M 1

Patrick AntolinPatrick Antolin

Department of AstronomyGraduate School of ScienceKyoto University

Shock heating by Fast/Slow MHD waves along plasma loops

OutlineOutline

Introduction

Shock wave theory

Important previous work

Results

Conclusions and objectives

IntroductionIntroductionFor over 50 years it has been known that For over 50 years it has been known that coronal temperatures in the Sun exceed coronal temperatures in the Sun exceed photospheric temperatures by a factor of 200.photospheric temperatures by a factor of 200.

→ → Coronal Heating problemCoronal Heating problem

Golub & Pasachoff 1997Golub & Pasachoff 1997

Non-constant coronal Non-constant coronal structure:structure:

Active RegionActive Region

Quiet SunQuiet Sun

Coronal HoleCoronal Hole

Different Different heating heating mechanism?mechanism?

Active RegionActive Region (AR)(AR)

Quiet Sun RegionQuiet Sun Region

Coronal HoleCoronal Hole

(QS)(QS)

(CH)(CH)

→ → Different Different magnetic magnetic structurestructure

EIT/SoHO: red: 200 M K, green: 150 M K, blue:100 M KEIT/SoHO: red: 200 M K, green: 150 M K, blue:100 M K

Heating mechanismsHeating mechanisms Among all the heating mechanisms proposed so far Among all the heating mechanisms proposed so far

the most promising are:the most promising are:AC modelAC modelDC modelDC modelAcoustic heatingAcoustic heatingChromospheric reconnectionChromospheric reconnection

In which way do they differ? In which way do they differ?

Energy transport from photosphere to coronaEnergy transport from photosphere to coronaDissipation of energy in the coronaDissipation of energy in the corona

Fast/Slow MHD wave generationFast/Slow MHD wave generation

How are these waves generated?How are these waves generated? Convective motions of plasma at the of plasma at the

footpoints of magnetic field lines.footpoints of magnetic field lines. Nanoflares (reconnection events) (reconnection events)

→ → MHD waves propagate: MHD waves propagate: Fast//Slow MHD MHD mode and mode and Alfven Mode. Mode.

MHD wavesMHD waves

Fast MHD mode can Fast MHD mode can transport energy in transport energy in any direction

Slow MHD mode can only Slow MHD mode can only transport energy to transport energy to directions directions close to the magnetic field line

Alfven mode’s transported Alfven mode’s transported energy can’t be dissipated energy can’t be dissipated

R.J.Bray et al. 1991R.J.Bray et al. 1991

First step: 1DFirst step: 1D→ → What happens along the magnetic field line?What happens along the magnetic field line?

It can be perturbed mainly by 2 different ways:It can be perturbed mainly by 2 different ways:

Longitudinal and and transversal oscillations oscillations

Longitudinal wave Longitudinal wave (slow mode)(slow mode)

11

11

22

22Transversal wave Transversal wave

(fast mode)(fast mode)

Propagation along Propagation along the magnetic field the magnetic field lineline

How do these waves dissipate? As How do these waves dissipate? As they propagate, their non-linear nature makes them steep they propagate, their non-linear nature makes them steep into shocksinto shocks

Shock wave theoryShock wave theory

A shock is the result of different parts of a A shock is the result of different parts of a wave traveling at different speeds.wave traveling at different speeds.

Suzuki 2004

For longitudinal waves we For longitudinal waves we have have N waves type of shock type of shock

For transversal waves we For transversal waves we have have Switch-on shock trains

Alfven waves don’t steep Alfven waves don’t steep into shocksinto shocks

1D model1D model

Objective: create a 1D model of a loop being Objective: create a 1D model of a loop being heated by Fast and Slow MHD waves. heated by Fast and Slow MHD waves.

→ → Can such a model produce and maintain a Can such a model produce and maintain a corona (taking into account radiation and corona (taking into account radiation and conduction losses) ?conduction losses) ?

