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Dispersion diagrams of chromospheric MHD waves in a 2D simulation

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. Outline. Solar atmosphere - motivating questions

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Dispersion diagrams of chromospheric MHD waves in a 2D simulation

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  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 ~106 K • Network regions: strongly magnetic regions, e.g. near sunspots • Magnetic canopy: a region where the plasma pressure and magnetic pressure are comparable A diagram of the Sun, courtesy NASA sohowww.nascom.nasa.gov/explore/images/layers.gif Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  4. Motivating questions • Why is the coronal temperature 106 K while the underlying photosphere is less than 104 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 B k Acoustic waves, sound waves, p-modes (pressure oscillations) Magnetic waves: Alfven waves travel along magnetic field lines Magnetosonic waves travel across field lines B k B k Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  6. B wave f Characteristic speeds • Sound speed cs = 8.49 km/s • Alfven speed vA= B0/(4πρ0)1/2 • Magnetohydrodynamic (MHD) waves can have hybrid speeds, depending on their angle of propagation f with respect to the magnetic field: Vfast= V=vA cos f Vslow= Insert these eqns if you like Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  7. Characteristic regions of plasma Plasma beta: Let β = Pplasma/Pmagnetic ~ P/B2 ~ cs2/vA2 “low b” means strong field: Pmagnetic > Pplasma : vA2 > cs2 fast magnetic waves “high b” means weak field: Pplasma> Pmagnetic : cs2 > vA2 fast acoustic waves z = altitude, x = position along photosphere Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  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 3D 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. 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. Example: p-modes (acoustic waves) in the solar atmosphere have w2 = stuff k Couldn’t find one easily. Suggestions? Scan Foukal? Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

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

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

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

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

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

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

  19. Technique: different plots of w(k) ../show3RhoSliceH50.jpg amplitude of w(k)  projection  contour plot of w(k) Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

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

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

  22. Analyze resultant w(k) diagrams Look at at heights: z = 0.2165 Mm, 1.0825 Mm Variables: fractional density Δρ(x,z)/ρ0(z)tracks acoustic (and magnetosonic) waves perpendicular velocity uperp tracks magnetic waves, whether Alfvenic or magnetosonic Vertical velocity uz tracks both acoustic and magnetic waves Waves are generally hybrid, not pure! Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  23. Fractional density oscillations at z = 0.2165 Mm Dr(x,t)/r_0w(k) of density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  24. Fractional density oscillations at z = 1.0825 Mm Dr(x,t)/r_0w(k) of density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  25. Patterns in fractional density oscillations w(k) of fractional density oscillations at both heights Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  26. Perpendicular velocity at z = 0.2165 Mm u(x,t)/c_0 w(k) of density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  27. Perpendicular velocity at z = 1.0825 Mm u(x,t)/c_0 w(k) of density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  28. Patterns in perpendicular velocity w(k) of perpendicular velocity oscillations at both heights Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  29. Vertical velocity at z = 0.2165 Mm u_z(x,t)/c_0 w(k) of density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  30. Vertical velocity at z = 1.0825 Mm u_z(x,t)/c_0 w(k) of density oscillations Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  31. Patterns in vertical velocity w(k) of vertical velocity oscillations at both heights Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

  32. Interpreting patterns • 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. Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

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

  34. Possible future work • Normalize w(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

  35. References and acknowledgements Foukal, P.,Solar Astrophysics, John Wiley & Sons, 1990 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/ Cairns, R.A., Plasma Physics, Blackie, 1985 Priest, E.R. 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 Chris Dove, presentation at HAO/NCAR, Thursday 29 July 2004

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