Chromospheric and coronal heating and jet acceleration due to reconnection driven by flux cancellation. I. At a three-dimensional current sheet
AAstronomy & Astrophysics manuscript no. main © ESO 2021January 13, 2021
Chromospheric and coronal heating and jet acceleration due toreconnection driven by flux cancellation
I. At a three-dimensional current sheet
E.R. Priest and P. Syntelis School of Mathematics and Statistics, University of St. Andrews, Fife KY16 9SS, Scotland, UKe-mail: [email protected]
Received ; accepted
ABSTRACT
Context.
The recent discovery of much greater magnetic flux cancellation taking place at the photosphere than previously realised hasled us in our previous works to suggest magnetic reconnection driven by flux cancellation as the cause of a wide range of dynamicphenomena, including jets of various kinds and solar atmospheric heating.
Aims.
Previously, the theory considered energy release at a two-dimensional current sheet. Here we develop the theory further byextending it to an axisymmetric current sheet in three dimensions without resorting to complex variable theory.
Methods.
We analytically study reconnection and treat the current sheet as a three-dimensional structure. We apply the theory tothe cancellation of two fragments of equal but opposite flux that approach each another and are located in an overlying horizontalmagnetic field.
Results.
The energy release occurs in two phases. During Phase 1, a separator is formed and reconnection is driven at it as it risesto a maximum height and then moves back down to the photosphere, heating the plasma and accelerating a plasma jet as it does so.During Phase 2 the fluxes cancel in the photosphere and accelerate a mixture of cool and hot plasma upwards.
Key words.
Sun: chromosphere – Sun: corona – Sun: magnetic fields – Magnetic reconnection – Methods: analytical
1. Introduction
Observations of the photospheric magnetic field at a resolutionof 0.15 arcsec from the Sunrise balloon (Solanki et al. 2010,2017) have shown that the rate of magnetic flux emergence andcancellation is an order of magnitude higher than previouslythought (Smitha et al. 2017). In addition, coronal loops havebeen found to be invariably rooted in mixed polarity fragmentsthat are cancelling at a rate of typically 10 Mx sec − , and theloops brighten when photospheric magnetic flux cancels (Tiwariet al. 2014; Chitta et al. 2017, 2018; Huang et al. 2018). Fluxcancellation has also been associated with the acceleration ofjets on a variety of scales (Sterling et al. 2015; Sterling & Moore2016; Sterling et al. 2017; Panesar et al. 2018; Samanta et al.2019; Panesar et al. 2020).The Sunrise observations led Priest, Chitta, & Syntelis(2018) to propose a ‘cancellation nanoflare model’ for heat-ing the chromosphere and corona, not just X-ray bright points,for which flux cancellation had previously been suggested as amechanism (Priest et al. 1994; Parnell & Priest 1995). The modelwas supported and extended by numerical simulations, whichalso showed that various kinds of reconnection-driven jets canform during flux cancellation (Syntelis et al. 2019; Syntelis &Priest 2020).Priest, Chitta, & Syntelis (2018) considered in particular amodel in which two magnetic fragments of flux F and − F ap-proach each other and cancel in an overlying horizontal mag-netic field of strength B . Initially, when far apart, the magneticsources are not connected, but separator reconnection (Priest &Titov 1996; Longcope & Cowley 1996; Galsgaard & Nordlund 1996; Parnell et al. 2008) starts to occur when the half-separation(d) of the sources becomes less than the flux interaction distance(Longcope 1998), d = (cid:32) F π B (cid:33) / . (1)As d decreases and the flux sources approach each other, theseparator rises to a maximum height of 0 . d and then falls tothe solar surface as the sources come into contact. The maxi-mum height at which reconnection occurs therefore depends onthe flux ( F ) and field strength ( B ) through the parameter d ,and it may be located in the chromosphere, transition region, orcorona. Numerical experiments studying this scenario have sincereinforced the validity of the model (Syntelis et al. 2019; Peteret al. 2019; Syntelis & Priest 2020), and as have recent observa-tions (Park 2020).In this series of papers, we plan to develop the basic theoryin several directions. Here we remedy a deficiency in the theory,namely, that the properties of the current sheet have so far usu-ally been analysed using complex variable theory, according towhich the input magnetic field at the entrance to a current sheetof length L is related to the sheet length ( L ) and the field gradient( k ) near the initial null point or separator by B i = kL , (2)which is a key result used in the theory.Thus, since complex variable theory applies only in two di-mensions (2D), the theory so far is valid only in 2D. In the Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . S R ] J a n & A proofs: manuscript no. main present paper, we therefore develop a corresponding theory fora current sheet in three dimensions (3D), in particular for an ax-isymmetric current sheet, and derive a generalisation of the resultin Eq. (2). We also apply this new result to the case of reconnec-tion driven by the approach of equal and opposite flux sources inan overlying uniform horizontal magnetic field studied in Priest,Chitta, & Syntelis (2018). First of all, we consider a 2D currentsheet and show how the expression for the magnetic field aroundit can be obtained by the new method without using complexvariable theory (Sect. 2.2). Then we generalise this method to thefield of a 3D axisymmetric current sheet (Sect. 3.2), and finallywe apply it to the creation of such a sheet by the flux cancellationof two flux sources (Sect. 4). (a) xz B i L ½L-½L
O (b) xz
Fig. 1.
