Wave Generation and Propagation in the Solar Atmosphere

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Wave Generation and Propagation in the Solar
Atmosphere

• Zdzislaw Musielak
• Physics Department
• University of Texas at Arlington (UTA)

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OUTLINE

• Theory of Wave Generation
• Theory of Wave Propagation
• Solar Atmospheric Oscillations
• Theory of Local Cutoff Frequencies
• Applications to the Sun

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The H-R Diagram
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Solar structure
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Model of the Solar Atmosphere
Averett and Loeser (2008)
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Energy Input

• From the solar photosphere
• acoustic and magnetic waves
• Produced in situ
• reconnective processes
• From the solar corona
• heat conduction

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Generation of Sound
Lighthill (1952)
James M. Lighthill
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Acoustic Sources
Monopole
Dipole
Quadrupole
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Efficiency of Acoustic Sources
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Lighthill Theory of Sound Generation
(Lighthill 1952)
The inhomogeneous wave equation
with
and the source function
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Lighthill-Stein Theory of Sound Generation
(Lighthill 1952 Stein 1967)
The inhomogeneous wave equation
with
and
and the acoustic cutoff frequency
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Lighthill-Stein Theory of Sound Generation
The source function is given by
where
and
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Applications of Lighthill-Stein Theory

• Generation of acoustic and magnetic flux tube
• waves in the solar convection zone
• Collaborators Peter Ulmschneider and Robert
  Rosner
• also Robert Stein, Peter Gail and Robert
  Kurucz
• Graduate Students Joachim Theurer, Diaa Fawzy,
  Aocheng Wang, Matthew Noble, Towfiq
  Ahmed, Ping Huang and Swati Routh

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Acoustic Wave Energy Fluxes
log g 4
Ulmschneider, Theurer Musielak (1996)
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Generation of Magnetic Tube Waves
Fundamental Modes
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Generation of Longitudinal Tube Waves I
The wave operator
with
,
and the cutoff frequency (Defouw 1976)
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Generation of Longitudinal Tube Waves II
The source function is given by
or it can be written as
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Generation of Transverse Tube Waves

The wave operator
with
,
,
The source function
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PROCEDURE

• Solution of the wave equations
• - Fourier transform in time and space
• Wave energy fluxes and spectra
• - Averaging over space and time
• - Asymptotic Fourier transforms
• - Turbulent velocity correlations
• - Evaluation of convolution integrals

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Description of Turbulence

• The turbulent closure problem
• - spatial turbulent energy spectrum
• (modified Kolmogorov)
• - temporal turbulent energy spectrum
• (modified Gaussian)
• (Musielak, Rosner, Stein Ulmschneider 1994)

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Solar Wave Energy Spectra
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Wave Energy and Radiative Losses
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Current Work

• Modifications of the Lighthill and
• Lighthill-Stein theories to include
• temperature gradients.

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Chromospheric Models

• Purely Theoretical
• Two-Component
• Self-Consistent
• Time-Dependent
• Collaborators Peter Ulmschneider, Diaa Fawzy,
• Wolfgang Rammacher,
  Manfred
• Cuntz and Kazik Stepien

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Models versus Observations

• Base - acoustic waves
• Middle - magnetic tube waves
• Upper other waves and / or non-wave heating

Fawzy et al. (2002a, b, c)
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Solar Chromospheric Oscillations

• Response of the solar chromosphere to propagating
  acoustic waves 3-min oscillations (Fleck
  Schmitz 1991, Kalkofen et al. 1994, Sutmann et
  al. 1998)
• Oscillations of solar magnetic flux tubes
  (chromospheric network) 7 min oscillations
  (Hasan Kalkofen 1999, 2003, Musielak
  Ulmschneider 2002, 2003)

Chromospheric oscillations are not cavity modes!
P-modes
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Applications of Fleck-Schmitz Theory

• Propagation of acoustic and magnetic flux tube
• waves in the solar chromosphere
• Collaborator Peter Ulmschneider
• Graduate Students Gerhard Sutmann, Beverly
  Stark, Ping Huang, Towfiq Ahmed, Shilpa
  Subramaniam and Swati Routh

