G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality of the solar corona G. Aulanier ( Observatoire de

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA

Complexity & non-potentiality

of the solar corona

G. Aulanier

( Observatoire de Meudon, LESIA )


Overview of this tutorial

1. Basic physics

2. Learning from 3D MHD simulations

3. Magnetic field extrapolations

4. Toward SDO & other future missions

1.1 Introduction

1.2 Non-potentiality

1.3 Complexity

2.1 Current sheets & reconnection in quasi-separatrices

2.2 Breakout model of an observed eruption

3.1 Motivation

3.2 Potential & linear force-free fields

3.3 Non-linear force-free fields

1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





Complexity & non-potentiality

Yohkoh SXT, SXR11:48 UT

TRACE, FeXI 171AJuly 14 1998, 12:05 UT – 14:00 UT

At the origin of all solar flares & eruptions

Among the major goals of all upcoming solar instruments


Magnetic energy : storage & release

Magnetically driven activity

Corona : ~ ETh / EB ~ 2P / B² < 1

Long-duration energy storage phase

a few days (flares) to a few weeks (prominence eruptions)

Sudden energy release & triggering of active phenomenon

Alfvénic timescales ~ a few minutes



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





Pre-eruptive B : force-free fields

Conservation of momentum : dt ( u )= 0

dt u = – (u .) u + ()–1 (x B) x B + P + g tA²/t² = u²/cA² + 1 + + L / HP

Slow evolution : t ~ days >> tA ~ minutes Photospheric velocities : u ~ 0.1 km/s << cA ~ 1000 km/s « Cold » plasma : = 0.0001 – 0.1 << 1 Loop sizes : L~ 10 – 100 Mm ~ Hp ~ 50 Mm

J x B = 0 & x B = J

Field-aligned currents : x B = B


Force-free fields : three classes

Potential fields : = 0

Linear force-free fields : = cst

Non-linear force-free fields : = varying

x (x B = B ) ² B + ²B = 0

Helmoltz equation has analytical solutions

x B = 0 B =

B defined by a scalar potential

(x B = B ) ( B )= 0

A field line is defined by its value


Non potentiality : free magnetic energy

Potential field : x B0 = 0 ; B0 = 0 ; B0 =

EB0 = III ½ B0² dV

EB = III ½ B0² dV + III ½ B1² dV + III B0B1 dV

= EB0 + EB1 + III B1 dV

= EB0 + EB1 + III ( B1) dV

= EB0 + EB1 + II B1 dS = EB0 + EB1 > EB0

Same as Kelvin’s theorem for incompressible fluids

Potential field = lower bound of energy

Non potential field : B = B0+ B1 ; B1 = 0 ; II B1 dS = 0


How to store energy in the corona

Wavelengths L of coronal waves with C = CA ~ cst :

Energy burst during dt : L ~ CA dt ~ 10 Mm(for CA= 200 km/s & dt = 50 s)

Slow & continuous motion of a footpoint : L ~ Lcoronal loop > 10 Mm

Corona / photosphere interface (assuming equal B) :

CAcor / CA

phot ~ (phot / cor)½ ~ (1017

cm-3 / 109 cm-3 )½ ~ 10

4

Lwavelength / HP scale-height > 104 km / 10

2 km > 10

2

Paradigm :

The Sun has no experimental-like well-defined confining boundaries But energy stored for t >> tAlfvén


When an Alfvén waves reaches the photosphere

At the wave-front, over 1% only of the whole wavelength Propagation speed by a factor 104

Velocity amplitude by a factor 108

This leads to a quasi-complete reflexion back into the corona

- This is not only the result of strong differences, it requires a sharp interface ! - Its is not always valid : e.g. steep waves & shocks, short loops, very short energy bursts

Energy storage : line-tying

Line-tying = extreme assumption = full reflexion


Origin of Energy : emergence & motions

Sub-photospheric emergence

Current carrying flux tube from convection zone

Flux tubes traveling the whole CZ twist necessary

Slow photospheric motions

Twisting of 1 or 2 of the polarities

Shearing motions // inversion line

Energy stored in closed field lines only

Evacuation of EB at Alfvénic speeds in open fields



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





Non-bipolar fields : complex topologies

2.5-D & 3D models :