X-point magnetic field (a), which collapses into (b) a field witha current sheet of length L and input magnetic field B i at x = + , z =
2. Relationship between B i and L for a 2D currentsheet Here we calculate the relationship between the input magneticfield ( B i ) to a current sheet and its length ( L ) by firstly the tra-ditional complex variable technique (Sect. 2.1) and secondly anew method that does not rely on complex variable theory (Sect.2.2) and so can be generalised to 3D (Sect. 3.2). We consider a potential magnetic field of the form B x = kz , B z = kx , that contains an X-type neutral point at the origin, where k is aconstant (Fig. 1a). This may be written in terms of the complexvariable Z = x + iz as simply B z + iB x = kZ . Now we suppose the distant sources of the magnetic fieldmove in such a way that the field collapses to a configurationcontaining a current sheet of length L stretching along the z -axis,as shown in Fig. 1b. Then an elegant way of writing the field thatis outside the current sheet is B z + iB x = k ( L + Z ) / , (3)so that the sheet is a cut in the complex plane from Z = − iL to Z = iL . In particular, it can be seen that the magnetic field( B i ) in Eq. (2) at the entrance ( x = + , z =
0) to the currentsheet can be obtained from Eq. (3) by putting z = x tend to zero through positive values. This method was first discovered by Green (1965) and later used by many authors, in-cluding Priest & Raadu (1975), Tur & Priest (1976), Somov &Syrovatsky (1976), Low (1987, 1991), Titov (1992).Within a 1D current sheet with magnetic field B z ( x ), the elec-tric current density is given by µ j = − ∂ B z ∂ x , which may be integrated across a sheet of width l to give a re-lationship between the current ( J ( z )) in the sheet at distance z along it and the magnetic field B S ( z ) = B z ( l , z ) at the edge ofthe sheet as µ J = (cid:90) l / − l / µ jdx = − (cid:90) l / − l / ∂ B z ∂ x dx = − [ B z ] l / − l / = − B z ( l , z ) , or µ J ( z ) = − B S ( z ) . (4)In particular, for the current sheet in Eq. (3), by letting x approach zero through positive values, we find B S ( z ) = k ( L − z ) / , (5)and so the total current ( I ) in the sheet is given by µ I = µ (cid:90) L / − L / Jdz = − (cid:90) L / − L / k ( L − z ) / dz or µ I = π kL = − π B i L . B i O B N N (a) xz ^x~^z~ B i NQ P r (b)z Fig. 2.
Notation used for the 2D magnetic field. (a) Magnetic fieldof the current sheet alone, where the values at the points O(0 + ,
0) andN(0 , L ) are denoted by B i ˆ z and B N ˆ x , respectively. (b) Representationof the current sheet by a set of line currents, denoted by dots, in whichthe magnetic field at a point P is calculated due to a line current at Q ata distance z along the z -axis. We note that the magnetic field of the current sheet alone(Fig. 2) is obtained by subtracting the background field from Eq.(3) to give B z + iB x = k ( L + Z ) / − kZ , (6)which implies that the field at ( x = + , z =
0) is B z = kL , whilethe field at the end N(0 , L ) of the current sheet is B x = − kL ,namely, minus the X-point field at N, since the field of the currentsheet plus background (Eq. (3)) vanishes at N. Article number, page 2 of 10.R. Priest and P. Syntelis: Heating by Flux Cancellation
We suppose the field has components( B x , B z ) = ( kz , kx ) + ( b S x ( x , z ) , b S z ( x , z )) (7)due to the X-field together with the field of the current sheet.We write the z -component of the field at the edge of the sheet asbefore, as B S ( z ) = b S z (0 + , z ) , such that B S (0) = B i . Then the current in the sheet is given as before by µ J ( z ) = − B S ( z ) . Now we suppose the current sheet consists of an infinite setof line currents J ( z (cid:48) ) dz (cid:48) at points Q a distance z (cid:48) along the currentsheet, each of them producing a magnetic field, at a point P, of b S = µ Jdz (cid:48) π r ˆ φ , where ( r , φ, z ) are cylindrical polar coordinates measured locallyrelative to Q (Fig. 2b).Therefore, the field at P due to the whole current sheet hasan x -component b S x ( x , z ) = − π (cid:90) L / − L / B S ( z (cid:48) ) sin φ [ x + ( z − z (cid:48) ) ] / dz (cid:48) , where B s = µ J and sin φ = ( z − z (cid:48) ) / [ x + ( z − z (cid:48) ) ] / .In terms of the dimensionless variables ¯ z = z / L , ¯ x = x / L and ¯ B S = B S / ( kL ), this becomes¯ b S x ( ¯ x , ¯ z ) = − π (cid:90) − ¯ B S (¯ z (cid:48) )(¯ z − ¯ z (cid:48) )¯ x + (¯ z − ¯ z (cid:48) ) d ¯ z (cid:48) . (8)Now the condition that our infinite set of line currents com-prises a current sheet is that the tangential field vanish at its sur-face, meaning, that B x ( ¯ x , ¯ z ) = k ¯ z + b S x ( ¯ x , ¯ z ) vanish as ¯ x tends tozero. In dimensionless variables this becomeslim ¯ x → (¯ b S x ( ¯ x , ¯ z )) = − ¯ z when ¯ x < , which, after substituting for ¯ b S x from Eq. (8), becomeslim ¯ x → π (cid:90) − ¯ B S (¯ z (cid:48) )(¯ z − ¯ z (cid:48) )¯ x + (¯ z − ¯ z (cid:48) ) d ¯ z (cid:48) = ¯ z . (9)This is an integral equation to solve for the unknown function¯ B S (¯ z (cid:48) ).The way we solve it is to consider N + z = − , ¯ z , ¯ z , ...., ¯ z N − , ¯ z N = +
1, such that¯ B S (¯ z n ) = B n and B = B N =
0. We approximate ¯ B S (¯ z (cid:48) ) by Nlinear functions in these N intervals stretching between ¯ z (cid:48) n and¯ z (cid:48) n + , namely, f n (¯ z (cid:48) ) = a n ¯ z (cid:48) + b n = a n (¯ z (cid:48) − ¯ z ) + c n , where n = , , , ... N − c n = B n + a n ¯ z . The constants thatmake this piecewise linear function continuous are a n = B n + − B n ¯ z n + − ¯ z n , b n = B n ¯ z n + − B n + ¯ z n ¯ z n + − ¯ z n , Fig. 3.
