Twisting and Writhing
with George Ellery Hale
Magnetic Helicity: from Turbulent
Convection to Space Weather
Richard C. Canfield
Department of Physics
Montana State University, Bozeman
Data courtesy Solar Dynamics Observatory AIA & HMI
AR 11158 video courtesy Daniel Brown
 Loops and Turbulence
National Geographic / Universe Today
Credit unknown
Convective Turbulence shreds  loops
•
•
•
•
Assume Ω loops are rigid and rise through
the photosphere with constant velocity
Stack 800 HMI magnetograms
1 mtgm every 7.5 min for 100 hours)
Red / blue: negative / positive polarity
Chintzoglou and Zhang ApJ 2013
Hale’s “Vortices (1908)”
Hale discovered:
•
•
Vortex direction
depends on
hemisphere.
Vortex direction does
not depend on cycle
Hale, ApJ 1927
Northern Hemisphere
(left handed upwelling)
Southern Hemisphere
(right handed upwelling)
?
Magnetic Helicity + Force-Free Fields
H   A  B d3 x
where
B=A
“We then, by the way, find … that
A  B is a constant of the motion.”
Elsasser, Rev Mod Phys 1956
------------------------------------------------------------------------Minimizing E = (1/8) {  B2 -   A  B d3x }
where  is a Lagrange multiplier, leads to the equation
B=B
for “force-free” ( j x B = 0 ) fields.
N.B.:  is a measure of twist
(constant when E is minimum).
Woltjer, Proc Nat Acad Sci 1958; Berger, AGU Geophys. Mon. 111, 1999
Twist, Writhe, Linking, Self &
Mutual Helicity and Conservation
Writhe
(W)
Twist
(T)
Magnetic
Helicity
Conservation
H=T+W
Moffatt & Ricca Proc. R.
Soc. Lond. A, 1992
Video courtesy Dana Longcope
N
N
i=1
j=1
H = Σ Σ Lij Φi Φj
Lij : Linking Number
i = j : Self Helicity
i ≠ j : Mutual Helicity
Wiegelmann & Sakurai LRSP
Hale’s Vortices Today:
The Hemispheric Twist Rule
Solar Cycle 22
203 Active
Regions
Solar Cycle 23
263 Active
Regions
B=B
Pevtsov, Canfield, Latushko ApJ 2001
Flux-Tube Twist from Helical
Convective Turbulence
Northern hemisphere
 Effect Model:
Southern hemisphere
•Horizontal thin flux tube
•Helical turbulence
•Mixing length theory
•Helicity conservation
•Monte Carlo approach
Glatzmeier ApJ 1985
Longcope, Fisher, Pevtsov ApJ 1998
Model Predicts:
•Hemispheric
sign trend
•Amplitude of
scatter
•Lack of cycle
dependence
Pevtsov, Canfield, Metcalf ApJ 1996
Longcope, Fisher, Pevtsov ApJ 1998
“THE SOLUTIONS ALL ARE SIMPLE …
AFTER YOU’VE ALREADY ARRIVED AT THEM.
BUT THEY’RE SIMPLE ONLY WHEN
YOU ALREADY KNOW
WHAT THEY
ARE”
Robert M. Pirsig
Zen and the Art of Motorcycle Maintenance
Measuring Coronal Magnetic
Energy and Helicity
Related quantities
•Free Magnetic Energy
•Relative Magnetic Helicity
Woltjer (Proc Nat Acad Sci 1958)
Berger & Field (JFM 1984)
Finn & Antonsen (CPPCF 1985)
Kusano et al (ApH 2002)
Tziotziou, Georgoulis, & Raouafi ApJ 2012
Two alternative approaches
1. Construct the instantaneous 3D coronal magnetic
field using a photospheric vector magnetogram
2. Calculate the flux of magnetic helicity and energy
into the corona using magnetogram sequences
Data courtesy Solar Dynamics Observatory Helioseismic and Magnetic Imager
AR 11158 video courtesy Lucas Tarr
Helicity and Energy Fluxes
from Magnetogram Sequences
dH
= 2 ò [(AP × Bt )v n - (AP × v t ) Bn ]ds
dt S
S
dE
1
= ò [Bt2 v n - (Bt × v t )Bn ]ds
dt S m0 S




Emergence
Shearing &
Braiding
Emergence
Shearing &
Braiding
Berger & Field (JFM 1984)
Finn & Antonsen (CPPCF 1985)
Kusano et al (ApJ 2002)
Chae (ApJ 2001)
November & Simon (ApJ 1988)
Demoulin & Berger (SP 2003)
Pariat et al (A&A 2005)
ADS Citations to Berger & Field (JFM 1984)
Sea Change: SDO HMI Data
Vector Magnetogram Sequences
Liu & Schuck ApJ 2012
Liu & Schuck ApJ 2012
Reconnection: Mutual Helicity 
Self Helicity  Flux Rope
Amari, Aly, Mikic, Linker ApJ L 2010
Coronal Mass Ejections
(CMEs) Are Flux Ropes
Vourlidas et al ApJ 2012
Courtesy of Ben Lynch
The Space Weather Challenge
for the helicity enthusiast:
Predict the leading-field orientation when these
flux ropes arrive at Earth – i.e.,
predict the Magnetic Cloud helicity
Courtesy of Thomas Zurbuchen
Li et al Solar Physics 2011
The most severe space weather is associated with flux ropes whose leading magnetic
field points South, pushing a sheath magnetic field that also points South.
