1. Dispersion diagrams of chromospheric MHD waves in a 2D simulation Chris Dove The Evergreen State College Olympia, WA 98505 with Tom Bogdan and E.J. Zita presented at HAO/NCAR, Boulder, CO Thursday 29 July 2004

2. Outline Solar atmosphere - motivating questions Background – qualitative picture 2D MHD code models dynamics Methods to get clearer pictures Analysis of results Patterns Interpretations Future work References and acknowledgments Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

3. Observations of solar atmosphere Photosphere: ~5700 K, this is where our driver excites waves. It lies between the chromosphere and the convection zone Chromosphere: this region is where our waves live Corona: extends millions of kilometers into the solar atmosphere and reaches temperatures of ~10 6 K Network regions: strongly magnetic regions, e.g. near sunspots Magnetic canopy: a region where the plasma pressure and magnetic pressure are comparable Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 A diagram of the Sun, courtesy NASA sohowww.nascom.nasa.gov/explore/images/layers.gif

4. Motivating questions Why is the coronal temperature 10 6 K while the underlying photosphere is less than 10 4 K? If surface sound waves die off as they rise, what transports energy up through the chromosphere? How can magnetic waves transport energy? How can sound waves transform into magnetic waves? What waves are evident in the chromosphere? Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

5. Waves in the solar atmosphere Acoustic waves, sound waves, p-modes (pressure oscillations) Magnetic waves: Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Alfven waves travel along magnetic field lines Magnetosonic waves travel across field lines B k k B Bk

6. Characteristic speeds Sound speed c s = 8.49 km/s Alfven speed v A = B 0 /(4πρ 0 ) 1/2 Magnetohydrodynamic (MHD) waves can have hybrid speeds, depending on their angle of propagation  with respect to the magnetic field: Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 B wave 

7. Characteristic regions of plasma Plasma beta: Let β = P plasma /P magnetic ~ P/B 2 ~ c s 2 /v A 2 Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 z = altitude in the solar atmosphere x = position along photosphere “low  ” means strong field: P magnetic > P plasma : v A 2 > c s 2 fast magnetic waves “high  ” means weak field: P plasma > P magnetic : c s 2 > v A 2 fast acoustic waves

8. 2D MHD code models chromospheric dynamics and waves Written by Ǻke Nordlund, edited and run by Mats Carlsson + team at Institute for Theoretical Astrophysics in Oslo Starts with “network” magnetic field in stratified chromosphere (density drops with altitude, constant temperature) and sound-wave “driver” at photosphere Self-consistently evolves velocities v(x,z,t) and changes in magnetic field B(x,z,t) and density r(x,z,t) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

9. A 2D MHD simulation code Magnetic field follows roughly Gaussian distribution: flux concentrated near driver, spreads out with increasing altitude Atmosphere is isothermal pressure/density drops with e -z/H Radial driver (400km-wide piston) models convection  p-modes x:500 steps by 15.8 km per step for total 7.90 Mm z:294 steps by 4.33 km per step for total 1.26 Mm t:161 steps by 1.3 s per step for total Driving frequency = 42.9 mHz Spatial extent scaled down from realistic values by a factor of 10, and driver frequency scaled up Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

10. Waves propagate and transform Sound waves channel up field lines Driver bends field lines  excites Alfvén waves Driver compresses field lines  excites magnetosonic waves Waves change identity, especially near  ~1 surface (mode-mixing) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

11. Goal: get a clearer picture of waves Trying to figure out what kind of waves are where, and how they transform Learning how to characterize waves by their structures in  (k) METHODS Look at 2D slabs (x,t) of 3D data (x,z,t) Find wavenumbers k = 2  / by Fourier- transforming signals in x Find frequencies  = 2  /T by Fourier-transforming signals in t Look for waves’ signatures in  (k) diagrams Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

12. Method: 3D data to 2D plots Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 3D data (x,z,t)  2D slices (x,t) at a given altitude z

13. Method: time  frequency Fourier transformation (FFT) can tell you what frequencies (  ) make up a signal varying in time (t) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Ex: y(t) =sin(2  t) + 2 sin(4  t) + 3 sin(7  t)  Three FFT peaks, at 2 , 4  and 7 

14. Method: space  wavenumber Fourier transformation (FFT) can tell you what wavenumbers (k) make up a signal varying in space (x) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 g(x) = cos(3 kx) + 2 cos (5kx)  Two FFT peaks, at 3k and 5k

15. Result of method: x(t)  k(  ) 2D FFT wavenumbers (k) and frequencies  describe a signal varying in space (x) and time (t) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

