Pros and Cons of Various Magnetic Field Extrapolation Techniques

Marc DeRosaLockheed 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) ExtrapolationsForce-Free Field ExtrapolationsMHD 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 coronaCan be computed faster than real-timeCan 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 Bwhite lines indicate closed fieldgreen and magenta lines are open to heliosphere, with color indicative of polarity

Topological Separation Maps

July 1996 July 1997 July 1998 July 1999

July 2000 July 2001 July 2002 July 2003

July 2004 July 2005 July 2006 July 2007

(SOHO vacation)

Model predicts active region field open to the heliosphere

TRACESOHO

Sources of the Heliospheric Field

from Schrijver & DeRosa (2003)

see also Luhmann et al. (2002), Neugebauer et al. (2002), and Wang & Sheeley (2003)

TRACESOHO

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)

Ecliptic field lines

(north pole tipped 40° toward observer)

(close to solar maximum)

Wind speed / polarity comparison

(WSA model, courtesy Nick Arge)

(sector boundary)

see also Arge & Pizzo (2000)see also Schrijver (2005)

NLFFF* modelStrengths:

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 3003-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 modelWe* have tested the candidate methods on both

analytic and simulated fields, finding: Correct solution is largely recovered by all methods Correct solution is largely recovered by all methods

when a “chromospheric” vector magnetogram is usedwhen 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 Correct solution is not recovered when a “photospheric” vector magnetogram is used“photospheric” vector magnetogram is used (i.e., a magnetogram containing forces and torques).

““Photospheric” boundary data can be pre-processedPhotospheric” boundary data can be pre-processed to remove forces and torques. However, getting getting accurate measurements of physical quantities (such accurate measurements of physical quantities (such as free energies) remains difficult.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 2006-12-13)

2006.12.12 2030 case

2006.12.12_2030

Hinode/XRT overlay

fieldlines contained within a 320×320×64 pixel volume2006.12.12_2030

MHD* modelStrengths: 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 Br , 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)

ConclusionsB 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.