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:


x B


za

x B


za

k

Two independent subsystems:

and


Alfvén speed

sound speed

Incompressive, transverse

Alfvén waves:

• The phase speed can be oblique, but the group speed is


• 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


• 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


Cf = CA2 + Cs

2

“fast” speed

Development of an MHD perturbation in a uniformmedium


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


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:


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)


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


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-)


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:


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:


Standing Running

Kink (m=1) modes:


Standing Running


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:


Phase mixing:

Alfven waves are not collective!

Main MHD modes:

• sausage (|B|, r)

• kink(almost incompressible)

Magnetoacoustic modes of a plasmacylinder:


• 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


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):


How we analyse it:


E.g.: Path G,Period 338 s,Amplitude 750km



• Oscillationperiod,

• Decay time

Estimation of the magnetic field:


One of the aims of SDO/AIA


SD

O/A

IA171

A possible mechanism: mechanical displacement of the loopby LCE from the equilibrium


Statistics of kink oscillations:


Goddard et al., in press, 2015


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)


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:


Why is it always about 3-5??

Decay time vs Period:


More statisticsis needed

An oscillatory pattern occurs before the onset of the main oscillation:


Decayless regime of kink oscillations:

Wang

etal.

ApJ

751,

L27,

2012


Anfinogentov et al., in press, 2015

Wang

etal.

2. Sausage modes:

m=0 mode


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


(directly, through |B| or throughthe modulation of the mirror

ratio)

Trapped mode:


Leaky mode:


3. Longitudinalmodes:


Standard analysis: time-distance plot


(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)


1999, 2000)

Standing longitudinal modes (T.J. Wang et al.;recently: Kumar et al.; Kim et al.)


Evolution of a typical two-ribbon flare: quasi-periodicpulsation sites progressalong the neutral line:

Slow magnetoacoustic wavesin coronal arcades


Grigis & Benz, ApJ 625, L143, 2005


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


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


“Whipping-like” and“zipping-like” asymmetricfilament eruption.

But, why are the observed speedsalways essentially subsonic?

What is the cause of the quasi-periodicity?


periodicity?

t=0 t=2

t=1 t=3


t=1 t=3

25-28° to B


Modulation of plasma waves by MHD waves:


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


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:


Yu

etal.

2013

Modelling of the QPP in Zebra Patterns as Fast Waves:


Propagatingfast kinkwave?


Association of fast wave trainswith impulsive energy releases:


Yuan et al. 2013

Recent event:


2D

num

erica

lm

odelli

ng


2D

num

erica

lm

odelli

ng

Creation of fast wave trains by waveguide dispersion:


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


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.

mhd and plasma waves - mathematics at leeds...mhd and plasma waves 09:00 – 10:30 wednesday 9...

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