1. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 MHD Wave Phenomena in Space Plasmas 2 July, 2006 Youra Taroyan y.taroyan@sheffield.ac.uk y.taroyan@sheffield.ac.uk SPARG, Department of Applied Mathematics, The University of Sheffield (UK)

2. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Part I Wave-flow interactions

3. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave-flow interactions Magnetosheath/magnetosphere

4. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave-flow interactions Heliosphere/Interstellar medium

5. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave-flow interactions Slow/fast solar wind

6. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave-flow interactions Relativistic jets

7. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave-flow interactions Sunspots

8. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave-flow interactions Solar dynamo

9. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Kelvin-Helmholtz Instability Flow

10. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Kelvin-Helmholtz Instability

11. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 KHI in Saturn’s atmosphere

12. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Discontinuous Stratified Medium KH Instability (with exponentially increasing wave amplitude)

13. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Continuous Stratified Medium KH Instability Ri - Richardson number

14. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 K-H Instability in a magnetic environment KH Instability (Chandrasekhar 1961)

15. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 flow outer boundary inner reflecting boundary upper boundary lower boundary Fast and Alfven waves in a uniform waveguide

16. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Upstream and downstream fast waves V ω KH Instability

17. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Fast and Alfven waves in a nonuniform waveguide flow outer boundary inner reflecting boundary upper boundary lower boundary Resonant surface

18. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Resonant Flow Instability V ω Res. Flow Instability Alfvén continuum KH Inst. Threshold <

19. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Equation of wave energy W – wave energy density ( kinetic+magnetic+thermal) S – wave energy flux

20. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Wave energy flux S x (x=+0) > 0 and S x (x=-0) < 0 S y (x=x A, y>0) > 0 and S y (x=x A, y<0) < 0

21. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Energy transfer by fast and Alfven waves flow outer boundary inner reflecting boundary upper boundary lower boundary x =0 x =x A

22. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 KH and RF instabilities: comparison RFI occurs at lower flow speeds KH unstable fast waves extract more energy KH unstable fast waves deposit the energy near the outer boundary Resonantly unstable fast waves deposit the energy at the resonant surface

23. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006

24. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006

25. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Other examples We have considered two examples: upstream/downstream fast waves (KHI) upstream fast waves/Alfvén waves (RFI) Other examples: interaction of MHD waves with viscosity, thermal conduction and other sinks of energy

26. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Part II Standing acoustic waves in coronal loops

27. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Observations of standing waves in coronal loops fast sausage (n=0) kink (n=1) slow sausage (acoustic) Asai et al. (2001) Aschwanden et al. (1999) Wang et al. (2002) VzVz

28. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Standing Acoustic Waves in Coronal Loops b) Solar atmosphere a) Solar interior Acoustic waves Plasma pressure Magnetic pressure slow sausage waves - c T

29. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Imaging Telescopes Transition Region & Coronal Explorer (TRACE) SOHO Extreme ultraviolet Imaging Telescope (EIT) Yohkoh Soft X-Ray Telescope (SXT) Ground based telescopes Observing Instruments Spectrometers SOHO Solar Ultraviolet Measurements of Emitted Radiation (SUMER) SOHO Coronal Diagnostic Spectrometer (CDS)

30. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Launch date Apr. 2, 1998 Mass 250 kg Angular resolution 1 arcsec  725 km on the Sun Filed of view 512 x 512 arcsec  1/10 of the Solar disc Average cadence 30 seconds TRACE Magnetic field morphology Tracking inhomogeneities Overlap in temperature coverage; Sensitivity of EUV band ratios to temperature changes => Coronal temperature diagnostics VL UV EUV

31. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006

32. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 SUMER quiet Sun spectrum 800-1500 Å Wavelength range 500-1610 Å Spatial resolution ~ 1.5 arcsec Minimum temporal resolution 3 s Spectral resolution 0.045 Å Doppler velocities 1 km/s Line profile measurements – line intensities, line widths, Doppler velocities

33. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Yohkoh/SXT TRACE 195A SUMER slit Standing acoustic waves in hot loops (Yohkoh, TRACE)

34. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Standing acoustic waves in hot loops (Yohkoh, TRACE)

35. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Observations in Fe XIX ~ 6.3 MK

36. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Standing acoustic waves on low mass M type stars

37. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Analytical approach s = 0 s = L Homogeneous loop Initial conditionsBoundary conditions

38. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Analytical approach BC I Laplace transform Solution of the transformed equation

39. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Analytical approach BC II Standing Wave Solution for t>P Propagating waves for P 1 =P/2 or P 2 =2P

40. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Stratified atmosphere Analytical approach - Cut-off frequency Klein-Gordon equation

41. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Analytical approach Propagation of weakly nonlinear, weakly dispersive waves in an isolated tube: F – integral operator Slab: Benjamin-Ono equation Cylinder: Leibovich-Roberts equation

42. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Thin flux tube equations Thermal conduction Heating rate Radiation

43. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Radiation L = n 2 Λ(T)

44. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Thermal conduction - Conductive flux along the field - Thermal conduction Can be positive or negative!!

45. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Scaling Laws

46. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Damping Maximum amplitude => amount of heat Period => heating duration

47. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Standing waves and loop diagnostics

48. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Motivation

49. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Corona? Major problems: Damping Localisation Limited observation duration

50. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Simple loop model: linear ideal (M)HD (Wave operator) v =Heating(t,s) Boundary conditions at s=0, s=L 0 < s < L Heating(t,s) = Σ Dirac delta or finite duration random pulses

51. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 The heating function

52. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Parker’s nanoflare model Initial B uniform / motions braiding

53. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 The Green’s function (Wave operator) G(t,s;τ,ξ) =δ(t-τ) δ(s-ξ) Boundary conditions at s=0, s=L Solution v uniquely determined by the Green’s function, BCs and the random Heating(t,s)

54. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Results of the analytical study The power spectrum for the time series of v has peaks at the frequencies of the standing waves(!) The height and sharpness of the peaks depends on the duration of the observations Finite duration pulses: peak at zero frequency; peak heights decrease faster with increasing harmonic number

55. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Loop model in a non-linear stratified atmosphere Continuity equation Momentum equation (gravity) Energy equation (field aligned thermal conduction, radiative losses, random heating)

56. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006

57. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Mean temperature Standard deviation

58. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Line synthesis Ne VIII (~0.794 MK), Mg X (~1.2 MK) resonant lines Doppler width of the line Projection of bulk velocity on the line of sight Line broadening function Emissivities along the loop Total intensity Line profiles

59. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Disc centre: the slit is transverse to the loop plane Youra’s impression 1

60. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Power spectra for Doppler shift time-series in Ne VIII

61. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Solar limb: the slit is transverse to the loop plane Youra’s impression 2

62. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Power spectra for Doppler shift time-series in Ne VIII

63. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Disc centre: the slit is aligned with the loop plane Youra’s impression 3

64. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Power spectra for Doppler shift time-series in Ne VIII

65. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Solar limb: the slit is aligned with the loop plane Youra’s impression 4

66. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Power spectra for Doppler shift time-series in Ne VIII

67. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 Multi-temperature loops Youra’s impression 5 Power Frequency

68. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006 A new diagnostic method for randomly heated loops is proposed Analysis of the power spectra of the Doppler shift time series for Ne VIII shows the presence of distinct peaks at the a) Fundamental mode frequency for loops on the limb b) Fundamental mode + 1 st harmonic for loops on the disc centre The method allows us to a) Determine average loop temperature b) Distinguish uniformly heated loops from loops heated at their footpoints The power spectra could be used to a) Estimate the energies involved in the random pulses b) Study the multi-thermal structure of the loops Conclusions Summary The power spectrum analysis of the Doppler shift time-series for coronal line profiles is a powerful tool for the diagnostics of the solar coronal plasma.

69. University of Sheffield Solar Physics & upper-Atmosphere Research Group Waves & Turbulence Phenomena in Space Plasmas, Kiten 2006