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Space and Astrophysics

Short period Short period MHD waves MHD waves in the solar in the solar

coronacoronaValery M. NakariakovValery M. Nakariakov

University of WarwickUniversity of Warwick

http://www.astro.warwick.ac.uk/~valery/


3-200 s3-200 s period MHD period MHD waves in the waves in the solar coronasolar corona

Valery M. NakariakovValery M. Nakariakov

University of WarwickUniversity of Warwick



Observational evidence of coronal MHD waves is abundant: Periods from 1 s to several min.

(Quasi) Periodicity:

• Resonance (characteristic spatial scales)

• Dispersion

• Nonlinearity / self-organisation

Characteristic scales: 1 Mm-100 Mm,

Alfvén speed 1 Mm/s, sound speed 0.2 Mm/s

→ periods 1 s – several min - MHD waves


Theory: MHD Modes of Plasma Structures

Two main building blocks:

Magnetic slab: Magnetic flux tube:

B. Roberts and colleagues, 1981-


Roberts 1981: 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

2 2 2 2 2 20 0 0 0

tanh( ) ( ) ( ) 0

cothe Ae A ek C m m a k C m

In a magnetic slab the dispersion relations are a little simpler:

Dispersion relations are transcendental equations: an infinite number of roots


Five main MHD modes in corona:

• sausage (|B|, )

• kink (almost incompressible)

• torsional (incompressible)

• acoustic (, V)

• ballooning (|B|, )

Dispersion curves of coronal loop: phase speed Dispersion curves of coronal loop: phase speed (longitudinal wave number)(longitudinal wave number)


m=0

sausage

m=1

kink

m>1

flute or ballooning


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)

MHD modes already identified in solar coronal structures:


This talk:

Theory-led attempts to identify

• Standing second acoustic harmonics,

• Global sausage harmonics,

• Propagating fast wave trains

in X-ray, radio and VL data.


1. A typical X-ray flare light curve:

From Terekhov et al. 2002; GRANAT, 8-20 keV, M1.7

Period=143 s

Similar periodicities are often observed by other X-ray observatories and in radio.


e.g. Farnik et al. 2003, MTI/HXRS: Period=25 s


(Nak

aria

kov

et a

l. 20

04)

1D Numerical Modelling:

• full nonlinearity

• RTV radiation

• thermal conduction

• chromosphere

• footpoint background heating

• flaring heating at the apex


t_dur=100s t_dur=100s

t_dur=1s


Acoustic oscillations in flaring loops:

a second spatial harmonics,

(c.f. global acoustic mode observed by SUMER)

P=L/Cs :

Periods: 20-300s

0

2 2( , ) cos sin ,

2 2( , ) sin cos

s

s

s

CV s t A t s

L L

A Cs t t s

C L L

/ s 6.7 ( / Mm) / / MK;P L T


What about decay?

c.f. SUMER loop oscillations:

Wan

g et

al.

(200

3)

Observations:

Modelling:

Ofm

an &

Wan

g (2

003)

Hmmm…


Autowaves?

• Thermal over-stability (heating + radiation)

• Thermal conductivity

• Finite nonlinearity

• Activity

• Dissipation

• Nonlinearity

A tool for the determination of the heating positioning

I’ll figure it out.

I am going to use all the power of my brain.

The Simpsons


2. Global sausage 2. Global sausage mode of a coronal loopmode of a coronal loop

02 /GSM AP a C

2 /GSM kP a C

Previous estimations:

or

e.g. Roberts 1984, Aschwanden 1987, 1999, 2001, 2003


0

0 0 0

2 / ,

2 2s

.62

GSM P A P Ae

A A

P L C C C C

a aP

j C C

External medium, finite wave length

The correct The correct estimation:estimation:

(Roberts, 1983)ck a

Every time I learn something new it pushes some old stuff out of my brain!

The Simpsons


Observational example:Observational example:

Nobeyama NoRH observations (Melnikov et al. 2003)

5” and 10”

0.1 s


Spectra at different parts of the loop:


0

0

0

23,200 km/s, and it must be <

2.62 2.62 2.62 3 524 k

3,200 km/s; 524

m/s.1

k /

5

m s

p Ae

AA

Ae A

LC C

P

a aP C

C

C P

C

See Nakariakov, Melnikov & Reznikova 2003 for details.


Sausage modes are essentially compressible and can modulate X-

ray and radio emission

A tool for determination of the Alfvén speed and the magnetic field outside the

loop

An unbreakable toy is good for breaking other toys

Jason's Law


4. Propagating fast waves4. Propagating fast wavesTheory: Group speed VS wave number

cutoffsDifferent density contrasts

Roberts et al., 1983

Adam was the only man who, when he said a good thing, knew that nobody had said it before him. Mark Twain


1,sech)()( 20

p

d

xx

p

p’

p=1

Nakariakov & Roberts 1995:

Analytical solutions


Impulsively excited fast pulse at distance 70 from the source, the density contrast is 5; Alfvén speed ratio is 2.3

“a crazy tadpole -wavelet”

/ Aez C 0/ Az C


Impulsively excited fast pulse at distance 70 from the source, the density contrast is 14.3; Alfvén speed ratio is 3.8

/ Aez C 0/ Az C


Impulsively excited fast pulse at distance 70 from the source, the density contrast is 5; Alfvén speed ratio is 2.3; smooth profile:

0/ Az C/ Aez C


What should we measure?• dP/dt,• dI/dt,size of the tail and of the head• distance from the source (signals at different points)

It would give us:

• Loop width (sub-resolution)

• Loop profile (filling factor)

Roberts, Edwin & Benz 1984:

Simulations:

Nakariakov et al. 2004


SECIS: Williams et al. 2001, 2002; Katsiyannis et al. 2003


C.f. theory:

“Facts are stupid things” R. Reagan


When is about a: LOS effects become important

Cooper et al. 2003, 2004: 2

0

( )

( ) ( )LOS

I l l dl

The observed signal is actually affected by the LOS angle and the ration of the wave length and the loop cross-section radius.


Kink modes:Kink modes:

2 2

2 2 2 20

0

| |sech / , where , / ,

| |

AA

AA A

A

k wU x w C a a k

C

CC a k w a C

C


2 2

2 2 2 20

sinh / | |, where 1 , / ,

cosh ( / )

| | 2 3

| |

AA

A AA A

x w k wU C a a k

x w C

k wa C C a

C k w C

Sausage Sausage modes:modes:

c.f. incompressible modes


Kink modes:

Sausage modes:

Variation of the observed amplitude:

Let amplitude be constant

Cooper, Nakariakov & Williams 2003:


Conclusions:


• Short period MHD waves have to be observed.

• Well, they can be observed and spatially resolved (e.g. in X-ray, in radio - NoRH, in VL – SECIS).

• Short period waves are an ideal tool for determination of sub-resolution structuring, heating positioning and the magnetic field.

• A lot of open questions, e.g. what is the mechanism responsible for P ≤ few s?

• Solar B, SDO, Solar Orbiter, …

Download - Short period MHD waves in the solar corona

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