1 Introduction

The magnetic activity of the Sun has a high impact on Earth. As illustrated in Figure 1View Image, large coronal eruptions like flares and coronal mass ejections can influence the Earth’s magnetosphere where they trigger magnetic storms and cause aurorae. These coronal eruptions have also harmful effects like disturbances in communication systems, damages on satellites, power cutoffs, and unshielded astronauts are in danger of life-threatening radiation.1 The origin of these eruptive phenomena in the solar corona is related to the coronal magnetic field as magnetic forces dominate over other forces (like pressure gradient and gravity) in the corona. The magnetic field, created by the solar dynamo, couples the solar interior with the Sun’s surface and atmosphere. Reliable high accuracy magnetic field measurements are only available in the photosphere. These measurements, called vector magnetograms, provide the magnetic field vector in the photosphere.

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Figure 1: Magnetic forces play a key role in solar storms that can impact Earth’s magnetic shield (magnetosphere) and create colorful aurora. Image courtesy of External LinkSOHO (ESA & NASA).

To get insights regarding the structure of the coronal magnetic field we have to compute 3D magnetic field models, which use the measured photospheric magnetic field as the boundary condition. This procedure is often called “extrapolation of the coronal magnetic field from the photosphere”. In the solar corona the thermal conductivity is much higher parallel than perpendicular to the field so that field lines may become visible by the emission at appropriate temperatures. This makes in some sense magnetic field lines visible and allows us to test coronal magnetic field models. In such tests 2D projection of the computed 3D magnetic field lines are compared with plasma loops seen in coronal images. This mainly qualitative comparison cannot guarantee that the computed coronal magnetic field model and derived quantities, like the magnetic energy, are accurate. Coronal magnetic field lines which are in reasonable agreement with coronal images are, however, more likely to reproduce the true nature of the coronal magnetic field.

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Figure 2: Plasma β model over active regions. The shaded area corresponds to magnetic fields originating from a sunspot region with 2500 G and a plage region with 150 G. The left and right boundaries of the shaded area are related to umbra and plage magnetic field models, respectively. Atmospheric regions magnetically connected to high magnetic field strength areas in the photosphere naturally have a lower plasma β. Image reproduced by permission from Figure 3 of Gary (2001Jump To The Next Citation Point), copyright by Springer.

To model the coronal magnetic field B we have to introduce some assumptions. It is therefore necessary to get some a priori insights regarding the physics of the solar corona. An important quantity is the plasma β value, a dimensionless number which is defined as the ratio between the plasma pressure p and the magnetic pressure,

-p- β = 2μ0B2 . (1 )
Figure 2View Image from Gary (2001Jump To The Next Citation Point) shows how the plasma β value changes with height in the solar atmosphere. As one can see a region with β ≪ 1 is sandwiched between the photosphere and the upper corona, where β is about unity or larger. In regions with β ≪ 1 the magnetic pressure dominates over the plasma pressure (and as well over other non-magnetic forces like gravity and the kinematic plasma flow pressure). Here we can neglect in the lowest order all non-magnetic forces and assume that the Lorentz force vanishes. This approach is called the force-free field approximation and for static configurations it is defined as:
j × B = 0, (2 ) 1 j = --∇ × B is the electric current density, (3 ) μ0 ∇ ⋅ B = 0, (4 )
or by inserting Equation (3View Equation) into (2View Equation):
(∇ × B ) × B = 0, (5 ) ∇ ⋅ B = 0. (6 )
Equation (5View Equation) can be fulfilled either by:
∇ × B = 0 current-free or potential magnetic fields (7 )
or by
B ∥ ∇ × B force- free fields. (8 )

Current free (potential) fields are the simplest assumption for the coronal magnetic field. The line-of-sight (LOS) photospheric magnetic field which is routinely measured with magnetographs are used as boundary conditions to solve the Laplace equation for the scalar potential ϕ,

Δ ϕ = 0, (9 )
where the Laplacian operator Δ is the divergence of the gradient of the scalar field and
B = − ∇ ϕ. (10 )
When one deals with magnetic fields of a global scale, one usually assumes the so-called “source surface” (at about 2.5 solar radii where all field lines become radial): see, e.g., Schatten et al. (1969) for details on the potential-field source-surface (PFSS) model. Figure 3View Image shows such a potential-field source-surface model for May 2001 from Wiegelmann and Solanki (2004Jump To The Next Citation Point).

Potential fields are popular due to their mathematical simplicity and provide a first coarse view of the magnetic structure in the solar corona. They cannot, however, be used to model the magnetic field in active regions precisely, because they do not contain free magnetic energy to drive eruptions. Further, the transverse photospheric magnetic field computed from the potential-field assumption usually does not agree with measurements and the resulting potential field lines do deviate from coronal loop observations. For example, a comparison of global potential fields with TRACE images by Schrijver et al. (2005) and with stereoscopically-reconstructed loops by Sandman et al. (2009) showed large deviations between potential magnetic field lines and coronal loops.

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Figure 3: Global potential field reconstruction. Image reproduced by permission from Wiegelmann and Solanki (2004), copyright by ESA.

The B ∥ ∇ × B condition can be rewritten as

∇ × B = αB, (11 ) B ⋅ ∇ α = 0, (12 )
where α is called the force-free parameter or force-free function. From the horizontal photospheric magnetic field components (B ,B ) x0 y0 we can compute the vertical electric current density
μ j = ∂By0-− ∂Bx0- (13 ) 0 z0 ∂x ∂y
and the corresponding distribution of the force-free function α(x,y ) in the photosphere
jz0- α(x,y ) = μ0 B . (14 ) z0
Condition (12View Equation) has been derived by taking the divergence of Equation (11View Equation) and using the solenoidal condition (4View Equation). Mathematically, Equations (11View Equation) and (12View Equation) are equivalent to Equations (2View Equation) – (4View Equation). Parameter α can be a function of position, but Equation (12View Equation) requires that α be constant along a field line. If α is constant everywhere in the volume under consideration, the field is called linear force-free field (LFFF), otherwise it is nonlinear force-free field (NLFFF). Equations (11View Equation) and (12View Equation) constitute partial differential equations of mixed elliptic and hyperbolic type. They can be solved as a well-posed boundary value problem by prescribing the vertical magnetic field and for one polarity the distribution of α at the boundaries. As shown by Bineau (1972Jump To The Next Citation Point) these boundary conditions ensure the existence and unique NLFFF solutions at least for small values of α and weak nonlinearities. Boulmezaoud and Amari (2000) proved the existence of solutions for a simply and multiply connected domain. As pointed out by Aly and Amari (2007Jump To The Next Citation Point) these boundary conditions disregard part of the observed photospheric vector field: In one polarity only the curl of the horizontal field (Equation (13View Equation)) is used as the boundary condition, and the horizontal field of the other polarity is not used at all. For a general introduction to complex boundary value problems with elliptic and hyperbolic equations we refer to Kaiser (2000).

Please note that high plasma β configurations are not necessarily a contradiction to the force-free condition (see Neukirch, 2005, for details). If the plasma pressure is constant or the pressure gradient is compensated by the gravity force (∇p = − ρ∇ Ψ, where ρ is the mass density and Ψ the gravity potential of the Sun) a high-β configuration can still be consistent with a vanishing Lorentz force of the magnetic field. In this sense a low plasma β value is a sufficient, but not a necessary criterion for the force-free assumption. In the generic case, however, high plasma β configurations will not be force-free and the approach of the force-free field is limited to the upper chromosphere and the corona (up to about 2.5R ⊙).


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