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Coronal null points and solar ﬂares: an investigation of magnetic ﬁeld topology www-solar.mcs.st-and.ac.uk/∼williams W.M.R. Simpson 5th June - 17th July 2008 SOHO / EIT Image (Fe XII 195 Å) depicting the X17 Flare on 28th October 2003, 11:12 UT Contents 1 Introduction 1 2 Magnetic reconnection 1 3 Magnetic null points in 2 dimensions 2 3.1 The threshold current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 The ﬂux function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 The eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 Classifying 2d null points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4.1 Potential null points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4.2 Non-potential null points . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Magnetic null points in 3 dimensions 7 4.1 The geometry of a 3D null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 Tracing the ﬁeldlines round a null . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2.1 M is diagonalisable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2.2 M is not diagonalisable . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Reﬂections on the null’s positive or negative character . . . . . . . . . . . . . . 10 5 The data for analysis 11 5.1 The nature of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 The structure of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Flux and energy 11 7 Null spotting 13 7.1 Field interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7.1.1 1d interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 7.1.2 2d interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.1.3 3d interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.2 Calculating the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 7.3 Calculating the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 8 Null tracking 15 8.1 Identity criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.2 Forming a diachronic identity signature . . . . . . . . . . . . . . . . . . . . . . 17 8.3 Eigenvector plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 9 Discussion and conclusions 19 Note: Further essential data and movies are available online at the website address www-solar.mcs.st-and.ac.uk/∼williams to supplement the material in this report. 1 Introduction Solar ﬂares and Coronal Mass Ejections (CMEs) are two types of solar phenomena that occur in intense magnetic regions of our Sun. Both involve the fast release of magnetic energy and produce large-scale perturbations throughout the solar atmosphere. For the largest ﬂares, and for all CMEs, the eﬀects are far-reaching, moving beyond the surface of our Sun to aﬀect the interplanetary medium. The energy release mechanism generally believed to be responsible for both of these phenomena is magnetic reconnection. It is understood that, in two-dimensions, reconnection can only take place at magnetic null points whereas in three-dimensions they are no longer a necessary condition. However, certain mathematical models have been proposed for ﬂares and CMEs that do require the presence of a null point in the solar atmosphere. Unfortunately, given the current observation techniques, it is almost impossible to observe these coronal nulls directly. An alternative approach is to combine magnetic observations and 3D modelling methods to extrapolate the coronal magnetic ﬁeld from the ﬁeld of the photosphere of an active region. We can then determine its magnetic topology, in particular the locations of the null points. In this paper we are concerned with the topology of the observed magnetic ﬁeld of active- region AR0486 during the time of the X17.2 ﬂare. By considering an extrapolation of data obtained from SOHO/MDI and Hinode/SOT photospheric magnetograms using the magnetic e ﬁeld extrapolation tools developed by S. R´gnier et al. (2002, 2006), and by employing the null point ﬁnding procedures created by Haynes et al. (2007) in conjunction with various analysis programs developed during this project by the author (2008), we undertake an analysis of the physical character and behaviour of the magnetic null points discovered in that region. Our motivating question is whether coronal null points are a necessary requirement for ﬂares or CMEs. Our analysis of the results obtained in this investigation is detailed from sections 6 to 8, with concluding comments oﬀered in section 9. It is hoped that the details of this study may contribute towards an answer to this question. 