Quadru-polar fields

Null point B=0 separatrix surfaces

z

x

In 3D : spine field line & fan surface

Karpen et al. (1998)

Aulanier et al. (2000)


Complexity : current sheet formation

Quasi-spontaneous current sheet formation in 2.5-D :

z

x

y

x

Field line equation : y = Bydxz/Bxz = By dxz/Bxz

( Bxzxz)By= 0 since J x B = 0 & d/dy = 0

On each side of separatrix : y equal & dxz /Bxz different

Jump in By

y

x


Null point : magnetic reconnection

Basic principle in a current sheet :

dB/dt = ² B & field line equation reconnection

mass & energy conservations uin /CA = Lu -½ (Sweet-Parker regime)

The Switch-on problem :

shearing separatrix spontaneous J sheet no flare, but heating

Advect stronger B, increasing , stronger driving, other physics (Petscheck, Hall…)

Or separatrix-less reconnection…

(Aulanier, 2004, La Recherche)



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





no 3D null point

Quasi-separatrices

Quasi-separatrices

Generic four flux concentrations model

Topology / geometry :

Continuous field line mapping

Sharp connectivity gradients


Log Q = J / B J= x B

pas de symétrie 2.5D

Quasi-separatricesJ (z=0)

Gradual formation of current layers

Current layers & topology :

Along the pre-existing Quasi Separatrix Layer (QSL)

J sheet thinnest in Hyperbolic Flux Tube (HFT)

Thickness decreases with time in HFT



Formation of current layers : where & how

Current sheets :

In pre-existing QSL

For any boundary motion

Thickness of J ~ thickness of QSL



Slip-running reconnection in 3D

Field line dynamics :

Coronal reconnection

Alfvénic continuous footpoint slippage

Origin of apparent fast motion of particle impact along flare ribbons ?



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





Yohkoh SXT, SXR11:48 UT

TRACE, FeXI 171A12:05 UT – 14:00 UT

A case study : The July 14, 1998 eruption


Realistic model of B & line-tied motions

Coronal field :

B(Kitt Peak) modified to have |Bzmax|=2900 G

potential field extrapolation

view from earth

Local photospheric twisting :

1 polarity in -spot

satisfies dBz/dt = 0

umax = 2% CA (slow)

top view


-spot coronal configuration :

Null point aside of (not above) the sheared field lines at z = 3.9 Mm

Sheared fields beneath the fan surface

Projection view of The full 3D MHD domain

2.5-D MHD breakout model

(Antiochos et al. 1999)

A generalized magnetic breakout ?Aulanier et al. (2000)


Dynamics of sheared & complex coronal fields

Projection view (field lines & Bz[z=0] ) 2D cut of currents J

Potential field



Eruption

with continuous photospheric motions

& null point reconnection

t = 0000 s , ndump = 00




Eruption



t = 0090 s , ndump = 03




Eruption



t = 0180 s , ndump = 06




Eruption



t = 0270 s , ndump = 09




Eruption



t = 0360 s , ndump = 12




Eruption



t = 0450 s , ndump = 15




Eruption



t = 0540 s , ndump = 18




Eruption



t = 0630 s , ndump = 21




Eruption



t = 0720 s , ndump = 24




Eruption



t = 0810 s , ndump = 27

Null point reconnection




Eruption



t = 0900 s , ndump = 30

Null point reconnection




Eruption



t = 0990 s , ndump = 33

Null point reconnectionFlux tube acceleration




Eruption



t = 1080 s , ndump = 36

Flux tube acceleration




Eruption



t = 1170 s , ndump = 39





Eruption



t = 1260 s , ndump = 42





Eruption



t = 1350 s , ndump = 45





Eruption



t = 1440 s , ndump = 48




TRACE 171AView from Earth

( field lines & Jz[z=0] )

Eruption

with photospheric motions supressed

TRACE observations vs. MHD model


Physical & observational ingredients :

Photospheric twisting & slow expansion 0 < t < 750

Magnetic energy: EBfree = 9.5 % EB

potential field t = 990

Null point reconnection & leaning sideward of overlaying fields rooted in the -spot 750 < t < 1080

Fast eruption of sheared fields & 2-ribbon flare 990 < t < 1440 ( idem if photospheric driving supressed )

Moving brightenings observed in EUV = dtJz (z=0) during reconnection

A generalized magnetic breakout ?

