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1 This article was downloaded by: [Soward, Andrew] On: 19 March 2011 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Geophysical & Astrophysical Fluid Dynamics Publication details, including instructions for authors and subscription information: Structure and collapse of three-dimensional magnetic neutral points C. E. Parnell ab ; T. Neukirch a ; J. M. Smith a ; E. R. Priest a a School of Mathematical and Computational Sciences, University of St Andrews, Scotland b Center for Space Science and Astrophysics, Stanford University, Stanford, CA, USA To cite this Article Parnell, C. E., Neukirch, T., Smith, J. M. and Priest, E. R.(1997) 'Structure and collapse of threedimensional magnetic neutral points', Geophysical & Astrophysical Fluid Dynamics, 84: 3, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Geophys. Astrophys. Fluid Dynamlcs, Vol. 84, pp Reprints available directly from the publisher Photocopying permitted by license only C 1997 OPA (Oversods Puhliahcrfi As~ociatioii) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Scicncc Publishers I rintcd in India STRUCTURE AND COLLAPSE OF THREE-DIMENSIONAL MAGNETIC NEUTRAL POINTS C. E. PARNELL*, T. NEUKIRCH, J. M. SMITH and E. R. PRIEST School of Mathematical and Computational Sciences, University of St Andrews, St Andrews, Scotland, K Y16 9SS (Received 3 July 1996; In jinal form 21 October 1996) The structure and collapse of linear three-dimensional magnetic neutral points is studied by varying the four parameters (p, q. jll, j,) that define, in general, the linear field of a neutral point. The effect of these parameters on both the skeleton structure (i.e. the fan and spine) and the actual field line structure of the null is considered. It is found that one current component (j,) causes the skeleton structure of the null to fold up from its potential state, whereas the other current component ( jll) causes thc field lines to bend. The two other parameters (p,q) determine the potential structure of the null and cause the null to transform from a three-dimensional null to a two-dimensional null and from a positive (type B) null to a negative (type A) null. To investigate the collapse of three-dimensional nulls, solutions to the linear, low$, ideal magnetohydrodynamic equations are found. It is found that three-dimensional null points can collapse if the field line foot-points are free and energy can propagate into the system. Keywords: Magnetic neutral points; magnetic reconnection 1. INTRODUCTION Magnetic reconnection is one of the fundamental processes of energy release in astrophysical, solar, space and laboratory plasmas, and therefore its investigation deserves much attention. Until recently most of the studies of magnetic reconnection focussed on two-dimensional systems, *Present address: Center for Space Science and Astrophysics, Stanford University, ERL 306, Stanford, CA 94309, USA 245

3 246 C. E. PARNELL rl a!. where reconnection is always accompanied by the occurrence of magnetic null points and its definition is unambiguously tied to the existence of separatrices (Vasilyunas, 1975). In three dimensions, reconnection does not necessarily imply the existence of null points within the reconnection region or changes in magnetic topology (Hesse and Schindler, 1988; Schindler, er al., 1988; Priest and Forbes, 1989, 1992; Priest and Demoulin, 1995; Hornig and Schindler, 1996), and so more general definitions of reconnection have been proposed (Hesse and Schindler, 1988; Schindler, et ul., 1988). However, as in two dimensions, if magnetic null points exist they can be very important for magnetic reconnection because at a null point, or along one or more of the field lines connected with it, strong currents may be induced by perturbations of the field under ideal conditions which favour reconnection. This has been shown by analytical (Lau and Finn, 1990; Priest and Titov, 1996) and numerical work (e.g. Galsgaard and Nordlund, 1996; Galsgaard et ul., 1996) on null point reconnection in three dimensions. The investigation of the different classes of three-dimensional magnetic null points and their topological structure is therefore a worthwhile undertaking. Studies of this type have been done by Cowley (1973) and Fukao et ul. (1975) and recently a more systematic classification of three-dimensional null points has been developed by Parnell et al. (1996) (here after referred to as Paper 1). In this paper we shall be focussing on the evolution of the field line structure in linear three-dimensional magnetic neutral points. Knowing how neutral points transform as the current around them increases or as the field in which they occur varies is an important first step in understanding where current sheets may form and how reconnection is most likely to take place. First, we continue this introduction with a brief review of the main characteristics of linear three-dimensional magnetic neutral points and we also review the simplest general form for the magnetic field of a linear three-dimensional null taken from Paper 1. Then in Section 2 we extend this work to look at how the skeleton structure of the null (that is, the spine and fan) evolves as the current and surrounding field of the null vary. In Section 3 the effects on the field lines in the null are investigated as the four parameters defining the field vary. Having investigated the structure of a null we then consider its stability by following the linear collapse of potential nulls due to