For a 2D current sheet (a) the dependence of the maximum cur-rent sheet field ( B S (0) = B i ) on N and (b) the profile ( B S ( z )) as a func-tion of the number ( N ) of points in the sheet. where ¯ z n + − ¯ z n = / N .Eq. (9) is then approximated bylim ¯ x → π N − (cid:88) n = (cid:90) ¯ z n + ¯ z n − a n (¯ z − ¯ z (cid:48) ) − c n (¯ z − ¯ z (cid:48) )¯ x + (¯ z − ¯ z (cid:48) ) d ¯ z (cid:48) = ¯ z . After some manipulation, this can be evaluated to givelim ¯ x → π N − (cid:88) n = a n ¯ x arctan (cid:32) x / N ¯ x + (¯ z n + − ¯ z )(¯ z n − ¯ z ) (cid:33) − a n N −− c n log e (cid:32) ¯ x + (¯ z n + − ¯ z ) ¯ x + (¯ z n − ¯ z ) (cid:33) = ¯ z . (10)as shown in Fig. 3. Evaluating this at N points should determinethe N unknowns B , B , .... B N . For the N points we pick themidpoints ¯ z ∗ m = (¯ z m + ¯ z m + ) of the intervals, where B S = ( B m + B m + ), m = , , ... N − ¯ x → π N − (cid:88) n = a n ¯ x arctan x / N ¯ x + (¯ z n + − (¯ z m + ¯ z m + ))(¯ z n − (¯ z m + ¯ z m + )) − ( B n + − B n ) − ( B n + B n + )4 log e ¯ x + (¯ z n + − (¯ z m + ¯ z m + )) ¯ x + (¯ z n − (¯ z m + ¯ z m + )) = (¯ z m + ¯ z m + ) . In the limit as ¯ x →
0, the first term in the summation vanishes,while the second term reduces to B − B N , which also vanishes Article number, page 3 of 10 & A proofs: manuscript no. main and so we are left with N equations for each value of m of theform − π N − (cid:88) n = ( B n + B n + ) log e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ¯ z n + − (¯ z m + ¯ z m + )¯ z n − (¯ z m + ¯ z m + ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (¯ z m + ¯ z m + ) . For each value of N, the N values of B m = ¯ B S ( z m ) are calculatedby numerically solving these equations, with the result, as shownin Fig. 3, that the piecewise linear approximation to ¯ B S (¯ z ) tendsto the function ¯ B S (¯ z ) = √ − ¯ z = (cid:112) − z / L as N → ∞ ,as expected from the complex variable theory result, with B i = kL . (a) z Q(R, ,0) P(R, ,z) r (b) R +½LR -½L R R ' R R ' R ' R R ' I Fig. 4.
Notation in cylindrical polar coordinates ( R , φ, z ) for (a) an ax-isymmetric current sheet stretching from R = R − L to R = R + L and (b) a ring current of radius R (cid:48) indicating the positions of a generalpoint P( R , φ, z ) and a point Q( R , φ,
0) in the plane of the ring at a radius R = R (cid:48) + r .
