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• 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

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

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?


Waves in the solar atmosphere

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

kB

B k

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 with respect to the magnetic field:


Bwave

2 2 2 4 4 2 2

2 2 20

2 2 2 4 4 2 2

1[ 2 cos 2

2

cos

1[ 2 cos 2

2

A s A s A s

A

A s A s A s

v v c v c v c

v v

v v c v c v c

Characteristic regions of plasma

Plasma beta: Let β = Pplasma/Pmagnetic ~ P/B2 ~ cs2/vA

2


z = altitude in the solar atmosphere

x = position along photosphere

“low ” means strong field:

Pmagnetic > Pplasma : vA2 > cs

2

fast magnetic waves

“high ” means weak field:

Pplasma> Pmagnetic : cs2 > vA

2

fast acoustic waves

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)


BBvB 2)(

t

1 1( ) ( )

4p g

t

u

u u B B z

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


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)


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


Method: 3D data to 2D plots


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

Method: time frequency

Fourier transformation (FFT) can tell you what frequencies () make up a signal varying in time (t)


Ex: y(t) =sin(2t) + 2 sin(4t) + 3 sin(7t)

Three FFT peaks, at 2, 4and 7

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

Result of method: x(t) k()

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


Real data: x(t) (k)

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


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


Technique: first plot (k)


amplitude of (k) projection contour plot of (k)

../show3RhoSliceH50.jpg

Technique: average data over z


Average over width z in altitude Smoother (k) contour

../avgRhoSlabH50.jpg

Technique: remove edge effects


Multiply by Hanning window (k) without edge effects

../Hanning.jpg ../HngRho.jpg

Analyze resultant (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!


Density oscillations near photosphere (z = 0.2165 Mm)


(x,t)/ (k) and contours

Density oscillations high in chromosphere (z = 1.0825 Mm)


(x,t)/ (k) and contours

Patterns in density oscillations


Near photosphere and high in chromosphere

Perpendicular velocity near photosphere (z = 0.2165 Mm)


uperp(x,t)/c0 (k) and contours

Perpendicular velocity high in chromosphere (z = 1.0825 Mm)


uperp(x,t)/c0 (k) and contours

Patterns in perpendicular velocity



Vertical velocity near photosphere (z = 0.2165 Mm)


uz(x,t)/c0 (k) and contours

Vertical velocity high in chromosphere (z = 1.0825 Mm)


uz(x,t)/c0 (k) and contours

Patterns in vertical velocity



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.

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 (k) plots go from positive to negative slope at high altitudes?


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


References & acknowledgementsFoukal, 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. 2003Thomas, Magneto-Atmospheric Waves, Ann.Rev.Fluid Mech., 1983Johnson, 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, 1985Priest, E.R., Solar Magnetohydrodynamics, from Dynamic Sun, ed. Dwivedi, B.,

Cambridge, 2003Brigham, E. Oran, The Fast Fourier Transform, Prentice-Hall, 1974Press, 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


http://academic.evergreen.edu/z/zita/research/

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 interior have

2 ( )2n H

ln gk

where

dispersion diagrams of chromospheric mhd waves in a 2d simulation

Documents

surface sound waves

solar atmosphereacoustic

magnetic pressure

magnetic regions

network magnetic field

magnetic field bx

alfven waves travel

waves live corona