Dispersion diagrams of chromospheric MHD waves in a 2D simulation
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...
![Page 1: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/1.jpg)
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
![Page 2: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/2.jpg)
MHD waves in a uniform medium:
Ideal MHD equations:
Centre for Fusion, Space & Astrophysics
![Page 3: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/3.jpg)
x B
Centre for Fusion, Space & Astrophysics
za
![Page 4: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/4.jpg)
x B
Centre for Fusion, Space & Astrophysics
za
k
![Page 5: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/5.jpg)
Centre for Fusion, Space & Astrophysics
![Page 6: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/6.jpg)
Two independent subsystems:
and
Centre for Fusion, Space & Astrophysics
Alfvén speed
sound speed
![Page 7: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/7.jpg)
Incompressive, transverse
Alfvén waves:
• The phase speed can be oblique, but the group speed is
Centre for Fusion, Space & Astrophysics
• 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)
![Page 8: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/8.jpg)
Magnetoacoustic waves:
Essentially compressive, longitudinal
Centre for Fusion, Space & Astrophysics
• 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.
![Page 9: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/9.jpg)
Polar plots for phase speeds (ω/k)
B
k
b < 1
Centre for Fusion, Space & Astrophysics
Cf = CA2 + Cs
2
“fast” speed
![Page 10: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/10.jpg)
Centre for Fusion, Space & Astrophysics
![Page 11: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/11.jpg)
Development of an MHD perturbation in a uniformmedium
Centre for Fusion, Space & Astrophysics
Ver
wic
hte
,2
00
6
![Page 12: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/12.jpg)
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
Centre for Fusion, Space & Astrophysics
But, the situation changes dramatically in the presence ofa non-uniformity (e.g. coronal loops, fibrils, filaments,etc.).
![Page 13: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/13.jpg)
Development of an MHD perturbation along an inhomogeneity:
Centre for Fusion, Space & Astrophysics
![Page 14: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/14.jpg)
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)
Centre for Fusion, Space & Astrophysics
x
CAw / (ky + kz )
trapped
propagating
in Alfven (fast)speed act aswaveguides(resonator,cavities) forfast waves
![Page 15: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/15.jpg)
Consider a plasma cylinder:
Magnetohydrodynamic(MHD) equations
Equilibrium
Linearisation
Centre for Fusion, Space & Astrophysics
Linearisation
Boundary conditions
V(r,J, z) = Am
m
å exp(iwt - ikzz - imJ )
![Page 16: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/16.jpg)
Magnetohydrodynamic (MHD)equations
Equilibrium
Linearisation
Boundary conditions
“Standard theory”: interaction of MHD waves with magneticstructures (Zaitsev & Stepanov, 1975; B. Roberts andcolleagues, 1981-)
Centre for Fusion, Space & Astrophysics
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
![Page 17: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/17.jpg)
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:
Centre for Fusion, Space & Astrophysics
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
![Page 18: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/18.jpg)
Sausage (m=0) modes:
Centre for Fusion, Space & Astrophysics
Standing Running
![Page 19: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/19.jpg)
Kink (m=1) modes:
Centre for Fusion, Space & Astrophysics
Standing Running
![Page 20: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/20.jpg)
Centre for Fusion, Space & Astrophysics
From Fedun, 2007
![Page 21: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/21.jpg)
Shear Alfven waves is a non-uniform medium
Consider a 1D non-uniformity of the Alfven speed acrossthe magnetic field:
For an Alfven wave:
Centre for Fusion, Space & Astrophysics
Phase mixing:
Alfven waves are not collective!
![Page 22: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/22.jpg)
Centre for Fusion, Space & Astrophysics
![Page 23: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/23.jpg)
Centre for Fusion, Space & Astrophysics
![Page 24: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/24.jpg)
Main MHD modes:
• sausage (|B|, r)
• kink(almost incompressible)
Magnetoacoustic modes of a plasmacylinder:
Centre for Fusion, Space & Astrophysics
• torsional (incompressible)
• acoustic (r, V)
• ballooning (|B|, r)
(Case of a coronal loop)
![Page 25: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/25.jpg)
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
Centre for Fusion, Space & Astrophysics
0Torsional mode: 2 /tors AP L C
But, mind that in the leaky regime, long-wavelength sausage mode becomesindependent of L.
![Page 26: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/26.jpg)
1. Kink modes of coronal loops (EUV, TRACE):
Centre for Fusion, Space & Astrophysics
![Page 27: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/27.jpg)
How we analyse it:
Centre for Fusion, Space & Astrophysics
![Page 28: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/28.jpg)
E.g.: Path G,Period 338 s,Amplitude 750km
Centre for Fusion, Space & Astrophysics
![Page 29: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/29.jpg)
Centre for Fusion, Space & Astrophysics
• Oscillationperiod,
• Decay time
![Page 30: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/30.jpg)
Estimation of the magnetic field:
Centre for Fusion, Space & Astrophysics
One of the aims of SDO/AIA
![Page 31: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/31.jpg)
Centre for Fusion, Space & Astrophysics
SD
O/A
IA171
![Page 32: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/32.jpg)
A possible mechanism: mechanical displacement of the loopby LCE from the equilibrium
Centre for Fusion, Space & Astrophysics
![Page 33: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/33.jpg)
Centre for Fusion, Space & Astrophysics
![Page 34: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/34.jpg)
Centre for Fusion, Space & Astrophysics
![Page 35: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/35.jpg)
Statistics of kink oscillations:
Centre for Fusion, Space & Astrophysics
Goddard et al., in press, 2015
![Page 36: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/36.jpg)
Centre for Fusion, Space & Astrophysics
Goddard et al., in press, 2015
![Page 37: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/37.jpg)
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)
Centre for Fusion, Space & Astrophysics
In the loop there areregions where thekink motions are inresonance with thelocal torsional(Alfven)perturbations.