4 3-D MAGNETIC NEUTRAL POINTS 247 different perturbations (Section 4). Finally, in Section 5, we conclude this paper. In all figures of three-dimensional nulls the spine will be indicated by a thick solid line and the fan plane outlined by thin dashed lines Three-Dimensional Null Characteristics The skeleton structure of a three-dimensional neutral point consists of two main features: a spine and a fan surface (Priest and Titov, 1996). The spine is made up of two field lines that are directed into (or out of) the neutral point. The fan consists of a particular surface of field lines that are pointing away from (or into) the null. Other flux surfaces of field lines that lie in the vicinity of the neutral point consist of field lines that run almost parallel to the spine of the null before spreading out parallel to the plane of the fan (Fig. 1). If the field along the spine is directed into the null and the fan field lines spread out from the null, then the null is known as a positive null (type B); similarly, a negative null (type A) has field lines pointing towards the null in the fan and directed out along the spine. Field lines in the plane of a fan are often not evenly distributed over its surface, but may be preferentially aligned in one direction, known as the major axis of the fan. The direction of the spine and the orientation of the fan surface are determined locally by the directions of the vectors associated with the eigenvalues of the matrix which defines the linear field close to the null. These vectors are, generally, only the eigenvectors of the matrix if all three eigenvalues are real and distinct. Otherwise, if the eigenvalues are either complex or repeated then their associated eigenvectors do FIGURE 1 The magnetic field structure of a positivc three-dimensional null showing the field lines (solid) in the spine (thick) and fan (thin). The fan plane is outlined by a dashed line and a bundle of field lines near the null are also depicted by dashed lines.

5 248 C. E. PARNELL et al. not determine the skeleton, but can be used to calculate the skeleton structure vectors (further details may be found in Paper 1). The spine lies parallel to the eigenvector associated with the real eigenvalue, whereas the fan is defined by the vectors associated with the two eigenvalues having real parts of the same sign (which in turn are opposite to the sign of the spine eigenvalue). If the real parts of two of the eigenvalues are positive then the null will be a positive null with the spine eigenvalue necessarily negative, whereas in a negative null the real parts of two of the eigenvalues will be negative and the spine eigenvalue will be positive. The major axis of the fan is aligned along the vector associated with the fan eigenvalue with the larger real part, while the minor axis of the fan is directed along the vector associated with the eigenvalue having the least real part. Three-dimensional nulls may be either potential or non-potential. We know from Fukao et al. (1975) and Paper 1 that in potential nulls the fan and spine are perpendicular and that the field lines form either radial or improper radial nulls. If the null is non-potential, however, three possible types of nulls may occur: improper nulls, critical spiral nulls and spiral nulls (Paper 1). In special cases three-dimensional neutral points may reduce to continuous lines of two-dimensional nulls. When this happens one or more of the eigenvalues is zero and a null line forms that threads a set of either X-type or O-type nulls. Also a null plane may form when the field is essentially one-dimensional and planes of anti-parallel field lines are created Simplest General Form for the Magnetic Field To study the evolution of the structure of linear neutral points we consider the simplest general form for the magnetic field about a neutral point. Following Paper 1, we write the magnetic field B in terms of a 3 x 3 matrix M and the position vector r = (x, y, z)~, so that where B=M-r,

6 3-D MAGNETIC NEUTRAL POINTS 249 and p 2 max [ - l,(q2 - ji)/4] and q2 <jf + 4p. The current associated with the null is This form for the matrix was chosen because it insures that the spine of the null is always aligned with the z-axis, and that the components of current associated with the null divide simply into a component parallel to the spine and one perpendicular. This then allows us to study the effects of these two components without having to worry about changing coordinate systems as the spine moves. The matrix has also been simplified by normalising the first element to unity. An important parameter of the matrix, called the threshold current ( jthresh), is defined as The eigenvalues of our matrix M may then be written as The threshold current is important because its value relative to the magnitude of jll (the component of current parallel to the spine of the null) will decide whether the matrix has real distinct, real repeated or complex conjugate eigenvalues, thus determining whether the null is an improper null, a critical spiral or a spiral, respectively. 2. THE SKELETON STRUCTURE From work by Fukao et al. (1975) and Paper 1, it is clear that the fan and spine are not generally perpendicular. We, therefore, in this section ask the question, what affects the angle between the fan and spine. Using our general form for the matrix M and satisfying the two constraints that ensure that the spine is in the z-direction, we can

