# 3rd SOLAIRE School Solar Observational Data Analysis SODAS Coronal Magnetic Field Extrapolations Stéphane RÉGNIER University of St Andrews SODAS

Document Sample

```					           3rd SOLAIRE School
Solar Observational Data Analysis (SODAS)

Coronal Magnetic Field
Extrapolations

Stéphane RÉGNIER
University of St Andrews
SODAS Solaire School                                            Glasgow, 1-5 June 2009

What I will focus on

 Magnetic field extrapolation of active regions

 Potential and nlff magnetic extrapolations
 Cartesian coordinates

 Full disc potential extrapolations
 How to analyse magnetic configurations

What I will not talk about

 Quiet-Sun magnetic extrapolations
 Synoptic maps and cycle variations
 Ground-based observatories

 Linear force-free extrapolations (just briefly)
 Non force-free magnetic fields

SODAS Solaire School                                        Glasgow, 1-5 June 2009

Photospheric magnetic field as boundary conditions

Vector magnetic fields
Line-of-sight magnetic fields
in Cartesian coordinates
SODAS Solaire School                                                         Glasgow, 1-5 June 2009

Coronal equilibrium

Three characteristic times are important: eq time of equilibrium, A Alfvén transit time,   rec
reconnection time. Typically, the Alfvén transit time is several minutes and the
reconnection time is few seconds.

The equilibrium can relax to a minimum energy state
(Woltjer 1958)
Time evolution can be describe by a series of linear
force-free equilibria (Heyvaerts and Priest 1984)

The equilibrium will more likely relax to a nonlinear
force-free field

We have shown that we can successfully describe the time evolution of an active region
with a time series of nonlinear force-free equilibria with a time interval of 15 min (Régnier
and Canfield 2006)
SODAS Solaire School                                                     Glasgow, 1-5 June 2009

Why to study reconstruction methods?

• No 3D measurements of the magnetic field in the corona;

• Access only to the magnetic field components (line-of-sight and vector)
on the photosphere and/or in the chromosphere;

• Understanding the magnetic field in the corona gives inputs for flare and
CME models (e.g., storage of energy, site of magnetic reconnection)
SODAS Solaire School                                                        Glasgow, 1-5 June 2009

Why to focus on nonlinear force-free fields?

 Next stage after potential and linear force-free fields, and which contain

 The corona (< 2.5 R) has been thought to be force-free (Gold & Hoyle
1960); the scale-height of plasma pressure and gravity are large compared to
the magnetic field (plasma <1) in the corona;

 Dissipation of Hall and Pedersen currents in the chromosphere; only
parallel currents remain in the corona;

 Recent numerical simulations of flux emergence including partial ionization
(Leake and Arber 2006) have shown that the final state of the coronal
magnetic field is force-free.
SODAS Solaire School                                                          Glasgow, 1-5 June 2009

Force-free fields

We consider that the corona is dominated by the magnetic field (low- plasma) and therefore the
coronal magnetic field is assumed to be force-free.

Magneto-hydrostatic
                      
equilibrium                 p               g      j B 0
Force-free
                
equilibrium             j B               0

                      
       
B                                 
j        0              j
0
B             j             (r ) B
Potential Field                Linear Force-                     Nonlinear Force-
free Field                         free Field
SODAS Solaire School                                                       Glasgow, 1-5 June 2009

Properties of force-free magnetic fields

Necessary integral properties of a force-free field (not sufficient) from Molodensky
(1969) and Aly (1989)

No magnetic force                                  No magnetic torque

Molodensky (1969), Aly (1989)
Potential and lff magnetic fields

1. Equation and boundary conditions

2. Fourier transform: Gary’s method
3. PFSS
2.1 Equations

2.2 Applications

4. Example of potential field study
SODAS Solaire School                                                    Glasgow, 1-5 June 2009

1. Equation and boundary conditions

Potential field model

Boundary conditions      Bz from observations on the bottom boundary;
On the sides and top: open, periodic, closed, mixed

Potential magnetic energy
- Minimum energy state
- Total magnetic flux (   2)

- Distribution of charges
SODAS Solaire School                                                      Glasgow, 1-5 June 2009

2. Fourier transform: Gary’s method

Fourier transform of magnetic components

Boundary condition on the photosphere

Fourier equations on the photosphere (same at z)

Condition to get non trivial solutions

Important limit
2           where L is the characteristic length of the box
L
SODAS Solaire School                                              Glasgow, 1-5 June 2009

3. PFSS: Potential Field Source Surface

Potential in spherical harmonics

where

Corresponding magnetic field components
SODAS Solaire School                                               Glasgow, 1-5 June 2009

3. PFSS: Potential Field Source Surface

Boudary conditions

component of the magnetic field on
the photosphere (full sphere)

The magnetic field is radial at the
source surface
SODAS Solaire School                                         Glasgow, 1-5 June 2009

4. Example of potential field study
Nonlinear force-free fields

1. Equations and boundary conditions
2. Methods

2.1 Existing methods

2.3 More comparison

3. Applications to the corona
SODAS Solaire School                                                       Glasgow, 1-5 June 2009

1. Equations and boundary conditions

Some analytical and semi-analytical solutions

• analytical solutions in cylindrical, spherical and toroidal coordinates
have been derived (e.g. Chandrasekhar 1956, Gold-Hoyle 1960, Buck
1965, Low 1973, Titov and Démoulin 1999, Török et al. 2004);

• applications to thin twisted flux tubes in the corona and to magnetic
clouds have been performed with the Gold & Hoyle solutions;

• well-known semi-analytical solutions were derived by Low & Lou
(1990); these solutions have been used to test nonlinear force-free
reconstruction techniques in Cartesian coordinates.
