1. Quelques développements récents sur la physique
du chauffage coronal.
T. Passot
UNS, CNRS, OCA
Hinode/XRT
Thanks to: E. Buchlin, S. Galtier. Some viewgraphs have been adapted from Cranmer’s, Buchlin’s
and Bigot’s talks.

2. Outline Description of the solar outer regions
Overview of coronal properties
The heating problem
Heating of the chromosphere
Heating of the corona:
Exoshperic model
AC vs. DC mechanisms (reconnection / waves)
Phase mixing
Properties of MHD waves
Turbulence
Model of closed coronal loops and of open regions
Ion heating (cyclotron resonance)
Need for a more complete model : fluid + kinetic

3. The Sun’s outer atmosphere
The solar photosphere radiates like a blackbody; its spectrum gives T, and dark “Fraunhofer lines” reveal its chemical composition.
Total eclipses let us see the vibrant outer solar corona: but what is it?
1870s: spectrographs pointed at corona:
Is there a new element (“coronium?”)
1930s: Lines identified as highly ionized ions: Ca+12 , Fe+9 to Fe it’s hot!
Fraunhofer lines
unknown bright lines

4. Overview of solar interior and atmosphere
Priest (1995)

6. Overview of coronal observations
Plasma at 106 K emits most of its spectrum in the UV and X-ray . . .
Coronal hole (open)
“Quiet” regions
Active regions
TRACE (171Å) Résolution : 350 km
Very intermittent structures: myriad of coronal loops; crucial role of magnetic field

7. Organization and complexity
Constant and impulsive reorganization of magnetic structrures
Anisotropic bipolar structures (coronal loops)
Inhomogenity of temperatures (7.105K to 2.107K)
Non-thermal velocities along loops: 50 km/s
Non-resolved spatial (350 km) and temporal (1s) scales
Reynolds number 1012
Magnetic field: 10-4 to 0.3 T
Solar corona is the site of a complex nonlinear dynamics
Coronal nanoflares
are related to
intermittent dissipative
events
in the MHD turbulence

8. UVCS results: over the poles (1996-1997 )
The fastest solar wind flow is expected to come from dim “coronal holes.”
In June 1996, the first measurements of heavy ion (e.g., O+5) line emission in the extended corona revealed surprisingly wide line profiles . . .
Off-limb profiles: T > 200 million K !
On-disk profiles: T = 1–3 million K

9. Solar minimum UVCS results
UVCS has led to new views of the collisionless nature of solar wind acceleration.
Key results include:
The fast solar wind becomes supersonic much closer to the Sun (~2 Rs) than previously believed.
In coronal holes, heavy ions (e.g., O+5) flow faster and are heated hundreds of times more strongly than protons and electrons, and have anisotropic temperatures. (Kohl et al. 1997, 1998, 2006)

10. Particles are not in “thermal equilibrium”
…especially in the high-speed wind.
mag. field
WIND at 1 AU
(Steinberg et al. 1996)
Helios at 0.3 AU
(e.g., Marsch et al. 1982)
WIND at 1 AU
(Collier et al. 1996)

11. Energy budget overview
What sets the temperature?
Photosphere: optical depth ~unity, with radiation dominating heating/cooling:
Chromosphere: optically thin, radiation cools the plasma (all photons escape!) Heating is provided “mechanically,” by irreversible damping of kinetic motions
Transition region & low corona: complicated balance of radiation, mechanical heating, downward conduction, and upward advection (enthalpy flux)
Extended corona: direct heating balances upward advection (adiabatic cooling)
Heliosphere: advection (adiabatic cooling) balances outward conduction

13. Chromospheric heating
Energy is transported by radiative diffusion through the chromosphere, which reveals itself most strongly in the light of Ha and CaII K. Views of the chromosphere show convective cell patterns similar to those in the photosphere, but much larger. This large scale convection is known as super-granulation.

14. Convection excites waves
All cool stars with sub-photospheric convection undergo “p-mode” oscillations:
Lighthill (1952) showed how turbulent motions generate acoustic power.
These ideas have been more recently generalized to MHD. . .

15. “Traditional” chromospheric heating
Vertically propagating acoustic waves conserve flux (in a static atmosphere):
Amplitude eventually reaches Vph and wave-train steepens into a shock-train.
Shock entropy losses go into heat; only works for periods < 1–2 minutes…
Bird (1964) ~
New idea: “Spherical” acoustic wave fronts from discrete “sources” in the photosphere (e.g., enhanced turbulence or bright points in inter-granular lanes) may do the job with longer periods.

