1. MHD of the solar corona :
Non-potentiality, complex topologies & line-tying
Guillaume Aulanier
( Observatoire de Meudon, LESIA )
n° 1

2. Complexity & non-potentiality of coronal B
At the origin of all solar activity
TRACE, FeXI 171A
July , 12:05 UT – 14:00 UT
n° 3
Yohkoh SXT, SXR
11:48 UT
Among the major goals of all upcoming solar instruments
But not easily measurable…

3. Activity : storage & release of magnetic energy
Magnetically driven activity
Corona : b ~ ETh / EB ~ 2mP / 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
n° 4

4. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion
1.) Physics of the solar corona
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion
1.) Physics of the solar corona
2.) 3D MHD simulations
3.) 3D MHS filament models
5.) Conclusion
n° 2

5. Pre-eruptive B : field-aligned currents
Conservation of momentum : dt ( r u )= 0
h dt u = – (u .s) u + (mr)–1 (sx B) x B + sP + rg
tA²/t² = u²/cA² b + b L / HP
In the solar corona :
h Slow evolution : t ~ days >> tA ~ minutes
Photospheric velocities : u ~ 0.1 km/s << cA ~ 1000 km/s
« Cold » plasma : b = – 0.1 << 1
Loop sizes : L~ 10 – 100 Mm ~ Hp ~ 50 Mm
n° 5
h J x B = 0 & sx B = mJ  sx B = aB

6. Force-free fields : three classes
h sx B = 0  B = sY
h B defined by a scalar potential
Potential fields : a = 0
Linear force-free fields : a = cst
h sx (sx B = a B )  s² B + a² B = 0
h Helmoltz equation has analytical solutions
h sh(sx B = a B )  ( B hs)a = 0
h A field line is defined by its a value
n° 6
Non-linear force-free fields : a = varying

7. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion
does not count

8. Non-potentiality |a| > 0 : free magnetic energy
Potential field : sx B0 = ; sh B0 = 0
h EB0 = III ½ B0² dV
; B0 = sF
; II B1h dS = 0
Non potential field : B = B0+ B1
; sh B1 = 0
h EB = III ½ B0² dV + III ½ B1² dV + III B0hB1 dV
= EB EB III (sF) hB1 dV
= EB EB III [ sh(F B1) – F (shB1) ] dV
= EB EB III sh(F B1) dV
= EB EB II F B1 h dS
n° 7
= EB EB > EB0
h Same as Kelvin’s theorem for incompressible fluids
h Potential field = lower bound of energy for a given Bzphot

9. How to store energy in the corona
Paradigm :
h The Sun has no experimental-like well-defined confining boundaries
h But energy stored for Dt >> tAlfvén
Wavelengths L of coronal waves with C = CA ~ cst :
h Energy burst during dt : L ~ CA dt ~ 10 Mm
(for CA= 200 km/s & dt = 50 s)
h Slow & continuous motion of a footpoint : L ~ Lcoronal loop > 10 Mm
n° 8
Corona / photosphere interface (assuming equal B) :
h CAcor / CAphot ~ (rphot / rcor)½ ~ (1017 cm-3 / 109 cm-3 )½ ~ 104
h Lwavelength / HP scale-height > km / km > 102

10. Energy storage : line-tying
When an Alfvén waves reaches the photosphere
h At the wave-front, over 1% only of the whole wavelength
h Propagation speed m by a factor 104
h Velocity amplitude m by a factor 108
h This leads to a quasi-complete reflexion back into the corona
- Not only the result of strong r differences, it requires a sharp interface !
- Not always valid : e.g. steep waves & shocks, short loops, very short energy bursts
Line-tying = extreme assumption = full reflexion
n° 9

11. Origin of Energy : emergence & motions
Sub-photospheric emergence
h Current carrying flux tube from convection zone
h Flux tubes traveling the whole CZ  twist necessary
Slow photospheric motions
h Twisting of 1 or 2 of the polarities
h Shearing motions // inversion line
n° 10
Energy stored in closed field lines only
h Evacuation of EB at Alfvénic speeds in open fields

12. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion
does not count

13. Definition of magnetic shear angle for a = constant
Bz = B° sin(kx x) exp[-(kx² – a²)1/2 z]
Single Fourier mode By = a/kx B° cos(kx x) exp[-(kx² – a²)1/2 z]
Bx = -(kx² – a²)1/2/kx B° cos(kx x) exp[-(kx² – a²)1/2 z]
Potential vs. linear force-free field (at x=z=0) :
h Bx = -(kx² – a²)1/2/kx B° & By = -a/kx B°  By = 0 for potential field only
h q = atan( By / Bx ) = atan( a / (kx² – a²)1/2 )  Magnetic shear x y z x y z q Bx Bx By
Bz>0
Bz<0
Bz>0
Bz<0

14. Localized line-tied shear : direct & return currentsy
a = 0 potential fields
a = 0 potential fields
a > 0 direct currents
a < 0 return currents
- to confine current-induced B
so as to keep external B potential
- Biot-Savart  Sj = 0 x z y x
Bz<0
Bz>0

15. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion
does not count

16. Non-bipolar fields : complex topologies
2.5-D & 3D models :
h Quadru-polar fields
h Null point B=0  separatrix surfaces
h In 3D : spine field line & fan surface z x
n° 11
Karpen et al. (1998)
Aulanier et al. (2000)

17. Complexity : current sheet formation
Quasi-spontaneous current sheet formation in 2.5-D :
h Field line equation : Dy = S By dxz/Bxz = By S dxz/Bxz
h ( Bxzhsxz) By = 0 since J x B = 0 & d/dy = 0
h On each side of separatrix : Dy equal & dxz /Bxz different
 Jump in By z x y x y x
n° 12

18. Null point : magnetic reconnection
Basic principle in a current sheet :
h dB/dt = hs² B & field line equation  reconnection
(Aulanier, 2004, La Recherche)
h mass & energy conservations  uin /CA = Lu -½ (Sweet-Parker regime)
The Switch-on problem :
h shearing separatrix  spontaneous J sheet  no flare, but heating
n° 13
h Advect stronger B, increasing h , stronger driving, other physics
(Petscheck, Hall…)
h Or separatrix-less reconnection…

19. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion
does not count

20. MHD equations in the LESIA code : zero-b
i Continuity
dt r = – (u .s) r – r (s.u)
[ advection ; compressible term ]
i Momentum
dt u = – (u .s) u + (mr)–1 (sx b) x b + n ru
[ advection ; Lorentz force ; pseudo-viscous term ]
i Induction
dt b = – (u .s) b – b (s.u) + (b .s) u + h rb + z s(s.b)
[ ideal term ; resistive diffusion ; divb cleaner ] ~ ~

21. Numerical scheme Finite differences : 4th order (5 point derivatives)
Predictor-Corrector : step dams-Bashorth & 2-step Adams-Moulton
explicit time-step with CFL condition
Numerical mesh : non-uniform, structured, fixed
stronger concentration where b stronger
Explicit diffusion terms Laplacian for b
Filter adapted to the mesh of u

22. Boundary conditions : line-tied & open
bx, by, bz, ux, uy z r uz
ghost
cells
domain
| | | | | |
r, bx, by, bz z
ghost
cells
domain
ux, uy, uz
| | | | | | z y x z y x

23. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion

24. Twisting flux tubes : rotating sunspots

25. Twisting flux tubes : sub-Alfvénic line-tied motions
3D projection view
Top view
uxy/cA = 0.02
Aulanier, Démoulin & Grappin (2005)

26. Dynamics & equilibrium properties
0.2
0.4
0.6
uz/cA
Equilibria z
F /2p
Dynamic phase
Quiet phase
Dynamic phase : even with sub-alfvénic boundary driving
Equilibrium curves : ln(z/z0) = cst F²
Why is there no kink ? Expansion  L k  F/L m  stabilizing effect

27. Electric currents : photosphere & corona
Jz (z=0)
S (J2 dz)
potential
return
direct

28. Sigmoïd topology : sheared lines vs. twisted flux rope
Twisted B lines around vortex centers
Whirled B lines on sigmoïd ends
Sheared B lines on brightest parts

29. Twisted sunspots : Non-radial magnetic fields
MHD model
Aulanier, Démoulin & Grappin (2005)
THEMIS / MTR
courtesy G. Molodij

30. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion

31. Confined flare topologies from cst-a extrapolations
Yohkoh/SXT
lfff extrapolation
Yohkoh/SXT
H (DPSM / Pic du Midi)
h a chosen to best match - large SXR loops
- transverse Bphot if available
h small connectivities weakly depend on a

32. Sharp mappings : quasi-separatrix layers (QSL’s)
Arch Filament System
H (DPSM / Pic du Midi)
SXR loops
Yohkoh/SXT
Démoulin et al. (1997), Schmieder et al. (1997)

