2.4 Null points

For a continuous magnetic field the field line equation (1

) is singular only where the magnetic field vector vanishes (Arnold, 1973 ). In a general field, $B(x)$ will vanish only at isolated points called null points $x0$ , in the vicinity of which it has the generic form

where $Mij =_ @Bi/@xj |x0$ is the field’s Jacobian matrix. A null point for which the matrix $Mij$ vanishes entirely is termed higher order, and will occur only in special circumstances. In the generic case⁶ where $Mij$ does not vanish identically the field lines have simple behavior in the neighborhood of the null, analogous the behavior of a general vector field in the vicinity of null points. The behavior may be characterized entirely from the eigenvectors and eigenvalues of the Jacobian matrix. In particular, it is possible to assign any null (excepting non-generic cases) to one of two categories, positive or negative, according to the number of its eigenvalues with positive real parts. A thorough analysis of this categorization is given by Parnell et al. (1996 ), from which we briefly report some of the main conclusions. The matrix $M$ will have three eigenvalues which may be all real or may include complex eigenvalues. In the first case the eigenvalues may be ordered, $c1 < c2 < c3$ ; in the second there must be one real eigenvalue $cr$ and a complex conjugate pair $cc$ and $c*c$ . Since $sum \~/ .B = iMii = 0$ , the three eigenvalues must sum to zero. A null point for which one eigenvalue is negative and the other two are positive (or have positive real parts) is called a positive null point (Priest and Titov, 1996 )⁷ . (The null is therefore called positive if $det M < 0$ , which is confusing until one considers the sense of field lines in the fan surface.) The real and imaginary parts of the eigenvectors from the two positive eigenvalues span a plane within which all field lines originate at the null point. Following these field lines beyond the immediate neighborhood they form a surface called the fan surface of the null. There are also two spine field lines which terminate at the null in directions both parallel and anti-parallel to the eigenvector of the negative eigenvalue (see Figure 4

). A negative null is one with the opposite structure: one positive eigenvalue and two with negative real parts, a fan surface of field lines ending at the null, and two spine field lines originating at the null.

Figure 4:

Schematic depictions of positive null points. Two spine field lines directed toward the null (white circle) appear as dark lines with arrow heads next to the null. The central portion of the horizontal fan surface is colored grey, and contains fan field lines directed outward like spokes on a wheel. The left case is the simplest: a potential null point which is cylindrically symmetric ( $c2 = c3$ ). Thinner lines show a few of the field lines on either side of the fan surface. The right case is a non-potential null whose fan is spanned by eigenvectors of complex eigenvalues and whose spines are not orthogonal to the fan surface.

Cases where one or more eigenvalue real parts vanish cannot be classified as either positive or negative. Such cases are not generic (they will not survive a small but arbitrary perturbation to the field) but do occur in cases of symmetry, such as two-dimensional models, or at the instant of bifurcation (discussed in Section 7). An X-type null is one where one eigenvalue vanishes, namely $c2$ , and the other two have equal magnitude and opposite sign: $c1 = - c3$ . This is the standard heteroclinic point in two-dimensional fields, however, as alluded to, they do not generally occur as such in three-dimensional fields. If the real parts of two eigenvalues vanish then so must the third, since they must sum to zero, Barring a higher-order null (all three eigenvalues are identically zero) this must be an O-type null with two purely complex eigenvalues $c = ig c$ .

Fan field lines and spine field lines are notable exceptions to the general tenet that field lines have no beginning or ending - it seems that certain field lines terminate at null points. A fan surface divides the volume of field lines in two regions (or domains), thereby serving as one form of separatrix in three-dimensional magnetic fields. The spines, on the other hand, are one-dimensional curves and therefore do not form separatrices. This makes three-dimensional null points topologically different from two-dimensional X-points, since all four field lines connecting to an X-point are dubbed separatrices, regardless of their orientation. If a three-dimensional null is transformed continuously into an X-point by taking its intermediate eigenvalue to zero, then the spine and fan both become separatrices in the limit $c2 = 0$ . Which of the X-point’s separatrices was formerly a spine depends on the former null type and which direction $c --> 0 2$ .

An even more exotic kind of field line occurs when the fan surface of a positive null intersects the fan surface of a negative null, as shown in Figure 5 . When the intersection is transversal it forms a null-null line (Lau and Finn, 1990 ). Intersections of separatrices are called separators, in general, and since a fan surface is one form of separatrix, null-null lines are one form of separator. There are, however, other forms of separators including finite-width TDs called current ribbons when the separatrix intersection is not transversal (Longcope and Cowley, 1996 ).

