Download Coronal heating: a theoretical approach

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PPARC, Adv. Summer
School, Palma 2006
Coronal heating: a
theoretical approach
Istvan Ballai
SPARG, University of Sheffield
PPARC, Adv. Summer
School, Palma 2006
Introduction
•The eclipse of 1869
revealed emission line in the
green part of the corona- was
named coronium.
•Grotrian in 1939 finally
showed that this emission
line to be due to Fe XIV at
5303Å.
•This demonstrated that the
corona has a temperature >
1MK, and so the coronal
heating problem began….
PPARC, Adv. Summer
School, Palma 2006
Publications/year (ADS for coronal heating)
(more than 4,600 publications)
350
Yohkoh
300
TRACE
250
SoHO
200
150
Skylab
100
50
0
1960 1964 1968 1972 1976 1989 1984 1988 1992 1996 2000 2004
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Introduction
• Problem: very high temperature of the upper atmosphere
• Question: what heating mechanism(s) do operate?
PPARC, Adv. Summer
School, Palma 2006
Multi-temperature vision of the Sun
 Blue: EIT 171 A (0.95 MK)
 Green: EIT 195 A (1.5MK)
 Red: EIT 284 A (2MK)
PPARC, Adv. Summer
School, Palma 2006
Introduction
“The literature of coronal heating is primarily theoretical.
Observations are often cited in support of a proposed theory
or another, ..but…neither existing observations nor the
current generation of models are sufficiently detailed to test
any mechanism critically.” (Zirker, 1993)
• We now have an explosion of high resolution space datasets
(Yohkoh, SOHO, TRACE, RHESSI and more to come) –
that are providing constraints on theory and distinguishing
between possible models.
The coronal heating is still an unsolved problem in the solar
and stellar physics
PPARC, Adv. Summer
School, Palma 2006
Observational facts
• Highly inhomogeneous
• Rôle of magnetic field
PPARC, Adv. Summer
School, Palma 2006
Observational facts
• Highly inhomogeneous
• Role of magnetic field
PPARC, Adv. Summer
School, Palma 2006
Observational facts
• Consists of myriads of coronal loops
PPARC, Adv. Summer
School, Palma 2006
Observational facts
• length scale: from resolution up to 700
Mm
Flux
tubes
• radius: from resolution up to 10 Mm
• temperature: from 1-2x104 K to 2x106 K
• magnetic field strength: 1- 104 G
PPARC, Adv. Summer
School, Palma 2006
• equilibrium bulk motion
The complex problem of coronal heating
Conversion
mechanism
Energy source
Heating
Plasma response
Radiation
Observables
PPARC, Adv. Summer
School, Palma 2006
Klimchuk, 2006
The energy requirement
Parameter
(erg cm-2s-1)
Coronal
hole
(open)
Chromospheric 4 ×106
radiation loss
Active
region
(closed)
2 ×107
Radiation
104
< 106
Conduction
5 ×104
105 – 106
Solar wind
5
×
(5-10) 10
PPARC, Adv. Summer
School, Palma 2006
5
( < 10 )
The energy source
Widely accepted: mechanical motions in and below the photosphere
Footpoint motions can generate stresses (DC currents) and waves
(AC currents) depending on the time-scale of the motion compared to
the Alfven time
EUV, UV, X-ray coronal images and magnetograms firmly established
that coronal heating is a magnetic phenomenon (e.g. Vaiana and
Rosner 1978)
Heating models
DC models
AC models
(Kinetic,
turbulences)
PPARC, Adv. Summer
School, Palma 2006
tdr>tA
tdr<tA
Hybrid AC/DC models
The energy source – DC heating
•
Footpoint motions perform work on the coronal magnetic field and increase its
free energy at a rate given by the Poynting flux through the base
F
•
•
•
•
1

Bv B h  Vh
Magnetic field concentrated in small tubes (~kG) which expand out in the
chromosphere and transition region
Small loops form a low-laying “magnetic carpet” and they do not penetrate
into the corona
Part of the inter-network flux extends above the carpet and spreads out in the
corona  the magnetic field in the quiet Sun is a mixture of network field and
surviving inter-network field.
Bv~100 G (AR), 5-10 G (QS), Vh~105 cm/s, assume Bh~Bv: F≈108 erg/cm2 s
PPARC, Adv. Summer
School, Palma 2006
Thin tubes merge into corona
Peter (2001)
Tu et al.
