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1. Patrick Antolin M 1 Shock heating by Fast/Slow MHD waves along plasma loops Department of Astronomy Graduate School of Science Kyoto University

2. Outline • Introduction • Shock wave theory • Important previous work • Results • Conclusions and objectives

3. Introduction • For over 50 years it has been known that coronal temperatures in the Sun exceed photospheric temperatures by a factor of 200. → Coronal Heating problem Non-constant coronal structure: • Active Region • Quiet Sun • Coronal Hole Different heating mechanism? Golub & Pasachoff 1997

4. Active Region (AR) Quiet Sun Region (QS) Coronal Hole (CH) → Different magnetic structure EIT/SoHO: red: 200 M K, green: 150 M K, blue:100 M K

5. Heating mechanisms • Among all the heating mechanisms proposed so far the most promising are: • AC model • DC model • Acoustic heating • Chromospheric reconnection In which way do they differ? • Energy transport from photosphere to corona • Dissipation of energy in the corona

6. Fast/Slow MHD wave generation How are these waves generated? • Convective motions of plasma at the footpoints of magnetic field lines. • Nanoflares (reconnection events) → MHD waves propagate: Fast/Slow MHD mode and Alfven Mode.

7. MHD waves • Fast MHD mode can transport energy in any direction • Slow MHD mode can only transport energy to directions close to the magnetic field line • Alfven mode’s transported energy can’t be dissipated R.J.Bray et al. 1991

8. 1 1 2 2 First step: 1D → What happens along the magnetic field line? It can be perturbed mainly by 2 different ways: Longitudinal and transversal oscillations Longitudinal wave (slow mode) Propagation along the magnetic field line Transversal wave (fast mode) • How do these waves dissipate? As they propagate, their non-linear nature makes them steep into shocks

9. Suzuki 2004 Shock wave theory • A shock is the result of different parts of a wave traveling at different speeds. • For longitudinal waves we have N waves type of shock • For transversal waves we have Switch-on shock trains • Alfven waves don’t steep into shocks

10. 1D model • Objective: create a 1D model of a loop being heated by Fast and Slow MHD waves. → Can such a model produce and maintain a corona (taking into account radiation and conduction losses) ?

11. Energy budget • Coronal Hole: 　＜10＾4 • Quiet Sun: 10＾4 ~ 10＾5 • Active Region: 10＾5 ~ 10＾6 (erg/cm^2/s^1) Aschwanden 2001b

12. Wave energy budget • Slow mode MHD wave: • Fast mode MHD wave: → Enough for heating CH, QS and a portion of AR loops

13. Limits of the mechanism • Reflection of the Fast mode MHD wave has a high probability (stratification of the atmosphere) • Dissipation of the waves in the corona is difficult (dissipation occurs mostly in the photosphere, chromosphere and TR) (Stein & Schwartz 1972)

14. Model • Suppositions: • Steady atmosphere (/ t = 0) • Gravity ignored for the propagation of the N Waves • Weak shock approximation. α: amplitude of the shock → indicates amount of dissipation • No viscosity • WKB approximation • Dissipation occurs only through the shocks

15. Variation of amplitude of shock: degree of dissipation • N Waves:（Suzuki，ApJ 578, 2002） 1 2 3 4 1: stratification 2: shock heating 3: geometrical expansion 4: temperature variation Switch-on Shock Trains（Suzuki,MNRAS 349,2004）  Switch-on shock trains are less dissipative than N waves

16. MHD equations and geometry • Variation of the Area along the loop • Mass continuity • Momentum equation Moriyasu et al. 2004 • Ideal gas equation

17. MHD equations (2) • Heat equation • Conservation of magnetic flux • Volumetric heating at the shocks for the waves (longitudinal waves) (transversal waves)

18. Important previous work: open magnetic field • Propagation of acoustic shocks along open magnetic fields • Dissipation of wave depends on: • Period • Height of generation • Sound speed → Cannot heat the corona Chromosphere Corona Foukal&Smart S.Ph.69,1981

19. Important previous work: open magnetic field (2) • Case of Switch-on shock trains: • Case of weak magnetic field: dissipation in a few solar radii. • Case of strong magnetic field: low dissipation Hollweg, ApJ 254,1981: Strong magnetic field. Low dissipation Weak, High dissipation → Possible heating mechanism for Coronal Holes and Quiet Sun regions.

20. Important previous work: open magnetic field (3) • Coronal heating and solar wind acceleration by Fast/Slow MHD waves: • For heating of CH and QS regions of the inner corona the N Waves are more important than the Switch-on shock trains • Inverse for the outer corona Suzuki、MNRAS 349 (2004)

21. Results: static case, N waves ρ (0)= 1.5x10^-12 g cm^-3 α(0) = 0.3 F(0) = 34000 erg cm^-2 s^-1 T(0) = 6440 K

22. Results: steady case (subsonic), N waves ρ (0)= 2.8x10^-14 g cm^-3 α(0) = 0.8 F(0) = 10^5 erg cm^-2 s^-1 T(0) = 14000 K

23. Results: steady case (subsonic), N waves (2) v(0) = 2km/s

24. Conclusions for open and closed magnetic field cases • Important parameters for the formation of shocks: • Height of generation of the wave • Period of the wave • Initial amplitude of the wave: dependent on local physical parameters, as density, temperature, magnetic field and geometrical expansion • For the open field case: • the N waves seem to be enough for heating the CH and QS regions of inner corona (1.5-2 R) • Switch-on shock trains dissipate less rapidly: they are important for the heating of the outer corona

25. Closed magnetic field case • For N Waves: • The static case reproduces well the physical conditions from the low chromosphere to the corona. • The steady case is only able to reproduce the conditions from the upper chromosphere. • In the steady case the flow seems to play a cooling effect in the upper part of the loop.

26. Future work • Extend the steady case for the N waves to the low chromosphere • Static and steady (subsonic) cases for the Switch-on shock trains • Transonic cases for N waves and Switch-on shock trains • Dynamical treatment of both cases (time-dependent)