3.2 Nonlinear force-free field models

While potential field solutions are straightforward to compute, the assumption of vanishing electric current density (j = 0) in the volume renders them unable to model magnetic structures requiring non-zero electric currents. Observations reveal such structures both in newly-emerged active regions, e.g., X-ray sigmoids, and outside active latitudes, including long-lived Hα filament channels and coronal magnetic flux ropes. A significant limitation of the potential field is that it has the lowest energy compatible with given boundary conditions. Yet many important coronal phenomena derive their energy from that stored in the magnetic field; this includes large-scale eruptive events (flares and CMEs), but also small-scale dynamics thought to be responsible for heating the corona. We still lack a detailed understanding of how these events are initiated. Such an understanding cannot be gained from potential field models, owing to the lack of free magnetic energy available for release. The models in this section, based on the force-free assumption, allow for electric currents and, hence, free magnetic energy.

In the low corona of a star like the Sun, the magnetic pressure dominates over both the gas pressure and the kinetic energy density of plasma flows. Thus to first approximation an equilibrium magnetic field must have a vanishing Lorentz force,

j × B = 0, (14 )
where j = ∇ × B ∕μ0 is the current density. Such a magnetic field is called force-free. It follows that
∇ × B = αB (15 )
for some scalar function α (r) which is constant along magnetic field lines (this follows from ∇ ⋅ B = 0). The function α(r) depends only on the local geometry of field lines, not on the field strength; it quantifies the magnetic “twist” (Figure 7View Image). The potential field j = 0 is recovered if α = 0 everywhere.
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Figure 7: Twisted magnetic fields. Panels (a) and (b) show the direction of twist for a force-free field line with (a) positive α, and (b) negative α, with respect to the potential field α = 0. Panels (c) and (d) show X-ray sigmoid structures with each sign of twist, observed in active regions with the Hinode X-ray Telescope (SAO, NASA, JAXA, NAOJ). These images were taken on 2007 February 16 and 2007 February 5, respectively.

Unfortunately, for non-zero α, the extrapolation of coronal force-free magnetic fields is not straightforward. If α is constant everywhere, then taking the curl of (15View Equation) leads to the vector Helmholtz equation

2 2 (∇ + α )B = 0, (16 )
which is linear and may be solved analytically in spherical harmonics (Durrant, 1989Jump To The Next Citation Point). While some early attempts were made to model the global corona using such linear force-free fields (Nakagawa, 1973; Levine and Altschuler, 1974), their use is limited for two main reasons. Firstly, the constant α in such a solution scales as − 1 L, where L is the horizontal size of the area under consideration. So for global solutions, only rather small values of α, close to potential, can be used. Secondly, observations of twist in magnetic structures indicate that both the sign and magnitude of α ought to vary significantly between different regions on the Sun. In addition, a mathematical problem arises if one attempts to apply the same boundary conditions as in the potential field model (namely a given distribution of Br on r = R⊙, and B𝜃 = B ϕ = 0 on r = Rss). A strictly force-free field satisfying B 𝜃 = Bϕ = 0 and Br ⁄= 0 on an outer boundary must have α = 0 on that boundary (Aly and Seehafer, 1993), so that α = 0 on all open magnetic field lines. Unless α = 0, this is incompatible with the linear force-free field.5 From this, it is clear that the use of linear force-free fields is restrictive and will not be discussed further. Instead, we will consider nonlinear force-free fields where α is a function of position.

The problem of extrapolating nonlinear force-free fields from given photospheric data is mathematically challenging, with open questions about the existence and uniqueness of solutions. Nevertheless, several numerical techniques have been developed in recent years, though they have largely been applied to single active regions in Cartesian geometry (Schrijver et al., 2006; DeRosa et al., 2009). Applications to global solutions in spherical geometry are in their infancy: we describe here the three main approaches tried.

3.2.1 Optimisation method

Wiegelmann (2007Jump To The Next Citation Point) has developed a method for numerically computing nonlinear force-free fields in spherical geometry, based on the optimisation procedure of Wheatland et al. (2000Jump To The Next Citation Point) and using the vector magnetic field in the photosphere as input. The idea is to minimise the functional

∫ ( ) |(∇ × B ) × B |2 2 L [B ] = --------2------+ |∇ ⋅ B | dV, (17 ) V B
since if L = 0 then j × B = 0 and ∇.B = 0 everywhere in the volume V. Differentiating (17View Equation) with respect to t, Wheatland et al. (2000) show that
1 dL ∫ ∂B ∮ ∂B -----= − ----⋅ FdV − ---⋅ GdS, (18 ) 2 dt V ∂t ∂V ∂t
F = ∇ × (Ω × B ) − Ω × (∇ × B ) − ∇ (Ω ⋅ B) + Ω (∇ ⋅ B ) + Ω2B, (19 ) G = n × (Ω × B) − n (Ω ⋅ B ), (20 ) 1 [ ] Ω = --2 (∇ × B ) × B − (∇ ⋅ B )B . (21 ) B
Evolving the magnetic field so that
∂B ----= μF (22 ) ∂t
for some μ > 0, and imposing ∂B ∕∂t = 0 on the boundary ∂V, will ensure that L decreases during the evolution. To apply this procedure, Wiegelmann (2007Jump To The Next Citation Point) takes an initial potential field extrapolation, and replaces B𝜃 and Bϕ on r = R⊙ with the measured horizontal magnetic field from an observed vector magnetogram. The 3D field is then iterated with Equation (22View Equation) until L is suitably small.

