MHD waves

Space and Astrophysics

Generation of quasi-Generation of quasi-periodic pulsations in periodic pulsations in solar flares by MHD solar flares by MHD

waveswaves

Valery M. NakariakovValery M. NakariakovUniversity of WarwickUniversity of Warwick

United KingdomUnited Kingdomhttp://www.warwick.ac.uk/go/cfsa/

Nobeyama, Japan 16/03/2006


A hypothesis or theory is clear, decisive, and positive but it is believed by none but the man who created it. Experimental findings, on the other hand are messy, inexact things which are believed by everyone except the man who did the work.

Harlow Shapley


Observational evidence of quasi-periodic pulsations in solar flares is abundant.

(Quasi) Periodicity:

• Resonance (characteristic spatial scales)

• Dispersion

• Nonlinearity / self-organisation

Characteristic scales: 1 Mm-100 Mm,

Alfvén speed 1-2 Mm/s, sound speed 0.3-0.5 Mm/s

→ periods 1 s – several min - MHD waves

Seismological information


Dispersion relations of MHD modes of a magnetic flux tube:

2 2 2 2 2 200 0 0

0

'( ) '( )( ) ( ) 0

( ) ( )m m e

e z Ae z A em m e

I m a K m ak C m k C m

I m a K m a

Magnetohydrodynamic (MHD) equations

Equilibrium

Linearisation

Boundary conditions

Zaitsev & Stepanov, 1975- B. Roberts and colleagues, 1981-

Standard theoretical model:


Main MHD modes in observed in the corona:• sausage (|B|, )

• kink (weakly compressible)

• torsional (incompressible)

• acoustic (, V)

• ballooning (|B|, )

Dispersion curves of coronal loop:Dispersion curves of coronal loop:


1. Kink oscillations of coronal loops (Aschwanden et al. 1999, Nakariakov et al. 1999)

2. Propagating longitudinal waves in polar plumes and near loop footpoints (Berghmans & Clette, 1999; Nakariakov et al. 2000, De Moortel et al. 2000-2004)

3. Standing longitudinal waves in coronal loops (Kliem at al. 2002; Wang & Ofman 2002; Nakariakov et al. 2004)

4. Global sausage mode (Nakariakov et al. 2003)

5. Propagating fast wave trains. (Williams et al. 2001, 2002; Cooper et al. 2003; Katsiyannis et al. 2003; Nakariakov et al. 2004, Verwichte et al. 2005)

Observed wave phenomena (to 2006):


Kink mode: θ(t) modulation If(t)

Sausage mode: B(t) modulation If(t)

Longitudinal mode can also modulate the GS emission

E.g., the optically thin gyrosynchrotron emission intensity If at a frequency f can be estimated as

In general, MHD waves should be well seen in microwave:


But, often QPP are seen in both microwave (GS) and hard X-ray : e.g. Asai et al. (2001)


Suppose that QPP are connected with some MHD oscillations. The model has to explain:

• the modulation of both microwave and hard X-ray (and possibly WL) emission simultaneously and in phase;

• the modulation depth (> 50% in some cases, while the amplitudes of known coronal MHD waves are usually just a few percent);

• the mechanism responsible for the periodicity;

• the “2D” structure of the pulsations.


Electron acceleration

Nonthermal electron dynamics

Emission in WL, microwave, X-rays

MHD waves

In general, MHD waves can affect the whole chain responsible for the emission:


Flaring loop

Faint cool loop

At the reconnection site: B ~ cos(Ωt) Ne ~ cos(Ωt)

Kink oscillation

Quadrupolar magnetic configuration

What if the modulating wave is magnetically disconnected with the flaring loop?

1. Long period pulsations (> 60 s)


MHD oscillation in the external

loop (very small amplitude)

Fast wave perpendicular to B approaches X-point

Electric currents build up (time variant)

Current driven micro-instabilities

Anomalous resistivity

Triggers fast reconnection

Acceleration of non-thermal electrons

(Nakariakov et al. A&A, 2006, in press)


Full MHD 2.5D simulations of the interaction of a periodic fast wave with a magnetic X-point.

• The fast wave energy is accumulated near the separatrix.

• The current density near the X-point experiences periodic building up.

The absolute value of the velocity perturbation.

run

The electric current density.

run

The electric current density, side view

run


Thus, the electric current at the x-point varies periodically in time:

The amplitude of the source fast wave is just 1%.


Current-driven plasma microinstabilities were suggested as a triggering mechanism for fast reconnection:

classical anomalousthreshold classical

threshold anomalous ,,,

jjjj

thersholdj

Periodic variation of the current density causes periodic triggering of fast reconnection


There is some observational evidence:

(Foullon et al. 2005)

Unseen kink oscillations of the faint trans-equatorial EUV loop cause modulation of the hard X-ray emission near the magnetically conjugate points.


2. Medium period case (10-60 s)

Sausage modes:

-for trapped modes

here L is the loop lengthAepA

p

CCC

CLP

0

,/2

The commonly used expressionis incorrect,(here a is the loop minor radius)

0/2 ACaP

(Nakariakov et al., 2003;

Aschwanden et al., 2004)


What about leaky modes?

Full MHD simulations:

Thus, the period of leaky sausage modes is also determined by the loop length, not by the loop minor radius.

(Pascoe et al., 2006)

trapped

leaky


3. Short period case (< 10 s)

• Higher spatial harmonics (but the problem is the selectivity of the excitation);

• Fast wave trains formed by dispersion:

(But, they should have period modulation - the “crazy tadpole wavelet spectra …)


Conclusions:• There are simple mechanisms for modulation of microwave and hard

x-ray emission by the modes.

• The longer periodicities can be connected with small-amplitude oscillations of an external cool loop. In this case the cool loop acts as a resonator and determines the period of oscillations.

• Sausage mode periods are prescribed by the flaring loop length, even in the leaky mode regime.

• Shorter periodicites can be connected with dispersion. However, there should be period modulation.

• In general, shorter periodicities (1-5 s) are still without appropriate interpretation (the “fast mode formula” does not work).

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MHD waves

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