3. Relationship between B i and L for a 3Daxisymmetric current sheet Complex variable theory applies only to 2D. However, the anal-ysis of Sect. 2.2 may be extended into 3D in order to calculatethe magnetic field of a 3D axisymmetric current sheet (Fig. 4a),as first suggested by Tur (1977) and explored briefly by Long-cope & Cowley (1996). The technique we propose here is builton their ideas. Using cylindrical polar coordinates ( R , φ, z ), wewrite the radial and axial components of the magnetic field at P ( R , φ, z ) near the current sheet in the form( B R , B z ) = ( kz , kr ) + ( b S R ( R , z ) , b S z ( R , z )) , (11)where the first term represents the field of a ring of X-points near R = R , while ( b S R , b S z ) is the field of the current sheet itself. Thetangential (namely, R -) component of the field at the edge of thesheet is then, say, B S ( R ) = b S R ( R , + ) , such that at the centre of the sheet B S (0) = B i . Also, the integral form of Ampère’s law may be used to showthat the current J ˆ φ in the sheet is related to B S by µ J ( R ) = B S ( R ) . (12) The aim is to deduce what profile of B S and therefore ofcurrent J in the sheet makes the normal component ( B z ( R (cid:48) )) ofmagnetic field vanish at the current sheet so that, according toEq. (11), kR = − lim z → b S z ( R , z ) for z = , − L < R (cid:48) < L . (13)The plan is therefore to calculate the magnetic field due to a cur-rent ring (Sect. 3.1), and then to sum over an infinite set of in-finitesimal current rings to find the magnetic field of the currentsheet (Sect. 3.2). We consider a ring of current ( I ( R (cid:48) )) of radius R (cid:48) in the z = F ( R , φ )] at a point P ( R , φ, z ) may be calculated as follows(Jackson 1999). In general, the vector potential ( A ) is such that B = ∇ × A and satisfies Poisson’s equation ∇ A = − µ j , which has general solution A = µ π (cid:90) j ( r (cid:48) ) | r − r (cid:48) | dV (cid:48) . For our current ring j dV (cid:48) = I δ ( R (cid:48) − R (cid:48) ) δ ( z (cid:48) ) R (cid:48) d φ (cid:48) dz (cid:48) ˆ φ , andat P ( R , φ, z ) the only component of A is A φ ( R ), giving a fluxfunction F ≡ RA φ of F ( R , φ, z ) = µ I R (cid:48) R π (cid:90) π cos φ (cid:48) d φ (cid:48) ( R + R (cid:48) + z − R (cid:48) R cos φ (cid:48) ) / , (14)where s (cid:48) = [( R − R (cid:48) cos φ (cid:48) ) + ( R (cid:48) ) sin φ (cid:48) + z ] / is the distance between the points ( R , , z ) in the plane φ = R (cid:48) , φ (cid:48) ,
0) on the ring. The corresponding field components are B R = − R ∂ F ∂ z , B z = R ∂ F ∂ R . After some manipulation, Eq. (14) may be written as F ( R , φ, z ) = µ I π k (cid:113) R (cid:48) R [(2 − K ) M ( K ) − E ( K )] , (15)in terms of K defined by K = R (cid:48) R ( R + R (cid:48) ) + z , (16)and the complete elliptic integrals of the first and second kind M ( K ) = (cid:90) π/ [1 − K sin x ] − / dx , E ( K ) = (cid:90) π/ [1 − K sin x ] / dx . The flux function at P( R , φ, z ) near the current ring (Fig. 4b)can be found by writing R = R (cid:48) + r and expanding Eq. (16) in Article number, page 4 of 10.R. Priest and P. Syntelis: Heating by Flux Cancellation
B weaker B stronger B I Fig. 5.
Magnetic field lines near a segment of a circular current loop ofcurrent I . powers of r / R (cid:48) (cid:28) z / R (cid:48) (cid:28)
1, using Gradshteyn & Ryzhik(1980), to give F = RA = µ I π (cid:40) R (cid:48) (cid:32) log e R (cid:48) ( z + r ) / − (cid:33) + r (cid:32) log e R (cid:48) ( z + r ) / − (cid:33)(cid:41) + .... (17)The corresponding magnetic field ( B z = (1 / R ) ∂ F /∂ R ) close tothe ring becomes, to lowest order in ( r + z ) / / R (cid:48) , B z ≈ − µ I π (cid:40) rr + z + R (cid:48) log e ( z + r ) / R (cid:48) (cid:41) , (18)where r = R − R (cid:48) . The first term is simply the field of a straightcurrent, and the second term gives the correction due to the cur-vature of the current ring. This correction lowers the magnitudeof the field on the outside of the ring and increases it on the in-side of the ring, as expected, since the field lines are further aparton the outer edge of the ring, as illustrated in Fig. 5. (c) z(a) (b) + zR L B N R O +½LR O -½LR O R O ' NC ^z~ B i ^R~B N R O +½LR O -½LR O R O ' NC ^z~ B i ^R~ Fig. 6.