7 250 C. E. PARNELL etnl calculate the equation for the plane of the fan from the vectors associated with the eigenvalues Awl and A2. It is interesting to find that the equation of the fan plane is independent of whether the null is an improper null, a critical spiral or a spiral and is equal to where p 2 max (- l,(q2 -,jf)/4) and jiresh = ( p - 1) + 4. Clearly when,j,=o the equation for the fan reduces to z=o, so that the fan and spine become perpendicular. Therefore, it is not only potential nulls that have their spine orthogonal to the fan plane, but also any null with no perpendicular component of current. We may calculate the angle (0) between the fan and the spine which is defined as 42 minus the angle between the spine and the normal to the fan plane, namely where 8 +jf = -- arccos (7) 71 (9jp + -.Lcsh 2 lnlanl As expected, if j, = 0 the fan is inclined at 7112 to the spine. As j, increases from zero, 0 decreases and the fan and spine close up. When j, decreases from zero (becomes negative) 8 increases, but this is just equivalent to the spine and fan closing up by tilting the other way. j, is not the only parameter that effects the angle between the fan and spine, since p,j,, and,j,,,,,, (or q) do too. In Figure 2a we have plotted 8 versus j, for three different values of p (2,4 and 9) and in each of these cases for,j,, equal to (O.OS[solid], 3[dashed], 6[dot-dashed]) while fixing j,,,,,, equal to 3. Here we can see that the larger.ill is, the closer the

8 3-1) MAGNETIC NEUTRAL POINTS 25 1 FIGURE2 A plot of the angle (0) between the fan plane and spine against jl, the component of current perpendicular to the spine for: (a) jthresh = 3, p = 2,4,9 and for each value of pj,, = 0.05 (solid), j,,,= 3 (dashed) and.ill = 6 (dot-dashed); (b) p = 2, j,,,,,, = 1,3 and for each value of jthtesh J,, = 0.05 (solid), j,, = 3 (dashed) and j,, = 6 (dot-dashed). (c) The angle 4 about which the fan tilts is plotted against p for jthresh = 1,1.25 and for each value of jthrerh for j,, = 0.8 (solid), j,, = jthrerh (dashed) and j,, = 6 (dot-dashed). fan and spine are to being orthogonal. This implies that the parallel component of current counteracts the tilting of the fan. We can also see from this graph that increasing p also causes the angle 8 to tend to n/2 so that the fan and spine become more perpendicular. Figure 2b shows a plot of 8 versus ji, as before, with this time p held constant at 2. The curves j,, = 0.05, jthrcsh and 6 (shown by solid, dashed and dot-dashed curves, respectively) are plotted for two different values of,jthresh (1 and 3). Here it can be clearly seen that H increases with jlhresh. However, if j,, and jthresh remain equal as jthresh varies, then there is no change in the angle of the fan plane. For example, the lines jj =jthresh for jthresh equal to 1 and 3 are exactly the same. In studying how the global structure of the null varies we can also consider how the angle $, about which the fan plane tilts in the xy-plane, varies with respect to the direction of the perpendicular component of current (the x-axis). The line about which the fan plane

9 252 C. E. PARNELL et al. tilts is given by It is interesting to notice that this line does not depend on jl, the parameter that causes the plane to incline, but it is instead dependent on p, j;, and jthresh. If Jjkresh - ( p - 1) = -j,, (that is, if q = -j,,), then the fan tilts only along the x-axis (the direction of the perpendicular component of current). However, if this relation does not hold, then neither j,, nor j, nor the resultant current lie in the plane of the fan. The line in the xy-plane about which the fan plane tilts is inclined at an angle 4 with respect to the x-axis, such that If the parallel component of current is so small that the fan contains improper nulls, then 4 is also small; however, ifjll is larger so that the fan contains spirals then 4 will be larger (Fig. 2c). As p increases the angle 4 initially increases, reaching a maximum before it decreases. A larger jthresh enables 4 to reach a greater maximum value, as does a larger value of jll. Finally, we note that the major axis of the fan plane (if one exists) is not necessarily along the line of tilt of the fan plane, but it may be any line in the fan plane. This implies that the major axis of the fan is not associated with the direction of the components of current or the resultant current. 3. THE FIELD LINE STRUCTURE OF THE NULL As the four parameters ( jl, j!, p and q) that define the field of the null point vary, not only do the angles between the fan and spine change, but also the actual field lines within the fan evolve. To determine how each parameter affects the field we just vary one parameter in turn

10 3-D MAGNETlC NEUTRAL POINTS 253 and keep all the others fixed, First, in Section 3.1, we investigate the effect of increasing the perpendicular current on the field line structures, and then in Section 3.2 we consider the parallel current. Section 3.3 studies the potential parameters p and q The Effect of the Component ci,) of Current Perpendicular to the Spine The effect of varying j, is studied by first considering a magnetic field of the form Firstly, we fix p and j,, = 0, so that all the nulls are positive (Fig. 3, top row). When j, < 0 the fan of the null point is inclined about the x-axis such that the fan faces away from us, The field lines in the fan are -m < ji < 0 j, = 0 p= 1, j,, = -2 FIGURE 3 The evolution of the magnetic null B = (x- j,, y/2,j,, x/2 + py, j, y - ( p + 1)z) as j, varies from - x to 00, with p = 1, j,, = 0 (top row) and p = 1, j,, = - 2 (bottom row).