16. Runaway to the transition region (TR)
Whatever the mechanisms for heating, they must be balanced by radiative losses to maintain chromospheric T.
When shock strengths “saturate,” heating depends on density only:
Why then isn’t the corona 109 K? Downward heat conduction smears out the “peaks,” and the solar wind also “carries” away some kinetic energy. Conduction also steepens the TR to be as thin as it is.

17. Coronal heating Sun, Soho EIT , Fe XII
Fe XII-XVIII emission is formed in the low corona (2x106 K) of the Sun and is due to recombination of elecrons with ionized Fe.

19. Exospheric models Collisionless Collision dominated where
M. Maksimovic et al. 2004

20. Filtrage des vitesses (Scudder,1992)
Use Liouville theorem

21. The coronal heating problem
We still don’t understand the physical processes responsible for heating up the coronal plasma. A lot of the heating occurs in a narrow “shell.”
Most suggested ideas involve 3 general steps:
1. Churning convective motions that tangle up magnetic fields on the surface.
2. Energy is stored in tiny twisted & braided “magnetic flux tubes.”
3. Collisions between ions and electrons (i.e., friction?) release energy as heat.
Heating Solar wind acceleration!

22. Coronal heating mechanisms
Many ideas! (Mandrini et al. 2000; Aschwanden et al. 2001)
Where does the mechanical energy come from?
How rapidly is this energy coupled to the coronal plasma?
How is the energy dissipated and converted to heat?
vs.
waves
shocks
eddies
(“AC”)
twisting
braiding
shear
(“DC”)
vs.
interact with
inhomog./nonlin.
turbulence
reconnection
collisions (visc, cond, resist, friction) or collisionless

23. Reconnection
Also involves kinetic effects

26. The observed distribution of nanoflares in the quiet Sun is about
Heating by numerous localized events due to reconnection as a result of, e.g. the continuous shuffling of the roots of coronal fields is possible
A more global heating mechanism
should be responsible for the heating
The observed distribution of nanoflares in the quiet Sun is about
one order of magnitude less than the theoretical requirement
which means a deficit of 3 times in energy (R. Erdelyi and I. Ballai, 2007).

27. Wave mechanisms

28. Made from SOT images, this movie shows prominences above an active region at the limb. Detailed analysis of high-resolution images attributes the waving motion of a prominence to Alfvén waves in the corona.
From : Coronal Transverse Magnetohydrodynamic Waves in a Solar Prominence,
T. Okamoto, S. Tsuneta, T. E. Berger, K. Ichimoto, Y. Katsukawa, B. W. Lites, S. Nagata, K. Shibata, T. Shimizu, R. A. Shine, Y. Suematsu, T. D. Tarbell, and A. M. Title.
Science, 318, 1577 (2007)

29. Chromospheric Alfvénic Waves Strong Enough to Power the Solar Wind
B. De Pontieu,1* S. W. McIntosh,2,3 M. Carlsson,4 V. H. Hansteen,4,1 T. D. Tarbell,1 C. J. Schrijver,1 A. M. Title,1 R. A. Shine,1 S. Tsuneta,5 Y. Katsukawa,5 K. Ichimoto,5 Y. Suematsu,5 T. Shimizu,6 S. Nagata7
Science, 318, 1574 (2007)
Alfvén waves have been invoked as a possible mechanism for the heating of the Sun's outer atmosphere, or corona, to millions of degrees and for the acceleration of the solar wind to hundreds of kilometers per second. However, Alfvén waves of sufficient strength have not been unambiguously observed in the solar atmosphere. We used images of high temporal and spatial resolution obtained with the Solar Optical Telescope onboard the Japanese Hinode satellite to reveal that the chromosphere, the region sandwiched between the solar surface and the corona, is permeated by Alfvén waves with strong amplitudes on the order of 10 to 25 kilometers per second and periods of 100 to 500 seconds. Estimates of the energy flux carried by these waves and comparisons with advanced radiative magnetohydrodynamic simulations indicate that such Alfvén waves are energetic enough to accelerate the solar wind and possibly to heat the quiet corona.
Fig. 2. Illustration of the ubiquity of Alfvén waves in the chromosphere. (A) A Hinode SOT Ca II H 3968 Å image showing how thin spicules that outline the magnetic field dominate the chromosphere. (B) A space-time plot [along the cut labeled 1 in (A)] of the Ca intensity processed to enhance 200-km-wide structures. The plot is dominated by a multitude of criss-crossed short linear tracks caused by spicular motion transverse to the magnetic field direction. (D) A similar cut for the line labeled 2 in (A). Image enhancement is unnecessary at these heights because of the smaller number of spicules. Similar linear tracks are visible, as well as swings. The general characteristics (linear tracks and swings) of (B) and (D) are well reproduced by cuts that are generated from Monte Carlo simulations [(C) and (E)] (12) in which spicules carry Alfvén waves.