Wiegelmann (2007Jump To The Next Citation Point) demonstrates that this method recovers a known analytical force-free field solution (Figure 8View Image), although the most accurate solution is obtained only if the boundary conditions at both the photosphere and upper source surface are matched to the analytical field. The method is yet to be applied to observational data on a global scale, largely due to the limitation that vector magnetogram data are required for the lower boundary input. Such data are not yet reliably measured on a global scale, though SDO or SOLIS should be a key step forward in obtaining this data. Another consequence is that even where they are measured there is the complication that the photospheric magnetic field is not force-free. Pre-processing techniques to mitigate these problems are under development (Tadesse et al., 2011).

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Figure 8: Test of the optimisation method for nonlinear force-free fields with an analytical solution (figure adapted from Wiegelmann, 2007). Panel (a) shows the analytical solution (Low and Lou, 1990) and (b) shows the PFSS extrapolation used to initialise the computation. Panel (c) shows the resulting NLFFF when the boundary conditions at both r = R ⊙ and r = Rss are set to match the analytical solution, while panel (d) shows how the agreement is poorer if the analytical boundary conditions are applied only at the photosphere (the more realistic case for solar application).

3.2.2 Force-free electrodynamics method

Contopoulos et al. (2011Jump To The Next Citation Point) have recently proposed an alternative method to compute global nonlinear force-free field extrapolations, adapted from a technique applied to pulsar magnetospheres. It has the advantage that only the radial component Br is required as observational input at the photosphere. The computation is initialised with an arbitrary 3D magnetic field such as B = 0 or that of a dipole, and also with zero electric field E. The fields E and B are then evolved through the equations of force-free electrodynamics,

∂B ----= − c∇ × E, (23 ) ∂t ∂E- = c∇ × B − 4πj, (24 ) ∂t ∇ ⋅ B = 0, (25 ) 1 ρeE + -j × B = 0, (26 ) c
where ρe = ∇ ⋅ E ∕(4π) is the electric charge density. The force-free condition (26View Equation) enables j to be eliminated from (24View Equation) since it follows that
( ) -c- E-×--B- -c- B-⋅ ∇-×-B-−--E-⋅ ∇-×-E- j = 4π ∇ ⋅ E B2 + 4π B2 B. (27 )
Equations (23View Equation) and (24View Equation) are integrated numerically with time-dependant E 𝜃, E ϕ imposed on the lower boundary r = R ⊙. These are chosen so that B r gradually evolves toward its required distribution. The photospheric driving injects electrodynamic waves into the corona, establishing a network of coronal electric currents. The outer boundary is chosen to be non-reflecting and perfectly absorbing, mimicking empty space. As the photosphere approaches the required distribution, the charge density and electric fields diminish, but the coronal currents remain. As E → 0 a force-free field is reached.

The authors have tested their method with an observed synoptic magnetogram: an example force-free field produced is shown in Figure 9View Image. They find that active region magnetic fields are reproduced quite rapidly, but convergence to the weaker fields in polar regions is much slower. Slow convergence is undesirable because numerical diffusion was found to erode the coronal currents before the photosphere reached the target configuration. However, the convergence rate could be improved by choosing the initial condition to be a dipole, approximating the average polar field in the magnetogram. This highlights an important feature of the method: non-uniqueness. The force-free field produced is not defined solely by the photospheric boundary condition, but depends both on (i) the choice of initialisation and (ii) the path followed to reach the final state. Contopoulos et al. (2011Jump To The Next Citation Point) suggest that one way to choose between possible solutions would be to incorporate measurements from vector magnetograms. The model described in the next section takes a different approach in treating the construction of the coronal magnetic field as an explicitly time-dependent problem.

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Figure 9: A force-free magnetic field produced by the force-free electrodynamics method for Carrington rotation 2009. Field line colours show the twist parameter α, coloured red where α > 0, blue where α < 0, and light green where α ≈ 0. Image reproduced by permission from Contopoulos et al. (2011), copyright by Springer.

