3.3 Footpoint mapping as a dynamical constraint

A field line anchored to the photospheric surface is either closed (it has two footpoints) or open (it has only one). A model photosphere may be partitioned into regions of open-field footpoints and closed-field footpoints; some models contain a third type of region where $Bn = 0$ , in which there are no footpoints. The coronal field provides a mapping between positive and negative portions of the closed-field photospheric regions.

This mapping maps one footpoint, $x = (x ,y ) + + +$ in a positive photospheric region to a point $X - = (X - ,Y-)$ in a negative portion. ( These are assumed to be photospheric footpoints, so they are given by $x$ and $y$ coordinates in the $z = 0$ plane. ) Topological coronal field models generally concern the footpoint mapping, meaning the entire function $X - (x+)$ , or its inverse mapping from negative to positive regions, $X+(x - )$ .

The photospheric mapping can be found from the coronal field $B$ , however, the inverse is not possible; one cannot deduce the coronal field based on its footpoint mapping alone. Such a problem can have no unique solution since coronal field lines can be deformed in many ways without moving their footpoints. This hypothetical process for physically exploring multiple magnetic configurations also suggests a method for finding an equilibrium coronal field corresponding to a given mapping. The method was proposed by Arnold (1974 ) (see also Moffatt, 1985 ) for un-anchored fields and adopted by various authors in the present coronal context (Parker, 1987 ; Antiochos, 1987 ; van Ballegooijen, 1988 ). Beginning with some coronal field consistent with the mapping, allow the coronal plasma to undergo dissipative dynamics, such as with viscosity, without pressure. During this relaxation the magnetic field must evolve according to ideal induction and line-tying $v = 0$ at $z = 0$ , so that the field line mapping is preserved. When the system ceases to evolve it will be in a force-free equilibrium consistent with the original footpoint mapping.

A numerical implementation of this scheme, called the magneto-frictional method (Yang et al., 1986 ; Craig and Sneyd, 1990 ), uses friction rather than viscosity since it provides simpler dynamics which are still dissipative. Such schemes may be designed to preserve the frozen field lines exactly, but still suffer from limitations due to representing the continuous mapping with a finite number of variables. It is still difficult to know if a given field line mapping might admit more than one equilibrium coronal field. The existence of equilibria subject to ideal instabilities do, however, imply that some mappings admit multiple equilibria.

An even more subtle problem, proposed by Parker (1972 ) and subsequently dubbed “The Parker Problem”, concerns when continuous mappings admit discontinuous coronal magnetic fields. If the vector field $B(x)$ is discontinuous across some TD, it follows from Equation (1 ) that the field lines will be discontinuous across that surface. This fact was used by several authors (van Ballegooijen, 1990 ; Longcope and Strauss, 1994 ) to argue that a continuous footpoint mapping admits only continuous coronal equilibria. While it is clear that a tangential discontinuity introduces a discontinuity into the incomplete field line mapping - the mapping to some surface in the corona - it has been argued that it is possible for this discontinuity to be “mended” into a continuous mapping at the opposite photosphere (Parker, 1990 ). After all it is possible to tear a curtain without affecting the top or bottom hems. Indeed, some examples have been found of discontinuous equilibria with continuous footpoint mappings (Parker, 1994 ). The present focus of this ongoing line of work concerns whether such equilibria are special cases, rather common, or almost ubiquitous, within the hypothetical space of “typical photospheric mappings” (Parker, 2004 ).