(2005)
PPARC, Adv. Summer
School, Palma 2006
Heating by DC currents
2D reconnection theories
¤ 2D
reconnection: Xpoint collapses to
a singular sheet
¤ Magnetic energy  heat+K.E.+ fast particles
¤ Well understood
¤ Source of heating and of many dynamic processes (flares, EEs, TRBs)
PPARC, Adv. Summer
School, Palma 2006
2D reconnection theories
¤ In 2D well-developed
Slow Sweet-Parker reconnection (1958); rec. rate ≈R-1/2
Fast Petschek reconnection (1964)rec. rate ≈1/ln R
Many other fast regimes (depend on B.C.’s)
Almost uniform (Priest &Forbes, 1986)
Non-uniform (Priest & Lee, 1992)
Excellent review by Priest and Forbes (Magnetic reconnection, CUP, 2000)
PPARC, Adv. Summer
School, Palma 2006
3D reconnection theories
Key question: structure of null-point
¤ Simplest: B=(x,y,-2z)
¤ Two families of field lines through
null-point:
Spine field lines
Fan surface
PPARC, Adv. Summer
School, Palma 2006
3D reconnection theories
Three types of reconnection at Null
¤ Spine reconnection
¤ Fan reconnection
¤ Separator reconnection
Double 3D null-point topology
(courtesy of K. Garlsgaard)
PPARC, Adv. Summer
School, Palma 2006
3D reconnection theories
Spine reconnection
PPARC, Adv. Summer
School, Palma 2006
Fan reconnection
3D reconnection theories
¤ So, can reconnection heat the corona?
¤ Yes, possibly, in different ways…but observations are
needed to see which way!
Examples:
 Reconnection at null-point, e.g., XBP interpreted as
converging flux
(Parnell et al. 1993,
Priest et al. 1994)
PPARC, Adv. Summer
School, Palma 2006
The energy source – AC heating
•
•
•
•
•
The turbulent convection that stresses the coronal magnetic field generates a
large flux of upwardly propagating waves (acoustic, Alfvén, slow and fast
magnetosonic)
Mode coupling and other processes transfer energy between different types of
waves, so the mix of waves changes as a function of height.
Theoretical and observational estimates suggest energy fluxes at the top of
convection zone of several 107 erg/cm2s (Narain & Ulmschneider,1996)
more than adequate to heat the corona
Only a small fraction of the flux is able to pass through the very steep density
and temperature gradients in the chromosphere and transition region.
Acoustic and slow waves steepen into shock waves and are strongly damped,
while fast waves are strongly refracted and reflected, only Alfvén waves are
able to penetrate into the corona. The do not form shocks since they are
transversal and their energy is ducted along the magnetic field rather than
being refracted across it.
PPARC, Adv. Summer
School, Palma 2006
Behaviour of acoustic waves
Chromospheric heating by acoustic waves
• Convection generates acoustic waves
propagating upwards, steepens into shock
waves or are reflected by the density
gradients in the TR
FM   CS v 2
   0e
h
H
Ae
2

k 
2
v
vA2
k
2
h
PPARC, Adv. Summer
School, Palma 2006
H  160 Km
,
h
2H
k 0
2
v
v 2  A2
evanescent
waves
Behaviour of Alfvén waves
•
•
•
Significant transmission of Alfvén waves is possible only within narrow
frequency bands centered on discrete values where loop resonance
conditions are satisfied (Hollweg, 1981)
Enough flux may pass through the base of long (>100 Mm) active regions
loops to provide their heating (Hollweg, 1985); in the case of short loops this
does not apply.
Waves can be generated in the corona itself by, e.g. magnetic reconnection
and change of the equilibrium (AC/DC heating mechanism)
PPARC, Adv. Summer
School, Palma 2006
Heating by AC currents
• Recent high resolution observations show undoubtful evidence for
waves in the corona
• Prominences
• Plumes
• Corona (EIT/SoHO, TRACE)
– Flare excited waves in loops-fast kink modes (Aschwanden et al.
1999, Nakariakov and Ofman 1999)
– Feet of long loops-slow waves (De Moortel et al. 2002ab,
Aschwanden et al. 2002 )
– CME/flare excited global waves (EIT waves) –fast waves
(Thompson et al. 1999, Ballai and Erdélyi 2003a,b, Ballai et al.
