Short period MHD waves in the solar corona
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Space and Astrophysics
Short period Short period MHD waves MHD waves in the solar in the solar
coronacoronaValery M. NakariakovValery M. Nakariakov
University of WarwickUniversity of Warwick
http://www.astro.warwick.ac.uk/~valery/
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Space and Astrophysics
3-200 s3-200 s period MHD period MHD waves in the waves in the solar coronasolar corona
Valery M. NakariakovValery M. Nakariakov
University of WarwickUniversity of Warwick
http://www.astro.warwick.ac.uk/~valery/
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Space and Astrophysics
Observational evidence of coronal MHD waves is abundant: Periods from 1 s to several min.
(Quasi) Periodicity:
• Resonance (characteristic spatial scales)
• Dispersion
• Nonlinearity / self-organisation
Characteristic scales: 1 Mm-100 Mm,
Alfvén speed 1 Mm/s, sound speed 0.2 Mm/s
→ periods 1 s – several min - MHD waves
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Space and Astrophysics
Theory: MHD Modes of Plasma Structures
Two main building blocks:
Magnetic slab: Magnetic flux tube:
B. Roberts and colleagues, 1981-
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Space and Astrophysics
Roberts 1981: Dispersion relations of MHD modes of a magnetic flux tube:
2 2 2 2 2 200 0 0
0
'( ) '( )( ) ( ) 0
( ) ( )m m e
e z Ae z A em m e
I m a K m ak C m k C m
I m a K m a
2 2 2 2 2 20 0 0 0
tanh( ) ( ) ( ) 0
cothe Ae A ek C m m a k C m
In a magnetic slab the dispersion relations are a little simpler:
Dispersion relations are transcendental equations: an infinite number of roots
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Space and Astrophysics
Five main MHD modes in corona:
• sausage (|B|, )
• kink (almost incompressible)
• torsional (incompressible)
• acoustic (, V)
• ballooning (|B|, )
Dispersion curves of coronal loop: phase speed Dispersion curves of coronal loop: phase speed (longitudinal wave number)(longitudinal wave number)
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Space and Astrophysics
m=0
sausage
m=1
kink
m>1
flute or ballooning
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Space and Astrophysics
1. Kink oscillations of coronal loops (Aschwanden et al. 1999, Nakariakov et al. 1999)
2. Propagating longitudinal waves in polar plumes and near loop footpoints (Berghmans & Clette, 1999; Nakariakov et al. 2000, De Moortel et al. 2000-2004)
3. Standing longitudinal waves in coronal loops (Kliem at al. 2002; Wang & Ofman 2002; Nakariakov et al. 2004)
4. Global sausage mode (Nakariakov et al. 2003)
5. Propagating fast wave trains. (Williams et al. 2001, 2002; Cooper et al. 2003; Katsiyannis et al. 2003; Nakariakov et al. 2004, Verwichte et al. 2005)
MHD modes already identified in solar coronal structures:
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Space and Astrophysics
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Space and Astrophysics
This talk:
Theory-led attempts to identify
• Standing second acoustic harmonics,
• Global sausage harmonics,
• Propagating fast wave trains
in X-ray, radio and VL data.
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Space and Astrophysics
1. A typical X-ray flare light curve:
From Terekhov et al. 2002; GRANAT, 8-20 keV, M1.7
Period=143 s
Similar periodicities are often observed by other X-ray observatories and in radio.
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Space and Astrophysics
e.g. Farnik et al. 2003, MTI/HXRS: Period=25 s
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Space and Astrophysics
(Nak
aria
kov
et a
l. 20
04)
1D Numerical Modelling:
• full nonlinearity
• RTV radiation
• thermal conduction
• chromosphere
• footpoint background heating
• flaring heating at the apex
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Space and Astrophysics
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Space and Astrophysics
t_dur=100s t_dur=100s
t_dur=1s
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Space and Astrophysics
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Space and Astrophysics
Acoustic oscillations in flaring loops:
a second spatial harmonics,
(c.f. global acoustic mode observed by SUMER)
P=L/Cs :
Periods: 20-300s
0
2 2( , ) cos sin ,
2 2( , ) sin cos
s
s
s
CV s t A t s
L L
A Cs t t s
C L L
/ s 6.7 ( / Mm) / / MK;P L T
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Space and Astrophysics
What about decay?
c.f. SUMER loop oscillations:
Wan
g et
al.
(200
3)
Observations:
Modelling:
Ofm
an &
Wan
g (2
003)
Hmmm…
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Space and Astrophysics
Autowaves?
