Wave Generation and Propagation in the Solar Atmosphere

1. Wave Generation and Propagation in the Solar Atmosphere Zdzislaw Musielak Physics Department University of Texas at Arlington (UTA)

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

3. The H-R Diagram

4. Solar structure

5. Model of the Solar Atmosphere Averett and Loeser (2008)

6. Energy Input From the solar photosphere: acoustic and magnetic waves Produced in situ: reconnective processes From the solar corona: heat conduction

7. Generationof Sound Lighthill (1952) JamesM. Lighthill

8. Acoustic Sources Monopole Dipole Quadrupole

9. Efficiency of Acoustic Sources

10. Lighthill Theory of Sound Generation (Lighthill 1952) The inhomogeneous wave equation with and the source function

11. Lighthill-Stein Theory of Sound Generation (Lighthill 1952; Stein 1967) The inhomogeneous wave equation with and and the acoustic cutoff frequency

12. Lighthill-Stein Theory of Sound Generation The source function is given by where and

13. 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

14. Acoustic Wave Energy Fluxes log g = 4 Ulmschneider, Theurer & Musielak (1996)

15. Generation of Magnetic Tube Waves Fundamental Modes

16. Generation of Longitudinal Tube Waves I The wave operator with , and the cutoff frequency (Defouw 1976)

17. Generation of Longitudinal Tube Waves II The source function is given by or it can be written as

18. Generation of Transverse Tube Waves The wave operator with , , The source function

19. 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

20. Description of Turbulence The turbulent closure problem: - spatial turbulent energy spectrum (modified Kolmogorov) - temporal turbulent energy spectrum (modified Gaussian) (Musielak, Rosner, Stein & Ulmschneider 1994)

21. Solar Wave Energy Spectra

22. Wave Energy and Radiative Losses

23. Current Work Modifications of the Lighthill and Lighthill-Stein theories to include temperature gradients.

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

25. Models versus Observations • Base - acoustic waves • Middle - magnetic tube waves • Upper – other waves and / or non-wave heating Fawzy et al. (2002a, b, c)

26. 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

27. 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

28. Excitation of Oscillations by Tube Waves I The wave operator for longitudinal tube waves is with , and the cutoff frequency (Defouw 1976)

29. Excitation of Oscillations by Tube Waves II The wave operator for transverse tube waves is with , and the cutoff frequency (Spruit 1982)

30. Initial Value Problems and IC: and BC: and Laplace transforms and inverse Laplace transforms

31. Solar Flux Tube Oscillations Longitudinal tube waves Transverse tube waves

32. Theoretical Predictions Solar Chromosphere: 170 – 190 s (non-magnetic regions) 150 – 230 s (magnetic regions Maximum amplitudes are 0.3 km/s

33. Solar Atmospheric Oscillations • Solar Chromosphere: 100 – 250 s • Solar Transition Region: 200 – 400 s • Solar Corona: 2 – 600 s TRACE and SOHO

34. Lamb’s 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

35. 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 Euler’s equation allow finding the acoustic cutoff frequency! Musielak, Musielak & Mobashi Phys. Rev. (2006)

36. The Oscillation Theorem Consider with periodic solutions Another equation If for all x then the solutions of the second equation are also periodic

37. Euler’s Equation and Its Turning Point Periodic solutions Turning point Evanescent solutions

38. 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

39. Torsional Tube Waves I Isothermal and ‘wide’ magnetic flux tubes Introducing and , we have and x and y are Hollweg’s variables

40. Torsional Tube Waves II Using the method, we obtain and where

41. Torsional Tube Waves III Eliminating the first derivatives, we obtain Klein-Gordon equations and where and

42. Torsional Tube Waves IV Making Fourier transforms in time, the Klein-Gordon equations become and Using Euler’s equation and the oscillation theorem, the turning-point frequencies can be determined. The largest turning-point frequency becomes the local cutoff frequency.

43. 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)

44. 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)

45. 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

46. 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