2005)
For an effective damping these waves require small scales
PPARC, Adv. Summer
School, Palma 2006
Resonant absorption
ωdriver =
ωlocal
Ideal MHD equations singular dissipation  heating
Concept of Connection Formulae
PPARC, Adv. Summer
School, Palma 2006
(Ionson 1978, Rae &
Roberts 1982, Hollweg
1984, Poedts et al. 1989,
Goossens 1991,
Ruderman et al 1997ab,
Ballai et al. 1998ab,
Ballai and Erdélyi
1998,2000ab, etc,etc)
Why resonant absorption ?
• Inhomogeneous plasmas: natural behaviour
• Easy wave energy transfer resulting in heating
• Condition to occur: ωdriver = ωlocal
• Could/may/viable to explain:
- local/atmospheric heating
- power loss of acoustic waves in sunspots
- damping of helioseismic (p/f/g) eigenmodes
- energisation of MHD waves in magneto/heliosphere
PPARC, Adv. Summer
School, Palma 2006
Resonant absorption
•
•
•
•
High frequency Alfvén waves are
able to reach corona
They are incompressible and
transversal subject to damping
due to ohmic and/or shear
viscosity
In the corona ν/µ≈1011 and
η0/η1≈105 , so they have a very
weak damping.
For effective damping small
trasversal scales are
requiredresonant absorption
PPARC, Adv. Summer
School, Palma 2006
d (r r )
 C1r r  C2 rP1
dr
dP
D 1  C3 r  C1 P1
dr
D   (c 2  v A2 )( 2   A2 )( 2   C2 )
D
Concept of connection formulae
• Driven problem  ω is prescribed
• Eigenvalue problem  ω is searched for
g BC A
 r   i 2 sgn  
B 
2 BzTC A
P1   i 2
sgn  
B  A 
C A  const
Jumps are independent of dissipative coefficient
PPARC, Adv. Summer
School, Palma 2006
Resonant absorption
PPARC, Adv. Summer
School, Palma 2006
Internal background motion
5-6%
vA
•Steady large-scale flows (e.g., Doyle et al. 1997)
•Flow has a major influence on resonant absorption
PPARC, Adv. Summer
School, Palma 2006
But…
•
•
•
ε– the dimensionless amplitude of the
perturbations; R– total Reynolds number;
f—any large variable
  1
 1
 linear theory
 nonlinear theory

f f
2

 f
z   R 2 3
r 2
2
  1   R 3  1
• Suppose
for   102  R  103
Resonant absorption is a nonlinear phenomena
(Ruderman et al.1997, Ballai et al. 1998,1999, 2000, Ballai and Erdélyi
1998)
PPARC, Adv. Summer
School, Palma 2006
But…
• Nonlinearity gives just a small correction to the net absorption
coefficientlinear theories give acceptable solutions (Ruderman
2000)
• Nonlinearity in dissipative layers generate a mean flow outside the
layer
• The mean (turbulent) flow can locally enhance the dissipative
coefficients
• The observation of the generated mean flow could be a first
evidence of the resonant absorption
PPARC, Adv. Summer
School, Palma 2006
(Ofman and Davila
1995)
PPARC, Adv. Summer
School, Palma 2006
Resonant absorption/phase mixing
• To have a heating for the entire loop, we have to suppose that
waves are not monochromatic or stochastic processes have to be
taken into account (Tsiklauri and Nakariakov 2002, Ruderman 2003)
• Dissipative layer the oscillations are in phase as long as ω and
kvA are in phase
• If they start to be out of phase phase mixing (Heyvaerts and
Priest 1983, Browning and Priest 1984, Hood et al 1997, Nakariakov
et al 1997, Ruderman et al. 1998, De Moortel et al. 2000, Tsiklauri et
al. 2003, etc.)
PPARC, Adv. Summer
School, Palma 2006
Energy conversion-conclusion
•
•
Through energy conversion, the magnetic stress energy and wave energy is
transformed into heat.
Since classical dissipation coefficients are small in the corona, significant
heating requires the formation of steep gradients and small length scales.
Magnetic gradients  heating by reconnection and Ohmic
dissipation
Velocity gradients  heating by viscous dissipation
• Gradients are formed through slow quasi-static evolution and through
dynamical processes
• Possible scenarios: instabilities, turbulences, loss of equilibrium, simple and
complex flow patterns at the base of complex coronal magnetic fields (DC)
and resonant absorption, phase mixing (AC)
PPARC, Adv. Summer
School, Palma 2006
Energy conversion and microphysics
•
•
•
•
Microphysics is likely to play a key role in the energy conversion process,
e.g. anomalously large (nonclassical) transport coefficients are required for
significant heating even in the presence of steep gradients.