So far difficulties to calculate full eruption :

numerical instabilities in current sheets calculation halts

work in progress



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





To link models & observations

2.5-D MHD breakout model SoHO/EIT 195 filament eruption

Magnetic field extrapolations :

Model of Bcoronal using observed Bphot as boundary conditions

To better analyze observed events knowing Bcorona

To test models & provide Binitial

for 3D MHD simulations

(Antiochos et al. 1999)


1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation






Potential & linear force-free field extrapolations

Assumption : = cst (=0 for potential)

² B + ²B = 0 : Helmoltz equation : analytical solutions

Fourier, Bessel functions, spherical harmonics(Nakagawa & Raadu 1972, Alissandrakis 1981, Démoulin et al. 1997, Chiu and Hilton 1977, Semel 1988, Altschuler & Newkirk 1969, Schrijver & DeRosa 2003 …)

Advantages & limits :

+ Fast computation low computer memory & power

+ Based on analytical formulas low dependance on algorithm

+ Do not require full Bphot vector magnetograms rare & noisy

+ Overall topology most topological regimes are stable

– Lower bounds on EB & HB poor estimation of free energy & helicity

– Small-scale shear largest field lines most affected by – limits cannot treat highly stressed fields

– = cst no mixed sheared & potential fields& no return currents


Setting force-free parameter cst

Yohkoh/SXT lfff extrapolation

Yohkoh/SXT

H (DPSM / Pic du Midi)

chosen to best match - large SXR loops- transverse Bphot if available

small connectivities weakly depend on


Yohkoh/SXT

H (DPSM / Pic du Midi)

SXR loops

Arch Filament System

Démoulin et al. (1997), Schmieder et al. (1997)

Flare ribbons : footpoints of QSLs = gradients of connectivites

Confined flare topologies in active regions


Pre-CME topologies in active regions


Eruption precursor : shear Alfvén wave along null spine because of reconnection


Pre-CME topologies between hemispheres

Delannée et al. (1999)

Large-scale dimmings : footpoints of TIL’s pushed from below during eruption


Pre-eruptive topologies : potential & lfff sufficient

Major results :

Topology (skeletons & QSL’s) of overlaying weakly stressed fields

Location of associated current sheets reconnection particle acceleration sites particle impacts confined flare ribbons dimmings during CMEs

Some codes already « available » to the community :

Potential Field Source Surface (PFSS) on SSWIDLSchrijver & DeRosa (2003)

FRench Online MAGnetic Extrapolations (FROMAGE) on the WWWDémoulin et al. (1997), Aulanier et al. (1999)

Such extrapolations well addess the issue of global connectivity


Aulanier & Schmieder (2002)


Topology of filaments with constant- extrapolations


Full field lines magnetic dips

H/ EUV filament bodies (feet) : magnetic dips within (aside)

a flux tube of low twist <1.5


9-hour evolution on Sep 25, 1996

VTT/MSDP 08:43 UT

12:14 UT 17:04 UT

15:57 UTSoHO/MDI 07:40 UTSoHO/MDI 07:40 UT 15:59 UT15:59 UT

17:35 UT17:35 UT12:53 UT12:53 UT


Evolving filaments with constant- extrapolations


9-hour evolution on Sep 25, 1996

12:14 UT 17:04 UT

VTT/MSDP 08:43 UT 15:57 UT


Moving parasitic polarities : destruction / formation of dips & evolution of barbs

