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

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

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

→ 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

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

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.

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

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

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

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) ?

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

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

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)

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

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

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

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

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

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.

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)

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

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

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

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

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.

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)