Energy and Helicity Storage
on Topological Separators
MINIMUM CURRENT CORONA (MCC)
•
•
•
•
•
As photospheric magnetic flux elements
shuffle around (shearing and braiding), the
flux content of each domain changes only
when reconnection occurs an/or when flux
emerges or submerges.
The constraint that fluxes in each
individual domain do not change causes
currents to build along separators:
Physics: The separator currents follow
from Faraday's Law & Ohm's Law in the
presence of moving magnetic point
changes -> separator energy and helicity.
When flare reconnection occurs, these
separator currents are fully dissipated.
Use flare imaging to identify which
separators flare.
Sweet IAU Symposium 6,1958
Courtesy of Dana Longcope
Predicted (MCC) & observed flare energy
(SDO/EVE) and MC helicity (ACE), 4 events
Canfield & Kazachenko 2013
Some interesting questions
in AR Helicity / Energetics
1. What will data-driven nonlinear force-free models
tell us about where energy and helicity is released
in flares, when compared to the topological models?
2. What is the contribution of distributed currents, as
opposed to separator currents, to CME energetics?
3. Will continuous vector magnetogram sequences
and helicity flux maps find dynamo effects, as
opposed to just convection zone effects ( effect)?
4. What is the axial variation of magnetic helicity per
unit length in Magnetic Clouds?
Summary
The conservation properties of magnetic helicity
link the solar dynamo and convection zone
to eruptive solar events and their interaction
with the magnetic field of Earth, an important
component of Space Weather.
Continuous high temporal and spatial resolution
vector magnetogram sequences and new
modeling techniques enable quantitative
measurement and modeling of energy and
helicity in erupting active regions, which is
now and will continue to be a rich research
field for decades to come.
• Thanks to the SPD, for this honor
• Thanks to you, for your attention
• Thanks to the following students &
Postdocs, of whom I am very proud:
PhD Students
•Robert E. Stencel, Ph.D. (Astronomy), University of Michigan, 1977
•Richard C. Puetter, Ph.D. (Physics), University of California, San Diego, 1980
•Paul J. Ricchiazzi, Ph.D. (Physics), University of California, San Diego, 1982
•Todd A. Gunkler, Ph.D. (Physics), University of California, San Diego, 1984
•George H. Fisher, Ph.D. (Physics), University of California, San Diego, 1984
•David H. Tamres, Ph.D. (Physics), University of California, San Diego, 1989
•Kenneth G. Gayley, Ph.D. (Physics), University of California, San Diego, 1990
•Thomas R. Metcalf, Ph.D. (Physics), University of California, San Diego, 1990
•Kimberly D. Leka Ph.D. (Astronomy), University of Hawaii, Honolulu, 1995
•Angela C. Des Jardins, Ph.D. (Physics), Montana State University, 2007
•Maria D. Kazachenko, Ph.D. (Physics), Montana State University, 2010
Postdoctoral Fellows
•Chang-Hyuk An, Ph.D. (Physics), University of Tennessee, 1979
•Stanley Owocki, Ph.D. (Physics), University of Colorado, 1981
•Jean-Pierre Wuelser, Ph.D. (Physics), University of Bern, 1988
•Thomas R. Metcalf, Ph.D. (Physics), University of California, San Diego, 1990
•Jean-Francoise de La Beaujardiere, Ph.D. (Astrophysical, Planetary, and
Atmospheric Sciences), University of Colorado, 1990
•Edward Lu, Ph.D. (Physics), Stanford University, 1990
•Gianna Cauzzi, Ph.D. (Astronomy), University of Florence, 1992
•Alexei Pevtsov, Ph.D. (Solar and Planetary Physics), Institute of SolarTerrestrial Physics, Irkutsk, 1992
•Tetsuya Magara, Ph.D. (Solar Physics), University of Kyoto, Japan 1998
•Robert Leamon, Ph.D. (Physics), University of Delaware, Newark, 1999
•Stephane Regnier, Ph.D. (Physics), Inst. d'Astrophysique Spatiale, Orsay, 2001
•Dibyendu Nandi, Ph.D. (Physics), Indian Institute of Science, Bangalore, 2003
Graduate and Undergraduate Research Students
•Marc Allen, University of Michigan
•James Rhoads, Harvard University
•Sean Sandborg, Montana State University
•Keith Lambkin, University College Dublin
•Tanya Freeman, Union College
•Zachary Holder, Montana State University
•Ji Son, University of California, Los Angeles
•Crystal Fordyce, Clemson University
•Emily McLinden, University of Chicago
•Scott Waitukaitis, University of Chicago
•Michael Hahn, Columbia University
•Stacy Gaard, University of Indiana
•Alexander Russell, University of St Andrews
•Thomas Schad, University of Notre Dame
•Christopher Lowder, Georgia Institute of Technology
•Meghan Cassidy, University of Maryland
•William Simpson, University of St. Andrews
Dedicated to the memory of Thomas R. Metcalf
Finis