Notation for the magnetic field near a current sheet in the Rz -plane due to the sum of (a) a ring of nulls at radius R (cid:48) and (b) a currentsheet of length L . The sheet may be replaced by (c) a set of current ringsof radius R (cid:48) between R = R − L and R = R + L . Just to the rightof the centre C of the sheet, the magnetic field is B i ˆR , while at the outeredge N of the current sheet the field is B N ˆz . We consider a current sheet of length L in the R -direction andcentred at a radius R with L (cid:28) R , so that the current sheetis short compared with its mean distance R from the z -axis, asshown in Fig. 6. It is made up of an infinite set of infinitesimal current rings that are located at radius R (cid:48) , say, with currents J ( R (cid:48) ) δ ( R − R (cid:48) ) dR (cid:48) ˆ φ , where R (cid:48) = R + r (cid:48) ranges between R − L and R + L , as indicated in Fig. 6c, where r (cid:48) (cid:28) R . Each ringhas a flux function of the form of Eq. (17) and a z -component ofmagnetic field of the same form as Eq. (18).The magnetic field of the current sheet may now built up anintegral of infinitesimal current rings J ( R (cid:48) ) dz at R (cid:48) = R + r (cid:48) ,where µ J = B S and each current ring gives rise to a magneticfield of the form of Eq. (18) with the distance between the currentring and P( R , φ, z ) being obtained by replacing r in Eq. (18) by( r − r (cid:48) ). The resulting z -component of magnetic field is b S z ( R , z ) ≈ − π (cid:90) L / − L / B S ( r (cid:48) ) r − r (cid:48) ) z + ( r (cid:48) − r ) + R log e [ z + ( r (cid:48) − r ) )] / R dr (cid:48) . (19)Then, after using this expression, the condition (Eq. (13))that the normal magnetic component vanish at the current sheetbecomes kR = π lim z → (cid:90) L / − L / B S ( r (cid:48) ) r − r (cid:48) ) z + ( r (cid:48) − r ) + R log e [ z + ( r (cid:48) − r ) )] / R dr (cid:48) , or, in terms of dimensionless variables¯ z = z / L , ¯ r (cid:48) = r (cid:48) / L , ¯ r = r / L , ¯ B S = B S / ( kL ) , ¯ r = π lim ¯ z → (cid:90) − ¯ B S (¯ r (cid:48) ) (cid:40) r − ¯ r (cid:48) )¯ z + (¯ r (cid:48) − ¯ r ) + (cid:15) log e (cid:18) (cid:15) z + (¯ r (cid:48) − ¯ r ) ] / (cid:19)(cid:41) d ¯ r (cid:48) , (20)where (cid:15) = L / (2 R ) (cid:28)
1. After taking the limit as ¯ z tends to zero,this reduces to¯ r = π (cid:90) − ¯ B S (¯ r (cid:48) ) (cid:40) r − ¯ r (cid:48) ) + (cid:15) log e (cid:15) + (cid:15) log e | ¯ r (cid:48) − ¯ r | (cid:41) d ¯ r (cid:48) . (21)Using the fact that the last term is important only where | ¯ r (cid:48) − ¯ r | (cid:46) (cid:15)/
8, so that ¯ B S (¯ r (cid:48) ) ≈ ¯ B S (¯ r ), we may evaluate its integral to give¯ r = π (cid:90) − ¯ B S (¯ r (cid:48) ) (cid:40) r − ¯ r (cid:48) ) + (cid:15) log e (cid:15) (cid:41) d ¯ r (cid:48) + π ¯ B S (¯ r ) (cid:15) e (cid:15) . (22)This is an integral equation for the unknown function¯ B S (¯ r (cid:48) ) = B S ( r (cid:48) ) / ( kL ), which may be solved, as in the 2D case,by approximating the function by a piecewise linear function of N equally spaced straight lines ( ¯ B S ( s ) = a n s + b n ) with N con-stants B n , and evaluating it at N − S i = ( s m + s m + ),where i = , , , .... ( N − (cid:15) log e (cid:15)/ (cid:28) (cid:15) , and so we may write¯ B S ≈ ¯ B S + (cid:15) log e ( (cid:15)/
8) ¯ B S , Article number, page 5 of 10 & A proofs: manuscript no. main where ¯ B S is the straight-field contribution and ¯ B S is thetoroidal correction, where (cid:15) = L / (2 R ) (cid:28)
1. The zeroth andfirst order parts of Eq. (22) become¯ r = (cid:90) − B S (¯ r (cid:48) ) | ¯ r (cid:48) − ¯ r | d ¯ r (cid:48) , (23)which is the same form as Eq. (9), and0 = (cid:90) − (cid:40) B S (¯ r (cid:48) ) | ¯ r (cid:48) − ¯ r | + ¯ B S (¯ r (cid:48) ) (cid:41) d ¯ r (cid:48) + B S (¯ r ) , (24)which determines ¯ B S (¯ r (cid:48) ). The solutions for ¯ B S and ¯ B S areshown in Fig. 7a,b.The main aim of this section is to determine the inflow field( B i ) to the current sheet. In the 2D case, it is just kL . For thetoroidal current sheet, it becomes B i = B S (0) = kL ¯ B S (0) { + (cid:15) log e (cid:15) ¯ B S (0) / ¯ B S (0) } , where ¯ B S = and ¯ B S (0) / ¯ B S (0) = − . ≈ − /
25, so that B i = kL (1 − . (cid:15) log e (cid:15) ) . (25)The resulting variation of B i with (cid:15) = L / (2 R ) is plotted in Fig.7c, which shows how B i increases as the radius R decreases andindicates the excellence of the 7 /
25 approximation.
4. Reconnection driven by the approach of twomagnetic fragments in a uniform horizontal field
Here we develop, in several ways, the theory for reconnectiondriven by the approach and cancellation of two photosphericmagnetic fragments that was proposed in Priest, Chitta, & Syn-telis (2018). The fragments have equal but opposite magneticflux ( ± F ) and are situated in an overlying uniform horizontalmagnetic field of strength B . They are separated by a distance2 d and approach each other at speeds ± v (Fig. 8a). The theoryso far has concerned ‘Phase 1’ of heating and jet acceleration,during which a separator forms in the photosphere at a criticalseparation, d = d ≡ (cid:32) F π B (cid:33) / , (26)called the interaction distance (Longcope 1998) (Fig. 8b). Theseparator is located at a height R = R S (Fig. 8c) which increasesto a maximum value of 0 . d and then moves back downwards,reaching the photosphere as d →
0. During the rise and fall ofthe separator, separator reconnection is driven at a current sheetof length L , where the input flow speed and magnetic field tothe current sheet are v i and B i , respectively (Fig. 8d). The theoryestimates the values of L , v i , and B i in terms of the imposedparameters v , B and F , and shows that the heating is likely tobe su ffi cient to heat the chromosphere and corona by a so-called‘cancellation nanoflare mechanism’.The two ways we extend the theory are: using the above anal-ysis for a 3D current sheet during the Phase 1 (Sect. 4.1) ratherthan a 2D one; and considering briefly the nature of the heatingduring a new Phase 2, namely, the ‘cancellation phase’ duringwhich the polarities are very close to each other (Sect. 4.2), andthe two photospheric fragments actually cancel with one another.For simplicity, we formulate the analysis in terms of cylindricalpolar rather than rectangular Cartesian coordinates. Fig. 7.
