3.1 Potential field source surface models

The most straightforward and therefore most commonly used technique for modeling the global coronal magnetic field is the so-called Potential Field Source Surface (PFSS) model (Schatten et al., 1969; Altschuler and Newkirk Jr, 1969Jump To The Next Citation Point). We note that an implementation of the PFSS model by Luhmann et al. (2002Jump To The Next Citation Point), using photospheric magnetogram data from Wilcox Solar Observatory, is available to run “on demand” at External Linkhttp://ccmc.gsfc.nasa.gov/. The key assumption of this model is that there is zero electric current in the corona. The magnetic field is computed on a domain R βŠ™ ≤ r ≤ Rss, between the photosphere, R βŠ™, where the radial magnetic field distribution4 is specified and an outer “source surface”, Rss, where the boundary conditions B πœƒ = B Ο• = 0 are applied. The idea of the source surface is to model the effect of the solar wind outflow, which distorts the magnetic field away from a current-free configuration above approximately 2R βŠ™ (once the magnetic field strength has fallen off sufficiently). Distortion away from a current-free field implies the presence of electric currents at larger radii: a self-consistent treatment of the solar wind requires full MHD description (Section 4.4), but an artificial “source surface” boundary approximates the effect on the magnetic field in the lower corona. The PFSS model has been used to study a wide variety of phenomena ranging from open flux (see Section 4.1), coronal holes (Section 4.2), and coronal null points (Cook et al., 2009) to magnetic fields in the coronae of other stars (Jardine et al., 2002a).

The requirement of vanishing electric current density in the PFSS model means that ∇ × B = 0, so that

B = − ∇ Ψ, (4 )
where Ψ is a scalar potential. The solenoidal constraint ∇ ⋅ B = 0 then implies that Ψ satisfies Laplace’s equation
2 ∇ Ψ = 0, (5 )
with Neumann boundary conditions
|| || || ∂Ψ-| = − Br (R βŠ™,πœƒ,Ο•), ∂Ψ-| = ∂Ψ-| = 0. (6 ) ∂r |r=RβŠ™ ∂ πœƒ|r=Rss ∂ Ο•|r=Rss
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 (∫ V B2 βˆ•(2μ0)dV) for these boundary conditions.

Solutions of Laplace’s equation in spherical coordinates are well known (Jackson, 1962). Separation of variables leads to

∑∞ ∑l [ ] Ψ (r,πœƒ, Ο•) = flmrl + glmr −(l+1) Pml (cosπœƒ)eimΟ•, (7 ) l=0m= −l
where the boundary condition at Rss implies that 2l+1 glm = − flmR ss for each pair l, m, and m P l (cos πœƒ) are the associated Legendre polynomials. The coefficients flm are fixed by the photospheric Br (RβŠ™, πœƒ,Ο•) distribution to be
( l− 1 − l−2 2l+1) −1 flm = − lR βŠ™ + (l + 1)RβŠ™ R ss blm, (8 )
where blm are the spherical harmonic coefficients of this Br (R βŠ™,πœƒ,Ο•) distribution. The solution for the three field components may then be written as
∑∞ ∑l Br(r,πœƒ,Ο•) = blmcl(r)P ml (cos πœƒ)eim Ο•, (9 ) l=0 m= −l ∞ l ∑ ∑ dPlm(cosπœƒ-) im Ο• Bπœƒ(r,πœƒ,Ο•) = − blmdl (r) dπœƒ e , (10 ) l=0 m= −l ∑∞ ∑l im B Ο•(r,πœƒ,Ο•) = − -----blmdl(r)P ml (cos πœƒ)eim Ο•, (11 ) l=0 m= −lsinπœƒ
where
( )− l−2 [ 2l+1 ] c (r) = -r-- -l +-1 +-l(rβˆ•Rss)---- , (12 ) l R βŠ™ l + 1 + l(R βŠ™βˆ•Rss)2l+1 ( )− l−2 [ 2l+1 ] dl(r) = -r-- ---1-−-(rβˆ•Rss)------- . (13 ) R βŠ™ l + 1 + l(R βŠ™βˆ•Rss)2l+1
In practice, only a finite number of harmonics l = 1, ...,Lmax are included in the construction of the field, depending on the resolution of the input photospheric Br distribution. Notice that the higher the mode number l, the faster the mode falls off with radial distance. This implies that the magnetic field at larger r 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 L max. An example PFSS extrapolation is shown in Figure 10View Imagea.
View Image

Figure 6: Comparison between (a) a drawing of the real eclipse corona, (b) the standard PFSS model, and (c) the Schatten (1971Jump To The Next Citation Point) current sheet model. The current sheet model better reproduces the shapes of polar plumes and of streamer axes. Image adapted from Figure 1 of Zhao and Hoeksema (1994Jump To The Next Citation Point).

The PFSS model has a single free parameter: the radius Rss 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 Rss = 2.5R βŠ™ is commonly used, following Hoeksema et al. (1983), but values as low as Rss = 1.3R βŠ™ have been suggested (Levine et al., 1977). Moreover, it appears that even with a single criterion, the optimal Rss 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 6View Imagea), as illustrated by Schatten (1971Jump To The Next Citation Point), 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 (1971Jump To The Next Citation Point) showed that the PFSS solution could be improved by replacing the source surface boundary Rss with an intermediate boundary at Rcp = 1.6R βŠ™, and introducing electric currents in the region r > Rcp. To avoid too strong a Lorentz force, these currents must be limited to regions of weak field, namely to sheets between regions of Br > 0 and Br < 0. These current sheets support a more realistic non-potential magnetic field in r > Rcp. The computational procedure is as follows:

  1. Calculate B in the inner region r ≤ Rcp from the observed photospheric Br, assuming an “exterior” solution (vanishing as r → ∞).
  2. Re-orientate B (Rcp, πœƒ,Ο•) to ensure that Br > 0 everywhere (this will temporarily violate ∇ ⋅ B = 0 on the surface r = Rcp).
  3. Compute B in the outer region r > Rcp using the exterior potential field solution, but matching all three components of B to those of the re-orientated inner solution on Rcp. (These boundary conditions at Rcp are actually over-determined, so in practice the difference is minimised with a least-squares optimisation).
  4. Restore the original orientation. This creates current sheets where Br = 0 in the outer region, but the magnetic stresses will balance across them (because changing the sign of all three components of B leaves the Maxwell stress tensor unchanged).

This model generates better field structures, particularly at larger radii (Figure 6View Imagec), 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 External Linkhttp://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 Br βˆ•|B| = 0.97 in their MHD model (Section 3.4), Riley et al. (2006Jump To The Next Citation Point) locate the true “source surface” at r ≈ 13R βŠ™, 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. (2006Jump To The Next Citation Point) model. In fact, a modified PFSS model allowing for a non-spherical source surface (though still at 2 –3R βŠ™) 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 |B | for the unbounded potential field solution.


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