Coronal Loop Model Including Ion Kinetics

1. Coronal Loop Model Including Ion Kinetics S. Bourouaine Max-Planck-Institut für Sonnensystemforschung (Lindau) Solar Cycle 24, NAPA, December 2008

2. Outline • Observations of EUV/SXT loops • Collisional heating mechanisms • Kinetic model for coronal loop • Multi-ions + electron plasma • wave absorption mechanism • Conclusion

3. Loop observations (warm loops): + (a), (diamond) (b), (triangles) (c), (squares) (d). • 4 loop systems observed at the limb of the sun (TRACE)‏ • Observations in the 171 A° and 195 A° Lenz et. al (1999)‏

4. Hydrostatic model Footpoint heating Apex heating Uniform heating Loop density: Warm loops Hot loops • 59 studied loops • It turned out that warm (<2MK) loops are ‘‘overdense“, while hot (>2 MK) loops are ‘‘underdense ’’ compared to the static solutions with uniform heating. Winebarger et al. (2003)‏

5. Loop width determination: Loops observed by SXT Loops observed by TRACE Total number of the observed loops = 24 (TRACE) + 43 (SXT)‏ : The expansion factor • Most of the expansion factors of the observed loops range between 1 and 1.5. • The mean expansion factor of hot loops (>1.5 MK ) observed by TRACE (28.4 nm) and SXT is about 1.13 i.e. the cross-section is about 30% wider at their mid-points than at their footpoints. • The mean expansion factor of warm loops (<1.5MK) observed by TRACE (17.1 and 19.5 nm) is about 1 (homogeneous cross section). Watko and Klimchuk/ Klimchuk (2000)‏

6. The modeling of the heating of coronal loops should adress the following steps: Energy source Conversion mechanism Plasma response Plasma emission The generation mechanism, and the amount of the energy released, and the location of this energy deposition in the coronal loop. The dissipation mechanism of the released energy along the coronal loop. The heating and dynamic profiles of the loop resulting from the converted energy. The synthesized loop emission as it would appear in the available observational instruments.

7. Heating mechanisms of coronal loops Microphysics m Macrophysics small-scale fluctuations AC and DC Collisionless dissipation mechanisms Collisional dissipation mechanisms • To avoid the complexity of the microphysics: Can the collisional heating mechanisms provide enough thermal energy to balance the energy lost by radiation?

8. Can the coronal loops be heated by collisions? • Let’s assume the typical values of the different plasma parameters in upper chromosphere: Ne = 1010 cm-3 , ∆Te = 1000 K , L = 200 km, ∆B = 1G, ∆V= 1 km, c = 1-10 km • The dissipation rates can be estimated (in cgs) : QV = η (∆V/L)2 = 2 x 10-8 (Viscosity) , Qc= k (∆T / L)2 = 3 x 10-8 (thermal conduction) Qj = j2 /  = (c/4)2 (∆B/L)2 /  = 7 x 10-7 (Ohmic dissipation) Have to balance But they are smaller by ~ 6 order of magnitude • Radiative cooling rate: QR = Ne 2(T) = 10-1 erg cm-3 s-1 • We may need to decrease L to solve the problem, but L >>  (Chapmann-Enskog regime) has to be satisfied Impossble to heat the upper chromosphere by collisions

9. Why Kinetic description of the coronal loop plasma is needed? • Generally the corona is weakly collisional and far from LTE (mean free path larger than 1 Mm). The coronal loop is denser than the surrounding plasma, but might also not be in LTE. • The loop heating mechanism presumably acts on a scale smaller than the mean free path of the particles. Consequently, HD or MHD models will not be able to really address the heating mechanism of coronal loops. • The classical collision coefficients are not large enough to dissipate the waves or the electric currents within the loop scales. (Typical mean free path Mm)‏ • Wave-particle interactions with resonant absorption of ion-cyclotron waves (with wavelengths of about 10-100 m) might be a possible scenario for the heating of the loop plasma.

