the heights ¯ y and ¯ y is ψ = (cid:82) ¯ y ¯ y ¯ B x d ¯ y , which for our magnetic field (Equa-tion (36)) becomes: ψ ¯ y ¯ y = 2 Fπ (cid:20) arctan ¯ y − ¯ y ¯ d − (¯ y − ¯ y ) − (cid:18) arctan ¯ y − ¯ y ¯ d − (¯ y − ¯ y ) (cid:19)(cid:21) . (A1)Thus, the total flux from the sources to the upper boundary ( y max ) is ψ ¯ y max ¯ y = 2 Fπ arctan ¯ y max − ¯ y ¯ d , (A2)which depends on ¯ d and ¯ y max . For a semi-infinite domain, such as considered in Sec. 2, ¯ y max → ∞ and so the fluxbecomes F , which is independent of ¯ d . However, the simulation has a finite region, and so the dependence of flux on¯ d and ¯ y max has to be taken into account in order to compare with theory.Consider the fluxes above and below the null point. The flux from the sources (¯ y ) to the null (¯ y (cid:48) N = (cid:112) ¯ d − ¯ d + ¯ y )is: ψ ¯ y (cid:48) N ¯ y = 2 Fπ (cid:16) arctan ¯ y N ¯ d − ¯ y N (cid:17) , (A3)whereas the flux from the null point to the upper boundary of the numerical domain is ψ ¯ y max ¯ y (cid:48) N = 2 Fπ (cid:20) arctan ¯ y max − ¯ y ¯ d − (¯ y max − ¯ y ) − (cid:16) arctan ¯ y N ¯ d − ¯ y N (cid:17)(cid:21) . (A4)As the sources cancel (and ¯ d decreases from 1 to 0), the flux below the null point changes from 0 to F , resulting in atotal cancelled flux of ∆ ψ ¯ y (cid:48) N ¯ y = F. (A5)The flux above the null up to ¯ y max changes by∆ ψ ¯ y max ¯ y (cid:48) N = − Fπ arctan (¯ y max − y ) , (A6)which becomes ∆ ψ ∞ ¯ y (cid:48) N = − F as ¯ y max → ∞ . Therefore, in a semi-infinite domain, when moving the sources from ¯ d = 1to ¯ d = 0, there is flux balance between the fluxes below and above the null. However, for a finite ¯ y max , there is extraflux above ¯ y > ¯ y max which we do not take into account. Thus, in a finite domain, it is not possible to fully cancel thetwo magnetic sources, to give a configuration with a uniform horizontal field, since a flux of | F | would be cancelledbelow the null while adding less than −| F | above the null.In the simulation, the rates of change of flux from the sources up to the null and from the null up to ¯ y max are˙ ψ ¯ y (cid:48) N ¯ y = 2 Fπ v d (cid:114) d − , (A7)and ˙ ψ ¯ y max ¯ y (cid:48) N = ˙ ψ f − ˙ ψ ¯ y (cid:48) N ¯ y , (A8)where ˙ ψ f = 2 Fπ v d ¯ y max − ¯ y (¯ y max − ¯ y ) + ¯ d . (A9)7 f Figure 13.
