Coronal null points and solar flares an investigation of by ruu17521

VIEWS: 8 PAGES: 24

									   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

								
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