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Excitation of Oscillations by Tube Waves I
The wave operator for longitudinal tube waves is
with
,
and the cutoff frequency (Defouw 1976)
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Excitation of Oscillations by Tube Waves II
The wave operator for transverse tube waves is
with
,
and the cutoff frequency (Spruit 1982)
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Initial Value Problems
and
IC
and
BC
and

Laplace transforms and inverse Laplace transforms
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Solar Flux Tube Oscillations
Longitudinal tube waves
Transverse tube waves
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Theoretical Predictions

• Solar Chromosphere
• 170 190 s (non-magnetic regions)
• 150 230 s (magnetic regions

Maximum amplitudes are 0.3 km/s
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Solar Atmospheric Oscillations

• Solar Chromosphere 100 250 s
• Solar Transition Region 200 400 s
• Solar Corona 2 600 s
• TRACE and SOHO

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Lambs Original Approach (1908)
Acoustic wave propagation in a stratified and
isothermal medium is described by the following
wave equation
With
, one obtains
Klein-Gordon equation
is the acoustic cutoff frequency
where
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A New Method to Determine Cutoffs
General form of acoustic wave equation in a
medium with gradients
i 1, 2, 3
Transformations
with
and
give
Using the oscillation theorem and Eulers
equation allow finding the acoustic cutoff
frequency!
Musielak, Musielak Mobashi Phys. Rev. (2006)
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The Oscillation Theorem
Consider
with periodic solutions
Another equation
If
for all x
then the solutions of the second equation are
also periodic
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Eulers Equation and Its Turning Point
Periodic solutions
Turning point
Evanescent solutions
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Applications of the Method

• Cutoff frequencies for acoustic and magnetic
• flux tube waves propagating in the solar
• chromosphere
• Collaborator Reiner Hammer
• Graduate Students Hanna Mobashi, Shilpa
  Subramaniam and Swati Routh

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Torsional Tube Waves I
Isothermal and wide magnetic flux tubes
Introducing
and
, we have
and
x and y are Hollwegs variables
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Torsional Tube Waves II
Using the method, we obtain
and
where
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Torsional Tube Waves III
Eliminating the first derivatives, we obtain
Klein-Gordon equations
and
where
and
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Torsional Tube Waves IV
Making Fourier transforms in time, the
Klein-Gordon equations become
and
Using Eulers equation and the oscillation
theorem, the turning-point frequencies can be
determined. The largest turning-point frequency
becomes the local cutoff frequency.

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Torsional Tube Waves V
Exponential models
where m 1, 2, 3, 4 and 5
The model basis is located at the solar
temperature minimum

Routh, Musielak and Hammer (2007)
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Torsional Tube Waves VI
Since
and
For isothermal and thin magnetic flux tubes, we
have
, which gives

cutoff-free propagation!
Musielak, Routh and Hammer (2007)
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Current Work

• Acoustic waves in non-isothermal media
• Waves in wide magnetic flux tubes
• Waves in wine-glass flux tubes
• Waves in inclined magnetic flux tubes

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CONCLUSIONS

• Lighthill-Stein theory of sound generation was
  used to calculate the solar acoustic wave energy
  fluxes. The fluxes are sufficient to explain
  radiative losses observed in non-magnetic regions
  of the lower solar chromosphere.
• A theory of wave generation in solar magnetic
  flux tubes was developed and used to compute the
  wave energy fluxes. The obtained fluxes are large
  enough to account for the enhanced heating
  observed in magnetic regions of the solar
  chromosphere.
• Fleck-Schmitz theory was used to predict
  frequencies and amplitudes of the solar
  atmospheric oscillations. The theory can account
  for 3-min oscillations in the lower chromosphere.
  
• A method to obtain local cutoff frequencies was
  developed. The method was used to derive the
  cutoffs for isothermal and wide flux tubes and
  to show that the propagation of torsional waves
  along isothermal and thin magnetic flux tubes is
  cutoff-free.

Supported by NSF, NASA and The Alexander von
Humboldt Foundation