Energy budgetEnergy budget

Aschwanden 2001b

Coronal Hole: Coronal Hole: 　＜　＜ 1010 ＾^ 4 4

Quiet Sun:Quiet Sun: 1010 ＾^ 4 4 ~~ 10 10 ＾^ 55

Active Region: Active Region: 　 　 1010 ＾^ 5 5 ~~ 10 10

＾^ 6 6 (erg/cm^2/s^1)(erg/cm^2/s^1)

Wave energy budgetWave energy budget

Slow mode MHD wave:Slow mode MHD wave: ,2|||| scvF

1615311 s cm10,scm10,cmg10 scv125

|| scmerg10 F

Fast mode MHD wave:Fast mode MHD wave: ,2AvvF

1715 scm10,scm10 Avv

126 scmerg10 F

→ → Enough for heating CH, QS and a portion of AR loopsEnough for heating CH, QS and a portion of AR loops

Limits of the mechanismLimits of the mechanism

Reflection of the Fast mode MHD wave has a Reflection of the Fast mode MHD wave has a high probability (stratification of the high probability (stratification of the atmosphere)atmosphere)

Dissipation of the waves in the corona is Dissipation of the waves in the corona is difficult (dissipation occurs mostly in the difficult (dissipation occurs mostly in the photosphere, chromosphere and TR) (Stein & photosphere, chromosphere and TR) (Stein & Schwartz 1972)Schwartz 1972)

ModelModelSuppositions:Suppositions:

Steady atmosphere (Steady atmosphere (/ / t = 0t = 0))

Gravity ignored for the propagation of the N WavesGravity ignored for the propagation of the N Waves

Weak shock approximation. Weak shock approximation. α: amplitude of the shock α: amplitude of the shock → → indicates amount of dissipationindicates amount of dissipation

No viscosityNo viscosity

WKB approximationWKB approximation

Dissipation occurs only through the shocksDissipation occurs only through the shocks

1||||

sc

v

1

Av

v

Variation of amplitude of shock: Variation of amplitude of shock: degree of dissipationdegree of dissipation

N Waves:N Waves: 　　（（ SuzukiSuzuki ，， ApJ 578, 2002ApJ 578, 2002 ））

ds

dc

cds

dA

Acds

d

ds

ds

ss

31)1(21

2 ||

||||||

1 2 3 4

1: 1: stratification stratification 2:2: shock heating shock heating 3:3: geometrical expansion geometrical expansion 4:4: temperature variation temperature variation

Switch-on Shock TrainsSwitch-on Shock Trains （（ Suzuki,MNRAS Suzuki,MNRAS

349,2004349,2004 ））

ds

dv

vvds

dv

vvv

vv

ds

dA

Avcvvv

B

ds

d

ds

d

A

A

AA

A

AsAA

2

)(

31

)/1)((8

31

2 222

2||

Switch-on shock trains are less dissipative than N wavesSwitch-on shock trains are less dissipative than N waves

MHD equations and geometryMHD equations and geometry

Variation of the Area Variation of the Area along the loopalong the loop

Mass continuity Mass continuity constvA

Moriyasu et al. 2004Momentum equation Momentum equation

ds

dp

ds

dp

ds

dp

LsL

R

LsGM

ds

dvv

111

/sin

/cos ||2

Ideal gas equation Ideal gas equation Tm

kp B

constvA

MHD equations (2)MHD equations (2)

RQQAFds

d

Am

Tk

ds

dv c

B

||1

1

1

Heat equation Heat equation

Conservation of magnetic flux Conservation of magnetic flux constvA

Volumetric heating at the shocks for the waves Volumetric heating at the shocks for the waves

222

42||

)/1(16 As vc

BQ

||

3||

|| 3

)1(2

p

Q

(longitudinal waves)(longitudinal waves)

(transversal waves)(transversal waves)

Important previous work: open magnetic field

Propagation of acoustic Propagation of acoustic shocks along open shocks along open magnetic fieldsmagnetic fields

Dissipation of wave Dissipation of wave depends on:depends on:

PeriodPeriod Height of generationHeight of generation Sound speedSound speed

→ → Cannot heat the coronaCannot heat the coronaFoukal&Smart S.Ph.69,1981

ChromosphereChromosphere CoronaCorona

Important previous work: open magnetic field (2)

Case of Switch-on Case of Switch-on shock trains:shock trains:

Case of weak Case of weak magnetic field: magnetic field: dissipation in a few dissipation in a few solar radii.solar radii.

