PX420 Solar MHD
2015-2016
Coronal Seismology
Valery M. Nakariakov
Centre for Fusion, Space & Astrophysics
1. Motivation
Wave and oscillatory processes in the solar
corona:
•  Possible relevance to coronal heating and solar wind
acceleration problems.
•  Possible role in the physics of solar flares.
•  Plasma diagnostics tools - coronal seismology.
•  Perspectives of stellar coronal seismology.
•  Observational evidence of coronal
(or quasi-periodic pulsations) is abundant.
oscillations
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“The Seven Sisters
Flare” – ISEE-3 and
Nobeyama
Radiopolarimeter,
(+ very faint pulsations
in < 17 GHz)
Period about 8 s.
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Typical periods from 1 s to several min.
Mechanisms for (Quasi) Periodicity:
•  Resonance (characteristic spatial scales)
Seismological
information
•  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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2. Revision: MHD waves in a uniform medium:
Two characteristic
speeds:
•  Alfvén speed:
•  Sound speed:
Alfven waves:
Magnetosonic
waves:
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Verwichte, 2006
Development of an MHD perturbation in a uniform
medium
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Characteristic speeds:
Sound speed:
CS ∝ T , - gradient of gas pressure
Alfve′n speed: C A ∝ B /
ρ , - magnetic tension,
Fast speed:
C F = C A2 + CS2 - gradient of (magnetic pressure + gas pressure)
Tube speed:
CS C A
CT =
C A2 + CS2
2
2
⎛ ρ0C A0
⎞
+ ρeC Ae
Kink speed: C K = ⎜
⎟
ρ
+
ρ
⎝
⎠
0
e
1/ 2
; in low-β : C K = C A0
2
1 + ρ e / ρ0
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Global Coronal Waves
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The Global Coronal Wave
•  propagates generally across the field;
•  is seen as the variation of the brightness of the coronal EUV
emission, i.e. the wave is compressive.
Thus, this wave must be a fast magnetoacoustic wave, with the
speed
CF = C + C
2
A
2
s
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Measuring the speed of the Global Coronal Wave at different
heights, we can estimate the vertical structure of the absolute value
of the magnetic field and plasma density (as they define the Alfvén
speed) and the temperature (as it defines the sound speed).
CF (z) = C (B0 (z), ρ0 (z)) + C (T (z))
2
A
2
s
(Of course, provided we are able to estimate some of these
parameters independently, e.g. the density could be stratified
exponentially, the field could be estimated by extrapolation, the
temperature could be measured spectroscopically).
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In the uniform or weakly non-uniform medium:
•  Along the field, there two propagating waves, Alfvén and slow
(degenerated into pure sound waves);
•  Across the field, there is only the fast wave.
But, the situation changes dramatically in the presence of a
non-uniformity (e.g. coronal loops, fibrils, filaments, etc.).
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Development of an MHD perturbation along an inhomogeneity:
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E.g., in the
zero-beta
plasma:
2
⎛
d V⊥
ω
2
2⎞
+⎜ 2
− kz − k y ⎟ V⊥ = 0
2
dx
⎝ CA (x)
⎠
2
c.f. the stationary Schrodinger Eq. in quantum mechanics
CA
ω / (k y + kz )
propagating
trapped
Regions with the
decrease in
Alfven (fast)
speed act as
waveguides
(resonator,
cavities) for fast
waves
x
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Consider a magnetic flux tube:
Magnetohydrodynamic
(MHD) equations à
Equilibrium à
Linearisation à
Boundary conditions
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Dispersion relations of MHD modes of a
magnetic flux tube:
I m '(m0 a)
K m '(me a)
2
2 2
ρe (ω − k C )m0
− ρ0 (ω − k z C A0 )me
=0
I m (m0 a)
K m (me a)
2
2
z
2
Ae
Zaitsev & Stepanov, 1975B. Roberts and colleagues, 1981-
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Trapped mode:
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Leaky mode:
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From Fedun, 2007
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Main trapped MHD modes of
coronal structures:
sausage (|B|, ρ)
•  kink
incompressible)
(almost
•  torsional (incompressible)
•  acoustic (ρ, V)
•  ballooning (|B|, ρ)
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GLOBAL MODES:
Sausage mode: Psaus = 2 L / C p , where C A0 < C p < C Ae
Kink mode: Pkink = 2 L / CK ,
Longitudinal mode: Plong = 2 L / CT 0
Torsional mode: Ptors = 2 L / C A0
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1. Kink modes of coronal loops (EUV, TRACE):
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How we analyse it:
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•  Oscillation
period,
•  Decay time
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Longitudinal modes:
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Running sausage
wave?
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Distance along slit
time
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Standing longitudinal wave:
The period is
determined by the
plasma temperature
and the length of the
loop,
P ≈ 2L / Cs
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3. Sausage modes:
m=0 mode
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Sausage modes are essentially
compressible and can modulate Xray and radio emission
(directly, through |B| or through the
modulation of the mirror ratio)
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First observational
identification:
Signals at different
parts of the loop:
17 GHz, 34 GHz + SXT
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Spectra at different
parts of the loop:
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2L
Cp =
≈ 3, 200 km/s, and it must be < C Ae
P
2.62a
P<
C A0
⇒
C A0
∴ C Ae > 3, 200 km/s;
2.62a 2.62 × 3
<
≈
≈ 524 km/s.
P
15
C A0 < 524 km/s
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