![Page 38: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/38.jpg)
Mathematically, it corresponds to the appearance of thesingularity in the governing equations:
Centre for Fusion, Space & Astrophysics
Why is it always about 3-5??
![Page 39: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/39.jpg)
Decay time vs Period:
Centre for Fusion, Space & Astrophysics
More statisticsis needed
![Page 40: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/40.jpg)
An oscillatory pattern occurs before the onset of the main oscillation:
Centre for Fusion, Space & Astrophysics
![Page 41: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/41.jpg)
Decayless regime of kink oscillations:
Wang
etal.
ApJ
751,
L27,
2012
Centre for Fusion, Space & Astrophysics
Anfinogentov et al., in press, 2015
Wang
etal.
![Page 42: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/42.jpg)
Centre for Fusion, Space & Astrophysics
![Page 43: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/43.jpg)
2. Sausage modes:
m=0 mode
Centre for Fusion, Space & Astrophysics
m=0 mode
In solar corona:
P = 10-120 s
![Page 44: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/44.jpg)
Sausage modes are essentiallycompressible. Can it modulate
X-ray and radio emission?
(directly, through |B| or through
Centre for Fusion, Space & Astrophysics
(directly, through |B| or throughthe modulation of the mirror
ratio)
![Page 45: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/45.jpg)
Trapped mode:
Centre for Fusion, Space & Astrophysics
![Page 46: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/46.jpg)
Leaky mode:
Centre for Fusion, Space & Astrophysics
![Page 47: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/47.jpg)
Centre for Fusion, Space & Astrophysics
![Page 48: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/48.jpg)
Centre for Fusion, Space & Astrophysics
![Page 49: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/49.jpg)
3. Longitudinalmodes:
Centre for Fusion, Space & Astrophysics
![Page 50: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/50.jpg)
Standard analysis: time-distance plot
Centre for Fusion, Space & Astrophysics
(Yuan & Nakariakov, 543, id.A9, 2012)
![Page 51: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/51.jpg)
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)
Centre for Fusion, Space & Astrophysics
1999, 2000)
![Page 52: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/52.jpg)
Standing longitudinal modes (T.J. Wang et al.;recently: Kumar et al.; Kim et al.)
Centre for Fusion, Space & Astrophysics
![Page 53: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/53.jpg)
Evolution of a typical two-ribbon flare: quasi-periodicpulsation sites progressalong the neutral line:
Slow magnetoacoustic wavesin coronal arcades
Centre for Fusion, Space & Astrophysics
Grigis & Benz, ApJ 625, L143, 2005
![Page 54: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/54.jpg)
Centre for Fusion, Space & Astrophysics
Bogachev et al. 2005
![Page 55: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/55.jpg)
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
Centre for Fusion, Space & Astrophysics
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
![Page 56: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/56.jpg)
Liu et al. 2009
Centre for Fusion, Space & Astrophysics
“Whipping-like” and“zipping-like” asymmetricfilament eruption.
![Page 57: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/57.jpg)
But, why are the observed speedsalways essentially subsonic?
What is the cause of the quasi-periodicity?
Centre for Fusion, Space & Astrophysics
periodicity?
![Page 58: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/58.jpg)
Centre for Fusion, Space & Astrophysics
![Page 59: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/59.jpg)
t=0 t=2
t=1 t=3
Centre for Fusion, Space & Astrophysics
t=1 t=3
![Page 60: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/60.jpg)
Centre for Fusion, Space & Astrophysics
![Page 61: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/61.jpg)
25-28° to B
Centre for Fusion, Space & Astrophysics
![Page 62: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/62.jpg)
Modulation of plasma waves by MHD waves:
Centre for Fusion, Space & Astrophysics
Zebra
-patt
ern
s
![Page 63: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/63.jpg)
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
Centre for Fusion, Space & Astrophysics
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
![Page 64: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/64.jpg)
QPP in radio zebra patterns:
Centre for Fusion, Space & Astrophysics
Yu
etal.
2013
![Page 65: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/65.jpg)
Centre for Fusion, Space & Astrophysics
![Page 66: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/66.jpg)
Modelling of the QPP in Zebra Patterns as Fast Waves:
Centre for Fusion, Space & Astrophysics
![Page 67: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/67.jpg)
Propagatingfast kinkwave?
Centre for Fusion, Space & Astrophysics
![Page 68: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/68.jpg)
Centre for Fusion, Space & Astrophysics
![Page 69: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/69.jpg)
Centre for Fusion, Space & Astrophysics
![Page 70: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/70.jpg)
Association of fast wave trainswith impulsive energy releases:
Centre for Fusion, Space & Astrophysics
Yuan et al. 2013
![Page 71: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/71.jpg)
Recent event:
Centre for Fusion, Space & Astrophysics
![Page 72: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/72.jpg)
2D
num
erica
lm
odelli
ng
Centre for Fusion, Space & Astrophysics
2D
num
erica
lm
odelli
ng
![Page 73: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/73.jpg)
Creation of fast wave trains by waveguide dispersion:
Centre for Fusion, Space & Astrophysics
![Page 74: MHD and Plasma Waves - Mathematics at Leeds...MHD and Plasma Waves 09:00 – 10:30 Wednesday 9 September STFC 2015 Valery M. Nakariakov Centre for Fusion, Space & Astrophysics ...](https://reader030.fdocuments.in/reader030/viewer/2022040902/5e747da5e611c206796c46dc/html5/thumbnails/74.jpg)
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
Centre for Fusion, Space & Astrophysics
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.