2 Magnetic reconnection Magnetic reconnection involves a restructuring of a magnetic ﬁeld caused by a change in the connectivity of its ﬁeld lines1 . It is considered the most important process for explaining large-scale, dynamic, fast releases of magnetic energy. The matter that makes up our Sun is in a plasma state. It is the existence of magnetic ﬁelds in the presence of plasma ﬂows that leads to magnetic reconnection. Such changes in the structure of a magnetic ﬁeld allow the release of stored magnetic energy, which is typically the dominant source of stored energy in plasmas. Reconnection is believed to be a wide-spread phenomenon. Priest cites such examples as laboratory fusion machines, terrestrial aurorae produced in the Earth’s magnetosphere, and disconnection events in comet tails as probable instances of magnetic reconnection2 . In the Sun, reconnection is held to be responsible for the motion of chromospheric ﬂare ribbons and coronal ﬂare loops during solar ﬂares, and the enormous energy release associated with ﬂares and CMEs. The ejection of magnetic ﬂux from the Sun during CMEs requires reconnection. 1 [Priest and Forbes 2000] 2 Ibid. 1 3 Magnetic null points in 2 dimensions A magnetic null point is a point in a magnetic ﬁeld where all the components of the ﬁeld are zero3 . In 2 dimensions: Bx = By = 0 This tells us little in itself about the local magnetic structure; the topology of the ﬁeld in the immediate vicinity of one null point may be quite diﬀerent near another. However, if we assume that the magnetic ﬁeld near a null point approaches zero linearly, we can approximate the components of the magnetic ﬁeld in this region by means of a two variable, ﬁrst order Taylor expansion about the neutral point X0 , Y0 . Consider the x component: ∂BX ∂BX BX = (X − X0 ) + (Y − Y0 ) + O(X 2 , Y 2 , ...) ∂X (X0 ,Y0 ) ∂Y (X0 ,Y0 ) Retain only the ﬁrst order, linear terms: ∂BX ∂BX BX ≈ (X − X0 ) + (Y − Y0 ) ∂X (X0 ,Y0 ) ∂Y (X0 ,Y0 ) Choose an origin such that X0 = Y0 = 0. ∂BX ∂BX BX ≈ X+ Y ∂X (0,0) ∂Y (0,0) Similarly for y: ∂BY ∂BY BY ≈ X+ Y ∂X (0,0) ∂Y (0,0) We may then express the magnetic ﬁeld near a null point (to lowest order) as B = M.r ∂BX ∂BX ∂X ∂Y where M = and r = (X, Y )T . ∂B ∂BY Y ∂X ∂Y This is the Jacobian. For simplicity, we rewrite the above matrix as a11 a12 M= a21 a22 However, this matrix can be simpliﬁed and rewritten in a form that will lend itself more readily to meaningful analysis. First, we impose the solenoidal constraint, ∂BX ∂BY ∇.B = 0 ⇒ + = 0 ⇒ a11 + a22 = 0 ⇒ a11 = −a22 ∂X ∂Y The diagonal entries are associated with the potential part of the ﬁeld (they do not show up in the expression for the current below), so we let a11 = p, a22 = −p. 3 In this section and the next I will be drawing heavily on the theory developed by [Parnell et al. 1996], but justifying some of the results in greater mathematical detail. It is possible that, at some points, Parnell may have a diﬀerent approach in mind to the one I have adopted. 2 Consider now the current, 1 1 p a12 1 J= ∇×B= ∇× r= (0, 0, a21 − a12 ) µ0 µ0 a21 −p µ0 We can conveniently rewrite 1 1 a12 = (q − Jz ), a21 = (q + Jz ) 2 2 q Thus for a current-free neutral point, where Jz = 0, a12 = a21 = 2 . Therefore the parameter q is associated with the potential ﬁeld. The matrix M may now be stated in its ﬁnal form: 1 p 2 (q − Jz ) M= 1 (1) 2 (q + Jz ) −p 3.1 The threshold current From the square root of the discriminant of the characteristic equation of the symmetric part of M, we deﬁne a threshold current, Jthresh = 4p2 + q 2 (2) which we note is only dependent on parameters associated with the potential part of the ﬁeld. The proof proceeds as follows: Take the symmetric part of M, 1 MS = (M + MT ) 2 1 1 1 1 p 2 (q − Jz ) p 2 (q + Jz ) 1 p 2 q = 1 + 1 = 1 2 2 (q + Jz ) −p 2 (q − Jz ) −p 2 2 q −p q2 q2 det(MS − λI) = −p2 + λ2 − = 0 ⇒ λ2 − (p2 + ) = 0 4 4 This yields a discriminant d = 4p2 + q 2 Thence, as given above: √ Jthresh = d= 4p2 + q 2 3.2 The ﬂux function We now determine the ﬂux function A – an expression that characterises the geometry of the magnetic ﬁeld, deﬁned to obey the solenoidal constraint. It satisﬁes ∂A ∂A BX = , BY = − ∂Y ∂X Since B = M.r, BX = pX + 1 (q − Jz ), and BY = 2 (q + Jz ) − pY . Hence: 2 1 3 1 A= BX dY = pXY + (q − Jz )Y 2 + f (X) 4 1 A=− BY dY = − (q + Jz )X 2 − pXY + f (Y ) 4 Therefore, 1 (q − Jz )Y 2 − (q + Jz )X 2 + pXY A= 4 This expression can be further simpliﬁed by a rotation of the XY axes, allowing us even- tually to rewrite it as 1 A=(Jthresh − Jz )y 2 − (Jthresh + Jz )x2 , (3) 4 ie. a function of the two parameters Jthresh and Jz . The proof proceeds as follows: Rotate XY-axes through an angle θ, X cosθ −sinθ x = Y sinθ cosθ y 1 A= (q − Jz )(xsinθ + ycosθ)2 − (q + Jz )(xcosθ − ysinθ)2 + 4 p(xcosθ − ysinθ)(xsinθ + ycosθ) Expanding, and factorising in x2 , y 2 and xy, yields 1 1 A = x2 (q − Jz )sin2 θ − (q + Jz )cos2 θ + psinθcosθ + 4 4 1 1 y2 (q − Jz )cos2 θ − (q + Jz )sin2 θ − psinθcosθ + 4 4 xy qsinθcosθ + p(cos2 θ − sin2 θ) Now let tan2θ = −2 p . First, consider the xy term: q q qsinθcosθ + p(cos2 θ − sin2 θ) = sin2θ + pcos2θ = −pcos2θ + pcos2θ = 0 2 ie. the xy term disappears. Next, consider the x2 term: 1 1 1 1 1 x2 (q − Jz ) (1 − cos2θ) − (q + Jz ) (1 + cos2θ) + psin2θ 4 2 4 2 2 x2 x2 4p2 =− (qcos2θ − 2psin2θ + Jz ) = − qcos2θ + cos2θ + Jz 4 4 q x2 1 2 2 x2 =− 1 (4p + q ) + Jz =− (Jthresh + Jz ) 4 (4p2 + q 2 ) 2 4 Finally, consider the y 2 term: 4 y2 [(q − Jz )(1 + cos2θ) − (q + Jz )(1 − cos2θ) − 4psin2θ)] 8 y2 y2 4p2 = (qcos2θ − 2psin2θ − Jz ) = qcos2θ + cos2θ − Jz 4 4 q y2 1 2 2 y2 = 1 (4p + q ) − Jz = (Jthresh − Jz ) 4 (4p2 + q 2 ) 2 4 And hence, 1 A= (Jthresh − Jz )y 2 − (Jthresh + Jz )x2 4 3.3 The eigenvalues Finally, we determine a general expression for the eigenvalues of M. 1 1 2 det(M − λI) = λ2 − (4p2 + q 2 − Jz ) = λ2 − (Jthresh − Jz ) 2 2 4 4 Hence, 1 λ=± J2 2 − Jz (4) 2 thresh 3.4 Classifying 2d null points 3.4.1 Potential null points A potential ﬁeld is ’current free’, ie. J = 0 ⇒ Jz = 0. It follows that M is symmetric in the potential case, and from (4) the eigenvalues are given by 1 λ = ± Jthresh 2 Thus we have two real non-zero eigenvalues. From (3) it is apparent that the ﬂux function is simply Jthresh 2 A= (y − x2 ) 4 From the ﬂux function we can quickly discover the geometry of the ﬁeld lines. ∂A Jthresh ∂A Jthresh Bx = = y, By = − = x. ∂y 2 ∂x 2 dy By = → ydx = xdy → y 2 − x2 = c dx Bx √ Plainly, for c = 0 → y = ±x, and for c = 0 → y = ± c + x2 . The ﬁeld lines are thus hyperbola with separatrices that intersect at an angle of π . We 2 call this an X-type neutral point (see ﬁg. 1, left), and it is the only possible conﬁguration in 2d for a neutral point in a potential ﬁeld. 5 3.4.2 Non-potential null points Although we are concerned with a potential ﬁeld extrapolation in this investigation, we brieﬂy consider here the case of 2d neutral points in a non-potential ﬁeld. 2d neutral points with current are classiﬁed by the magnitude of Jz and Jthresh . 1. |Jz | < Jthresh : Here, (4) implies the eigenvalues are real, equal in magnitude, and opposite in sign, and (3) implies A = ay 2 − bx2 where a, b > 0, ie. the ﬁeld lines are hyperbolae with separatrices that intersect at an angle of 2 2 1 −1 (Jthresh − Jz ) 2 tan Jz Again, we have an X-type neutral point, which reduces to the potential case (rectangular hyperbolae) as Jz → 0. 2. |Jz | = Jthresh : (4) implies the eigenvalues are both zero. (3) implies − 1 Jthresh x2 , for Jthresh = Jz 2 A= 1 J 2 thresh y 2 , for Jthresh = −Jz Hence, ∂A 0, for Jthresh = Jz Bx = = ∂y Jthresh y, for Jthresh = −Jz ∂A Jthresh x, for Jthresh = Jz By = − = ∂x 0, for Jthresh = −Jz Therefore, dx Case J = Jz : = 0 → x = const dy dy Case J = −Jz : = 0 → y = const dx Thus this conﬁguration produces anti-parallel ﬁeld lines with a null line either along the x-axis or the y-axis. 3. |Jz | > Jthresh : Here, (4) implies the eigenvalues are complex conjugates. Jz 2 Case Jthresh = 0 : (3) → A = − (x + y 2 ) 4 ie. the ﬁeld lines are circular and centred around the origin. 1 Case Jthresh = 0 : (3) → A = − (ay 2 + bx2 ) 4 (where a, b are constants, both greater than zero), ie. the ﬁeld lines are concentric ellipses. In both cases, then, we have an O-type neutral point (see ﬁg. 1, right). 6 Figure 1: Depictions of X-type and O-type nulls [Simpson, 2008] 4 Magnetic null points in 3 dimensions Generalising from the above, we ﬁnd that the magnetic ﬁeld, B, near a 3d neutral point may be expressed to lowest order as B = M.r ∂BX ∂BX ∂BX ∂X ∂Y ∂Z ∂BY ∂BY ∂BY and r = (X, Y, Z)T where M = ∂X ∂Y ∂Z ∂B Z ∂BZ ∂BZ ∂X ∂Y ∂Z For simplicity, rewrite the above matrix as a11 a12 a13 M = a21 a22 a23 a31 a32 a33 Again imposing the solenoidal constraint we ﬁnd that a11 + a22 + a33 = 0. Since the diagonalised (or Jordan normalised) form has the same trace, it follows that λ1 + λ2 + λ3 = 0 (5) The associated eigenvectors are x1 , x2 , x3 . 4.1 The geometry of a 3D null It is apropos at this point to brieﬂy comment on the two basic components that make up the skeleton of any 3-dimensional neutral point: the fan of a null point is the surface formed by the set of ﬁeld lines radiating out of or into the null point; the spine is formed by the two ﬁeld lines that enter or leave the null point (not in the plane of the fan). The spine is directed away from the null if the ﬁeld lines in the fan are directed towards it, and vice-versa. 7 We will make reference to these features and demonstrate how we can identify them in the mathematical investigation of the ﬁeld lines that follows. 