3.2.3 Flux transport and magneto-frictional method

Recently, van Ballegooijen et al. (2000Jump To The Next Citation Point) and Mackay and van Ballegooijen (2006Jump To The Next Citation Point) have developed a new technique to study the long-term evolution of coronal magnetic fields, which has now been applied to model the global solar corona (Yeates et al., 2007Jump To The Next Citation Point, 2008bJump To The Next Citation Point). The technique follows the build-up of free magnetic energy and electric currents in the corona by coupling together two distinct models. The first is a data driven surface flux transport model (Yeates et al., 2007Jump To The Next Citation Point). This uses observations of newly emerging magnetic bipoles to produce a continuous evolution of the observed photospheric magnetic flux over long periods of time. Coupled to this is a quasi-static coronal evolution model (Mackay and van Ballegooijen, 2006Jump To The Next Citation Point; Yeates et al., 2008bJump To The Next Citation Point) which evolves the coronal magnetic field through a sequence of nonlinear force-free fields in response to the observed photospheric evolution and flux emergence. The model follows the long-term continuous build-up of free magnetic energy and electric currents in the corona. It differs significantly from the extrapolation approaches which retain no memory of magnetic flux or connectivity from one extrapolation to the next.

The photospheric component of the model evolves the radial magnetic field Br on r = R⊙ with a standard flux transport model (Section 2.2.1), except that newly emerging bipolar active regions are inserted not just in the photosphere but also in the 3D corona. These regions take an analytical form (Yeates et al., 2007Jump To The Next Citation Point), with parameters (location, size, flux, tilt angle) chosen to match observed active regions. An additional twist parameter allows the emerging 3D regions to be given a non-zero helicity: in principle this could be determined from vector magnetogram observations, but these are not yet routinely available.

The coronal part of the model evolves the large-scale mean field (van Ballegooijen et al., 2000Jump To The Next Citation Point) according to the induction equation

∂A ----= v × B + ℰ, (28 ) ∂t
where B = ∇ × A is the mean magnetic field (with the vector potential A in an appropriate gauge). The mean electromotive force ℰ describes the net effect of unresolved small-scale fluctuations, for example, braiding and current sheets produced by interaction with convective flows in the photosphere. Mackay and van Ballegooijen (2006Jump To The Next Citation Point) and Yeates et al. (2008bJump To The Next Citation Point) assume the form
ℰ = − ηj, (29 )
( ) |j| η = η0 1 + 0.2 B (30 )
is an effective turbulent diffusivity. The first term is a uniform background value 2 −1 η0 = 45 km s and the second term is an enhancement in regions of strong current density, introduced to limit the twist in helical flux ropes to about one turn, as observed in solar filaments.

Rather than solving the full MHD equations for the velocity v, which is not computationally feasible over long timescales, the quasi-static evolution of the coronal magnetic field is approximated using magneto-frictional relaxation (Yang et al., 1986), setting

1-j ×-B- v = ν B2 . (31 )
This artificial velocity causes the magnetic field to relax towards a force-free configuration: it may be shown that the total magnetic energy decreases monotonically until j × B = 0. Here, the magneto-frictional relaxation is applied concurrently with the photospheric driving so, in practice, a dynamical equilibrium between the two is reached. At the outer boundary r = 2.5R ⊙, Mackay and van Ballegooijen (2006Jump To The Next Citation Point) introduced an imposed radial outflow, rather than setting B 𝜃 = B ϕ = 0 exactly. This allows magnetic flux ropes to be ejected after they lose equilibrium (simulating CMEs on the real Sun), and simulates the effect of the solar wind in opening up magnetic field lines radially above this height.
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Figure 10: (a) Example of a Potential Field Source Surface extrapolation. (b) Example of a non-potential coronal field produced by the global coronal evolution model of Mackay and van Ballegooijen (2006) and Yeates et al. (2008bJump To The Next Citation Point). The grey-scale image shows the radial field at the photosphere and the thin lines the coronal field lines. Image reproduced by permission from Yeates et al. (2010bJump To The Next Citation Point), copyright by AGU.

Figure 10View Imageb shows an example nonlinear force-free field using this model, seen after 100 days of evolution. Figure 10View Imagea shows a PFSS extrapolation from the same photospheric Br distribution. The coronal field in the non-potential model is significantly different, comprising highly twisted flux ropes, slightly sheared coronal arcades, and near potential open field lines. Since this model can be run for extended periods without resetting the coronal field, it can be used to study long-term helicity transport across the solar surface from low to high latitudes (Yeates et al., 2008aJump To The Next Citation Point).

There are several parameters in the model. As in the PFSS model the location of the upper boundary R ss is arbitrary, although it has less influence since the non-potential magnetic field strength falls off more slowly with radius than that of the potential field. In the magneto-frictional evolution, the turbulent diffusivity η0 and friction coefficient ν are arbitrary, and must be calibrated by comparison with observed structures or timescales for flux ropes to form or lose equilibrium. Finally a 3D model is required for newly-emerging active regions: the simple analytical bipoles of the existing simulations could in fact be replaced with more detailed extrapolations from observed photospheric fields in active regions.

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