Transcript Shock heating by Fast/Slow MHD waves along plasma loops Patrick Antolin

Shock heating by Fast/Slow
MHD waves along plasma loops
Patrick Antolin
M1
Department of Astronomy
Graduate School of Science
Kyoto University
Outline
Introduction
Shock wave theory
Important previous work
Results
Conclusions and objectives
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
Golub & Pasachoff 1997
Different
heating
mechanism?
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
→ What happens along the magnetic field line?
It can be perturbed mainly by 2 different ways:
Longitudinal and transversal oscillations
1
2
Longitudinal wave
1
Propagation along
(slow mode)
the magnetic field
2
Transversal wave
(fast mode)
line
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.
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
Suzuki 2004
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
Active Region:
10＾5 ~ 10＾6
(erg/cm^2/s^1)
Aschwanden 2001b
Wave energy budget
Slow mode MHD wave: F||  v c ,
11
3
5
1
6
1
  10 g cm , v  10 cm s , cs  10 cm s
2
|| s
2 1
F||  10 erg cm s
5
Fast mode MHD wave: F  v v A ,
2
v  105 cm s 1 , v A  107 cm s 1
2 1
F  10 erg cm s
6
→ 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.
v||
|| 
 1
α: amplitude of the shock
cs
→ indicates amount of dissipation   v  1

vA
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 d 2(  1)|| 1 dA 3 dcs 
 




ds
2   ds
cs ||
A ds cs ds 
d||
1
1: stratification
2: shock heating
2
3
3: geometrical expansion
4
4: temperature variation
Switch-on Shock Trains （Suzuki,MNRAS 349,2004）
3B|| 2
d     1 d
1 dA
3v A  v dv A
2 dv 








ds
2   ds 8  v A  ( v A  v )(1  cs2 / v A2 ) 2 A ds v A ( v A  v ) ds v A  v ds 
 Switch-on shock trains are less dissipative than N waves
MHD equations and geometry
Variation of the Area
along the loop
Mass continuity
vA  const
vA  const
Momentum equation
dv
GM coss / L 
v

ds
Moriyasu et al. 2004
1 dp 1 dp|| 1 dp



2
 ds  ds  ds
L


 R  sin s / L 



Ideal gas equation
kB
p
T
m
MHD equations (2)
Heat equation
  d  k BT 
1 d

 AFc   Q||  Q  R
v 
 1 
  1 ds  m 
A ds
Conservation of magnetic flux
vA  const
Volumetric heating at the shocks for the waves
Q|| 
Q 
2 (  1) p ||
B||
2
3 ||
3
(longitudinal waves)
 4
16 (1  c / v )
2
s
2
A
2
(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 Switchon shock trains
 Inverse for the outer
corona
Suzuki、MNRAS 349 (2004)
Results: static case, N waves
α(0) = 0.3
ρ (0)= 1.5x10^-12 g cm^-3
F(0) = 34000 erg cm^-2 s^-1
T(0) = 6440 K
Results: steady case (subsonic), N
waves
α(0) = 0.8
ρ (0)= 2.8x10^-14 g cm^-3
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 (timedependent)