30. Phase mixing But: At large magnetic Reynolds number, S:
tdiss = S 1/3 (Petkaki et al., 1998) .
Works better with chaotic magnetic topology: tdiss =log( S) (Malara et al. 2004)
But:

31. In the collisionless case (Tsiklauri et al. 2005)
Collisionless phase mixing leads to electron acceleration.
Progressive distortion of the AW front, due to the differences in local Alfvén speed, generates electrostatic fields nearly parallel to the magnetic field, which accelerate electrons via Landau damping. Surprisingly, the amplitude decay law in the inhomogeneous regions, in the kinetic regime, is the same as in the MHD approximation described by Heyvaerts and Priest (1983 Astron. Astrophys ).
Formation of filaments
also possible in homogeneous
media due to transverse
Instability of Alfvén waves
This process however requires strong inhomogeneities (300%).
In the case where the density fluctuations are moderate (10%), dispersive effects
can arrest the phase front steepening and lead to magnetic filaments (Borgogno et al. PoP ‘08).
MHD
Hall-MHD

32. In the case of inhomogeneities of 100%, a new phenomenon, intrinsically
3D, takes places. The rollup of very thin magnetic filaments can increase
global heating with respect to dispersionless case. In this case, there is
preferential increase of perpendicular temperature.

33. AC versus DC heating? Are they really so very different?
Waves cascade into MHD turbulence (eddies), which tends to:
break up into thin reconnecting sheets on its smallest scales.
accelerate electrons along the field and generate currents.
Coronal current sheets are unstable in a variety of ways to growth of turbulent motions which may dominate the energy loss & particle acceleration.
e.g., Dmitruk et al. (2004)
Onofri et al. (2006)

34. Alfvén waves: from Sun to Earth
Velocity amplitudes of fluctuations measured (mainly) perpendicular to the background magnetic field.

35. Properties of MHD waves
In the absence of a magnetic field, acoustic waves propagate at the sound speed (restoring force is gas pressure)…
B-field exerts “magnetic pressure” as well as “magnetic tension” transverse to the field. The characteristic speed of MHD fluctuations is the Alfvén speed…
Plasma β = (gas pressure / magnetic pressure) ~ (cs/VA)2
“high beta:” fluid motions push the field lines around
“low beta:” fluid flows along “frozen in” field lines

36. Properties of MHD waves
Phase speeds: Alfven, fast, slow mode; ● = sound speed, ● = Alfven speed
β = 12
β = 2.4
β = 1.2
β = 0.6
F/S modes damp collisionally in low corona; Alfven modes are least damped.
Standard MHD dispersion applies only for frequencies << particle Larmor freq’s.
For high freq & low β, Alfven mode → “ion cyclotron;” fast mode → “whistler.”

37. Alfvén wave evolution Energy density & flux: Static medium:
Non-zero wind speed (“wave action conservation”):
Alfvén waves also reflect & refract as the background properties change…
A(r)

38. Turbulence
It is highly likely that somewhere in the outer solar atmosphere the fluctuations become turbulent and cascade from large to small scales.
The original Kolmogorov (1941) theory of incompressible fluid turbulence describes a constant energy flux from the largest “stirring” scales to the smallest “dissipation” scales.
Largest eddies have kinetic energy ~ v2 and a “turnover” time-scale  =l/v, so the rate of transfer of energy goes as v2/ ~ v3/l .
Dimensional analysis can give the spectrum of energy vs. eddy-wavenumber k: Ek ~ k–5/3

39. f -1 “energy containing range”
Fourier transform of B(t), v(t), etc., into frequency:
f -1 “energy containing range”
f -5/3
“inertial range”
The inertial range is a “pipeline” for transporting magnetic energy from the large scales to the small scales, where dissipation can occur.
Magnetic Power
f -3 “dissipation range”
few hours
0.5 Hz

40. MHD turbulence: 2 kinds of “anisotropy”
With a strong background field, it is easier to mix field lines (perp. to B) than it is to bend them (parallel to B).
Also, the energy transport along the field is far from isotropic: Z– Z– Z+
(e.g., Hossain et al. 1995; Matthaeus et al. 1999; Dmitruk et al. 2002)

41. Turbulent heating of coronal loops

42. Un modèle phénoménologique (Heayverts et Priest 1992)

43. Généralisation à la turbulence anisotrope
(Bigot, Galtier, Politano, 2008)
Utilisation de modèles en couche pour atteindre de grands nombres de Reynolds
(Buchlin et Velli 2007):

44. With direct numerical simulations

45. Heating the extended corona
Addtional energy deposition is necessary above 2 RO:
to accelerate the fast wind
produce the temperatures of ions and electrons measured in IP
accelerate heavy ions
The physics is different, as the plasma rapidly becomes collisionless

46. Open flux tubes: global model
Photospheric flux tubes are shaken by an observed spectrum of horizontal motions.
Alfvén waves propagate along the field, and partly reflect back down (non-WKB).
Nonlinear couplings allow a (mainly perpendicular) cascade, terminated by damping.
(Heinemann & Olbert 1980; Hollweg 1981, 1986; Velli 1993; Matthaeus et al. 1999; Dmitruk et al. 2001, 2002; Cranmer & van Ballegooijen 2003, 2005; Verdini et al. 2005; Oughton et al. 2006; many others!)