11 254 C. E. PARNELL eta/. straight lines though they are gathered along the major axis of the fan (the line y = 3z[j,). As I,jll decreases to 0 the field lines in the fan radiate out such that at j, = 0 they form a potential radial null, with the fan and spine perpendicular to one another. The fan plane once again tilts about the x-axis as jl increases from 0, but this time the plane faces towards us and the field lines gather along the line y = 3z/j,. Secondly, the parameters p and,jl, are held fixed at 1 and -2, respectively, so that j,hrcsh = 0 and the nulls formed are positive spirals (because lj,,l > jthrcsh). Again, as before, we vary,jl and find that when ji<,, the fan plane tilts one way and when,j, > 0 it tilts the opposite way (Fig. 3, bottom row). However, in this case the fan does not tilt about the x-axis, but tilts about the line y= -x/3. We can also see that, as the plane of the fan steepens, the field lines lying in it become more and more elongated in the z-direction, thus stretching the spirals along the major axis of the fan The Effect of the Component (j,,) of Current Parallel to the Spine To consider the effect of jli we consider the non-potential magnetic field Here we keep the parameter p fixed at 2 and just vary j,, (Fig. 4). The three-dimensional nulls formed are all positive. In the range - c/3 < j,l < - 1 clockwise spirals are created. At j,, = jthrerh = - 1 the field lines have untwisted to form a critical spiral, which then allows the null to become an improper radial null in the range - 1 < jl < 1. The major axis of the null is at first located along the x = y line, and then as j, becomes nearer to 1 the major fan axis revolves around to lie more along the x = -y line. Another critical null is formed atj,, = 1 permitting the null to return to a spiral configuration for 1 <j, < CG; this time though, the spirals are twisted in an anti-clockwise manner. So we see that increasing the magnitude of the parallel component of current allows the null to evolve from an improper null to a spiral.

12 3-D MAGNETIC NEUTRAL POINTS * FIGURE 4 The magnetic configurations through which a magnetic neutral point of the form B = (x- jll y/2, ill x/2 + py, - ( p + 1)z) evolves as jll varies ( p = 2). The transition state between these two forms of nulls is the critical spiral. If, however, we had considered a case where p = 1 and therefore jthresh = 0, then we would not have seen any improper nulls. Instead, for the whole range jll # 0 the nulls are three-dimensional null spirals. Only at j,l = 0 do we then find instead of a spiral a radial null, which is a particular type of critical null since it is the transition state between clockwise and anti-clockwise logarithmic spirals The Potential Parameters ( p and q) The parameters p and q determine the potential field of the null which is generated by the electric currents lying outside the considered domain. These currents are not unique, therefore, it is difficult to associate p and q with any real physical quantities apart from jthresh. However, for simplicity, in this analysis we use p and q as free parameters and not jthresh. To investigate the effects of p and q we relax the constraints which force the spine of the null to remain in the z-direction

13 256 C. E. PARNELL etal. and so consider what happens when parameters p and y vary between - a and 00. The components (j,, and j,) of current may no longer be defined as the parallel and perpendicular components of current once the z-axis is not the spine of the null, so we simply rename them as j, and j,, respectively. To investigate the effects of p we consider two cases with the magnetic field of the form Here the nulls have just one component of current along the z-axis. In the first case we assume that the magnetic field only has a weak component of current and so fix j, = 0.5. From Figure 5 we find that as p varies over its whole range the neutral point evolves from being positive (- 00 < p < - 1) to negative (- 1 < p < ) and back to being positive again ( <p < 00). We also find that at the points p = -1 and p = the neutral point reduces to a twodimensional configuration. If we look closely at one of these twodimensional nulls we can see how this may occur. For example, when p is just less than the null is three-dimensional, but its field lines in the fan plane have closed up along the z-axis to give the field an almost two-dimensional feel. Then, as p increases to the field lines close up even further until a null line is formed along the minor axis of the fan, with two-dimensional nulls formed in planes parallel to what would be the plane containing the major-axis of the fan and the spine in the original null. With a further increase of p the field lines reconfigure themselves to form a three-dimensional null again, but this time of opposite sign to the previous one. Field lines of the negative null begin to fan out from what was the spine of the positive null in the plane of the positive null's minor fan axis and spine, with the negative null spine now aligned with the previous null's major fan axis. We therefore find that for a three-dimensional null to transform from a positive null to a negative null and vice versa the null must evolve through a transitionary state of a two-dimensional null. From these figures we can also see that, as p increases, the null evolves from being a positive improper null to a negative improper null and back to a positive improper null. Then, as p increases further,