SODAS Solaire School                                                      Glasgow, 1-5 June 2009

2.1 NLFFF methods

Vertical integration method (Wu et al. 1990, Démoulin et al. 1992, Song et al. 2006)

Grad & Rubin method (Grad and Rubin 1958, Sakurai 1981, Aly 1988, Amari et al. 1997,
1999, Wheatland 2004, Amari et al. 2006, Inhester and Wiegelmann 2006, …)

Optimization method (Pridmore-Brown et al. 1981, Wheatland et al. 2000, Wiegelmann et
al. 2003)

Evolutionary techniques (Mikic and McClymont 1994)

“Stress-and-Relax” technique (Roumeliotis 1996)

Boundary element method (Yan and Sakurai 2000, Li et al. 2004)

Magneto-frictional method (Yang et al. 1986, van Ballegooijen et al. 2000, Mackay et al.
2000, 2001, Valori et al. 2005)

Finite element method (Amari et al. 2006)

(…)
SODAS Solaire School                       Glasgow, 1-5 June 2009

2.1 NLFFF methods
SODAS Solaire School                                                       Glasgow, 1-5 June 2009

Problem       To find the magnetic configuration the closest to a nonlinear force-free and
divergence free magnetic field which matches all components of the magnetic
field at the boundaries. This is an ill-posed boundary value problem

Minimization of a functional L:

Following Wheatland et al. (2000):
where

and

The iterative process starts from a potential field, then solves the above equation, and
updates the new magnetic field configuration following:

and increases t until the functional L is minimized.
SODAS Solaire School                                                          Glasgow, 1-5 June 2009

Problem        To find the nonlinear force-free field associated with those boundary conditions
corresponding to a well-posed boundary value problem

The Grad-Rubin (1958) scheme separates the nonlinear equations into two linear systems of
equations (Aly 1988, Amari et al. 1999):

Transport of     along field lines (hyperbolic):                     Properties
Existence and uniqueness of solution for
small (Bineau 1972) in simply connected
domains; extended to multiple connected
domains (Boulmezaoud and Amari 2000);
Updating the 3d magnetic field (elliptic):          Sakurai (1981) and Wheatland (2004)
schemes use the magnetic field (on the
left)
Amari et al. (1997, 1999), Inhester and
Wiegelmann (2006) and Amari et al. (2006)
use a scheme based on the vector
potential to preserve div.B;
SODAS Solaire School                                     Glasgow, 1-5 June 2009

2.3 More comparison

 In Schrijver et al. (2006) and Amari et
al. (2006), a quantitative comparison of
different numerical nonlinear force-free
fields was performed, based on the Low
& Lou solutions (1990);

 The different methods performed well
in strong field regions;

 The methods are compared in terms
of the speed of computation, and of the
accuracy with respect to the semi-
analytical model;

 The authors gave a ”ranking” of
methods as shown in the image on the
right.
SODAS Solaire School                         Glasgow, 1-5 June 2009

2.3 More comparison
SODAS Solaire School                                      Glasgow, 1-5 June 2009

2.3 More comparison

DeRosa et al. (2009)

Comparison of the integrated
current density along the vertical
dimension for all nlfff models
SODAS Solaire School                                                              Glasgow, 1-5 June 2009

3. Applications to the corona

Twisted flux bundles like sigmoids and filaments (Yan et al. 2000, Liu et al. 2002,
Régnier et al. 2002, Régnier et al. 2004)

Nature of filaments associated with eruptions   Evidence of twisted flux bundles supporting dense plasma,
(Yan et al. 2000)                   and not in quadrupolar-like magnetic dips (Régnier &
Amari, 2004)

Coronal loops (Yan et al. 2000, Bleybel et al. 2002, Liu et al. 2002, Wiegelmann et al. 2005,
Régnier and Canfield 2006)

black: observed loop, white: potential loop,
orange: linear force-free field, yellow:
nonlinear force-free field
Wiegelmann et al. (2005)
SODAS Solaire School                                                Glasgow, 1-5 June 2009

3. Applications to the corona

H features during a flare

From Mikic & McClymont (1994), the field
lines computed using the nonlinear force-
free field match the H features during a
M-class flares, while the potential field
lines do not match.