For a current sheet in 3D, the profiles as functions of radius R (cid:48) and the number N of points in the sheet of (a) the zeroth order field B S ( R (cid:48) ) and (b) the functional form ( B S ( R (cid:48) ) of the toroidal correction.(c) The inflow magnetic field to the current sheet, namely, B i = B S (0) asa function of (cid:15) = L / (2 R ) for large N and its analytical approximation(dashed). We note that another possibility has been suggested by Low(1991), namely, that of a ‘Phase 0’ such that, after the separatorappears in the solar surface in Fig. 8b, a current sheet grows up-wards from the solar surface rather than being localised arounda separator located above the photosphere. We shall analyse thispossibility in future and compare the energy release with the casewe are studying here. If the driving does not switch on and o ff ,so that the current sheet dissipates and then reforms at a di ff erentheight, or if reconnection is slow enough that the current sheetdoes not go unstable to tearing, it is possible that such a Phase 0exists for some time. Article number, page 6 of 10.R. Priest and P. Syntelis: Heating by Flux Cancellation o A B O2d(d) Current sheet v P(R,φ,z) v i r ~ r ~ L v (c) d < d o AS B S(b) d = d o A BR zB i Fig. 8.
Phase 1 of the cancellation process. (a) Two photospheric mag-netic sources of flux ± F , situated on the z -axis a distance 2 d apart in anoverlying uniform horizontal field B ˆz approach one another at speed ± v . (b) When d = d , a separator S is formed. (c) Reconnection isdriven at the separator S which rises in the atmosphere. (d) Energy isconverted at a current sheet of length L , where plasma flows in at speed v i carrying magnetic field B i . The magnetic field above the photosphere ( R >
0) may be writ-ten in terms of cylindrical polar coordinates ( R , φ, z ), with the z -axis being horizontal and situated in the photosphere, joiningthe two magnetic fragments located at (0 , , ± d ) (Fig. 8a) B = F ˆr π r − F ˆr π r + B ˆz , (27)where r = ( z − d ) ˆz + R ˆR , r = ( z + d ) ˆz + R ˆR are the vector distances from the two sources to a point P( R , φ, z ).We consider what happens when the distance 2 d between thetwo sources decreases from a large value. When the sources aretoo far apart, such that d > d , two separatrix surfaces com-pletely surround the flux that enters one source and leaves theother, so that no magnetic field lines link one source to another.On the other hand, when d = d a separator bifurcation occursin which these two separatrices touch at a separator field line(S) that lies in the photospheric plane ( R = d < d the sep-arator rises above z = d decreases from d to 0, we have Phase 1, during whichreconnection is driven at the separator that rises to a maximumand then falls to the photosphere. Finally, when d =
0, Phase 1is over, Phase 2 begins when the actual cancellation of the pho-tospheric fragments begins (Sect. 4.2).In the case of magnetic fragments of equal magnitude that weare considering here, the magnetic field is axisymmetric aboutthe z -axis and so there is a ring of null points at distance R S fromthe origin in every plane through the z -axis.Along the R -axis, B R = B z B = − d d ( d + R ) / + . (28) The location ( R = R S ) of the separator where B z vanishes istherefore given by R S = d / d / − d . (29)When d = d , the separator is located at the origin, and, as d decreases, it rises along the R -axis to a maximum height, andthereafter it falls back to the origin as d →
0. The maximumheight varies with B and F , but is typically about 0 . d , and so itlies in the chromosphere, transition region or corona dependingon the sizes of F and B (Priest, Chitta, & Syntelis 2018).When analysing flux cancellation, the natural parameters, foreach value of the source separation (2 d ), are the critical sourcehalf-separation distance ( d ), the flux source speed ( v ≡ ˙ d ≡ dd / dt ) and the overlying field strength ( B ). On the other hand,the parameters that determine the rate of release of energy at areconnecting current sheet (Fig. 8c) are the inflow speed ( v i ) andmagnetic field ( B i ) to the current sheet and the sheet length ( L ).We now therefore proceed to calculate them as functions of d , v and B .Firstly, to find B i calculate the potential field near the separa-tor, which can be shown from Eq. (28) to have the form B z = kR to lowest order, where k = − ( d / d ) / ] / ( d / d ) / B d . (30)When a current sheet forms, the magnetic field at the inflow tothe sheet then becomes, after substituting the above value of k into Eq. (25), B i B = − ( d / d ) / ] / d / d ) / Ld (1 − . (cid:15) log e (cid:15) ) , (31)where (cid:15) = L / (2 R ) (cid:28) v i from the rate of change ( ˙ ψ ≡ d ψ/ dt )of magnetic flux through the semicircle of radius R S out of theplane of Fig. 8c. This rate of change of flux becomes, after using E + v × B = and Faraday’s Law, d ψ dt = − π R S E = π R S v i B i . (32)However, ψ may be calculated from the magnetic flux below z S through the semicircle, namely, ψ = (cid:90) R S π R B z dR = F (cid:32) dd (cid:33) / − (cid:32) dd (cid:33) − , which vanishes when d = d and increases monotonically to avalue of F as the separation (2 d ) between the sources approacheszero. The rate of change of the flux then becomes d ψ dt = v Fd (cid:32) dd (cid:33) − / − dd . (33)After substituting into Eq. (32) together with the values of R S and B i from Eqs. (29) and (31), the required expression for v i becomes v i = v d L (cid:32) d d (cid:33) / { − . (cid:15) log e (cid:15) } − . (34)Then, the rate of conversion of inflowing magnetic energyinto heat can be written, following Priest, Chitta, & Syntelis(2018) as dWdt = . v i B i µ L π R S , (35) Article number, page 7 of 10 & A proofs: manuscript no. main
Fig. 9.