Coronal transport coefficients are not known with precision but indirect
techniques are used to infer values for, e.g. viscosity, thermal and electrical
conduction, etc.  CORONAL SEISMOLOGY (Nakariakov et al. 1999,
Ofman and Aschwanden 2002, Klimchuk et al. 2004, Ballai and Erdélyi
2005)
Collisionality of the coronal plasma: the collisionless effects are extremely
important for reconnection (Bhattacharjee 2004) and wave propagation
(Ballai et al. 2002)
Hybrid codes developed to take into account both the MHD and particle
aspects of the plasma
PPARC, Adv. Summer
School, Palma 2006
Plasma response
•
•
•
•
The fundamental principle: the close thermal and dynamic connection
between the corona and the lower atmosphere (coupled system)
In the case of static equilibrium, thermal conduction transports more than a
half of the coronal heating energy down to the transition region, where it is
more efficiently radiated
When heating is time-dependent, an increase in the heating rate causes the
coronal temperature to rise, producing an increase of the downward heat
flux. The TR is unable to radiate the additional energy, so heated plasma
flows into corona through “chromospheric evaporation”
If the upflow is fast, it can be explosive causing shocks, if the heating rate
then decreases, an inverse-like process occurs in which the plasma drains
from the loop and “condenses” back into the chromosphere.
PPARC, Adv. Summer
School, Palma 2006
Radiation
• We determine the radiation spectrum emitted by the heated corona
• If the plasma is in ionisation equilibrium, this task is relative simple
(see the CHIANTI software, Dere et al. 1997).
  emissivity  n G (T )
2
e
•
If the plasma is not in ionisation equilibrium the problem is much more
complicated. The equilibrium can be destroyed by, e.g.
– Rapid evolution of an impulsive heating
– Rapid cooling
– Flow through a steep temperature gradient
In this case we have solve the ionisation rate equation in order to
determine the radiation spectrum
PPARC, Adv. Summer
School, Palma 2006
Observation of heating events
•
•
•
Even the present high resolution satellites provide a minimum information
about the heating and the findings are often the result of averaging over
space, time and wavelength.
The best resolution at the moment is ≈ 350 km. In order to see heating at
work we would need 10-103 m (!!!)
Small-scale events have different names but they may turn out to belong to
identical physical processes.
- ephemeral regions
- nanoflares
- emerging flux events
- microflares
- flux cancellation
- soft X-ray jets
- events, blinkers
- AR transient brightening
- soft X-ray bright points
PPARC, Adv. Summer
School, Palma 2006
Small-scale phenomena and their occurrence domain (QS- quiet Sun,
AR-active region, Ph–photosphere, TR–transition region, C–corona)
Phenomenon
Horizontal
domain
Vertical domain
Wavelength
Ephemeral
regions
QS
Ph
Optical
Emerging flux
events
QS, AR
Ph
Optical
Flux cancellation
QS, AR
QS
QS, AR
QS, AR
Ph
TR
TR
C
Optical
EUV
EUV
EUV, SXR
X-ray
brightpoints
QS
C
SXR
Soft X-ray jets
QS, AR
AR
C
C
SXR
SXR
Explosive events
Blinkers
Nano- and
microflares
AR
brightenings
PPARC,
Adv. Summer
School, Palma 2006
Physical parameters of coronal small-scale phenomena (Lspatial scale, T-electron temperature, n-electron density)
Phenomenon
L [Mm]
T [MK]
n [x108 cm-3]
Nanoflares
2.8-7.9
1-1.4
2.9-4.4
QS transient
brightening
3.2-14.1
1.3-1.7
…..
QS heating event
4.5-7.9
1.2-1.5
7-20
AR transient
brightening
5-40
4-8
20-200
SXR jets
15-100
3-8
7-40
PPARC, Adv. Summer
School, Palma 2006
Open questions in the coronal heating
problem
• Are distinct coronal loops
heated differently from the
diffuse corona?
• Are there different classes of
loops that are heated in
different ways?
• Is quiet Sun heating similar to
active regions heating?
• How the AC/DC mechanisms
work together?
• Are stellar coronae heated in
the same way as the solar
corona?
PPARC, Adv. Summer
School, Palma 2006
UVCS results: solar minimum (1996-1997 )
• The fastest solar wind flow is expected to come from dim “coronal holes.”
• In June 1996, the first measurements of heavy ion (e.g., O+5) line emission in the
extended corona revealed surprisingly wide line profiles . . .