4.2 Tracing the ﬁeldlines round a null We can write a magnetic ﬁeld line near the null in terms of a position vector r = (x, y, z)T dependent on an arbitrary parameter k, dr(k) = M.r(k) = B (6) dk Now let r(k) = Pu(k) (7) where P is the matrix of the eigenvectors of M, P = (x1 , x2 , x3 ). Then (6) implies du(k) = P−1 MPu (8) dk There are two cases to consider: 4.2.1 M is diagonalisable In this case, M can be diagonalised to a matrix Λ. (8) implies du(k) = Λu (9) dk Hence, du = Λdk → u = Aexp(Λk) (10) u where A is also a diagonal matrix with entries A, B, C which are constant along a ﬁeld line. Now (7) and (10) imply r(k) = Aeλ1 k x1 + Beλ2 k x2 + Ceλ3 k x3 (11) So each ﬁeld line may be written in terms of the eigenvectors and eigenvalues of M. 1. Consider the case where all the eigenvalues are real. Suppose λ1 , λ2 > 0, λ3 < 0, ie. there is always one eigenvalue of opposite sign to the other two. We can trace a ﬁeld line backwards (away from the null point) by letting k → −∞, and forwards (away from the null point) by letting k → ∞. i. Let k → −∞. r(k) → Ceλ3 k x3 So all the ﬁeld lines that head in towards the null are parallel to x3 . It follows that x3 (ie. the eigenvector associated with the eigenvalue of opposite sign to the others) deﬁnes the path of the spine, and that for λ1 , λ2 > 0, λ3 < 0 the spine is heading towards the null. ii. Let k → ∞. r(k) → Aeλ1 k x1 + Beλ2 k x2 8 So all the ﬁeld lines that are directed away from the null lie parallel to the plane deﬁned by x1 , x2 . It follows that x1 , x2 (ie. the eigenvectors whose eigenvalues share the same sign) deﬁne the plane of the fan. Here, the fan ﬁeld lines are emanating radially outwards from the null. 2. Consider the case where we have two complex and one real eigenvalue, λ1 = η + iν, λ2 = ´ ´ ´ ´ η − iν, λ3 = −2η, with corresponding eigenvectors x1 = (x1 + ix2 )/2, x1 = (x1 − ix2 )/2, x3 . Then (11) implies 1 1 r(k) = (A + iB)e(η+iν)k (´ 1 + i´ 2 ) + (A − iB)e(η−iν)k (´ 1 − i´ 2 ) + Ce−2ηk x3 x x x x ´ (12) 2 2 (12) can be conveniently rewritten as r(k) = eηk Rcos(Θ + ν)k´ 1 − eηk Rsin(Θ + ν)k´ 2 + Ce−2ηk x3 x x ´ (13) where A and B have been rewritten in terms of the constants R and Θ. The proof: ´ ´ Consider the ﬁrst two terms, and separate for x1 , x2 . 1 ηk e (A + iB)eiνk + (A − iB)e−iνk x1 + i (A + iB)eiνk − (A − iB)e−iνk x2 ´ ´ 2 1 = eηk A(eiνk + e−iνk ) + iB(eiνk − e−iνk ) x1 + i A(eiνk − e−iνk ) + iB(eiνk + e−iνk ) x2 ´ ´ 2 = eηk [(Acos(νk) − Bsin(νk)) x1 − (Asin(νk) + Bcos(νk)) x2 ] ´ ´ Now, cos(θk + νk) = cos(θk)cos(νk) − sin(θk)sin(νk) sin(θk + νk) = sin(θk)cos(νk) + cos(θk)sin(νk) Letting sin(θk) = B/R, cos(θk) = A/R, we recover (13) from (12). i. Let k → −∞. ´ r(k) = Ce−2ηk x3 It follows that x3 deﬁnes the path of the spine. ii. Let k → ∞. r(k) → Reηk cos(Θ + ν)k´ 1 − Reηk sin(Θ + ν)k´ 2 x x It follows that x1 , x2 deﬁnes the fan plane, and the ﬁeld lines in the fan are spirals. 4.2.2 M is not diagonalisable In this case, two of the eigenvalues are repeated and the matrix M can only be reduced to the Jordan normal form, λ 1 0 Jn = 0 λ 0 0 0 −2λ 9 Recalling that, dr(k) = M.r(k) = B (14) dk we can again write r(k) = Pu(k), where this time P = (x1 , x∗ , x3 ), and x1 , x∗ , x3 satisfy 2 2 Mx1 = λx1 Mx∗ = x1 + λx∗ 2 2 Mx3 = −2λx3 Then, du(k) = P−1 MPu = Jn u (15) dk And the equation for a ﬁeld line is therefore: r(k) = (A + Bk)eλk x1 + Beλk x∗ + Ce−2λk x3 2 (16) where A, B, C are constant along a ﬁeld line. Here, assume λ > 0. i. Let k → −∞. ´ r(k) = Ce−2λk x3 It follows that x3 deﬁnes the path of the spine. ii. Let k → ∞. r(k) → (A + Bk)eλk x1 + Beλk x∗ 2 It follows that x1 and x∗ deﬁne the fan plane. It is not diﬃcult to see the reversal entailed 2 by making λ < 0. 4.3 Reﬂections on the null’s positive or negative character It is apparent from what has been discussed above that, in the general case, the spine is seen to lie along the eigenvector of M that relates to the single eigenvalue of opposite sign to the real parts of the remaining eigenvalues. The remaining eigenvectors deﬁne the plane of the fan. It is also apparent, with little need for further discussion, that 1. Real(λ1 , λ2 ) > 0 implies we have a positive neutral point. The ﬁeld lines in the fan are directed away from the null, whereas the spine points into the null. 2. Real(λ1 , λ2 ) < 0 implies we have a negative neutral point. The ﬁeld lines in the fan point into the null, whereas the spine is directed away from it. (Null Point Nature Criteria) This is as far as we need explore in this investigation; the non-potential cases (which are many) are not relevant for our purposes here, and we will resort to computational means to determine eigenvalues and eigenvectors. We shall simply note that, similarly to the 2d case, the matrix M in the 3d potential case can be reduced analytically to the form 10 1 0 0 M= 0 p 0 0 0 −(p + 1) where p ≥ 0. We also note that, for p = 1, we obtain a proper radial null ; for p > 0, p = 1, an improper null, where the ﬁeld is aligned predominantly in the direction of the eigenvector corresponding to the greatest fan plane eigenvalue; and for p = 0, the ﬁeld reduces to the 2d case with potential X-points lying in planes parallel to the xz-planes and forming a null-line along the y-axis. The interested reader is referred to [Parnell et al. 1996] for further details. 