Phase 1 properties for a 3D current sheet as functions of thehalf-separation ( d ) of the two magnetic sources in units of the interac-tion distance ( d ): (a) the height ( R S ) of the separator, (b) the rate ofchange ( d ψ/ dt ) of magnetic flux below the separator, (c) the currentsheet length ( L ), and (d) the energy conversion rate ( dW / dt ) in unitsof W / t = v B d /µ . Dash-dot curves show the results for a 2D sheet(Priest et al. 2018). where L is determined by the condition for fast reconnection thatthe inflow speed acquire any value up to a maximum of v i = α v Ai , (36)where α is likely to be a non-trivial function of the external pa-rameters (for example, Priest 2014) but, as discussed in Synteliset al. (2019) is typically 0.1. Then, after writing v Ai = v A B i / B ,where v A = B / √ µρ i , and substituting for v i from Eq. (34) and B i from Eq. (31), we obtain L d = M A α − ( d / d ) / ] / { − . (cid:15) log e (cid:15) } − , (37)where M A = v / v A . Thus, by substituting for v i , B i , L and R S from Eqns.(34), (31), (37), and (29), the energy conversion ratebecomes finally dWdt = . π v B d M A [1 − ( d / d ) / ]3 µα ( d / d ) / (1 − . (cid:15) log e (cid:15) ) . (38)The variations of L / d and dW / dt with d / d are shown inFig. 9 for both the 2D and 3D cases. The curves are cut o ff near d = d = d , where the analysis fails since it implies un-physically that d > R S . The 3D treatment of the current sheetproduces a correction of 9% in the total energy release. There has been a debate on the actual process of flux cancella-tion in the photosphere, dating back to Zwaan (1987) and Priest(1987) and others, as summarised in, for example, Priest, Par-nell, & Martin (1994). One suggestion was that it represents pureflux submergence (without reconnection nearby) and anotherwas that it is caused by magnetic reconnection. If reconnectionoccurs at the photosphere, then the photospheric cancellation isoccurring in the reconnection site. If, however, reconnection oc-curs just above the photosphere, then cancellation represents thesubmergence of inverted U-loops retracting down through thephotosphere after having been reconnected. The argument for (a) B i v i (b) (c) plasmoid/flux rope hot jet L (d) hot jet Fig. 10.
Phase 2 of cancellation. (a) Fluxes come into contact in thephotosphere and reconnection is driven there, creating (b) a flux rope(whose cross-section is a magnetic island or bubble). (c) Flux-ropegrows in size and erupts, carrying cool plasma upwards. (d) Close-upof the reconnection region where a hot jet is accelerated. reconnection in either location, which we favour here, is that itwould then naturally produce the energy release that is often ob-served in the form of heating and plasma acceleration.As can be seen in Fig. 10a, while the two polarities approachand eventually come in contact, the field above the polarity in-version line becomes non-potential, and will form another lo-calised current sheet (di ff erent from the one discussed in Phase1), which extends upwards from the photosphere or above. In avertical plane through the cancellation process (panel b), a mag-netic bubble or island is naturally produced by reconnection ator just above the photosphere. Indeed, this naturally carries coolplasma from the photosphere and chromosphere upwards, as hasbeen proposed by, for example, Sterling et al. (2015); Sterling &Moore (2016); Sterling et al. (2016, 2020). The cool plasma theydub a ‘mini-filament’. If there is an extra component of magneticfield out of the plane, as is usually the case, then the magneticisland is just the cross-section of a magnetic flux rope or a smallsheared arcade. We note that the initiation of Phase 2 of the can-cellation does not have to wait until Phase 1 ends. The cancel-lation process starts with Phase 1, but Phase 2 can occur whilePhase 1 is still on-going. The timing between the two phaseswill depend on the magnetic configuration, area, flux content,and distance between the two cancelling polarities.The physical properties of flux cancellation during Phase1 with reconnection in the atmosphere have been estimated inPriest et al. (2018) and Syntelis et al. (2019), and so we nowestimate the corresponding properties during Phase 2 with re-connection in the photosphere as follows. They vary hugely, de-pending on the size and field strength of the flux and of the lengthof the current sheet. The sheet length depends on the nature ofreconnection. For Sweet-Parker reconnection, the length of thecurrent sheet would be l = η v A / v i , where values of the magneticdi ff usivity η = m / sec, Alfvén speed 1 −
10 km s − and inflowspeed v i = m / sec would give a length of only L = . − . Article number, page 8 of 10.R. Priest and P. Syntelis: Heating by Flux Cancellation