10. Resonant wave-particle interactions (lower corona)‏ > The Solar corona is not in LTE (there is no thermodynamic temperature)‏ > The ions have different temperatures and are hotter than electrons > The ions are heated directly but not the electrons > Small fluctuations (ion-cyclotron waves) lead to preferential heating ~ q/m CH observations Dolla and Solomon 2008

11. : The expansion factor Loop geometry: Kinetic model for coronal loop TR Asymmetric heating Symmetric heating Boundary conditions . The loop length is L≈ 63 Mm . The height of the mid-point above the TR is about 30 Mm Bourouaine et al. 2008a

12. Vlasov equation for reduced VDFs of ions: Electron energy equation: Energy conservation: Transport equation of wave energy: Kinetic model for coronal loop Equations: Bourouaine et al. 2008a

13. Multi-ions plasma loop: Steady final state for and   1 N 10 N He p   3 N 10 N O p     15 3 N q N 5 10 m e i i i • The abandunces for • Remarkable temperature anisotropy in oxygen ions. • The temperature isotropy is well maintained for the other ion species. • In case of homogeneous loop the plasma is not in LTE only close to the left footpoint. • The plasma is not in LTE in the expanding loop. • The bulk flow speed is roughly constant in the steady state for • The wave-particle interactions lead for a perpendicular ion heating with respect to the magnetic field • The Coulomb collisions tend to thermalize the plasma (VDFs nearly Maxwellian)‏ Fig. (a),(b)‏ Fig. (c),(d)‏ (dashed lines)‏ (solid lines)‏ Bourouaine et al. 2008b

14. Wave absorption mechanism (1): • The ions are heated by the wave absorption. • Waves that interact with protons have frequency: • At higher altitudes far from the footpoints, only waves having frequencies will interact with ions (the interacting ions have negative velocity . Represents the frequency interval of the waves that interact with proton

15. 3.3 Mm (solid lines)‏ 5.3 Mm (dashed lines)‏ 15.6 Mm (dash-dotted lines)‏ 31.4 Mm (dot-dot-dashed lines)‏ Wave absorption mechanism (2): • The three ion species can simultaneously be heated • The ion species interact with a narrow interval of frequencies close to the ion gyrofrequency in case of nearly-homogeneous loop • Wider frequency interval of the spectrum resonate with ions in plasma in case of a more expanding loop • More preferential heating for minor oxygen ions in the perpendicular direction of the mean magnetic field Bourouaine et al. 2008b

16. Velocity distribution function of oxygen ion: VDFs (Maxwellian)‏ VDFs is not Maxwellian • Oxygen ions having on the tails of VDF are much more diffused • Oxygen ions moving inward fall in resonance with waves having frequencies . This is because all waves with frequencyies were already absorbed at lower heights. Bourouaine et al. 2008b

17. Conclusion • In our model, the ion heating has a strong spatial connection with the variation of the cross-section area along the loop. • The footpoint-type heating with small heating scale height is a consequence of a quasi-homogeneous flux tube the ion temperature remains nearly constant, and the plasma is nearly thermal in a large section of the loop. • If the magnetic field of the loop is more diverging from both footpoints to the apex, the ion heating scale height is larger, leading to more uniform loop heating, and the plasma is far from LTE. • Electrons are cooler than ions for reasons of relatively high radiative losses and strong heat conduction.

18. Thank you

19. e-p plasma loop(symmetric heating): (solid lines)‏ (dashed lines)‏ . The waves penetrate the loop from both footpoints and heat the protons simultaneously . At t = 0 the loop is cool and denser, and after 25000s the loop relaxes to a steady final state . At initial time the gravity forces dominate due to the high density in the loop . At the final state the wave pressure balances the gas pressure at all positions in the loop (static loop)‏ Plasma evolution only for Total pressure at steady final state

20. Kinetic model for coronal loop Assumptions and approximations : • Quasi-linear Theory (QLT) of Vlasov equation (wave-particle interaction and Coulomb collisions are included) has been used. • Gyrotropic VDF • Parallel motion along the mean magnetic field is the only spatial dimension • Reduced VDFs depends on parallel velocity component • Non-dispersive Alfven/ ion cyclotron waves (small amplitude) are assumed

21. Dissipation rate: • The waves enter the simulation domain with energy flux about J m s • In case of quasi-homogeneous loop, the dissipation rate is higher at the footpoints and then rapidly decreases at positions far from the footpoints (small heating scale length)‏ • In case when the loop expands with the heating rate is slowly decreasing, leading to more uniform heating (higher heating scale length)‏