1. Extrapolating Coronal Magnetic Fields T. Metcalf

2. Force-Free Field Extrapolations Three “classes” of force-free field extrapolations (J=αB): –Potential (α=0, J=0), B is uniquely described by a scalar potential –Linear Force-Free (α constant) Helmholtz equation readily solved for small α. –Non-linear Force-Free (α non-constant) α is constant along field lines.

3. Potential & Linear Force-Free Extrapolations Potential and Linear Force-Free Extrapolations: –Fast. –Do not require the full vector field. But... –There is a limit on α, so highly stressed field cannot be dealt with properly. –Long, high field lines are most affected by α, this is not what is observed. Typically the stresses are confined to AR cores and the large scale field is more potential in character. –α is known to be far from constant in active regions. Solar ARs typically do not even have a uniform sign of α. –The potential or LFF field gives a lower bound on the energy in the magnetic field, so, obviously cannot be used to determine the free energy available to power solar activity. –Note: potential field is unique in principle, but is dependent on details of how the BCs are implemented.

4. Dependence on Implementation of BC's - I Potential field with no guard band E=1.29e28 ergs

5. Dependence on Implementation of BC's - II Potential field with guard band E=1.33e28 ergs

6. Dependence on Implementation of BC's - III Potential field with perfectly conducting walls. E=1.47e28 ergs

7. Non-Linear Force-Free Extrapolations Are LFF extrapolations useless? –No! –A potential field may well get the global connectivity correct if stresses are confined to the core of ARs. We need to use NLFF extrapolations for detailed analyses. But.... –The equations must be solved numerically at great expense. –NLFF algorithms are very sensitive to the treatment of the boundary conditions. Great care is required. E.g. redundent BC's imply discontinuities. –Is the lower boundary force-free? What if it is not? Need to extrapolate through the thin, but significant, forced layer. An algorithm that guarantees a force-free solution will have trouble.

8. To address the NLFF field extapolation problem, several workshops have been held to compare existing algorithms applied to problems with known solutions. –See Schrijver, DeRosa, Metcalf, Liu, McTiernan, Regnier, Valori, Wheatland, Wiegelmann, Solar Physics, 2006.

15. Conclusions from NLFFF Workshops Optimization methods are most successful. –Simultaneously minimize the Lorentz force and the field divergence. –Can treat fields outside volume by computing field on the sides The fields in the outer part of the domain depend sensitively on the details of the BCs. The implementation of the BCs is the most important limitation for the field modeling. The best algorithm is able to recover the energy in the field to an accuracy of 4%. The range of computation speeds for the tested algorithms varies by over one million. –Sidenote: The difference between C and IDL is a factor of 50. For 512x512x512, the fastest algorithm would require 8000 hours.

16. What's Next... There will be another NLFFF workshop in June 2006. –More complex cases: solar and models. –Will the algorithm rankings hold up under the more complex test cases? –Further topics for discussion will include: variable grid spacing & resolution optimization and parallelization quantitative comparison with coronal observables. –Incorporate into model –Quantify model accuracy Force free Flux Rope Model A. van Ballegooijen

17. Recommendations for Solar-B Observe as large a FOV as possible. –Flux imbalance is a problem. If one end of a field line is not observed, no algorithm can be expected to model that line correctly. Embed region of interest in larger LOS field map. –Algorithms that treat the sides of the volume have a speed advantage here. –Make sure that MDI/SOLIS/HMI data are available and can be properly aligned. May need to invest in massively parallel computing sytem. –How often will a full extrapolation be required? –Can we get away with a coarse resolution in the vertical direction? Are extrapolations from the force-free chromospheric field more robust? –Probably, but needs research.