Evolving filaments with constant- extrapolations



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





Performing non-linear force-free extrapolations

Properties : = varying (as observed in vector magnetograms & in MHD models)

x B = B & ( B )= 0 must be computed numerically in general

Common feature in most algorithms :

Stress imposed at 1 photospheric footpoint or on the whole photosphericplane or on all faces via increments of bdry or Bbdry or ubdry

Relax toward force-free state by imposing some transport method (e.g. physical or numerical) & conditions (e.g. Emin or |JxB|min)

Great care with mathematical ill-posed methods !i.e. redundant BC’s at boundary(ies) discontinuities in domain

Simple vertical integration doomed to failure no side/upper BC’s existing k modes growing as ~exp(k.z)


Results : sheared loops & sigmoids

Bleybel et al. (2002)

Bz

PHOT

z

PHOT

Yohkoh/SXT

Jiao, McClymont & Mikic (1997)

Bz

PHOT

z

PHOT

Yohkoh/SXT

Pre-post eruption EB & HB ; non-homogeneous shear & return currents


Results : flux ropes & filaments

Régnier & Amari (2004)

van Ballegooijen (2004)

courtsesy van Ballegooijen

Filament bodies : magnetic dips within a flux tube of medium twist ~ or > 2


Some results, but difficulties at every level

PDE’s Growth (& instability) of non-physical spatial oscillations

Multiple solutions can exist for 1 same boundary condition

MHD-unstable solutions can exist

Numerical : for locally strong gradients diBj

Physical : for regions of strong values

Observational : spectro-polarimetry & magnetography

Inversion of Stokes profiles I,Q,U,V & solving 180o ambiguity B

Weak Q,U noise in Bxy & weak I errors in Bsunspots

Many non force-free regions where >1 & in (quasi-) separatrices

Limited field of view artificial flux imbalance


Finding the best method(s)

Some input models :

Low & Lou (1990) ()/ < (Bz)/Bz & weak & no return currents analytical solution

Valori (2005), Schrijver et al. (2006), Inhester & Wiegelmann (2006), Amari et al. (2006)

Best methods :

Optmization method better for analytical models (with BC’s on 6 faces imposed)

Grad-Rubin (& magneto-frictional) better on symmetric twisted flux tubes

Grad-Rubin a priori best (well posed, no redundant BC’s), but J x B not imposed convergence not ensured losses of convergence found for strong

3D MHD [Török&Kliem03] or

Photospheric bipolar Bz & 3D nlfff [Titov&Démoulin01] or

symmetries 2D arbitrary [Inhester&Wiegelmann06]

no analytical solution for most models


More realistic & demanding input models

()/> (Bz)/Bz phot strong & varies faster than Bz

phot

narrow non force-free layers with strong at the fan/spine footpoints 0 & 0 in one polarity return currents

from MHD simulation of the July 14, 1998 pre-eruptive B

= Jz/Bz (z=0)

~ 0 on all 5 coronal faces «open» field lines are potential

Bz (z=0)Bz (z=0) & field lines



1. Basic physics

1.1 Introduction


1.3 Complexity





3.1 Motivation





Toward SDO & other future missions

Non-linear fff extrapolations

Take BPhotObs from vector magnetogramCalculate potential field BPOT

Prescribe Stress on bounary(ies) and tend toward

Bcorona ~ EUV loops

Super-computer resources (high resol. needed)

Then analyze observations & analyze stability with MHD simulations

Forward modeling : 3D MHD

Take BzPhotObs from LOS magnetogram

Calculate potential field BPOT

Prescribe uxy

phot or Ephot and tend toward

Bcorona ~ EUV loops and/or BPhotMHD ~ BPhotObs

Super-computer resources

Then analyze observations & pursue MHD simulation to model coronal dynamics

Beyond potential and linear force-free fields analyses of observations :

MHD & nlfff algorithms to be validated & tested against observations

Observing full coronal field lines (multi- : SDO/AIA) Measuring photospheric full vector B (SDO/HMI & inversion techniques)

Documents

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality of the solar corona G. Aulanier ( Observatoire de