MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9...
Transcript of MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9...
MHD and Plasma Waves
09:00 – 10:30 Wednesday 9 September STFC 2015
Valery M. Nakariakov
Centre for Fusion, Space & Astrophysics
Valery M. Nakariakov
UniversityUniversity ofof WarwickWarwick
MHD waves in a uniform medium:
Ideal MHD equations:
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x B
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za
x B
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za
k
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Two independent subsystems:
and
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Alfvén speed
sound speed
Incompressive, transverse
Alfvén waves:
• The phase speed can be oblique, but the group speed is
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• The phase speed can be oblique, but the group speed isalways along the field.
• Displacement of the magnetic field lines in the wavesalways keeps the same distance between them.
• Dispersionless (phase and group speeds are independentof the frequency)
Magnetoacoustic waves:
Essentially compressive, longitudinal
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• A bi-quadratic equation: two modes, fast and slow.
• In low-beta plasma (typical for the solar corona), in the fastwave, perturbations of the gas and magnetic pressure are inphase; while in the slow wave – in anti-phase.
• Can propagate obliquely to the field.• Dispersionless.
Polar plots for phase speeds (ω/k)
B
k
b < 1
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Cf = CA2 + Cs
2
“fast” speed
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Development of an MHD perturbation in a uniformmedium
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Ver
wic
hte
,2
00
6
In the uniform medium:
• Along the field, there two propagating waves, Alfvén andslow (degenerated into pure sound waves;
• Across the field, there is only the fast wave.
But, the situation changes dramatically in the presence of
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But, the situation changes dramatically in the presence ofa non-uniformity (e.g. coronal loops, fibrils, filaments,etc.).
Development of an MHD perturbation along an inhomogeneity:
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E.g., in thezero-betaplasma:
d 2V^
dx2+
w 2
CA2 (x)
- kz2 - ky
2æ
èçö
ø÷V^ = 0
c.f. the stationary Schrodinger Eq. in quantum mechanics
Cw / (ky + kz )
Regions withthe decreasein Alfven (fast)
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x
CAw / (ky + kz )
trapped
propagating
in Alfven (fast)speed act aswaveguides(resonator,cavities) forfast waves
Consider a plasma cylinder:
Magnetohydrodynamic(MHD) equations
Equilibrium
Linearisation
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Linearisation
Boundary conditions
V(r,J, z) = Am
m
å exp(iwt - ikzz - imJ )
Magnetohydrodynamic (MHD)equations
Equilibrium
Linearisation
Boundary conditions
“Standard theory”: interaction of MHD waves with magneticstructures (Zaitsev & Stepanov, 1975; B. Roberts andcolleagues, 1981-)
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Dispersion relations of MHD modes ofa magnetic flux tube:
2 2 2 2 2 200 0 0
0
'( ) '( )( ) ( ) 0
( ) ( )m m e
e z Ae z A e
m m e
I m a K m ak C m k C m
I m a K m a
Boundary conditions
Sound speed: CS
µ T , - gradient of gas pressure
Alfv ¢e n speed: CA
µ B / r , - magnetic tension,
Fast speed:
CF
= CA
2 + CS
2 - gradient of (magnetic pressure + gas pressure)
Characteristic speeds:
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F A S
Tube speed:
CT
=C
SC
A
CA
2 + CS
2
Kink speed: CK
=r
0C
A0
2 + reC
Ae
2
r0
+ re
æ
èç
ö
ø÷
1/ 2
; in low-b : CK
= CA0
2
1+ re
/ r0
Sausage (m=0) modes:
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Standing Running
Kink (m=1) modes:
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Standing Running
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From Fedun, 2007
Shear Alfven waves is a non-uniform medium
Consider a 1D non-uniformity of the Alfven speed acrossthe magnetic field:
For an Alfven wave:
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Phase mixing:
Alfven waves are not collective!
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Main MHD modes:
• sausage (|B|, r)
• kink(almost incompressible)
Magnetoacoustic modes of a plasmacylinder:
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• torsional (incompressible)
• acoustic (r, V)
• ballooning (|B|, r)
(Case of a coronal loop)
0
0
0
GLOBAL MODES:
Sausage mode: 2 / , where
Kink mode: 2 / ,
Longitudinal mode: 2 /
Torsional mode: 2 /
saus p A p Ae
kink K
long T
tors A
P L C C C C
P L C
P L C
P L C
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0Torsional mode: 2 /tors AP L C
But, mind that in the leaky regime, long-wavelength sausage mode becomesindependent of L.
