Pros and Cons of Various Magnetic Field Extrapolation Techniques Marc DeRosa Lockheed Martin Solar and Astrophysics Lab WG5 - SHINE 2007

Some facts Coronal dynamics controlled by magnetic field B A large range of length and time scales are involved most energy input occurs on scale of granulation reconnection occurs in a very small space field is global, as field lines connect seemingly isolated regions energy can be released gradually (non-eruptive reconnection) or very quickly (impulsive eruptions) eruptive events often result in large-scale reconfiguration of B B is important, but it is difficult to measure directly in the corona! (However, clues are provided by EUV, X-rays, and possibly chromospheric lines.)

So, can we model B ? Models come in different types: Potential Field Source Surface (PFSS) Extrapolations Force-Free Field Extrapolations MHD Solutions I will provide an overview of the strengths and weaknesses of each type of model.

PFSS* model (and other current-free models) Strengths: Readily computed from Laplace equation, due to current-free assumption: B = −  →  2  = 0 since  ·B = 0 Reproduces many large-scale features of corona Can be computed faster than real-time Can be used in a predictive capacity Weaknesses: Agreement isn’t perfect (e.g. variation of heliospheric field with latitude is contradicted by Ulysses) The corona does have currents, especially in regions of interest (e.g., above active regions) Can only capture quiescent state (no transient phenomena) * PFSS = potential field source surface, introduced by Schatten et al. (1969)

“Hairy Sun” fieldline rendering lines of B white lines indicate closed field green and magenta lines are open to heliosphere, with color indicative of polarity

Topological Separation Maps

July 1996July 1997July 1998July 1999 July 2000July 2001July 2002July 2003 July 2004July 2005July 2006July 2007 (SOHO vacation)

Model predicts active region field open to the heliosphere TRACE SOHO Sources of the Heliospheric Field from Schrijver & DeRosa (2003) see also Luhmann et al. (2002), Neugebauer et al. (2002), and Wang & Sheeley (2003)

TRACE SOHO Sources of the Heliospheric Field e (white = open, black = closed) from Schrijver & DeRosa (2003) see also Luhmann et al. (2002), Neugebauer et al. (2002), and Wang & Sheeley (2003) Model predicts active region field open to the heliosphere

Coronal hole maps Black contour indicates coronal hole boundary on photosphere. Black-and-white contour denotes neutral line at source surface.

Helmet streamers PFSS fieldlines from Wang/Sheeley model overlaid on (edge- enhanced) photo of 2006 eclipse Some streamers overlie large loop arcades that separate open field having opposite polarity Others overlie interface between two regions of open field having like polarity from Wang et al. (2007)

Helmet streamers Quasi-steady evolution Only fieldlines around source-surface neutral line are drawn. High, miter-shaped structures in corona are evident. Seen face-on, these appear to be well correlated with locations of helmet streamers (e.g., Wang & Sheeley 2007).  = phase of cycle

Ecliptic field lines (north pole tipped 40° toward observer) (close to solar maximum)

Base field strengths for IMF (can be used to model wind speeds) BUT: Around solar maximum, the source strength of the heliospheric field can be accurately forecast only a few days ahead of time, because (a) active regions evolve quickly, and (b) active regions are “seen too late.” Noisy past, with smoothly evolving future Sub-Earth point PastFuture Past Future Carrington

Wind speed / polarity comparison (WSA model, courtesy Nick Arge) (sector boundary) see also Arge & Pizzo (2000)see also Schrijver (2005)

NLFFF* model Strengths: More physically realistic, allows currents as long as corona is (Lorentz-) force-free: J = αB →  ×B = αB. Computational demands are manageable, fastest method takes ~1 hr using 4 CPUs for a pixel domain Weaknesses: It has proven difficult to get accurate estimates of free energy above active regions (so far) Photospheric B not force-free (but maybe can be dealt with) No global models exist (yet), but coupled local NLFFF and global PFSS models in development * NLFFF = nonlinear force-free field

NLFFF model We* have tested the candidate methods on both analytic and simulated fields, finding: Correct solution is largely recovered by all methods when a “chromospheric” vector magnetogram is used Correct solution is largely recovered by all methods when a “chromospheric” vector magnetogram is used (i.e., a magnetogram containing no net Lorentz force or magnetic torque). Correct solution is not recovered when a “photospheric” vector magnetogram is used Correct solution is not recovered when a “photospheric” vector magnetogram is used (i.e., a magnetogram containing forces and torques). “Photospheric” boundary data can be pre-processed getting accurate measurements of physical quantities (such as free energies) remains difficult. “Photospheric” boundary data can be pre-processed to remove forces and torques. However, getting accurate measurements of physical quantities (such as free energies) remains difficult. * see Schrijver et al. (2006) and Metcalf et al. (2008)

Pre-flare magnetogram Hinode/SOT-BFI (before X-flare on )

case _2030

Hinode/XRT overlay fieldlines contained within a 320×320×64 pixel volume _2030

MHD* model Strengths: Most physically realistic, solves MHD equations either in a spherical shell or in a spherical wedge Can capture transient phenomena Compares well with large-scale characteristics (such as streamer observations) Weaknesses: Not all needed boundary conditions are measured (necessary BC’s include B, V, and two state variables at lower boundary) Energy equation / heating model in corona is also uncertain (polytropic is not good enough) Very computationally demanding, even at low resolution, and cannot be done in real-time *MHD = magnetohydrodynamic

Eclipse Predictions image copyright Koen van Gorp observed eclipse (edge enhanced)predicted brightness courtesy Zoran Mikić / Jon Linker

Observational Limitations Calibration, saturation, polar correction affect large-scale field. Current-free models need B r, others need full vector B at some lower boundary radius. Measurements of B at photosphere are (usually) used, but this may not be optimal. For global problem, need B everywhere, including at poles and around back! synoptic maps data assimilation models (for all models)

Surface flux assimilation An example: Schrijver model of B r MDI data assimilated into evolving model DR, MC advect flux random-walk dispersal of flux fragmentation and collisions allowed Polar caps develop naturally Note “jumps” in flux at edge of assimilation window from Schrijver & Title (2001)

Conclusions B controls dynamics in corona, but cannot be directly measured, so it is important to have models. A potential field captures many aspects of the large- scale corona reasonably well, but… Need more physically realistic models for many studies. Need to capture some global characteristics. Need MHD models for transient phenomena, though NLFFF models may be a faster way to provide an estimate of free energy without doing MHD. Evolving, data assimilation models (such as those in terrestrial weather forecasting) will likely be used for space weather forecasting in the future.