Magnetic Field Extrapolations And Current Sheets

B. T. Welsch,1 I. De Moortel,2 and J. M. McTiernan1

1Space Sciences Lab, UC Berkeley

2School of Mathematics & Statistics,

University of St. Andrews, Scotland

AbstractSolar flares and coronal mass ejections (CMEs) --- phenomena which impact our society, but are scientifically interesting in themselves --- are driven by free magnetic energy in the coronal magnetic field.

Since the coronal magnetic field cannot be directly measured, modelers often extrapolate the coronal field from the photospheric magnetograms --- the only field measurements routinely available.

The best extrapolation techniques assume that the field is force free (coronal currents parallel the magnetic field), but that currents are not simply a linear function of the magnetic field.

Recent tests (Metcalf et al. 2007), however, suggest that such non-linear force-free field (NLFFF) extrapolation techniques underestimate free magnetic energy.

We hypothesize that, since relaxation-based NLFFF techniques tend to smooth field discontinuities, such approaches will fail when current sheets are present.

Here, we test this hypothesis by applying the Optimization NLFFF method (Wheatland et al. 2000) to two configurations from an MHD simulation --- one with strong current concentrations, and one with weak concentrations.

This work is supported by a NASA Sun-Earth Connections Theory grant to SSL/UCB.

Free energy is the energy difference between

the actual and potential field energies. • For a given field B, the magnetic energy is

U dV (B · B)/8.

• The lowest energy the field could have would match the same boundary condition Bn, but would be current-free (curl-free), or “potential:”

B(P) = - , with 2 = 0, and U(P) dV |B(P) |2/8

• The difference U(F) = U – U (P) is the energy available to power flares and CMEs!

Non-linear force-free field (NLFFF) extrap-olations give B, allowing integration of B2/8.

• Strictly, NLFFF extrapolation should not be applied to non-force-free photospheric magnetograms.

• Wheatland et al. (2000) described the Optimization Method to determine a NLFFF, B(x,y,z), that matches a given magnetic boundary condition.

• McTiernan has implemented this extrapolation procedure in IDL, and distributed it via SSW, in the NLFFF package.

The Optimization Method minimizes an objective function, L:

Specification of B is required on all surfaces: for solar applications, magnetograms give B(x,y) at z=0, and B(P) is used on other boundaries.

L can be expressed as a functional of t B, providing a way to update B to minimize L.

and

The Optimization Method for NLFFF extrapolation has been tested in several ways.

• Wheatland et al. (2000) and Schrijver et al. (2006) used the analytic solution of Low & Lou (1990).

• Abbett et al. (2004) used MHD simulations from Magara et al. (2001).

• Metcalf et al. (2007) used a “hybrid” potential / non-potential reference model.

The Optimization method performed well with Magara’s (2004) MHD simulations in tests by Abbett et al. (2004).

Potential ActualNLFF

Chromosphere

Photosphere

Figure 1.

Tests by Metcalf et al. (2007), however, showed free energy is underestimated from photospheric magnetograms.

From Metcalf et al. 2007

Model E/Epot

OptimizationMethods

Why does the Optimization method fail at estimating free energy?

1. The non-force-free character of the boundary clearly plays a role. (Chromospheric extrapolations do better.)

2. But also: the Optimization method is a relaxation method, and might relax away current concentrations, as seen by Antiochos et al. 1999. (These currents store free energy.)

Antiochos et al. (1999) found that NLFF extrapolations missed current sheets in their “breakout” simulations, and therefore underestimated free energy.

We decided to test the Optimization method against MHD configurations with current sheets.

From De Moortel & Galsgaard (2006)

INITIAL STATE FINAL STATE

Figure 2.

We extrapolated two MHD states, Figure 2’s final state….

Current density in pink.

Figure 3.

…and an intermediate state.

Figure 4.

Current density in pink; density is lower than final state shown in Figure 3.

The NLFF extrapolations were initiated from potential fields.

Figure 5. Initial conditions for attempted extrapolation of field in Fig. 3 (left) and Fig. 4 (right).

The NLFF extrapolations did not accurately reproduce the topology of the MHD configurations.

Figure 6. NLFF extrapolations for configurations in Figure 3. (left) and 4 (right).

Conclusions

1. It’s a good idea to start the research described in one’s SPD Abstract sooner than the week before the meeting. (CONCLUSIONS ARE PRELIMINARY!)

2. Considering 1., operator error (by Welsch!) is not unlikely.

3. When the initial topology is far from the actual topology, Optimization Method extrapolations fail.

Future Work

• We plan to include trial “smoothing penalty” terms in L, to prevent smoothing away current concentrations.

• We plan to try a weighting function, w(x,y,z) (Wheatland et al. 2000, Wiegelmann et al. 2004), to limit the effect of forces on boundaries, e.g.,

References

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