The solutions that have been used to extrapolate photospheric data have a current density of the form

where and , are constant parameters. Thus, the current comprises a field-aligned part and a part perpendicular to gravity. For this form of , Neukirch (1995) shows that the solution takes the form where Here, and are Bessel functions of the first and second kinds, respectively. The coefficients are determined from the boundary conditions, as in the PFSS model. The plasma pressure and density then take the forms The functions and describe a spherically symmetric background atmosphere satisfying , and may be freely chosen. Solutions of this form have been applied to the coronal extrapolation problem by Zhao et al. (2000), Rudenko (2001), and
Ruan et al. (2008).^{6}
These solutions are of exterior type, i.e., with as . For , the latter condition
follows from the properties of both types of Bessel function.

There are two free parameters in the solution that may be varied to best fit observations: and . Broadly speaking, the effect of increasing is to inflate/expand the magnetic field, while the effect of increasing is to twist/shear the magnetic field. This is illustrated in Figure 11. Note that too large a value of would lead to zeros of the Bessel functions falling within the computational domain, creating magnetic islands that are unphysical for the solar corona (Neukirch, 1995). The (unbounded) potential field solution is recovered for , while if then and we recover the linear force-free field case. If then the current is purely horizontal and the solution reduces to Case III of Bogdan and Low (1986). Gibson and Bagenal (1995) applied this earlier solution to the Solar Minimum corona, although they showed that it was not possible to match both the density distribution in the corona and the photospheric magnetic field to observations. This problem is likely to be exacerbated at Solar Maximum. Similarly, Ruan et al. (2008) found that the strongest density perturbation in this model appears in active regions in the low corona. Preventing the density from becoming negative can require an unrealistically large background density at these radii, particularly for large values of the parameter .

Another limitation of this magnetic field solution is that the magnetic energy is unbounded, since the Bessel functions decay too slowly as . But, as pointed out by Zhao et al. (2000), the model is applicable only up to the cusp points of streamers, above which the solar wind outflow must be taken into account. So this problem is irrelevant in practical applications. For the solution of Bogdan and Low (1986), Zhao and Hoeksema (1994) showed how the model can be extended to larger radii by adding the “current sheet” extension of Schatten (1971) (described in Section 3.1). Instead of using an “exterior” solution in the external region they introduce an outer source surface boundary at , corresponding to the Alfvén critical point. The resulting model better matches the shape of coronal structures and the observed IMF (Zhao and Hoeksema, 1995) and is often termed the Current Sheet Source Surface (CSSS) model, though the authors term it HCCSSS: “Horizontal Current Current Sheet Source Surface”, to explicitly distinguish it from the Schatten (1971) model using potential fields. This CSSS model has also been applied by Schüssler and Baumann (2006) to model the Sun’s open magnetic flux (Section 4.1).

Finally, we note that Wiegelmann et al. (2007) have extended the numerical optimisation method (Section 3.2) to magnetohydrostatic equilibria, demonstrating that it reproduces the analytical solution of Neukirch (1995). This numerical technique offers the possibility of more realistic pressure and density profiles compared to the analytical solutions, although there is the problem that boundary conditions on these quantities must be specified.

Living Rev. Solar Phys. 9, (2012), 6
http://www.livingreviews.org/lrsp-2012-6 |
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