Figure 5:

The structure of a null-null line. Positive and negative null points $B$ and $A$ have fan surfaces $SB$ and $SA$ shown in light and dark shades of grey, respectively. These intersect transversally along the null-null line shown as a thick black line. Surface $S B$ is then bounded by the two spines from null point $A$ (blue), and similarly for $SA$ and the spines from $B$ (red). The neighborhood of the null-null line is illustrated by the inset. This shows the direction of $B$ within a plane pierced normally by the null-null line at the dark circle. The fan surfaces cross the plane along the dotted lines.

A null-null line has two termini since it must both begin at a positive null and end at a negative null. Since they lie at the intersection of separatrices, null-null lines are a more natural analog of two-dimensional X-points than are the three-dimensional null points themselves. In this analogy, however, it must be borne in mind that while an X-point may be identified locally, a null-null line is locally indistinguishable from nearby field lines; its uniqueness derives from only global topology. It is often, although not always, the case that the field vectors in the vicinity of the null-null line have an X-type shape (see inset of Figure 5

). The exact location of the X in such a slice depends critically on the orientation of the plane, and each field line in some neighborhood may play the same role in a different plane. This local criterion may not, therefore, be used to identify a null-null line; the only way to locate it is by following field lines in both directions. (Longcope, 1996

, presents a numerical algorithm for this.)

Substituting the local field (11 ) into the ideal induction equation (6 ) yields, to lowest order in distance from the null point, the requirement that the null point move with the flow: $x0 = v(x0)$ . Continuing to next order in distance yields an equation for the evolution of the Jacobian matrix

$3 ( ) | dMij- = sum @vk-- \~/ .v d || M . (12) dt @xi ik |x kj k=1 0$

The matrix in parentheses, call it $Vki$ , is related to the plasma’s local rate of strain at the null point. This can be used to show that each eigenvalue evolves according to $cn = Vnncn$ where $Vnn$ is a product of $V ki$ with the corresponding left and right eigenvectors. The most significant point is that, barring a singular flow field, an eigenvalue which is non-zero will remain non-zero and can never change sign. Under ideal induction, therefore, null points of a given type will move with the plasma flow but cannot change type and can be neither created nor destroyed (Hornig and Schindler, 1996

It is also possible for the edge of a tangential discontinuity (TD) to include a null point which is locally Y-shaped. Such a null cannot be characterized by its derivative matrix $Mij$ , since derivatives are not defined at the discontinuity. If the TD occurs on a smooth sheet, however, the Y-type null will locally resemble one from a field which is otherwise current-free. It will therefore resemble the two-dimensional configuration first studied by Green (1965 ) and Syrovatskii (1971 ), and shown in Figure 6 , except that field lines following the edge of the TD will either diverge away from or converge toward the null point in the erstwhile ignorable direction (making it a positive or negative Y-null). A pair of Y-type null points at the edges of a TD of finite breadth, as in Figure 6 , have the same topological degree (Greene, 1993 ) as a single regular null point which is either positive or negative. In both two and three dimensions it is possible to create the null pair continuously by deforming a regular null point (Syrovatskii, 1971 ; Longcope and Cowley, 1996 ).

The two-dimensional current sheet equilibrium proposed by Green (1965 ) and Syrovatskii (1971 ) is current-free everywhere except the current sheet. Such a structure can be described by a potential $A(x, y)$ which is the real (or imaginary) part of a complex potential analytic except at a branch cut defining the current sheet. This powerful technique has been used to construct equilibria resembling realistic coronal current sheets (Priest and Raadu, 1975 ; Tur and Priest, 1976 ; Hu and Low, 1982 ). General formulations developed by Aly and Amari (1989 ) and Titov (1992 ) permit the construction, and evolution, of equilibria of arbitrary complexity, containing numerous current sheets.

Figure 6:

A tangential magnetic discontinuity of the Greene-Syrovatskii type, in a two-dimensional magnetic field. Lines show magnetic field lines which are contours of a flux function $A(x, y)$ , arrows indicate the field’s direction. The field direction reverses across the current sheet of width $D$ . The vertical field along the $x$ -axis is plotted at the bottom. The Y-type null points are located at the tips of the current sheet: $(x,y) = (0,± D/2)$ .