N+N+N International Meeting for Young Scientists (a British Council initiative)

1. N+N+N International Meeting for Young Scientists (a British Council initiative) From our star to far stars: variation and variability Budapest (Hungary) 15-18 January, 2007 …details to follow soon (check the forthcoming issues of UK Solar Newsletter; if you are not subscribed, speak to Robertus ASAP) …for more info see http://astro.elte.hu/nnn2007

2. Coronal heating: a theoretical approach Istvan Ballai SPARG, University of Sheffield

3. Introduction • The eclipse of 1869 revealed emission line in the green part of the corona- was named coronium. • Grotrian in 1939 finally showed that this emission line to be due to Fe XIV at 5303Å. • This demonstrated that the corona has a temperature > 1MK, and so the coronal heating problem began….

4. Publications/year (ADS for coronal heating)(more than 4,600 publications) Yohkoh TRACE SoHO Skylab

5. Introduction • Problem: very high temperature of the upper atmosphere • Question: what heating mechanism(s) do operate?

6. Multi-temperature vision of the Sun • Blue: EIT 171 A (0.95 MK) • Green: EIT 195 A (1.5MK) • Red: EIT 284 A (2MK)

7. Introduction “The literature ofcoronal heating is primarily theoretical. Observations are often cited in support of a proposed theory or another, ..but…neither existing observations northe current generation ofmodels are sufficiently detailedto test any mechanism critically.” (Zirker, 1993) • We now have an explosion of high resolution space datasets (Yohkoh, SOHO, TRACE, RHESSI and more to come) – that are providing constraints on theory and distinguishing between possible models. • The coronal heating is still an unsolved problem in the solar and stellar physics

8. Observational facts • Highly inhomogeneous • Rôle of magnetic field

9. Observational facts • Highly inhomogeneous • Role of magnetic field

10. Observational facts • Consists of myriads of coronal loops

11. length scale: from resolution up to 700 Mm • radius: from resolution up to 10 Mm • temperature: from 1-2x104 K to 2x106 K • magnetic field strength: 1- 104 G • equilibrium bulk motion Flux tubes Observational facts

12. The complex problem of coronal heating Conversion mechanism Energy source Heating Plasma response Klimchuk, 2006 Radiation Observables

13. The energy requirement × × × ×

14. The energy source Widely accepted: mechanical motions in and below the photosphere Footpoint motions can generate stresses (DC currents) and waves (AC currents) depending on the time-scale of the motion compared to the Alfven time EUV, UV, X-ray coronal images and magnetograms firmly established that coronal heating is a magnetic phenomenon (e.g. Vaiana and Rosner 1978) DC models tdr>tA Heating models AC models tdr<tA Hybrid AC/DC models (Kinetic, turbulences)

15. The energy source – DC heating • Footpoint motions perform work on the coronal magnetic field and increase its free energy at a rate given by the Poynting flux through the base • Magnetic field concentrated in small tubes (~kG) which expand out in the chromosphere and transition region • Small loops form a low-laying “magnetic carpet” and they do not penetrate into the corona • Part of the inter-network flux extends above the carpet and spreads out in the corona the magnetic field in the quiet Sun is a mixture of network field and surviving inter-network field. • Bv~100 G (AR), 5-10 G (QS), Vh~105 cm/s, assume Bh~Bv: F≈108 erg/cm2 s

16. Thin tubes merge into corona Peter (2001) Tu et al. (2005)

17. Heating by DC currents 2D reconnection theories • 2D reconnection: X-point collapses to a singular sheet • Magnetic energy  heat+K.E.+ fast particles • Well understood • Source of heating and of many dynamic processes (flares, EEs, TRBs)

18. 2D reconnection theories • In 2D well-developed • Slow Sweet-Parker reconnection (1958); rec. rate ≈R-1/2 • Fast Petschek reconnection (1964)rec. rate ≈1/ln R • Many other fast regimes (depend on B.C.’s) • Almost uniform (Priest &Forbes, 1986) • Non-uniform (Priest & Lee, 1992) • Excellent review by Priest and Forbes (Magnetic reconnection, CUP, 2000)

19. 3D reconnection theories Key question: structure of null-point • Simplest: B=(x,y,-2z) • Two families of field lines through null-point: • Spine field lines • Fan surface

20. 3D reconnection theories Three types of reconnection at Null • Spine reconnection • Fan reconnection • Separator reconnection Double 3D null-point topology (courtesy of K. Garlsgaard)

21. 3D reconnection theories Spine reconnection Fan reconnection

22. 3D reconnection theories • So, can reconnection heat the corona? • Yes, possibly, in different ways…but observations are needed to see which way! • Examples: • Reconnection at null-point, e.g., XBP interpreted as converging flux (Parnell et al. 1993, Priest et al. 1994)

23. The energy source – AC heating • The turbulent convection that stresses the coronal magnetic field generates a large flux of upwardly propagating waves (acoustic, Alfvén, slow and fast magnetosonic) • Mode coupling and other processes transfer energy between different types of waves, so the mix of waves changes as a function of height. • Theoretical and observational estimates suggest energy fluxes at the top of convection zone of several 107 erg/cm2s(Narain & Ulmschneider,1996) more than adequate to heat the corona • Only a small fraction of the flux is able to pass through the very steep density and temperature gradients in the chromosphere and transition region. • Acoustic and slow waves steepen into shock waves and are strongly damped, while fast waves are strongly refracted and reflected, only Alfvén waves are able to penetrate into the corona. The do not form shocks since they are transversal and their energy is ducted along the magnetic field rather than being refracted across it.