Case of strong Case of strong magnetic field: low magnetic field: low dissipationdissipation

Hollweg, ApJ 254,1981:

Strong magnetic Strong magnetic field. Low field. Low dissipationdissipation

Weak, High Weak, High dissipationdissipation

→ → Possible heating mechanism for Coronal Holes and Possible heating mechanism for Coronal Holes and Quiet Sun regions.Quiet Sun regions.

Important previous work: open magnetic field (3)

Coronal heating and solar Coronal heating and solar wind acceleration by wind acceleration by Fast/Slow MHD waves:Fast/Slow MHD waves:

For heating of CH and QS For heating of CH and QS regions of the inner corona regions of the inner corona the N Waves are more the N Waves are more important than the Switch-important than the Switch-on shock trainson shock trains

Inverse for the outer Inverse for the outer coronacorona Suzuki 、 MNRAS 349

(2004)

Results: static case, N Results: static case, N waveswaves

αα(0) = 0.3(0) = 0.3 ρρ (0)= 1.5x10^-12 g cm^-3 (0)= 1.5x10^-12 g cm^-3

T(0) = 6440 KT(0) = 6440 K

F(0) = 34000 erg cm^-2 s^-1F(0) = 34000 erg cm^-2 s^-1

Results: steady case (subsonic), N Results: steady case (subsonic), N waveswaves

αα(0) = 0.8(0) = 0.8 ρρ (0)= 2.8x10^-14 g cm^-3 (0)= 2.8x10^-14 g cm^-3

T(0) = 14000 KT(0) = 14000 K

F(0) = 10^5 erg cm^-2 s^-1F(0) = 10^5 erg cm^-2 s^-1

Results: steady case (subsonic), N Results: steady case (subsonic), N waves (2)waves (2)

v(0) = 2km/sv(0) = 2km/s

Conclusions for open and closed Conclusions for open and closed magnetic field casesmagnetic field cases

Important parameters for the formation of shocks:Important parameters for the formation of shocks: Height of generation of the waveHeight of generation of the wave Period of the wavePeriod of the wave Initial amplitude of the wave: dependent on local Initial amplitude of the wave: dependent on local

physical parameters, as density, temperature, physical parameters, as density, temperature, magnetic field and geometrical expansionmagnetic field and geometrical expansionFor the open field case: For the open field case:

the N waves seem to be enough for heating the CH the N waves seem to be enough for heating the CH and QS regions of inner corona (1.5-2 R)and QS regions of inner corona (1.5-2 R)

Switch-on shock trains dissipate less rapidly: they are Switch-on shock trains dissipate less rapidly: they are important for the heating of the outer coronaimportant for the heating of the outer corona

Closed magnetic field caseClosed magnetic field case

For N Waves:For N Waves: The static case reproduces well the physical The static case reproduces well the physical

conditions from the low chromosphere to the conditions from the low chromosphere to the corona.corona.

The steady case is only able to reproduce the The steady case is only able to reproduce the conditions from the upper chromosphere.conditions from the upper chromosphere.

In the steady case the flow seems to play a In the steady case the flow seems to play a cooling effect in the upper part of the loop.cooling effect in the upper part of the loop.

Future workFuture work

Extend the steady case for the N waves to the Extend the steady case for the N waves to the low chromospherelow chromosphere

Static and steady (subsonic) cases for the Static and steady (subsonic) cases for the Switch-on shock trainsSwitch-on shock trains

Transonic cases for N waves and Switch-on Transonic cases for N waves and Switch-on shock trainsshock trains

Dynamical treatment of both cases (time-Dynamical treatment of both cases (time-dependent)dependent)