5 The data for analysis With the theoretical underpinnings of the essential topological elements of 2d and 3d analytical magnetic ﬁelds in place, our concern now is to study the topology and locate the null points of the observed active-region AR0486 magnetic ﬁeld extrapolation. The time interval is from 10:00 UT to 12:58 UT on Oct. 28th, 2003. The start time of the ﬂare is 11:01 UT, and 11:15 UT corresponds to the end time of photospheric changes. 5.1 The nature of the data As noted earlier, it is diﬃcult to measure the coronal magnetic ﬁeld directly. We will there- fore be using potential ﬁeld data computed from SOHO/MDI line-of-sight magnetograms, e performed by S. R´gnier. There are some points to note about this kind of extrapolation: 1. it assumes the coronal magnetic conﬁguration is in a magnetostatic equilibrium at the time of observation (J × B = 0). 2. it assumes, moreover, that the ﬁeld is current-free (J = 0). The magnetic energy of the potential ﬁeld conﬁguration is therefore the lowest magnetic energy a magnetic conﬁguration can reach with the same photospheric ﬁeld - it does not include the free energy. 5.2 The structure of the data For every minute of the interval we have at our disposal the x, y, and z components of the magnetic ﬁeld at uniformly spaced, discrete points in a volume region approximately 300 x 200 x 200 Mm (175 such magnetic conﬁgurations, in total). We will consider the region then to be divided into cubic pixels of volume δV , and surfaces δA, with each pixel assigned values for the ﬁeld components. 6 Flux and energy Our ﬁrst port of call is to determine the behaviour of the ﬂux and the magnetic energy during the interval. The observed magnetic ﬁeld is assumed to be essentially vertical on the photosphere, thus at time t the photospheric magnetic ﬂux is 11 Figure 2: Magnetic Flux and Energy plots from 10:00 UT to 12:58 UT [Simpson, 2008]. φtot (t) = B(t).da = Bz (x, y, z = 0, t)da A We employ the modulus, as we are after the total rather than the net ﬂux. Applied to our data, we approximate the integral as φtot ≈ |Bz (x, y, z = 0)| δA x,y The magnetic energy is B2 Em = dV V 8π Applied to our data, we approximate the integral as 1 2 2 2 Em = Bx + By + Bz δV 8π x,y,z Evidently the magnetic energy of the potential ﬁeld follows the same evolution as the unsigned photospheric magnetic ﬂux (see ﬁg. 2). We also observe an energy increase be- tween the start and ﬁnish of the plot. Further, the marked drop in the energy is what we would (naively) expect during the ﬂare due to the release of energy. However, we are working with the potential ﬁeld, and, as we noted earlier, it does not include the free energy. Conse- quently, it should not be dropping like this, since any energy given up to the ﬂare would come from the free energy. An explanation for this phenomenon has already been put forward by e [R´gnier & Fleck. (2004)]. Brieﬂy, it appears to result from an induced instrument response to the disturbances created by the ﬂare. The high speed particle ejection that occurs in this interval results in extreme Doppler shifts of the NiI line, confusing the instrument readings and resulting in the apparent detection of parasitic polarities inside a sunspot. This throws the ﬂux and energy calculations for this interval. However, since this problem is conﬁned to the period of intense solar ﬂare activity, we can at least detect this interval quite clearly from our energy and ﬂux plots, even if we cannot rely on their actual readings over this period. 12 Figure 3: Interpolation in 2d and 3d 7 Null spotting The photospheric changes above may lead us to expect small-scale changes in the magnetic ﬁeld conﬁguration of AR0486, but the large scale potential ﬁeld is not modiﬁed. Our concern now is to locate and classify any null points in the ﬁeld occurring during the evolution of the ﬁeld. To ﬁnd nulls we are, by deﬁnition, looking for points in the volume where the magnetic ﬁeld goes to zero (in x, y and z). Applying the 3d interpolation theory detailed in [Haynes and Parnell 2007]4 , and brieﬂy discussed below, the nulls in each time frame have been located and listed5 for all 175 conﬁgurations (see ﬁg. 4). We also seek to classify these null points as positive or negative. From the theory discussed earlier our ﬁrst task is evidently to determine the Jacobian of the ﬁeld at the position of the null point in question. For the ﬁrst element in the matrix, we can obviously write ∂Bx Bx (x0 + δx, y0 , z0 ) − Bx (x0 − δx, y0 , z0 ) ≈ ∂x 2δx for some small perturbation δx. But clearly the ﬁeld is only known at discrete points in the volume, and to set δx to the width of an entire pixel is unacceptable. In order to determine any of the Jacobian derivatives, we must use values of