47. Turbulent heating models in open regions
Cranmer & van Ballegooijen (2005) solved the wave equations & derived heating rates for a fixed background state.
New models: (preliminary!) self-consistent solution of waves & background one- fluid plasma state along a flux tube: photosphere to heliosphere
Ingredients:
Alfven waves: non-WKB reflection, turbulent damping, wave-pressure acceleration
Acoustic waves: shock steepening, TdS & conductive damping, full spectrum, wave-pressure acceleration
Rad. losses: transition from optically thick (LTE) to optically thin (CHIANTI + PANDORA)
Heat conduction: transition from collisional (electron & neutral H) to collisionless “streaming”

48. Turbulent heating models
For a polar coronal hole flux-tube:
Basal acoustic flux: 108 erg/cm2/s (equiv. “piston” v = 0.3 km/s)
Basal Alfvenic perpendicular amplitude: 0.4 km/s
Basal turbulent scale: 120 km (G-band bright point size!)
T (K)
reflection coefficient
Transition region is too high (8 Mm instead of 2 Mm), but otherwise not bad . . .

49. Alfven wave’s oscillating ion’s Larmor motion around radial B-field
Ion cyclotron waves in the corona?
UVCS observations have rekindled theoretical efforts to understand heating and acceleration of the plasma in the (collisionless?) acceleration region of the wind.
Alfven wave’s oscillating
E and B fields
ion’s Larmor motion around radial B-field
Ion cyclotron waves (10–10,000 Hz) suggested as a “natural” energy source that can be tapped to preferentially heat & accelerate heavy ions.

50. Ion cyclotron waves in the corona
Dissipation of ion cyclotron waves produces diffusion in velocity space along contours of ~constant energy in the frame moving with wave phase speed:
lower Z/A
faster diffusion

51. Where do cyclotron waves come from?
Alfvén waves with frequencies > 10 Hz have not yet been observed in the corona or solar wind, but ideas for their origin abound
(1) Base generation by, e.g., “microflare” reconnection in the lanes that border convection cells (e.g., Axford & McKenzie 1997).
Problem: “minor” ions consume base-generated wave energy before it can be absorbed by ions seen by UVCS.
(2) Secondary generation: low-frequency Alfvén waves may be converted into cyclotron waves gradually in the corona.
Problem: Turbulence produces mainly small-scale eddies in the direction transverse to the field; these don’t have high frequencies!

52. cyclotron resonance-like phenomena
In the corona, “kinetic Alfven waves” with high k heat electrons (T >> T ) when they damp linearly.
How then are the ions heated & accelerated?
Nonlinear instabilities that locally generate high-freq. waves (Markovskii 2004)?
Coupling with fast-mode waves that do cascade to high-freq. (Chandran 2006)?
KAW damping leads to electron beams, further (Langmuir) turbulence, and Debye-scale electron phase space holes, which heat ions perpendicularly via “collisions” (Ergun et al. 1999; Cranmer & van Ballegooijen 2003)?
Non-linear/non-adiabatic KAW-particle effects (Voitenko & Goossens 2004)?
If the corona is filled with thin collisionless shocks, ions can pass through them and aquire gyromotion when the background field changes direction (Lee & Wu 2000)?
Collisionless velocity filtration from intrinsically suprathermal velocity distributions (Pierrard et al. 2004)?
Larmor “spinup” in dissipation-scale current sheets (Dmitruk et al. 2004)?
cyclotron resonance-like phenomena
MHD turbulence
something else?
We can compute a net heating rate from the cascade, even if we don’t know how the energy gets “partitioned” to the different particle species.

53. Future directions for theory
Generation and nonlinear evolution of the solar wind fluctuation spectra must be understood.
Self-consistent kinetic models (from corona to wind) of protons, electrons, & ions are needed.
Because these processes interact with one another on a wide range of scales, their impact can only be evaluated when all are included together.
There’s a need for “phenomenological” terms that encapsulate what we learn from micro-scale simulations, so that macro-scale modeling can proceed!