4.3 Connectivity

MCT models characterize a field’s connectivity by quantifying the amount of flux interconnecting each pair of photospheric sources. This is not possible in either pointwise mapping models or submerged poles models. The domain graph of a field with D domains and S sources has D edges connecting S vertices (Longcope, 2001Jump To The Next Citation Point). This defines an incidence matrix Mar which is 1 when domain r connects to source a, and 0 otherwise. The domain matrix relates domain fluxes ψr to source fluxes Φa,
∑D Φa = Mar ψr. (18 ) r=1
These S − 1 relations15 leave only D − S + 1 domain fluxes undetermined (Longcope, 2001). Sweet’s configration, with D = 4 and S = 4, thus has only one domain flux not fixed by the sources. This one degree of freedom is set by the flux passing through its one separator, which can be varied through reconnection without changing any source fluxes. It can be shown in general that relation (18View Equation) augmented with the fluxes through all separators uniquely determines all domain fluxes (Longcope and Klapper, 2002Jump To The Next Citation Point). A direct consequence of this is that domain fluxes can only change from changing source fluxes, due to emergence or submergence, or from transferring field lines across separators, through separator reconnection.

There is not yet a method for enumerating the domains in a general MCT field, however, Longcope and Klapper (2002Jump To The Next Citation Point) present a method for enumeration in potential fields and those topologically equivalent to them, and Beveridge and Longcope (2005Jump To The Next Citation Point) generalize it. A field with X ′ separators, n nulls, nuf with unbroken fans, and S sources has D ′ = S + X ′ − n + nuf domains, including those in the mirror corona16. Assuming there are no upright nulls (which introduce such complications as fans and separators in the photosphere) we set the number of coronal separators to ′ X = X ∕2 and subtract off half of the domains not present in the footprint (the number of footprints is given by Equation [17View Equation]) to find the total number of domains D not counting those in the mirror corona. Designating by n c the number of coronal nulls gives (Beveridge and Longcope, 2005)

D = S + X − nc − 1, (19 )
once np has been eliminated using Equation (16View Equation). Furthermore, the number of purely coronal domains, those without footprints, is D = X − (n − n ) + 1 c uf, equal to the number of independent circuits formed by coronal separators (Longcope and Klapper, 2002Jump To The Next Citation Point). This is a consequence of the fact that each purely coronal domain must be engirdled by a unique circuit of separators.
  Go to previous page Go up Go to next page