1. Short period MHD waves in the solar corona
Valery M. Nakariakov
University of Warwick

2. 3-200 s period MHD waves in the solar corona
Valery M. Nakariakov
University of Warwick

3. Observational evidence of coronal MHD waves is abundant: Periods from 1 s to several min.
Characteristic scales: 1 Mm-100 Mm,
Alfvén speed 1 Mm/s, sound speed 0.2 Mm/s
→ periods 1 s – several min - MHD waves
(Quasi) Periodicity:
Resonance (characteristic spatial scales)
Dispersion
Nonlinearity / self-organisation

4. Theory: MHD Modes of Plasma Structures
Two main building blocks:
Magnetic slab:
Magnetic flux tube:
B. Roberts and colleagues, 1981-

5. Roberts 1981: Dispersion relations of MHD modes of a magnetic flux tube:
In a magnetic slab the dispersion relations are a little simpler:
Dispersion relations are transcendental equations: an infinite number of roots

6. Five main MHD modes in corona:
Dispersion curves of coronal loop: phase speed (longitudinal wave number)
Five main MHD modes in corona:
sausage (|B|, r)
kink (almost incompressible)
torsional (incompressible)
acoustic (r, V)
ballooning (|B|, r)

7. m>1
flute or ballooning
m=0
sausage
m=1
kink

8. MHD modes already identified in solar coronal structures:
Kink oscillations of coronal loops (Aschwanden et al. 1999, Nakariakov et al. 1999)
Propagating longitudinal waves in polar plumes and near loop footpoints (Berghmans & Clette, 1999; Nakariakov et al. 2000, De Moortel et al )
Standing longitudinal waves in coronal loops (Kliem at al. 2002; Wang & Ofman 2002; Nakariakov et al. 2004)
Global sausage mode (Nakariakov et al. 2003)
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:

10. 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.

11. 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.

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

13. 1D Numerical Modelling:
(Nakariakov et al. 2004)
full nonlinearity
RTV radiation
thermal conduction
chromosphere
footpoint background heating
flaring heating at the apex

15. t_dur=100s
t_dur=100s
t_dur=1s

17. Acoustic oscillations in flaring loops:
a second spatial harmonics,
(c.f. global acoustic mode observed by SUMER)
P=L/Cs :
Periods: s

18. c.f. SUMER loop oscillations:
What about decay?
Ofman & Wang (2003)
Observations:
c.f. SUMER loop oscillations:
Wang et al. (2003)
Modelling:
Hmmm…

19. 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
Autowaves?
Thermal over-stability (heating + radiation)
Thermal conductivity
Finite nonlinearity
Activity
Dissipation
Nonlinearity
A tool for the determination of the heating positioning a

20. 2. Global sausage mode of a coronal loop
Previous estimations:
e.g. Roberts 1984, Aschwanden 1987, 1999, 2001, 2003 or

21. The correct estimation:
External medium, finite wave length 
Every time I learn something new it pushes some old stuff out of my brain!
The Simpsons
(Roberts, 1983)

22. Observational example:
Nobeyama NoRH observations (Melnikov et al. 2003)
5” and 10”
0.1 s

23. Spectra at different parts of the loop:

24. See Nakariakov, Melnikov & Reznikova 2003 for details.

25. 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

26. 4. Propagating fast waves
Adam was the only man who, when he said a good thing, knew that nobody had said it before him Mark Twain
4. Propagating fast waves
Theory: Group speed VS wave number
Different density contrasts
cutoffs
Roberts et al., 1983

27. Nakariakov & Roberts 1995:
p=1zz
Analytical solutions
p’z

28. “a crazy tadpole -wavelet”
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”

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

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

31. Roberts, Edwin & Benz 1984:
Simulations:
Nakariakov et al. 2004
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)

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

33. C.f. theory:
“Facts are stupid things” R. Reagan

34. When l is about a: LOS effects become important
Cooper et al. 2003, 2004:
The observed signal is actually affected by the LOS angle and the ration of the wave length and the loop cross-section radius.

35. Kink modes:

36. Sausage modes:
c.f. incompressible modes

37. Let amplitude be constant
Variation of the observed amplitude:
Kink modes:
Cooper, Nakariakov & Williams 2003:
Sausage modes:

38. 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, …