From Lee et al. (1998), good correlation
between the reconstructed field (evolutionary
method)   and    the   sources     of   radio
(microwaves) gyro-resonant frequencies at 4.9
GHz and at 8.4 GHz;
Radio peaks are C1-C2 at 4.9 GHz, and X1-X2
at 8.4 GHz (squares: 430 G, crosses: 750 G);
Sunspots are marked with a S letter.
How to analyse magnetic fields?

1. Characteristic parameters

1.1 PIL, Magnetic energy

1.2 Current density, CIL

3. 3D magnetic configurations
2.1 Magnetic Energy

2.2 Magnetic Helicity

2.3 Geometry of field lines

2.4 Magnetic Topology

2.5 Time series
SODAS Solaire School                                                       Glasgow, 1-5 June 2009

2.1 Magnetic Energy

Several quantities are of interest to understand the storage and release of energy:

• the total magnetic energy of a magnetic configuration:
• the free magnetic energy budget:
• the magnetic energy density along the vertical axis (see graphs on the left)

• the time evolution of those quantities
Change of the magnetic energy and
helicity in linear force-free fields

AR8151: decaying active region with high current
density exhibiting highly twisted flux bundles
SODAS Solaire School                                                        Glasgow, 1-5 June 2009

2.2 Magnetic Helicity

Relative magnetic helicity (Finn & Antonsen 1985, Berger & Field 1984):

where A is the vector potential associated to B, and the index 0 corresponds to a
reference field (usually taken to be the potential field).

If                        then

For this particular active region, the mutual helicity dominates due to the complex topology of
the field and no evidence of twisted flux bundles (Régnier et al. 2005)
SODAS Solaire School                                                         Glasgow, 1-5 June 2009

2.2 Magnetic Helicity

The minimum energy state of a nlff fields with helicity Hm is the linear force-free field with
the same helicity (Taylor’s relaxation theory)

AR 8151: old decaying active region with highly twisted bundles
AR 8210: newly emerged active region with complex topology (C-flares)
AR 9077: Bastille day flare 2000, post-flare phase
AR 10486: Halloween event 2003 on 28th OCt., before X17 flare
Δ
7.23 1032 erg
x                   Free Magnetic Energy
in Solar Active Regions above
the Minimum-Energy Relaxed State
Δ                            Régnier, S., Priest, E. R.

1.62 1032 erg              2007, ApJ, 669, L53

0.05 1032 erg       x
Δ
x           Δ
0.14 1032 erg                     Δ: Nonlinear force-free energy
x
x: Linear force-free energy
AR8151      AR8210 AR9077            AR0486
SODAS Solaire School                                                             Glasgow, 1-5 June 2009

2.3 Geometry of field lines

Potential vs. NLFFF
   changing connectivity
   shear (aligned with PIL)
   twist (e.g., highly twisted bundles)

Linear Force-free Field =-      Linear Force-free Field      Nonlinear Force-
E    Potential Field          2.6 10-3 Mm-1                =6.6 10-2 Mm-1              free Field

E
E
E
E
E
SODAS Solaire School                                                          Glasgow, 1-5 June 2009

2.4 Magnetic Topology

Magnetic topology: discontinuity of the mapping of magnetic field configurations
In 2D (Parnell et al. 1996, Phys. Plas.):
Topological elements: null points, separatricies
Types of null points: X-points, O-points
In 3D (Parnell et al. 1996):
Topological elements: null points, separatrix surfaces, separators
Types of null points: plenty of …, e.g., prone and upright photospheric
nulls

Skeleton
Definition: to find the skeleton corresponds to find all the topological
elements in a given magnetic configuration, including null points, separatrix surfaces,
spine field lines, separators.
Basics of reconnection:
In 2D, magnetic reconnection occurs at X-points.
In 3D, magnetic reconnection can occur at null points, along separatrix surfaces and along
separators where strong electric currents and current sheets can be created; and in
absence of null points (e.g. at quasi separatrix layers)
SODAS Solaire School                                                                  Glasgow, 1-5 June 2009

2.4 Magnetic Topology

We compare the topology of a magnetic configuration with a coronal null points using the
same boundary conditions
   (left) potential field
   (centre) linear force-free fields with   values between -1 and 1 Mm-1
   (right) nonlinear force-free fields for a ring distribution of Jz
SODAS Solaire School                                                           Glasgow, 1-5 June 2009

2.5 Time series

Method         describing the time evolution of an active region as successive nonlinear
force-free equilibria assuming that the evolution is slow enough (t > n.tA)

Moving Magnetic Features
Emergence of a parasitic polarity in a pre-existing magnetic configuration having a broken
fan-like topology.
Flare scenario: reconnection process along separatrix surface, due to (apparent)
horizontal motions of a parasitic polarity leading to small-scale re-organization of the
magnetic field
(Régnier & Canfield 2006)

Emerging and fast
NE                            moving parasitic
Pre-existing                                         polarity
magnetic
configuration

SW

Separatrix

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 44 posted: 2/4/2010 language: English pages: 33
How are you planning on using Docstoc?