The requirement of vanishing electric current density in the PFSS model means that , so that

where is a scalar potential. The solenoidal constraint then implies that satisfies Laplace’s equation with Neumann boundary conditions Thus, the problem reduces to solving a single scalar PDE. It may readily be shown that, for fixed boundary conditions, the potential field is unique and, moreover, that it is the magnetic field with the lowest energy () for these boundary conditions.Solutions of Laplace’s equation in spherical coordinates are well known (Jackson, 1962). Separation of variables leads to

where the boundary condition at implies that for each pair , , and are the associated Legendre polynomials. The coefficients are fixed by the photospheric distribution to be where are the spherical harmonic coefficients of this distribution. The solution for the three field components may then be written as where In practice, only a finite number of harmonics are included in the construction of the field, depending on the resolution of the input photospheric distribution. Notice that the higher the mode number , the faster the mode falls off with radial distance. This implies that the magnetic field at larger is dominated by the low order harmonics. So calculations of the Sun’s open flux (Section 4.1) with the PFSS model do not require large . An example PFSS extrapolation is shown in Figure 10a.The PFSS model has a single free parameter: the radius of the source surface. Various studies have chosen this parameter by optimising agreement with observations of either white-light coronal images, X-ray coronal hole boundaries, or, by extrapolating with in situ observations of the interplanetary magnetic field (IMF). A value of is commonly used, following Hoeksema et al. (1983), but values as low as have been suggested (Levine et al., 1977). Moreover, it appears that even with a single criterion, the optimal can vary over time as magnetic activity varies (Lee et al., 2011).

A significant limitation of the PFSS model is that the actual coronal magnetic field does not become purely radial within the radius where electric currents may safely be neglected. This is clearly seen in eclipse observations (e.g., Figure 6a), as illustrated by Schatten (1971), and in the early MHD solution of Pneuman and Kopp (1971). Typically, real polar plumes bend more equatorward than those in the PFSS model, while streamers should bend more equatorward at Solar Minimum and more poleward at Solar Maximum.

Schatten (1971) showed that the PFSS solution could be improved by replacing the source surface boundary with an intermediate boundary at , and introducing electric currents in the region . To avoid too strong a Lorentz force, these currents must be limited to regions of weak field, namely to sheets between regions of and . These current sheets support a more realistic non-potential magnetic field in . The computational procedure is as follows:

- Calculate in the inner region from the observed photospheric , assuming an “exterior” solution (vanishing as ).
- Re-orientate to ensure that everywhere (this will temporarily violate on the surface ).
- Compute in the outer region using the exterior potential field solution, but matching all three components of to those of the re-orientated inner solution on . (These boundary conditions at are actually over-determined, so in practice the difference is minimised with a least-squares optimisation).
- Restore the original orientation. This creates current sheets where in the outer region, but the magnetic stresses will balance across them (because changing the sign of all three components of leaves the Maxwell stress tensor unchanged).

This model generates better field structures, particularly at larger radii (Figure 6c), leading to its use in solar wind/space weather forecasting models such as the Wang–Sheeley–Arge (WSA) model (Arge and Pizzo, 2000, available at http://ccmc.gsfc.nasa.gov/). The same field-reversal technique is used in the MHS-based Current-Sheet Source Surface model (Section 3.3), and was used also by Yeh and Pneuman (1977), who iterated to find a more realistic force-balance with plasma pressure and a steady solar wind flow.

By finding the surface where in their MHD model (Section 3.4), Riley et al. (2006) locate the true “source surface” at , comparable to the Alfvén critical point where the kinetic energy density of the solar wind first exceeds the energy density of the magnetic field (Zhao and Hoeksema, 2010). The Alfvén critical points are known not to be spherically symmetric, and this is also found in the Riley et al. (2006) model. In fact, a modified PFSS model allowing for a non-spherical source surface (though still at ) was proposed by Schulz et al. (1978). For this case, before solving with the source surface boundary, the surface shape is first chosen as a surface of constant for the unbounded potential field solution.

Living Rev. Solar Phys. 9, (2012), 6
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