The flux correction factor ( f ) from Equation (A14), using the simulation’s values for ¯ d ( t ) and ¯ y max . The flux outside the finite domain changes at a rate ˙ ψ ∞ ¯ y max = − ˙ ψ f , and the rate of change of flux added to the regionabove the null is −| v i b i | . Therefore, below the null, from Equation (A8), the rate of change has to be:˙ ψ ≡ ˙ ψ ¯ y (cid:48) N ¯ y = | v i b i | + ˙ ψ f (A10)For a semi-infinite domain, ˙ ψ f →
0, and therefore ˙ ψ ¯ y max ¯ y (cid:48) N = − ˙ ψ ¯ y (cid:48) N ¯ y .This affects the theory in the following way. In Sec. 2, the inflow speed was found using the conservation of flux and B i : v i B i = ˙ ψ. (A11)To compare theory with simulation, we must use ˙ ψ from Equation (A10) to give v i = v d L (cid:32) − ¯ y max − ¯ y (¯ y max − ¯ y ) + ¯ d (cid:112) / ¯ d − (cid:33) (A12)or v i = f v d L , (A13)where f = 1 − ¯ y max − ¯ y (¯ y max − ¯ y ) + ¯ d (cid:112) / ¯ d − . (A14) f is a flux correction factor, which is plotted in Fig. 13 and which modifies several of the previous expressions, namely,changing L sp → f L sp , L → f L , dW /dt → f dW /dt . For a semi-infinite domain ( y max → ∞ ), f → v i → v d /L . REFERENCES Arber, T., Longbottom, A., Gerrard, C., & Milne, A. 2001,Journal of Computational Physics, 171, 151 ,doi: 10.1006/jcph.2001.6780Archontis, V., & Hood, A. W. 2010, A&A, 514, A56,doi: 10.1051/0004-6361/200913502 Avrett, E. H., & Loeser, R. 2008, ApJS, 175, 229,doi: 10.1086/523671Bhattacharjee, A., Huang, Y. M., Yang, H., & Rogers, B.2009, Phys. Plasmas, 16, 112102 Birn, J., & Priest, E. R. 2007, Reconnection of MagneticFields: MHD and Collisionless Theory and Observations(Cambridge, UK: Cambridge University Press)Birn, J., Drake, J., Shay, M., et al. 2001, J. Geophys. Res.,106, 3715, doi: 10.1029/1999JA900449Biskamp, D. 1986, Phys. Fluids, 29, 1520Chitta, L. P., Peter, H., & Solanki, S. K. 2018, A&A, 615,L9, doi: 10.1051/0004-6361/201833404Chitta, L. P., Peter, H., Young, P. R., & Huang, Y.-M.2017a, Astron. Astrophys., 605, A49,doi: 10.1051/0004-6361/201730830Chitta, L. P., Peter, H., Solanki, S. K., et al. 2017b,Astrophys. J. Suppl., 229, 4,doi: 10.3847/1538-4365/229/1/4Danilovic, S., Solanki, S. K., Barthol, P., et al. 2017, TheAstrophysical Journal Supplement Series, 229, 5,doi: 10.3847/1538-4365/229/1/5Forbes, T., & Priest, E. 1987, Rev. Geophys., 25, 1583,doi: 10.1029/RG025i008p01583Golub, L., Krieger, A., Silk, J., Timothy, A., & Vaiana, G.1974, Astrophys. J., 189, L93Hansteen, V. H., Archontis, V., Pereira, T. M. D., et al.2017, ApJ, 839, 22, doi: 10.3847/1538-4357/aa6844Harvey, K. L., & Martin, S. F. 1973, Solar Phys., 32, 389Heyvaerts, J., Priest, E., & Rust, D. 1977, Astrophys. J.,216, 123, doi: 10.1086/155453Hong, J., Ding, M. D., & Cao, W. 2017, ApJ, 838, 101,doi: 10.3847/1538-4357/aa671eHuang, Z., Mou, C., Fu, H., et al. 2018, Astrophys. J., 853,L26, doi: 10.3847/2041-8213/aaa88cHuba, J. D. 2003, in Space Simulations, ed. M. Scholer,C. Dum, & J. B¨uchner (New York: Springer), 170–197Huba, J. D., & Rudakov, L. I. 2004, Phys. Rev. Lett., 93,175003, doi: 10.1103/PhysRevLett.93.175003Johnston, C. D., Hood, A. W., Cargill, P. J., & De Moortel,I. 2017, A&A, 597, A81,doi: 10.1051/0004-6361/201629153Kim, Y.