1. Kink modes of coronal loops (EUV, TRACE):
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How we analyse it:
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E.g.: Path G,Period 338 s,Amplitude 750km
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• Oscillationperiod,
• Decay time
Estimation of the magnetic field:
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One of the aims of SDO/AIA
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SD
O/A
IA171
A possible mechanism: mechanical displacement of the loopby LCE from the equilibrium
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Statistics of kink oscillations:
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Goddard et al., in press, 2015
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Goddard et al., in press, 2015
Effect of resonant absorption of kink waves inthe corona
If the Alfven speed isnonuniform in theradial direction, CA(r),
In the loop there are
Ck=CA(r)
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In the loop there areregions where thekink motions are inresonance with thelocal torsional(Alfven)perturbations.
Mathematically, it corresponds to the appearance of thesingularity in the governing equations:
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Why is it always about 3-5??
Decay time vs Period:
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More statisticsis needed
An oscillatory pattern occurs before the onset of the main oscillation:
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Decayless regime of kink oscillations:
Wang
etal.
ApJ
751,
L27,
2012
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Anfinogentov et al., in press, 2015
Wang
etal.
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2. Sausage modes:
m=0 mode
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m=0 mode
In solar corona:
P = 10-120 s
Sausage modes are essentiallycompressible. Can it modulate
X-ray and radio emission?
(directly, through |B| or through
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(directly, through |B| or throughthe modulation of the mirror
ratio)
Trapped mode:
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Leaky mode:
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3. Longitudinalmodes:
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Standard analysis: time-distance plot
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(Yuan & Nakariakov, 543, id.A9, 2012)
Also:
• In coronal holes: (Ofman et al. 1997;Banerjee et al. SSR 158, 267, 2011)
• In non-sunspot loops: (Berghmans & Clette, 1998;De Moortel et al. 2000)
• In polar plumes: (De Forest & Gurman, 1998; Ofman et al.1999, 2000)
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1999, 2000)
Standing longitudinal modes (T.J. Wang et al.;recently: Kumar et al.; Kim et al.)
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Evolution of a typical two-ribbon flare: quasi-periodicpulsation sites progressalong the neutral line:
Slow magnetoacoustic wavesin coronal arcades
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Grigis & Benz, ApJ 625, L143, 2005
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Bogachev et al. 2005
Reznikova et al. 2010: V = 8 km/sTripathy et al. 2006: V = 3-39 km/sKrucker et al. 2003: V = 50 km/sKrucker et al. 2005: V = 20-100 km/sLi & Zhang 2009: V = 3-39 km/s
(by 124 two-ribbon flares)Grigis & Benz 2005: V = 50-60 km/s
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Grigis & Benz 2005: V = 50-60 km/sZimovets & Struminsky 2009: 60-90 km/s
QPP:Grigis & Benz 2005: P = 8-30 sZimovets & Struminsky 2009: P = 1-3 min
Liu et al. 2009
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“Whipping-like” and“zipping-like” asymmetricfilament eruption.
But, why are the observed speedsalways essentially subsonic?
What is the cause of the quasi-periodicity?
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periodicity?
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t=0 t=2
t=1 t=3
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t=1 t=3
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25-28° to B
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Modulation of plasma waves by MHD waves:
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Zebra
-patt
ern
s
280
300sfB
sfce
Double Plasma Resonance (DPR)
s
s+1
B
s-1
fuh = fp2 + fce
2 » fp = sfce
fp = e2N p m >> fce = eB 2pmc
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15 20 25 30 35 40
height, Mm
120
140
160
180
200
220
240
260
freq
uen
cy,
MH
z
fp
sfce
D f fce » LB LN - LB » LB LN
LB << LN D f << fce
QPP in radio zebra patterns:
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Yu
etal.
2013
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Modelling of the QPP in Zebra Patterns as Fast Waves:
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Propagatingfast kinkwave?
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Association of fast wave trainswith impulsive energy releases:
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Yuan et al. 2013
Recent event:
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2D
num
erica
lm
odelli
ng
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2D
num
erica
lm
odelli
ng
Creation of fast wave trains by waveguide dispersion:
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Conclusions:• A number of new results on MHD waves and oscillations inthe corona. The topic of MHD coronal seismology is becomingmore and more popular: thank SDO/AIA.
• Standing kink waves are found to be excited by LCE. How?
• The new regime of kink oscillations of coronal loops: low-amplitude decay-less standing oscillations: the driver?
http://www.warwick.ac.uk/go/cfsa
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amplitude decay-less standing oscillations: the driver?
• Fast wave trains are confidently interpreted as wave-guidedfast magnetoacoustic wave trains. A new toy to play with!
• Field-aligned filamentation of coronal plasma plays thedecisive role in MHD wave dynamics.