14 3-D MAGNETIC NEUTRAL POINTS 251 -m < p < -1 p=-1-1 < p < FIGURE 5 p = < p < 0.5 p = < p < 1.5 p = < p < w The stages through which the configurations of a linear three-dimensional magnetic null of the form B = (x - j, y/2, j, x/2 + py, - ( p + 1)z) with j, = 0.5 evolve as the parameter p varies from - 03 to co. the null transforms into a critical spiral at p = 0.5, then to a spiral in the range 0.5 c p < 1.5. At p = 1.5 a critical spiral is formed once again before the null reverts back to an improper null. As can be seen from the figures, the major axis of the fan has changed in this second improper null, so that it is now predominantly along the y-axis rather than the x-axis. In the second situation we assume a much higher value of the current (j, = -4). Again, if we look at what happens as p varies over the range - 00 < p 00 (Fig. 6). We find that the three-dimensional

15 258 C. E. PARNELL et a1 --m < p < -4 p= -4 p=-l -l<p<6-1<pc5 p=5 s<p<oo FlGURE 6 The stages through which the configurations of a linear three-dimensional magnetic null of the form B = (x - j, y/2. j, 42 + py, - ( p + 1) z) with j, = - 4 evolve as the parameter p varies from - co to co. null changes from being initially positive to negative and back to positive again by way of two-dimensional null transition states. However, in this case the first transition, p = -4, is of an X-point and leads from a positive improper null to a negative spiral null. The new negative spiral initially has a very strong major axis, but, as p increases, the spiral begins to open out until at p = - 1 concentric ellipses form: thus the second two-dimensional transition state is an 0-point. For p > - 1 we have a positive null with a spine axis in the z-direction. Spiral nulls exist for - 1 < p < 5, including at p = 1, the

16 3-D MAGNETIC NEUTRAL POINTS 259 special case of a logarithmic spiral. Then at p = 5 a critical spiral forms allowing the null to evolve into an improper null for the range 5<p<cO. The second potential parameter (q) also has an effect on the topology of the neutral point. To investigate its effect we consider a magnetic field of the form The parameters p,j, and,j, are fixed at 1, 1 and 2, respectively, with q varying from -a to 00 in Figure 7. It is clear that, as q varies, the null evolves through two two-dimensional transition states; thus, like the parameter p, the parameter q can change the sign of three-dimensional nulls. The q also behaves in a similar manner to p, in that it causes the null to deform from an improper null to a spiral and back to an improper null again, as seen in the range - 3 < q < fi when there exists a component of current parallel to the spine of the null. However, in these figures there is also a component of current perpendicular to the spine of the null: thus q not only causes the major fan axis to rotate about the spine, but the fan itself also to rotate. In the range - cc < q < -$ and fi < q < K there is a weak component of current along the spine, which decreases or increases as the spine rotates and q varies. When q = k,i% the three-dimensional null becomes a critical spiral, but for the rest of the range it remains an improper null, since the threshold current never becomes weak enough to cause the field lines to spiral. The potential parameters therefore affect the null in many ways. Their most important role is to allow three-dimensional nulls to transform from positive nulls to negative nulls through the transition state of a two-dimensional null. The parameters also affect the position of the spine and fan plane as well as the actual field line structure of the null. For example, in Figure 5, when p < - 1, the spine axis lies along the line x = [2(1- p )- m ] y and the fan lies in the plane x - [2(1 - p) y = 0. Finally, since the threshold current (jthre,h) is dependent on p and q they also influence the type of

17 260 C. E. PARNELL etal. q=-1 -liq<l q= I 1<q<fi q = f i \m<9<m FIGURE7 The evolution of a linear three-dimensional magnetic null of the form B=(x+(y-j,)~/2,(qtj,)x/2+py,,j,y-(p+l)z) with p=l, j,=1 and j,=2 as q vanes from - co to co. three-dimensional null that occurs, since their size determines whether a null is a spiral or an improper null ifj, remains fixed. 4. THE COLLAPSE OF A NULL 4.1. Two-dimensional Nulls It is well known that two-dimensional X-type neutral points are locally unstable and thus prone to collapse if the field lines are free and

18 3-D MAGNETIC NEUTRAL POINTS 26 1 energy can propagate into the system. The first person to denionstratc this from physical arguments was Dungey (1953). He started initially with the simplc potential field R = B,,(j,. s //(Bo and I are constants), which has an X-point at the origin. Field lines in the vicinity of this null form hyperbolae. It was then supposed that the field distorts to the form B = B,, [j; ( 1 + x ).~] :I. (15) whcre 2 > - 1. such that the separatrices (the local limiting field lines through the null) are no longer inclined at the angle n/2, but have closed up. The resulting current density in this new field is purely in the,--direction and equals so that the Lorentz forcc is The action of this force is to increase the original perturbation, and so the instability proceeds as x increases and the separatrices close up, so increasing the current density and Lorentz force. Priest (1985) demonstrated this instability formally by considering the linearised form of the ideal magnetohydrodynamic (MHD) equations and assuming that plasma pressure is much smaller than the magnetic pressure, so that the plasma p is small. Given that B=B,,+EB,, V=EV~ and p = po + cpl, where t:<< 1, the set of linearised equations may be written as JB 1 -y- = V x (vl x B"), fft The perturbation of the density p1 can be calculated from the linearised continuity equation, however. since it is decoupled from the