24. Behaviour of acoustic waves • Chromospheric heating by acoustic waves • Convection generates acoustic waves propagating upwards, steepens into shock waves or are reflected by the density gradients in the TR so evanescent waves

25. Behaviour of Alfvén waves • Significant transmission of Alfvén waves is possible only within narrow frequency bands centered on discrete values where loop resonance conditions are satisfied (Hollweg, 1981) • Enough flux may pass through the base of long (>100 Mm) active regions loops to provide their heating (Hollweg, 1985); in the case of short loops this does not apply. • Waves can be generated in the corona itself by, e.g. magnetic reconnection and change of the equilibrium (AC/DC heating mechanism)

26. Heating by AC currents • Recent high resolution observations show undoubtful evidence for waves in the corona • Prominences • Plumes • Corona (EIT/SoHO, TRACE) • Flare excited waves in loops-fast kink modes (Aschwanden et al. 1999, Nakariakov and Ofman 1999) • Feet of long loops-slow waves (De Moortel et al. 2002ab, Aschwanden et al. 2002 ) • CME/flare excited global waves (EIT waves) –fast waves (Thompson et al. 1999, Ballai and Erdélyi 2003a,b, Ballai et al. 2005) For an effective damping these waves require small scales

27. Resonant absorption ωdriver = ωlocal Ideal MHD equations singular dissipation  heating Concept of Connection Formulae (Ionson 1978, Rae & Roberts 1982, Hollweg 1984, Poedts et al. 1989, Goossens 1991, Ruderman et al 1997ab, Ballai et al. 1998ab, Ballai and Erdélyi 1998,2000ab, etc,etc)

28. Why resonant absorption ? • Inhomogeneous plasmas: natural behaviour • Easy wave energy transfer resulting in heating • Condition to occur: ωdriver = ωlocal • Could/may/viable to explain: • -local/atmospheric heating • - power loss of acoustic waves in sunspots • - damping of helioseismic (p/f/g) eigenmodes • - energisation of MHD waves in magneto/heliosphere

29. Resonant absorption • High frequency Alfvén waves are able to reach corona • They are incompressible and transversal subject to damping due to ohmic and/or shear viscosity • In the corona ν/µ≈1011 and η0/η1≈105 , so they have a very weak damping. • For effective damping small trasversal scales are requiredresonant absorption

30. Concept of connection formulae • Driven problem ω is prescribed • Eigenvalue problem ω is searched for Jumps are independentof dissipative coefficient

31. Resonant absorption Resonant absorption is working!

32. Internal background motion 5-6% vA • Steady large-scale flows (e.g., Doyle et al. 1997) • Flow has amajor influenceon resonant absorption

33. But… • ε– the dimensionless amplitude of the perturbations; R– total Reynolds number; f—any large variable •  linear theory •  nonlinear theory • Suppose for Resonant absorption is a nonlinear phenomena (Ruderman et al.1997, Ballai et al. 1998,1999, 2000, Ballai and Erdélyi 1998)

34. But… • Nonlinearity gives just a small correction to the net absorption coefficientlinear theories give acceptable solutions (Ruderman 2000) • Nonlinearity in dissipative layers generate a mean flow outside the layer • The mean (turbulent) flow can locally enhance the dissipative coefficients • The observation of the generated mean flow could be a first evidence of the resonant absorption

35. (Ofman and Davila 1995)

36. Resonant absorption/phase mixing • To have a heating for the entire loop, we have to suppose that waves are not monochromatic or stochastic processes have to be taken into account (Tsiklauri and Nakariakov 2002, Ruderman 2003) • Dissipative layer the oscillations are in phase as long as ω and kvA are in phase • If they start to be out of phase phase mixing (Heyvaerts and Priest 1983, Browning and Priest 1984, Hood et al 1997, Nakariakov et al 1997, Ruderman et al. 1998, De Moortel et al. 2000, Tsiklauri et al. 2003, etc.)