On-disk profiles: T = 1–3 million K
PPARC, Adv. Summer
School, Palma 2006
Off-limb profiles: T > 200 million K !
Heating of the open coronal structures
(Xing et
al. 2002)
Very strong perp. heating of the
oxygen
PPARC, Adv. Summer
School, Palma 2006
(Cranmer et
al. 1998)
The impact of UVCS
UVCS has led to new views of the collisionless nature of solar wind acceleration.
Key results include:
• The fast solar wind becomes supersonic
much closer to the Sun (~2 Rs) than
previously believed.
• In coronal holes, heavy ions (e.g., O+5)
both flow faster and are heated hundreds
of times more strongly than protons and
electrons, and have anisotropic
temperatures. (e.g., Kohl et al. 1997,1998)
PPARC, Adv. Summer
School, Palma 2006
Ion-cyclotron resonance
• SUMER and UVCS (SoHO) have provided very strict constraints on
heating of coronal holes
• H+ are mildly anisotropic (Tperp>Tparallel); O 5+ are strongly
anisotropic (Tperp/Tparallel=10-200) above 2-3 RSun
• At r=3RSun, Tperp for O5+ is 2x108K (vth=450 km/s), while H+ have
Tperp=3x106K (vth=225 km/s)
• At r=3.5 RSun the outflow speed of O5+ is twice the outflow speed of
H+
• These properties can be explained by the resonant interaction of coronal
ions with ion-cyclotron waves, i.e. by ion-cyclotron resonance
• Ion cyclotron waves (10-104 Hz) have not yet been observed in the solar
wind or corona (Cranmer et al. 1999)
• Some attempts to describe waves in collisionless plasmas (Nakariakov
and Oraevski 1995, Ballai et al. 2002)
PPARC, Adv. Summer
School, Palma 2006
Ion-cyclotron resonance
•
The condition of resonance
 k||   k||v||  i ,
•
•
•
•
i 
qi B
mi c
This mass-dependent mechanism is a wave-particle interaction
Ωi decreases with distance more and more energy injected at lower k is
swept into the high frequency domain, where is dissipated by the ions
Dissipation of ion-cyclotron waves produces diffusion in velocity space,
along contours of constant energy
Ions are accelerated along the field lines
PPARC, Adv. Summer
School, Palma 2006
Where do cyclotron waves come from?
(1) Base generation by, e.g., “microflare”
reconnection in the lanes that border
convection cells (e.g., Axford & McKenzie
1997).
(2) Secondary generation: low-frequency
Alfven waves may be converted into
cyclotron waves gradually in the corona.
Both scenarios have problems . . .
PPARC, Adv. Summer
School, Palma 2006
How the ion-cyclotron waves are
generated?
• Alfvén waves with frequencies > 10 Hz have not been observed in
the corona or solar wind
• Base generation: by, e.g. “microflare” reconnection in the lines that
border convection cells.
• Problem: Low Z/A ions consume base-generated wave energy
before it can be absorbed
• Secondary generation: The Sun is suspected to emit lowfrequency (<10 mHz) Alfvén waves. This source of “free energy”
may be converted into ion cyclotron waves gradually throughout the
corona (MHD turbulent cascade, instabilities seeded by nonMaxwellian distributions)
• Problem: Turbulence produces mainly high-kperp fluctuations (i.e. still
low frequency). Ion-cyclotron waves propagating parallel to B0 may
compromise only a small fraction of the total fluctuation power
PPARC, Adv. Summer
School, Palma 2006
Heating mechanisms
• A surplus of proposed ideas? (Mandrini et al. 2000; Aschwanden et al. 2001)
PPARC, Adv. Summer
School, Palma 2006
Conclusions: What do we need?
• Data analysis
• Direct observations
• Direct or indirect evidence for heating, e.g. mean flow
for resonant heating
• Observe reconnection driven resonant MHD waves
• Use the newly developed coronal seismology for
plasma and field parameters
PPARC, Adv. Summer
School, Palma 2006
Conclusions
• MHD heating occurs across S-STP
• Theories (waves, reconnection, turbulence) progressed
• “Candidates” are all natural for plasma heating/acceleration
• MHD heating is sensitive to flows
• Various structures may be heated by different mechanisms
• More observations are needed (Solar B, STEREO, SDO,
etc…) to establish the effects of magnetic carpet, and of zoo of
transients!
PPARC, Adv. Summer
School, Palma 2006