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

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

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

9. Shock wave theory
A shock is the result of different parts of a wave traveling at different speeds.
Suzuki 2004
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
　＜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) = 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) = 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)