33. QSL’s in four flux concentrations model
Quasi-separatrices
no 3D null point
Quasi-separatrices
Topology / geometry :
Continuous field line mapping
Sharp connectivity gradients

34. Current sheet formation in QSL’s
Log Q
a = J / B
J = sx B
Aulanier, Pariat & Démoulin (2005)
Current layers & topology :
J (z=0)
Along the pre-existing Quasi
Separatrix Layer (QSL)
J sheet thinnest in Hyperbolic
Flux Tube (HFT)
Thickness decreases with time
in HFT
pas de symétrie 2.5D
Quasi-separatrices

35. Formation of current sheets : where & how
In pre-existing QSL
For any boundary motion
Thickness of J ~ thickness of QSL
Aulanier, Pariat & Démoulin. (2005)

36. Slip-running reconnection in 3D
Aulanier, Pariat, Démoulin & DeVore (2006)
Field line dynamics :
Coronal reconnection
Alfvénic continuous footpoint slippage
Origin of apparent fast motion of
particle impact along flare ribbons ?

37. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion

38. Potential & linear force-free field extrapolations
Semi-analytical solutions
h s² B + a² B = 0 : Helmoltz equation
 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 a
– a limits  cannot treat highly stressed fields
– a = cst  no mixed sheared & potential fields
& no return currents

39. Extrapolation of filament channels
08:12 UT
07:52 UT

40. Plasma support in magnetic dipsh
Levitation of dense plasma
H (prominence) ~ Hg  no hydrostatic support
e (prominence) ~ 0.01L(Bphot)  Alfvén wave support requires to
be localized : why ?
u (plasma) ~ 0.1 cA  field-aligned motions : role debated
(B . s) B > 0
Bz = 0 h
Full field line
altitude (z)
Filling of the dip : h
dHa = Hg = 300 km
Simulation of filament observations = 3D ensemble of all dips h

41. Topology of filaments with constant-a extrapolations
Aulanier et al. (1999)
Aulanier et al. (2000)
Full field lines magnetic dips
Aulanier & Schmieder (2002)

42. Constant-a extrapolations vs. MHD models
Magnetic dips
(IP & NP)
Full field lines
Full field lines magnetic dips
Aulanier & Schmieder (2002)
DeVore & Antiochos (2000)
Full field lines
Amari et al. (1999)
Aulanier, DeVore & Antiochos (2006)

43. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion

44. Evolving filament barbs : observations
9-hour evolution on Sep 25, 1996
VTT/MSDP :43 UT
12:14 UT
17:04 UT
15:57 UT
SoHO/MDI :40 UT
15:59 UT
17:35 UT
12:53 UT
Aulanier et al. (1999)

45. Evolving filament barbs : constant-a extrapolations
9-hour evolution on Sep 25, 1996
VTT/MSDP :43 UT
15:57 UT
12:14 UT
17:04 UT
Aulanier et al. (1999)

46. Comparisons with B measured in prominences
Normal
Inverse
Aulanier & Démoulin (2003)
|B| ~ 14 G
d|B|/dz ~ G/km
1 isolated region
of normal polarity
same as in
Bommier et al. (1994)

47. – l + Magnetic dips measured in the photosphere i THEMIS Ha spine barb
constant-a model – i +
THEMIS Ha
THEMIS B//
barb
spine
THEMIS
observations – l +
parasitic
polarities
THEMIS
vector B
Lopez Ariste, Aulanier,
Schmieder & Sainz-Dalda
(2006)

48. Overview 1.) Physics of the solar corona 2.) 3D MHD simulations
1.1.) Force-free fields
1.2.) Free magnetic energy
1.3.) Extended currents & magnetic shear
1.4.) Current sheet formation & magnetic reconnection
2.) 3D MHD simulations
2.1.) The LESIA code
2.2.) Twisting flux tubes & X-ray sigmoïds
2.3.) Current sheets & reconnection in quasi-separatrices
3.) 3D MHS filament models
3.1.) Extrapolation applied to multi-l observations
3.2.) Comparison with magnetic field measurements
5.) Conclusion

49. MHD models of the solar corona are VERY MUCH
Conclusions
MHD models of the solar corona are
VERY MUCH
constrained by observations !
Many reasonable models of solar features
(even those that use the right & well-known orders of magnitudes)
FAIL
the observational test…