19 262 c. E. PARNELL el ar. above equations it is omitted. Solutions to these equations are of the form where B,, po, tia = Bo/(popo)l~z and E(<< 1) are constants. The growthrate is Other investigations have also been undertaken into the collapse of a two-dimensional X-point. Chapman and Kendall (1963) and Forbes and Speiser (1 979) found an exact incompressible solution whereas Imshennik and Syrovatsky (1967) discovered a nonlinear self-similar compressible solution with a uniform density, which shows that the collapse occurs over a few Alfven times. Papers by Syrovatsky in 1966, 1969 and 1971 studied the response to a small motion of the sources and included the idea in a solar flare model. In a more recent paper by Craig and McClymont (1991) magnetic diffusion was included in a linear analysis of a small perturbation of an X-type field Three-Dimensional Nulls In three-dimensions we use similar methods to show that three-dimensional nulls may collapse by looking for solutions of the linearised, low-j3, ideal MHD equations (18) and (19). Consider initially the potential field for a three-dimensional null, where p, B, and 1 are constants and, without loss of generality, assume that p > 0 such that the spine of the null is always along the z-axis. This field may be perturbed in several different ways: for example, if the initial perturbation is such that a component of current is created in the x-direction then the equations have solutions of the form

20 3-D MAGNETIC NEUTRAL POINTS 263 where,j,x, p = po, ua = B,,/(p,,po)1/2, E( << 1) are constants and the growth-rate is The current density 0) is uniform throughout the region and equals The structure of the magnetic field and the flow about the null is depicted in Figure 8 as t increases. The velocity (Fig. 8e) is just a stagnation-point flow in constant-x planes. Its profile does not change wl = log(soo), rew' = 0.3 velocity (4 (4 FIGURE 8 The collapse of a potential null due to an initial perturbation that produces a component of currcnt in the.\--direction. The same two bundles of field lines are drawn in each frame. The first has solid field lines and the second dashed ones.

21 264 C. E. PARNELL etnl. over time, but its strength does increase exponentially. We can see, from Figures 8a-d, that the fan and spine fold up towards one another due to the stagnation-point flow. This type of collapse will continue until the non-linear terms become significant, which occurs when tjx[jo' is of order 1. In each of the figures the same two bundles of field lines have been drawn in each frame. The solid field lines start off in the plane of the fan and remain in this plane as it rotates about the x-axis. The second bundle of field lines is shown by dashed lines and runs parallel to the spine before spreading radially out below the fan plane. In the frames shown the frozen-in-flux condition holds so that the field lines remain in the fan plane and below the spine (see Appendix for calculation of the equation of the foot-points of the field lines as they vary in time). Similarly, an initial perturbation causing a component of current in the!%-direction leads to an identical form of collapse, except that this time the stagnation-point flow is in constant-y planes and the fan plane rotates about the y-axis. However, if the potential null field is perturbed such that a component of current is created in the z-direction only, then (18 ) and (19) have solutions of the form B=Bo S- and where p( # l), B,), 1, jz, p = po, ra = Bo/(popo)'iZ, c(<< 1) are all constants and the growth-rate is

22 3-D MAGNETIC NEUTRAL POINTS 365 We note this time though that a solution does not exist for p = 1, i.e. when the potential null is radial. Again, in Figure 9, we show how the field line structure evolves as time passes. The flow is, as before, two-dimensional (Fig. 9e), but this time in constant-z planes. It is not a stagnation-point flow, but a rotating flow, which as t increases merely increases in strength and does not change its flow pattern. We consider two bundles of field lines (see Appendix for equations of the foot-points). The solid lines are initially lying in the fan plane and remain so as they move. However, they do not form a spiral, but form an improper null due to the differing speeds of rotation of the plasma. The second bundle of (dashed) field lines runs parallel to the spine before spreading out below the fan plane. This type of behaviour is valid provided that Eew'j, < 1.,.Here we have just considered the behaviour of potential three-dimensional nulls. This is because all other linear three-dimensional nulls are not force-free. Furthermore, it is not possible to create an *r ut = kg(ru), fe' = 0.25 velocity FIGURE 9 The collapse of a potential null duc to an initial perturbation that produces a component of current in the z-direction. The same two bundles of field lines arc drawn in each frame. The first has solid field lines and the second dashed ones.

