Dispersion diagrams of chromospheric MHD waves in a 2D simulation

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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

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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
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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
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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
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Waves in the solar atmosphere

• Acoustic waves, sound waves, p-modes (pressure
  oscillations)
• Magnetic waves

B
k
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
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Characteristic speeds

• Sound speed cs 8.49 km/s
• Alfven speed vA B0/(4p?0)1/2
• Magnetohydrodynamic (MHD) waves can have hybrid
  speeds, depending on their angle of propagation f
  with respect to the magnetic field

Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Characteristic regions of plasma

• Plasma beta Let ß Pplasma/Pmagnetic P/B2
  cs2/vA2

low b means strong field Pmagnetic gt Pplasma
vA2 gt cs2 fast magnetic waves high b means
weak field Pplasmagt Pmagnetic cs2 gt vA2 fast
acoustic waves
z altitude in the solar atmosphere x position
along photosphere
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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2D MHD code models chromospheric dynamics and
waves

• Written by ?ke 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
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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
• x500 steps by 15.8 km per step for total 7.90 Mm
• z294 steps by 4.33 km per step for total 1.26 Mm
• t161 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
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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
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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
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Method 3D data to 2D plots

• 3D data (x,z,t) ? 2D slices (x,t) at a given
  altitude z

Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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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 peaks, at 2w, 4w, and 7w
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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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
peaks, at 3k and 5k
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Result of method x(t) ? k(w)

• 2D FFT wavenumbers (k) and frequencies w describe
  a signal varying in space (x) and time (t)

Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Real data x(t) ? ??(k)

• 2D FFT wavenumbers (k) and frequencies w describe
  a signal varying in space (x) and time (t)

Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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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
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Technique first plot w(k)
../show3RhoSliceH50.jpg
amplitude of w(k) ? projection ? contour plot of
w(k)
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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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
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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
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Analyze resultant w(k) diagrams

• Look at at heights z 0.2165 Mm, 1.0825 Mm
• Variables
• fractional density ??(x,z)/?0(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
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Density oscillations near photosphere (z 0.2165
Mm)
Dr(x,t)/r0 w(k) and contours
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Density oscillations high in chromosphere (z
1.0825 Mm)
Dr(x,t)/r0 w(k) and contours
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Patterns in density oscillations
Near photosphere and high in chromosphere
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Perpendicular velocity near photosphere (z
0.2165 Mm)
uperp(x,t)/c0 w(k) and contours
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Perpendicular velocity high in chromosphere (z
1.0825 Mm)
uperp(x,t)/c0 w(k) and contours
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Patterns in perpendicular velocity
Near photosphere and high in chromosphere
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Vertical velocity near photosphere (z 0.2165 Mm)
uz(x,t)/c0 w(k) and contours
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Vertical velocity high in chromosphere (z
1.0825 Mm)
uz(x,t)/c0 w(k) and contours
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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Patterns in vertical velocity
Near photosphere and high in chromosphere
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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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
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Outstanding questions

• Where do the fjords in uperp 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 w(k) plots go from positive to negative
  slope at high altitudes?

Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004
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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
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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/
• 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
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Characteristic waves in plasmas

• Equation of motion ? wave equation ? dispersion
  relation between frequencies w2p/T and
  wavenumbers k2p/l
• Different waves have characteristic w(k)
  relations.

Example p-modes (acoustic waves) in the solar
interior have
where
Chris Dove, presentation at HAO/NCAR, Thursday 29
July 2004