37. Energy conversion-conclusion • Through energy conversion, the magnetic stress energy and wave energy is transformed into heat. • Since classical dissipation coefficients are small in the corona, significant heating requires the formation of steep gradients and small length scales. Magnetic gradients heating by reconnection and Ohmic dissipation Velocity gradients  heating by viscous dissipation • Gradients are formed through slow quasi-static evolution and through dynamical processes • Possible scenarios: instabilities, turbulences, loss of equilibrium, simple and complex flow patterns at the base of complex coronal magnetic fields (DC) and resonant absorption, phase mixing (AC)

38. Energy conversion and microphysics • Microphysics is likely to play a key role in the energy conversion process, e.g. anomalously large (nonclassical) transport coefficients are required for significant heating even in the presence of steep gradients. • Coronal transport coefficients are not known with precision but indirect techniques are used to infer values for, e.g. viscosity, thermal and electrical conduction, etc.  CORONAL SEISMOLOGY (Nakariakov et al. 1999, Ofman and Aschwanden 2002, Klimchuk et al. 2004, Ballai and Erdélyi 2005) • Collisionality of the coronal plasma: the collisionless effects are extremely important for reconnection (Bhattacharjee 2004) and wave propagation (Ballai et al. 2002) • Hybrid codes developed to take into account both the MHD and particle aspects of the plasma

39. Plasma response • The fundamental principle: the close thermal and dynamic connection between the corona and the lower atmosphere (coupled system) • In the case of static equilibrium, thermal conduction transports more than a half of the coronal heating energy down to the transition region, where it is more efficiently radiated • When heating is time-dependent, an increase in the heating rate causes the coronal temperature to rise, producing an increase of the downward heat flux. The TR is unable to radiate the additional energy, so heated plasma flows into corona through “chromospheric evaporation” • If the upflow is fast, it can be explosive causing shocks, if the heating rate then decreases, an inverse-like process occurs in which the plasma drains from the loop and “condenses” back into the chromosphere.

40. Radiation • We determine the radiation spectrum emitted by the heated corona • If the plasma is in ionisation equilibrium, this task is relative simple (see the CHIANTI software, Dere et al. 1997). • If the plasma is not in ionisation equilibrium the problem is much more complicated. The equilibrium can be destroyed by, e.g. • Rapid evolution of an impulsive heating • Rapid cooling • Flow through a steep temperature gradient In this case we have solve the ionisation rate equation in order to determine the radiation spectrum

41. Observation of heating events • Even the present high resolution satellites provide a minimum information about the heating and the findings are often the result of averaging over space, time and wavelength. • The best resolution at the moment is ≈ 350 km. In order to see heating at work we would need 10-103 m (!!!) • Small-scale events have different names but they may turn out to belong to identical physical processes. • ephemeral regions - nanoflares • emerging flux events - microflares • flux cancellation - soft X-ray jets • events, blinkers - AR transient brightening • soft X-ray bright points

42. Small-scale phenomena and their occurrence domain (QS- quiet Sun, AR-active region, Ph–photosphere, TR–transition region, C–corona)

43. Physical parameters of coronal small-scale phenomena (L-spatial scale, T-electron temperature, n-electron density)

44. Open questions in the coronal heating problem • Are distinct coronal loops heated differently from the diffuse corona? • Are there different classes of loops that are heated in different ways? • Is quiet Sun heating similar to active regions heating? • How the AC/DC mechanisms work together? • Are stellar coronae heated in the same way as the solar corona?

45. On-disk profiles: T = 1–3 million K Off-limb profiles: T > 200 million K ! UVCS results: solar minimum (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 . . .

46. Heating of the open coronal structures (Xing et al. 2002) Very strong perp. heating of the oxygen (Cranmer et al. 1998)

47. The impact of UVCS 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) both flow faster and are heated hundreds of times more strongly than protons and electrons, and have anisotropic temperatures. (e.g., Kohl et al. 1997,1998)

48. Ion-cyclotron resonance • SUMER and UVCS (SoHO) have provided very strict constraints on heating of coronal holes • H+ are mildly anisotropic (Tperp>Tparallel); O 5+ are strongly anisotropic (Tperp/Tparallel=10-200) above 2-3 RSun • At r=3RSun, Tperp for O5+ is 2x108K (vth=450 km/s), while H+ have Tperp=3x106K (vth=225 km/s) • At r=3.5 RSun the outflow speed of O5+ is twice the outflow speed of H+ • These properties can be explained by the resonant interaction of coronal ions with ion-cyclotron waves, i.e. by ion-cyclotron resonance • Ion cyclotron waves (10-104 Hz) have not yet been observed in the solar wind or corona (Cranmer et al. 1999) • Some attempts to describe waves in collisionless plasmas (Nakariakov and Oraevski 1995, Ballai et al. 2002)

49. Ion-cyclotron resonance • The condition of resonance • This mass-dependent mechanism is a wave-particle interaction • Ωi decreases with distance more and more energy injected at lower k is swept into the high frequency domain, where is dissipated by the ions • Dissipation of ion-cyclotron waves produces diffusion in velocity space, along contours of constant energy • Ions are accelerated along the field lines

50. Where do cyclotron waves come from? (1) Base generation by, e.g., “microflare” reconnection in the lanes that border convection cells(e.g., Axford & McKenzie 1997). (2) Secondary generation: low-frequency Alfven waves may be converted into cyclotron waves gradually in the corona. Both scenarios have problems . . .