the ﬁeld in the immediate vicinity of the relevant point. Since those values are unknown, we must interpolate the ﬁeld. 7.1 Field interpolation We make a brief excursus here to explain the trilinear interpolation adopted6 . In each case in this discussion, we shall consider just the x-component of the ﬁeld. 7.1.1 1d interpolation First, we consider the 1d case. The simplest form of interpolation between two known values on a line, Bx (0) and Bx (1), generates the equation 4 Haynes has developed a null point ﬁnding code, which has been used here. 5 See www-solar.mcs.st-and.ac.uk/∼williams 6 For further details, see [Haynes and Parnell 2007] 13 Bx (x) = Bx (0) + (Bx (1) − Bx (0))x which we can rewrite Bx (x) = (1 − x)b0 + xb1 7.1.2 2d interpolation This is easily expanded to 2d (see ﬁg. 3, left). Points along the bottom horizontal line are given by b0 (x) = (1−x)b00 +xb10 , and along the top horizontal line by b1 (x) = (1−x)b01 +xb11 , and the point at (x, y) is given by interpolating along the vertical line of constant x between b0 (x) and b1 (x), ie. Bx (x, y) = (1 − y)b0 (x) + yb1 (x) = (1 − x)(1 − y)b00 + x(1 − y)b10 + (1 − x)yb01 + xyb11 which we can rewrite Bx (x, y) = a + bx + cy + dxy where a = b00 , b = b10 − b00 , c = b01 − b00 , d = b11 − b10 − b01 + b00 . 7.1.3 3d interpolation Similarly, a trilinear equation can be obtained for the 3d case (see ﬁg. 3, right), Bx (x, y, z) = a + bx + cy + dxy + ez + f xz + gyz + hxyz where, a = b000 b = b100 − b000 c = b010 − b000 d = b110 − b100 − b010 + b000 e = b001 − b000 f = b101 − b100 − b001 + b000 g = b011 − b010 − b001 + b000 h = b111 − b110 − b101 − b011 − b100 + b010 + b001 − b000 7.2 Calculating the Jacobian Our approach, then, for determining the ﬁeld strength at some arbitrary point P in the ﬁeld is to form what we shall call a ’capture cube’ about P , where each corner is a point where the ﬁeld strength is known directly from the available data (see ﬁg. 3). We interpolate the ﬁeld using this cube to deduce its approximate value at P . Since we are attempting to calculate the Jacobian at a null point it will be necessary to determine the ﬁeld strength for positive and negative perturbations about the null in x, y and z. This may necessitate the use of more than one capture cube, depending on whether or not the point in question lies on a face or in one of the corners of its associated cube7 . 7 A calculation of the divergence of B is a sensible check to make, or equivalently the sum of the eigenvalues. Both should, analytically, come to zero. Obviously this is not going to be the case here - we can only reasonably expect that they should be small values. It is sensible to check i λi / i |λi |. 14 Figure 4: The null point locations at 10:00 UT [Simpson, 2008]. 7.3 Calculating the eigenvalues With the derivatives of the Jacobian thus determined by interpolating the ﬁeld and choosing appropriate perturbations about P , we turn the task of calculating the matrix’s eigenvalues over to IDL, and then examine the signs of the eigenvalues associated with the fan, according to the Null Point Nature Criteria. A table of classiﬁed null points can then be produced for all the nulls located in each frame8 . A movie of the evolution of AR0486, plotting the positions of positive and negative null points in time against a contour plot of the line-of-sight magnetic ﬁeld strength, is available for viewing online9 . 8 Null tracking We are on the lookout for stable null points - it is not enough to list and categorise nulls in each time frame. From the movie, it is clear that the majority of the apparent null points we have located do not persist through time, and are likely to be unphysical, arising most probably from noise in the MDI magnetograms as well as theoretical constraints, such as the boundary conditions imposed on the extrapolation and the assumption of the linearity of the ﬁeld. We note that the majority of these pseudo-nulls occur towards the periphery of the conﬁguration, where the magnetic ﬁeld strength is weaker and the noise has a more pronounced eﬀect. Any conclusions in this report will be based on nulls we hold to be physical and stable. (If a null is stable, we will assume it is physical). For a stable null, N , we will 8 All these tables are available at http://www-solar.mcs.st-and.ac.uk/∼williams 9 http://www-solar.mcs.st-and.ac.uk/∼williams 15 require that 1. N persists through time, 2. the nature of N remains unchanged, 3. the direction of the eigenvectors of N change minimally through time (Null Stability Criteria) Again, a visual appraisal of the movie suggests a small number of possible candidates satisfying criteria 1 and 2. Our concern here will be to establish null identity through time with more rigor. If some null N appears to persist through the interval, we would like to be able to track