23 266 C. E. PARNELL et a1 equilibrium by balancing the Lorentz force with a pressure gradient (VP) for the following reason. The curl of our momentum equation is Vx(jxB)=VxVP=O, (31) and since, B = M.r and j is constant, V x (j x B)= -(j.v)(m*r)= -M.j. From Paper 1 we know that for all true three-dimensional nulls M is nonsingular, and so the only solution to (31) is j = 0, i.e. the potential cases we have already considered. The matrix M is singular when its determinant is equal to zero, but the nulls formed then are two-dimensional in continuous parallel planes and have a null line rather than a null point. These solutions are not investigated here as they are the well-known two-dimensional ones. 5. CONCLUSION From our study of the structure of linear three-dimensional neutral points we have found that there are 3 important transitional nulls. l-irst, it has been shown that by varying the potential parameters (p, q) it is possible for a three-dimensional neutral point to reduce to a twodimensional one; this two-dimensional null is a transition state between positive and negative three-dimensional nulls. It is only possible to change from a positive null to a negative null or vice versa through this transition state. A natural consequence of the conversion from positive to negative is that the directions of the spine and fan of the three-dimensional nulls change dramatically. This cannot occur by only varying the current in the null. Another important transition state is the critical spiral which occurs between improper and spiral nulls. The final transitional null is the radial null which occurs at the interchange between clockwise and anti-clockwise spirals or between improper nulls as their major and minor fan axis switch.

24 3-D MAGNETIC NEUTRAL POINTS 267 The component of current parallel to the spine of the null and the potential parameters determine the actual field line structure in the null, depending on the size of the parallel current relative to that of the threshold current, which depends on the potential parameters. The perpendicular component of current affects the size of the angle between the fan and the spine. When it is zero the fan and spine are at right angles, and as it increases the fan and spine close up. The parallel component of current counteracts the closing up of the spine and fan, whereas the threshold current tends to increase the closing of the fan and spine. The angle about which the fan tilts varies with both of the potential parameters and the parallel component of current. It is, however, independent of the perpendicular component of current; thus, as the perpendicular current increases, the fan and spine close up, but they do not rotate unless one of the other three parameters is also varying. We find that the fan, in general, does not tilt about the axis of the perpendicular current or about the major axis of the fan. From our solutions to the linearised MHD equations we find that three-dimensional nulls may collapse provided that the field lines are not line-tied and energy can propagate into the system. The way in which they collapse depends on the initial perturbation. If a current is induced perpendicular to the spine of the null then the fan will start folding up towards the spine; however if a current is induced parallel to the spine then the field lines begin to close up in the fan plane. Both of these types of collapse suggest that the null is trying to collapse towards a two-dimensional X-point. However, we have only considered linear perturbations so we are unable to determine the exact state to which the null will finally collapse since non-linear effects will eventually come into play. Also, as the current grows resistive effects will become important and change the dynamics of the system. Furthermore, we have only considered first order terms in both the magnetic field and velocity, thus restricting the analysis to a finite region about the null in which the current is effectively uniform. A ckno wledgemeats C. E. Parnell, T. Neukirch and E.R. Priest are grateful to PPARC for their financial support and J. M. Smith wishes to thank the EPSRC.