16. Real data: x(t)   (k) 2D FFT wavenumbers (k) and frequencies  describe a signal varying in space (x) and time (t) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

17. Goal: get clearer  (k) diagrams Techniques: View contour plots of 2D FFT  (k) plots Average data over small width in altitude Window data to remove edge effects Analyze resultant  (k) diagrams Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

18. Technique: first plot  (k) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 amplitude of  (k)  projection  contour plot of  (k)../show3RhoSliceH50.jpg

19. Technique: average data over z Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Average over width  z in altitude  Smoother  (k) contour../avgRhoSlabH50.jpg

20. Technique: remove edge effects Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Multiply by Hanning window   (k) without edge effects../Hanning.jpg../HngRho.jpg

21. Analyze resultant  (k) diagrams Look at at heights: z = 0.2165 Mm, 1.0825 Mm Variables: fractional density Δρ(x,z)/ρ 0 (z) tracks acoustic (and magnetosonic) waves perpendicular velocity u perp tracks magnetic waves, whether Alfvenic or magnetosonic Vertical velocity u z tracks both acoustic and magnetic waves Waves are generally hybrid, not pure! Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

22. Density oscillations near photosphere (z = 0.2165 Mm) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004  (x,t)/    (k) and contours

23. Density oscillations high in chromosphere (z = 1.0825 Mm) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004  (x,t)/    (k) and contours

24. Patterns in density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Near photosphereand high in chromosphere

25. Perpendicular velocity near photosphere (z = 0.2165 Mm) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004  u perp (x,t)/c 0  (k) and contours

26. Perpendicular velocity high in chromosphere (z = 1.0825 Mm) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004  u perp (x,t)/c 0  (k) and contours

27. Patterns in perpendicular velocity Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Near photosphereand high in chromosphere

28. Vertical velocity near photosphere (z = 0.2165 Mm) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004  u z (x,t)/c 0  (k) and contours

29. Vertical velocity high in chromosphere (z = 1.0825 Mm) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004  u z (x,t)/c 0  (k) and contours

30. Patterns in vertical velocity Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Near photosphereand high in chromosphere

31. Interpreting patterns Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Ringing in k: Driver has sharp edges, so there are harmonics of the fundamental driving frequency and wavenumber. Spacing of harmonics in k becomes smaller with increasing height because the effective driver wavelength increases. Steep slopes and negative slopes: The apparent group velocity of a wave can approach infinity at the instant a wavefront breaks through our slab. Negative group velocity seems to indicate waves moving in the negative x- direction.

32. Outstanding questions Where do the “fjords” in u perp come from and what do they mean? What relationship do the S-curves have with the driver? Why do small-k (long-wavelength) S-curves in density  (k) plots go from positive to negative slope at high altitudes? Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

33. Possible future work Normalize  (k) signals, subtract out acoustic part, and get better resolution of magnetic waves Add right- and left-going waves for better signal to noise Analyze runs with weak magnetic field Analyze runs with better boundary conditions Analyze 2.5D simulations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

34. References & acknowledgements Foukal, P.,Solar Astrophysics, John Wiley & Sons, 1990 Chen, F.F., Introduction to Plasma Physics, Plenum Press, 1984 Bogdan et al., Waves in the magnetized solar atmosphere: II, ApJ, Sept. 2003 Thomas, Magneto-Atmospheric Waves, Ann.Rev.Fluid Mech., 1983 Johnson, M.,Petty-Powell,S., and E.J. Zita, Energy transport by MHD waves above the photosphere numerical simulations, http://academic.evergreen.edu/z/zita/research/ http://academic.evergreen.edu/z/zita/research/ Cairns, R.A., Plasma Physics, Blackie, 1985 Priest, E.R., Solar Magnetohydrodynamics, from Dynamic Sun, ed. Dwivedi, B., Cambridge, 2003 Brigham, E. Oran, The Fast Fourier Transform, Prentice-Hall, 1974 Press, W. et al., Numerical Recipes, Cambridge, 1986 We thank Tom Bogdan and E.J. Zita for their training, guidance, and bad jokes. This work was supported by NASA's Sun-Earth Connection Guest Investigator Program, NRA 00-OSS-01 SEC This talk is available online at http://academic.evergreen.edu /z/zita/research/summer2004/chromo/Chris2HAO.ppt Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

35. Characteristic waves in plasmas Equation of motion  wave equation  dispersion relation between frequencies w=2p/T and wavenumbers k=2p/l Different waves have characteristic w(k) relations. Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004 Example: p-modes (acoustic waves) in the solar interior have where