it, pull together all the information we know about its various instances in the diﬀerent time frames, and analyse the accumulated data. 8.1 Identity criteria Consider two consecutive time frames, I and J. We will regard a null i in time frame I to be associated with a null j in time frame J if the following criteria hold: 1. dij < dnj , ∀n = i 2. dij < dim , ∀m = j 3. dij < dmax 4. ni = nj (Diachronic Identity Criteria) where dnm is the distance between some null n in time frame I and some null m in time frame J, given by d2 = |xm − xn |2 + |ym − yn |2 + |zm − zn |2 nm and np is the nature (positive or negative) of a null, p. The logic of each criterion runs like this: ﬁrst, if nulls i and j are associated, then i (in time frame I) should be closer to j (in time frame J) than some other null n in time frame I; secondly, the distance between i and j should be smaller than the distance between i and some other null m in frame J; thirdly, the trajectory of a null should be continuous, and we have suﬃcient time snaps to expect to be able to track the motion without large ’leaps’ in distance, so we should place a constraint on the distance between consecutive instances of a null; ﬁnally, we preclude from association with the null we are tracking any null points that have a diﬀerent nature (thus absorbing the second criterion of the Null Stability Criteria). 16 Figure 5: The locations of the tracked nulls at 10:07 UT [Simpson, 2008]. 8.2 Forming a diachronic identity signature Some complications remain. Firstly, we should allow for the fact that there may be time frames in which an otherwise stable null does not appear, for whatever reason. From the movie, it is apparent that a null point near the origin of the ﬂare persists throughout the interval (175 time frames), but disappears momentarily from two of those time frames. We do not wish to dispose of part of its identity signature for that reason. Consequently, our approach here is to move through time frames, looking to make associations, but without insisting that an association should be limited to consecutive frames. When a null we have successfully tracked through several frames apparently disappears in a subsequent frame, the data of its last known position will be retained with a view to making an association in a later frame. This has been successfully implemented by the author. Secondly, our criteria as stated oﬀers (perhaps) necessary but certainly not suﬃcient con- ditions for identity through time: in ’noisy’ regions of the ﬁeld, where many null points appear to be forming and disappearing (many of them are surely unphysical), our null tracker will form false associations. Also, enabling it to successfully track a stable null in the midst of intense noise has not proven to be simple - it may be misled into making an association with a ’pseudo-null’, and then subsequent associations are likely to be mistaken as well. The author is conﬁdent that further development of the identity tracking algorithm will improve its reliability. For instance, the somewhat myopic preoccupation with isolated pairs of time frames could be replaced by a more complex comparison process that considers the longer term consequences of diﬀerent possible associations and determines the more likely identity signature, discounting alternatives that, say, lead us too far from our initial position. 17 Figure 6: Eigenvector Plot of a Null Point in AR0486 [Simpson, 2008]. The eigenvector associated with the spine is depicted in red, the eigenvectors associated with the fan are in blue, and the plane of the fan is marked out with a blue dotted line. This could involve a relaxation of the association criteria above to allow for other possible associations, but impose additional long-term criteria that would discriminate between them. For the purpose of this project we content ourselves with applying the ’myopic’ version of the algorithm. So far, this has resulted in 7 nulls being successfully tracked through time. An eighth null (N6), which appears in the movie to persist, has proven more diﬃcult to track with this algorithm.10 . The locations of the nulls are depicted in ﬁg. 5, and some of their properties tabulated in table 1. 