For fast reconnection, on the other hand, the energy releaseis much larger since the current sheet now refers not just to atiny Sweet-Parker sheet, but also to the bifurcated sheet includ-ing the slow shock waves for Petschek reconnection, or to a tur-bulent current sheet for impulsive bursty reconnection or a col-lisionless Hall sheet, as discussed in Syntelis, Priest, & Chitta(2019). If the sheet extends up to a height of, say, 1 Mm in theatmosphere, then the energy release is su ffi cient, as the followingestimates show, to account for microflares and subflares and onmuch smaller scales for nanoflares. Thus, for the various kindsof fast reconnection, most of the energy is not liberated in thecentral Sweet-Parker sheet but in the bifurcated or turbulent partof the sheet. Also, the observed decline in energy release as timeproceeds could be due to a decline in field strength B i and orcancellation speed v i .A magnetic flux tube of radius R and field strength B has aflux F = π R B , which may be written as F = R B × Mx , (39)where R is the radius in units of Mm and B is the magneticfield in units of 100 G. Thus, for example, a magnetic fragmentof radius 1 Mm and field of 1 kG has a flux of 3 × Mx,whereas if the radius is only 50 km, then the flux is 7 . × Mx.The velocity ( v i ) and duration ( τ ) of the cancellation of tubesof radius R are related by v i = R /τ or v i = R τ , (40)where τ is measured in units of 1000 sec, and so for a radius of0.7 Mm and a duration of 3000 sec, the cancellation speed wouldbe v i = . − .The energy released during cancellation may be estimated intwo ways as follows. The first estimate is to consider two mag-netic flux tubes of radius R , and field strength B , each with amagnetic energy of π R B / (4 π ) per unit length. If two such tubescancel over a length L , the energy released is W = R L B × erg , (41)where L and R are measured in Mm and B in hundreds ofGauss.The second estimate is to consider the rate of release of en-ergy in a sheet of width 2 R and height L , as given by dWdt = v i B i π R L , (42)where v i = R /τ , and so, during a time τ , an energy W = B i π L (2 R ) = R L B × ergis released, which is of the same form as Eq. (41) and dependscrucially on the length of the current sheet.We then consider first two tiny intense flux tubes of radius50 km with fields of 1 kG and a sheet length of 1 km. If thecancellation speed is 1 km s − , it will produce an energy of 10 erg over 100 sec, which is appropriate for a nanoflare. On theother hand tubes of radius 0 . − yield an energy of10 erg over 10 sec appropriate for a microflare, whereas tubesof radius 1 Mm with fields of 500 G and a length L = erg typical of a subflare over 200 sec. Also, we note that if the larger flux elements consist of many finer intenseflux tubes with persistent flux cancellation, or if the cancellationoccurs in fits and starts, then the total energy release may takeplace as a series of nanoflares or microflares over an extendedtime of hundreds or thousands of seconds, as reported in someobservations and simulations of flux cancellation such as Peteret al. (2019) and Park (2020).
5. Conclusions
Magnetic flux cancellation was previously realised to be impor-tant in heating tiny regions of the solar atmosphere, namely, X-ray bright points. However, the Sunrise observations have trans-formed our appreciation of its significance and shown flux can-cellation to be very much more widespread, and therefore poten-tially to be the dominant factor in heating the atmosphere andaccelerating various kinds of jets in di ff erent parts of the solaratmosphere. The aim of the present paper has been to furtherdevelop, in several directions, the basic theory for such energyrelease driven by flux cancellation.The first direction was a technical one, namely, to determinehow the previous simple theory of reconnection at a Cartesiancurrent sheet in 2D can be set up without using complex vari-able theory and how it can be extended to 3D. For an axisym-metric toroidal current sheet, we have found how the large-scalecurvature decreases the field outside the torus. Then we appliedthe theory to flux cancellation between two magnetic fragments,where we realised there are two stages, namely, (i) Phase 1, dur-ing which reconnection occurs at a separator that first moves upand then descends back to the photosphere and (ii) Phase 2, dur-ing which reconnection occurs in or just above the photospherebetween the two cancelling regions.Future possible developments include detailed computa-tional experiments that can produce more realistic models forthe process and can validate the basic theory that we have pro-posed here. In addition, we have focused here on conceptuallythe simplest building block of the theory, namely, the elemen-tary interaction of two magnetic fragments, but in future it willbe possible to apply the theory to a variety of more complex andrealistic geometries and flux systems. Acknowledgements.
ERP is grateful for helpful suggestions and hospitalityto Pradeep Chitta, Hardi Peter, Sami Solanki and other friends in MPSGöttingen, where this research was initiated. P.S. acknowledge support bythe ERC synergy grant “The Whole Sun”. The authors are most grateful fora thorough and insightful referee report that has substantially improved the paper.
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