example when one bipole is twisted around another (Longcope and Magara, 2004;Bev- eridge and Longcope,2006). Barnes and Leka(2006) have also shown how the statistical properties of magnetic charge topologies based on magnetogram data can be used to pre- dict solar flares. It is expected that more implications about the role of helicity in the onset of flares and coronal mass ejections will be discovered in future.
The chapter is ordered as follows: the Green’s function method for calculating the mag- netic field is detailed in section 4.2, while the specifics of the model are given in sec- tion4.3, along with an explanation of how the analysis was carried out. The results of the modelling are split up into the “2+2” case (section4.4) and the “3+1” case (section4.6). Each set of results is broken down into a complete specification of the topological states and bifurcations present, along with some discussion of the structure of the bifurcation diagrams and the effects of varying model parameters. Sections 4.5 and 4.7 provide a comparison of my results to the previously studied case of an equivalent set of sources on a flat photosphere. Finally in section4.8I conclude by discussing the significance of the results and how they could be built upon in the future.
4.2
Green’s function method to construct the magnetic field
For simplicity, I assume that the field is potential, and generated by a finite set of point magnetic sources located on the solar photosphere. The photosphere is taken to be a spherical surface of radius a, with the corona represented by the volume outside the sphere. There is no source surface; the field strength simply falls off towards zero far from the photosphere. However, I wish the photosphere outside the point sources to be a flux surface, so that fieldlines pass through it only at the sources; any fieldlines below it will be ignored. Aside from the spherical photosphere, these are assumptions commonly used in magnetic topology studies (see for exampleLongcope and Klapper,2002;Brown and Priest, 1999b). A potential field is assumed, as this strikes an appropriate balance between ease of calculation of the field and the presence of similar topological features to those found in more complex general force-free fields. These features will occur for sim- ilar parameter values so long as the non-potentiality is not too severe (Brown and Priest,
2000).
A magnetic field B is potential if it can be written as B = −∇Φ, whereΦ is a scalar potential satisfying4Φ(r) = 0. According to classical potential theory (Jackson,1962),
Φ is uniquely determined if one of two boundary conditions holds: either the value of
4.2 Green’s function method to construct the magnetic field 77
derivative ofΦnormal to the boundary is given, which is called the Neumann problem. The problem I face here is a so-called exterior Neumann problem, since I wish to find a Green’s function for the volume outside the boundary (the solar photosphere, a sphere of radiusa), and the boundary condition is that
−n· ∇Φ = Bn(θ, ϕ) onr=a, (4.1)
which is a Neumann-type boundary condition, withna unit vector normal to the bound- ary. An additional requirement is that the potential must fall to zero asrtends to infinity, in order to keep the model physically realistic.
This means that the Green’s function,G(r,r0), must satisfy the following conditions:
4G(r,r0) = 0 onr > a, (4.2)
−n· ∇G(r,r0) = δ2(r,r0) onr=a, (4.3)
G(r,r0) = 0 asr→ ∞, (4.4)
where δ2 is the two-dimensional delta function, whose integral over the whole photo-
sphere would give1. Such a Green’s function is given in Sakurai(1982) and derived in
Nemenman and Silbergleit(1999) as
G(r,r0) = 1 4π " 2 |r−r0|−ln a2−r·r0+a|r−r0| ra−r·r0 # , (4.5)
wherer0 is the position of a magnetic source. This expression forG(r,r0)can be found by writing the Green’s function in terms of Legendre functions and summing the infinite series that results.
Physically, this Green’s function represents the magnetic field due to a point source on the surface of the sphere and a line source inside the sphere, extending from the origin to touch the surface at the position of the point source. Figure 4.1 shows a plot of the resulting magnetic field. The point source has a strength of +2 units, and integrating along the line source shows that it has a strength of−1units in total, as expected. Hence one unit of flux extends from the point source into the corona, and one unit stays inside the photosphere to balance the flux from the line source.
Φcan be found fromG(r,r0), simply by integrating over the boundary:
Φ(r) = Z
r0=a
4.2 Green’s function method to construct the magnetic field 78
Figure 4.1: Plot of magnetic field given by the Green’s function for a magnetic source atθ= 0.
which in the case of a set ofmpoint magnetic sources on the boundary, each with strength i, becomes Φ(r) = m X i=1 iG(r,r0i). (4.7)
The Green’s function itself behaves liker−1 at large distances, but, provided the sources
are flux balanced, the resulting potential will decay like r−2 (or faster) and so will be physically allowable. Flux balance is also necessary as the model is based on a sum of magnetic monopoles. These monopoles do not exist in nature, but when the flux from photospheric sources and sinks is balanced,∇ ·B= 0holds in the corona and the model can be considered to be physical. So the assumption of balanced sources that is used throughout this and following chapters is actually not an arbitrary restriction placed on the system, but a necessity.
Thus, starting with a set of balanced magnetic sources distributed over the photosphere, the potential magnetic field at any point in the photosphere or corona resulting from these sources can be calculated using the Green’s function method. I now move on to an overview of the specific model setup chosen, and how it was analysed.