8.3 Eigenvector plots The third of the Null Stability Criteria we have imposed on our results requires us to consider the eigenvectors of each null (which we obtain from the Jacobian, along with the eigenvalues). To put it simply, we intend to discount as unstable (and possibly unphysical) any null whose geometry exhibits erratic behaviour with time. To this end movies have been produced of all the successfully tracked null points, depicting the spine and fan plane of each null for each time frame, and the behaviour of the eigenvector plots can be observed directly. It seems clear on viewing the movies that the null points so far identiﬁed are well divided between the fairly (geometrically) stable, where the spine and the fan plane remain at more or less the same orientation, and the obviously unstable. However, we would like to oﬀer a more mathematical means of characterising the geometric stability of a null. There are several ways we might do this. The approach we have settled on here involves calculating the angle between the normal of the fan plane and the z-axis, and observing how this angle changes with time. This can be achieved with a very little vector algebra. The normal of the fan plane is given by n = x1 × x2 10 Movies for each of these nulls are available at www-solar.mcs.st-and.ac.uk/∼williams. Further mod- iﬁcation of the algorithm is evidently necessary in order to form a correct identity signature for N6. A movie demonstrating the simulataneous tracking of all the nulls (but also showing our failure to track N6) is available for online viewing as well. 18 Null nature instances absences angle deviation N0 positive 173 2 1.19 N1 negative 175 0 2.77 N2 positive 175 0 8.51 N3 negative 175 0 37.1 N4 negative 102 73 13.6 N5 positive 113 62 2.18 N6 positive - - - N7 positive 148 27 9.2 Table 1: Null points tracked through time. Figure 7: Fan normal / z-axis angle variation through time of N0 and N3 [Simpson, 2008]. where x1 and x2 are the eigenvectors of the fan plane. The angle between the normal and the z-axis is simply given by the dot product: z n.ˆ n.ˆ = |n|cos(θ) → θ = cos−1 z |n| For a stable null, we anticipate the variation in the fan plane’s orientation through time to be small. Taking the standard deviation of the set of angles calculated (call it the angle deviation), we can clearly see that the geometric instability of nulls N2,3,4 and N7 are (or are close to being) of a higher order of magnitude than nulls N0, 1 and N5. By the Null Stability Criteria, we now set them aside from further analysis in this investigation. We also note that N1 and N5 are in regions of low magnetic ﬁeld density, outside of the region of the solar ﬂare. 9 Discussion and conclusions Since we have oﬀered an analysis of our results at each stage we need only summarise here and conclude. First, we enumerated the apparent null points for all 175 magnetic conﬁgurations. 19 Second, we obtained a set of nulls that persist through time (N0-7), requiring that the nature of a null (whether it is positive or negative) remains ﬁxed. Third, we imposed the requirement of geometric stability, discarding nulls that exhibit a large angle deviation (N2-4,7). This reduced our set of stable, physical nulls to 3. So there does, in fact, appear to be at least 3 stable coronal nulls in AR0486 during the time of the X17 Solar Flare; namely, N0, N1 and N5. However, N1 and N5 are less physically interesting, occurring in parts of the active region where the z component of the magnetic ﬁeld is low, at some distance from the location of the Solar Flare. The most signiﬁcant ﬁnding is null point N0, which occurs near the location of the Solar Flare in an area where the magnetic ﬁeld contours are dense. This is consistent with those theories that have posited [a] coronal null point[s] as a necessary requirement for ﬂares or CMEs; our investigation has not falsiﬁed that claim. Of course, one conﬁrmation is a far cry from establishing the truth of such a proposition - many more active regions exhibiting similar solar activity would have to be similarly investigated before we could hope to answer our motivating question. But there is good reason to think that the work we have done here could be extended to do just that. Acknowledgments I should like to oﬀer my thanks to The Royal Society of Edinburgh for a Cormack Vacation Research Scholarship, and the Solar and Magnetospheric Theory Group in the School of Mathematics and Statistics at the University of St Andrews for additional ﬁnancial assistance. e Also, my thanks to S. R´gnier, my supervisor, for advice and assistance, and to my other research colleagues, Blair, David, Martin, Clare, Lucy & Hania. It’s been an enjoyable six weeks. References [Haynes and Parnell 2007] Haynes, A. L, & Parnell, C. E. 2007, A trilinear method for ﬁnding null points in a three-dimensional vector space, Phys. Plasmas, 14 [Priest and Forbes 2000] Priest, E. R., & Forbes, T. G. 2000, Magnetic Reconnection, Cam- bridge University Press [Parnell et al. 1996] Parnell, C. E., Smith, J. M., Neukirch, T., Priest, E. R. 1996, The Struc- ture of Three-Dimensional Magnetic Neutral Points, Phys. Plasmas, 3, 759 e e [R´gnier & Fleck. (2004)] R´gnier, S., & Fleck, B. 2004 Magnetic Field Evolution of AR0486 before and after the X17 Flare on October 28, 2003, SOHO 15: Coronal Heating, ESA-SP 575 [Simpson, 2008] Simpson, W. M. R. 2008, website url: www-solar.mcs.st-and.ac.uk/ ∼williams 20