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, 2001

). 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 $yr$ to source fluxes $Pa$ ,

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 (18

) augmented with the fluxes through all separators uniquely determines all domain fluxes (Longcope and Klapper, 2002

). 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 (2002

) present a method for enumeration in potential fields and those topologically equivalent to them, and Beveridge and Longcope (2005

) generalize it. A field with $X'$ separators, $n$ nulls, $nuf$ with unbroken fans, and $S$ sources has $D'= S + X' - n + n uf$ 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 [17

]) to find the total number of domains $D$ not counting those in the mirror corona. Designating by $nc$ the number of coronal nulls gives (Beveridge and Longcope, 2005 )

once $n p$ has been eliminated using Equation (16

). Furthermore, the number of purely coronal domains, those without footprints, is $Dc = X - (n - nuf) + 1$ , equal to the number of independent circuits formed by coronal separators (Longcope and Klapper, 2002

). This is a consequence of the fact that each purely coronal domain must be engirdled by a unique circuit of separators.