5.2 Quasi-separatrix layers

Separatrices occur where the photospheric mapping is discontinuous. This discontinuity permits the generation, under line-tied coronal dynamics, of magnetic discontinuities at which the current density is formally infinite. If the footpoint mapping is continuous but severely distorted, it may preclude the formation of a genuine current sheet (pace Paker) but will not preclude current structures which are very thin and contain extremely large current densities (van Ballegooijen, 1985 ; Longcope and Strauss, 1994 ). This motivates the definition of a quasi-separatrix layer (QSL), defined as a region of large mapping distortion (Priest and Démoulin, 1995

; Démoulin et al., 1996

) or “squashing” (Titov et al., 2002

An inherent ambiguity in the definition of QSL stems from specifying how much deformation should be considered “large”. Most useful definitions are given in terms of derivatives of the mapping functions, $X+(x -)$ and $X - (x+)$ , collected into Jacobian matrices

|_ @X - @X - _| |_ @X+ @X+ _| ----- ----- ---------- @x+ @y+ @x - @y- D+(x+) =_ |_ @Y @Y _| , D - (x -) =_ |_ @Y @Y _| . (20) ---- ---- ---+ --+- @x+ @y+ @x - @y-

These matrices are defined in the positive and negative photospheric regions, respectively. A straightforward estimate for the degree of local distortion is given by the norm of the Jacobi matrix (Priest and Démoulin, 1995 ; Démoulin et al., 1996 )

A related but slightly more complicated definition, offered by Titov et al. (2002

)

is directly related to the degree of squashing imposed by the mapping¹⁷ . Moreover, this function takes the same value at conjugate footpoints of the same field line, $Q(x ) = Q[X (x )] - + -$ , which is not true of the norms. (The value of $Q$ on a field line is the product of the norms from its two footpoints.) Both the norm $N ±(x± )$ and the squashing function $Q(x ± )$ designate QSLs in the same basic way. They are dimensionless, are smallest when the mapping is a rigid translation, rotation or inversion, and become larger with the degree of deformation. Plots of their values, or the logarithms of them, exhibit narrow areas of exceptionally large distortions which are the QSLs. Figure 17

shows a plot of $Q(x)$ for a submerged poles model of Sweet’s configuration taken from Titov et al. (2002

). Two roughly horizontal strips have $6 Q -~ 10$ , roughly where the spine curves would be in an MCT model. For a precise definition one must define a threshold value of $N$ or $Q$ , and then all footpoints exceeding that value are part of a QSL.

Figure 17:

QSLs in a submerged pole model of Sweet’s configuration. Contours shows the photospheric field $Bz(x, y)$ , with the dark line designating the PIL (here labeled IL). (Left) Grey scale shows $Q(x)$ on a logarithmic scale. (Right) Grey scale shows the degree of flux tube expansion for each footpoint. Plusses and dots indicate the principal locations of the four interacting flux systems. (Reproduced from Titov et al. (2002

).)

The QSL is therefore a layer rather than an infinitesimal surface like a separatrix. Tracing the field lines of every footpoint at some threshold value of $Q(x)$ , defines a surface enclosing the three-dimensional QSL. QSL with a particular X-shaped cross section are common and have been dubbed hyperbolic flux tubes (HFTs, Titov et al., 2003

Submerged poles models, discussed in the following Section 6, offer a link between pointwise mapping models and MCT as the depth of the submerged poles is reduced to zero. Each model with truly submerged poles has a non-intermittent photospheric field, and is thus a pointwise mapping model, containing only QSLs. In the limit that the poles reach the surface, the QSLs become genuine separatrices and the footpoints of a QSL lie approximately where the spines of the prone nulls will appear in that limit. In this same limit, a hyperbolic flux tube becomes a pair of positive and negative fan surfaces along with the null-null line. Its two ends become the spines of the two nulls linked by the null-null line. The X-shaped coronal portion of the HFT becomes the pair of separatrices and the separator at their intersection (Titov et al., 2002 ).

Most instances where QSLs are invoked in reference to observations use a submerged poles model of the coronal magnetic field (see Démoulin et al., 1997 , for an example). Thus the QSLs provide a self-consistent explanation for the localization of current in an evolving active region field. These applications are discussed at the end of the following Section 6.