-H., Yurchyshyn, V., Bong, S.-C., et al. 2015, ApJ,810, 38, doi: 10.1088/0004-637X/810/1/38Lee, L.-C., & Fu, Z. 1986, J. Geophys. Res., 91, 6807,doi: 10.1029/JA091iA06p06807Libbrecht, T., Joshi, J., Rodr´ıguez, J. d. l. C., Leenaarts,J., & Ramos, A. A. 2017, A&A, 598, A33,doi: 10.1051/0004-6361/201629266Longcope, D. W. 1998, Astrophys. J., 507, 433Loureiro, N. F., Samtaney, R., Schekochihin, A. A., &Uzdensky, D. A. 2012, Phys. Plasmas, 19, 042303,doi: 10.1063/1.3703318Loureiro, N. F., Schekochihin, A. A., & Cowley, S. C. 2007,Phys. Plasmas, 14, 100703 Loureiro, N. F., Schekochihin, A. A., & Uzdensky, D. A.2013, Phys. Rev. E, 87, 013102,doi: 10.1103/PhysRevE.87.013102Martin, S. F., Livi, S., & Wang, J. 1985, Astrophys. J., 38,929Mart´ınez-Sykora, J., Hansteen, V., & Moreno- Insertis, F.2011, ApJ, 736, 9, doi: 10.1088/0004-637X/736/1/9Meyer, C. D., Balsara, D. S., & Aslam, T. D. 2012,MNRAS, 422, 2102,doi: 10.1111/j.1365-2966.2012.20744.xMoore, R. L., Cirtain, J. W., Sterling, A. C., & Falconer,D. A. 2010, Astrophys. J., 720, 757,doi: 10.1088/0004-637X/720/1/757Moreno-Insertis, F., & Galsgaard, K. 2013, Astrophys. J.,771, 20, doi: 10.1088/0004-637X/771/1/20Nelson, C. J., Doyle, J. G., & Erd´elyi, R. 2016, MNRAS,463, 2190, doi: 10.1093/mnras/stw2034Nelson, C. J., Freij, N., Reid, A., et al. 2017, TheAstrophysical Journal, 845, 16N´obrega-Siverio, D., Mart´ınez-Sykora, J., Moreno-Insertis,F., & Rouppe van der Voort, L. 2017, ApJ, 850, 153,doi: 10.3847/1538-4357/aa956cN´obrega-Siverio, D., Moreno-Insertis, F., &Mart´ınez-Sykora, J. 2018, ApJ, 858, 8,doi: 10.3847/1538-4357/aab9b9Nordlund, ˚A., & Stein, R. F. 1990, Computer PhysicsCommunications, 59, 119,doi: 10.1016/0010-4655(90)90161-SParnell, C. E., & Priest, E. R. 1995, Geophys. Astrophys.Fluid Dyn., 80, 255, doi: 10.1080/03091929508228958Peter, H., Tian, H., Curdt, W., et al. 2014, Science, 346,doi: 10.1126/science.1255726Petschek, H. 1964, in The Physics of Solar Flares(Washington: NASA Spec. Publ. SP-50), 425–439Priest, E. 1986, Mit. Astron. Ges., 65, 41Priest, E. 2014, Magnetohydrodynamics of the SunPriest, E., & Forbes, T. 1986, J. Geophys. Res., 91, 5579Priest, E., Parnell, C., & Martin, S. 1994, Astrophys. J.,427, 459Priest, E. R., Chitta, L. P., & Syntelis, P. 2018, ApJL, 862,L24, doi: 10.3847/2041-8213/aad4fcReid, A., Mathioudakis, M., Doyle, J. G., et al. 2016, ApJ,823, 110, doi: 10.3847/0004-637X/823/2/110Rezaei, R., & Beck, C. 2015, A&A, 582, A104,doi: 10.1051/0004-6361/201526124Rutten, R. J. 2016, A&A, 590, A124,doi: 10.1051/0004-6361/201526489Rutten, R. J., van der Voort, L. H. M. R., & Vissers, G.J. M. 2015, The Astrophysical Journal, 808, 1339