25 268 C. E. PARNELL et trl. References Chapman, S. and Kendall, P. C., Liquid instability and energy transformation near a magnetic neutral line: a soluble non-linear hydromagnetic problem. Proc. R. Soc. A 271, (1963). Cowley, S., A qualitative study of the region between the earths magnetic field and an interplanetary field of arbitrary orientation, Rudio Science 8, (1973). Craig, 1. J. D. and McClymont, A. N., Dynamic magnetic reconnection at an X-type neutral point, Astrophys. J L41LL44 (1991). Dungey, J. W., Conditions for the occurrence of electrical discharges in astrophysical systems, Phil. Mug. 44, (1953). Forbes, T. G. and Speiser, T. W., Temporal evolution of magnetic reconnection in the vicinity of a magnetic neutral line, J. Plasmn Phys (1979). Fukao, S., Ugai, M. and Tsuda, T., Topological study of magnetic field near a neutral point, Rep. ion. Spuce Res. Jop. 29, (1975). Galsgaard, K. and Nordlund, A,, The Heating and Activity of the solar corona 111: Dynamics of a low beta plasma with 3D null points, J. G~phys. Res. in press (1996). Galsgaard, K., Rickard, G. J., Reddy, R. V. and Nordlund, A., Dynamical properties of single and double 3D null points. Proc. Yohkoh Conference on Ubservtrtidns of Magnetic Reconnrction in the Soh Atniospliew, Bath (1996). Hesse, M. and Schindler, K., A theoretical foundation of general magnetic reconnection, J. Geophys. RKS. 93, (1988). Hornig, G. and Schindler. K., Magnetic topology and the problem of its invariant definition, Phys. Plusnuis 3, (1996). Imshennik, V. S. and Syrovatsky, S. I., Two-dimensional flow of an ideally conducting gas in the vicinity of the zero line of a magnetic field. Sorier Phys. JETP 25, (1967). Lau, Y. T. and Finn, J. M., Three-dimensional kinematic reconnection in the presence of field nulls and closed field lines, Asrrop/rj.,s. J. 350, (1990). Parnell, C. E., Smith, J. M., Neukirch. T. and Priest. E. R., The structure of three-dimensional magnetic neutral points, Phj,.s. P/t~srn~r.s 3, (1996). Priest, E. R., The magnetohydrodynamics of current sheets, Rep. Ploy. Pkys. 48, (1985). Priest, E. R. and Dtmoulin. P.. Three-dimensional magnetic reconnection without null points 1. Basic theory of magnetic flipping, J. Geoph~.~. Res. 100, (1995). Priest, E. R. and Forbes, T., Steady reconnection in three-dimensions, Solar Phys. 119, (1989). Priest, E. R. and Forbes, T., Magnetic flipping: Reconnection in three dimensions without null points. J. Geophys. Rrs. 97, I (1992). Priest, E. R. and Titov, V. S., Magnetic reconnection at three-dimensional null points, : Phil. Trans. R. Soc. Lonrl. in press (1996). Priest, E. R., Titov, V. S. and Rickard, G., The formation of magnetic singula-rities by time-dependent collapse of an X-type magnetic field, P Id. Trtrns. R. Soc. Lond. 351, 1-37 (1995). Schindler, K., Hesse, M. and Rim. J., General magnetic reconnection. parallel electric fields and helicity, J. Geopliys. Res. 93, ( 1988). Syrovatsky, S. I., Dynamic dissipation of a magnetic field and particle acceleration, Soviet Astron. 10, (1966). Syrovatsky, S. I., On the mechanism of solar flares, in: Solar Flares and Space Research (Ed. de Jager C. and Svestka Z.), Amsterdam: North Holland (1969). Syrovatsky, S. I., Formation of a current sheet in a plasma with a frozen-in strong magnetic field, Soviet Phys. JETP 33, (1971). Vasilyunas, V., Theoretical models of magnetic field line merging, Rrr. Geophjs. Spuce Phys (1975).

26 3-D MAGNETIC NEUTRAL POINTS 269 APPENDIX Equations of the Foot-points The motion of the foot-points of the field lines (or indeed any point on a field line) is simply described by the following equations with the initial condition that at t = 0, x = xu, y = yo and z = zo; this foilows from the fact that the flux is assumed to be frozen into the plasma. In the case where the initial perturbation causes a current in the x-direction the following equations arise: dx - dr = 0, where w = ca(2p + l)/l The initial conditions are x(0) = xo, y(0) = yo and z(o)=z,. Thus we can immediately see that the x-component remains constant over time, whist the substitution z = e''" reduces equations (AS) and (A.6) to dy dz -- - E L P+l (2p + 1)2 z3

27 270 C. E. PARNELL eta/. -= dz. P EJx dz (2p + 1)'" where y = yo and z = z,, when 7: = 1. Since p > 0, these two equations may be solved simultaneously to give the iinal equations motion of the foot-points: y(t) = A exp (ad'") + B exp ( - aewt), (A.lO) where A = ( y 0 P +,/( p + l)/pz0e-")/2, B = ( yoeu -,/( p + l)/pz,e")@ and c( = cjxjp( p + 1)/(2p + 1)2. Figure 10a shows the field line footpoints with initial start points -4<xo <4, y,=zo=o and - 4 <yo < 4, xo = zo = 0 at times ot equal to 0 (solid), log 100 (dashed), log 200 (dot-dashed) and log 300 (triple dot-dashed). The arrows show the direction of motion of the field lines Y X (4 (b) FIGURE 10 The foot-points of field lines at time (a) cut equal to 0, log 100, log 200 and log 300 and (b) ox equal to 0, log 75, log 150 and log 225 shown by solid, dashed, dot-dashed and triple dot-dashed lines respectively.

28 3-D MAGNETIC NEUTRAL POINTS 27 1 If, however, the initial perturbation creates a current in the z-direction then the equations defining the motion of the foot-points are (A.12) (A. 13) dz - = 0, dt (A.14) with x = xo, y =yo and z = zo when t =O. Using an analagous method to the above calculation we find that for p > 0 the solutions to these equations are ~(t) = A cos (ap) + B sin (re""), (A.15) where A = xocosz + fiyosina, B =.u,sinx - JGcosa and x = cj,$/(p - 1)2. The field line foot-points with initial start points x,, = r cosh, yo = r sin0 and zo = 0 (r constant, 0 < H < 271) are shown in Figure 10b at times cut equal to 0 (solid), log 75 (dashed), log 150 (dot-dashed) and log 225 (triple dot-dashed). Again the arrows show the direction of motion of the field lines.