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:

Two independent subsystems:

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)

Cf = CA2 + Cs

“fast” speed

Development of an MHD perturbation in a uniformmedium

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:

CA2 (x)

- kz2 - ky

èçö

ø÷V^ = 0

c.f. the stationary Schrodinger Eq. in quantum mechanics

Cw / (ky + kz )

Regions withthe decreasein Alfven (fast)

CAw / (ky + kz )

trapped

propagating

in Alfven (fast)speed act aswaveguides(resonator,cavities) forfast waves

Consider a plasma cylinder:

Magnetohydrodynamic(MHD) equations

Equilibrium

Linearisation

Boundary conditions

V(r,J, z) = Am

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

( ) ( )m m e

e z Ae z A 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:

2 + CS

2 - gradient of (magnetic pressure + gas pressure)

Characteristic speeds:

Tube speed:

2 + CS

Kink speed: CK

2 + reC

; in low-b : CK

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)

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

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

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

Statistics of kink oscillations:

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:

Anfinogentov et al., in press, 2015

2. Sausage modes:

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)

• 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

25-28° to B

Modulation of plasma waves by MHD waves:

300sfB

Double Plasma Resonance (DPR)

fuh = fp2 + fce

2 » fp = sfce

fp = e2N p m >> fce = eB 2pmc

15 20 25 30 35 40

height, Mm

D f fce » LB LN - LB » LB LN

LB << LN D f << fce

QPP in radio zebra patterns:

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:

odelli

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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