• Thermal over-stability (heating + radiation)
• Thermal conductivity
• Finite nonlinearity
• Activity
• Dissipation
• Nonlinearity
A tool for the determination of the heating positioning
I’ll figure it out.
I am going to use all the power of my brain.
The Simpsons
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Space and Astrophysics
2. Global sausage 2. Global sausage mode of a coronal loopmode of a coronal loop
02 /GSM AP a C
2 /GSM kP a C
Previous estimations:
or
e.g. Roberts 1984, Aschwanden 1987, 1999, 2001, 2003
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Space and Astrophysics
0
0 0 0
2 / ,
2 2s
.62
GSM P A P Ae
A A
P L C C C C
a aP
j C C
External medium, finite wave length
The correct The correct estimation:estimation:
(Roberts, 1983)ck a
Every time I learn something new it pushes some old stuff out of my brain!
The Simpsons
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Space and Astrophysics
Observational example:Observational example:
Nobeyama NoRH observations (Melnikov et al. 2003)
5” and 10”
0.1 s
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Space and Astrophysics
Spectra at different parts of the loop:
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Space and Astrophysics
0
0
0
23,200 km/s, and it must be <
2.62 2.62 2.62 3 524 k
3,200 km/s; 524
m/s.1
k /
5
m s
p Ae
AA
Ae A
LC C
P
a aP C
C
C P
C
See Nakariakov, Melnikov & Reznikova 2003 for details.
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Space and Astrophysics
Sausage modes are essentially compressible and can modulate X-
ray and radio emission
A tool for determination of the Alfvén speed and the magnetic field outside the
loop
An unbreakable toy is good for breaking other toys
Jason's Law
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Space and Astrophysics
4. Propagating fast waves4. Propagating fast wavesTheory: Group speed VS wave number
cutoffsDifferent density contrasts
Roberts et al., 1983
Adam was the only man who, when he said a good thing, knew that nobody had said it before him. Mark Twain
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Space and Astrophysics
1,sech)()( 20
p
d
xx
p
p’
p=1
Nakariakov & Roberts 1995:
Analytical solutions
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Space and Astrophysics
Impulsively excited fast pulse at distance 70 from the source, the density contrast is 5; Alfvén speed ratio is 2.3
“a crazy tadpole -wavelet”
/ Aez C 0/ Az C
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Space and Astrophysics
Impulsively excited fast pulse at distance 70 from the source, the density contrast is 14.3; Alfvén speed ratio is 3.8
/ Aez C 0/ Az C
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Space and Astrophysics
Impulsively excited fast pulse at distance 70 from the source, the density contrast is 5; Alfvén speed ratio is 2.3; smooth profile:
0/ Az C/ Aez C
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Space and Astrophysics
What should we measure?• dP/dt,• dI/dt,size of the tail and of the head• distance from the source (signals at different points)
It would give us:
• Loop width (sub-resolution)
• Loop profile (filling factor)
Roberts, Edwin & Benz 1984:
Simulations:
Nakariakov et al. 2004
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Space and Astrophysics
SECIS: Williams et al. 2001, 2002; Katsiyannis et al. 2003
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Space and Astrophysics
C.f. theory:
“Facts are stupid things” R. Reagan
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Space and Astrophysics
When is about a: LOS effects become important
Cooper et al. 2003, 2004: 2
0
( )
( ) ( )LOS
I l l dl
The observed signal is actually affected by the LOS angle and the ration of the wave length and the loop cross-section radius.
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Space and Astrophysics
Kink modes:Kink modes:
2 2
2 2 2 20
0
| |sech / , where , / ,
| |
AA
AA A
A
k wU x w C a a k
C
CC a k w a C
C
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Space and Astrophysics
2 2
2 2 2 20
sinh / | |, where 1 , / ,
cosh ( / )
| | 2 3
| |
AA
A AA A
x w k wU C a a k
x w C
k wa C C a
C k w C
Sausage Sausage modes:modes:
c.f. incompressible modes
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Space and Astrophysics
Kink modes:
Sausage modes:
Variation of the observed amplitude:
Let amplitude be constant
Cooper, Nakariakov & Williams 2003:
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Space and Astrophysics
Conclusions:
http://www.astro.warwick.ac.uk/~valery/
• Short period MHD waves have to be observed.
• Well, they can be observed and spatially resolved (e.g. in X-ray, in radio - NoRH, in VL – SECIS).
• Short period waves are an ideal tool for determination of sub-resolution structuring, heating positioning and the magnetic field.
• A lot of open questions, e.g. what is the mechanism responsible for P ≤ few s?
• Solar B, SDO, Solar Orbiter, …