Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Using the Butterfly Effect to Probe How the Sun Generates Acoustic Noise

Abstract

A major encumbrance to recognition of individual episodes of noise emission is the accumulation over hours of other noise emitted long before. This is true in simulations just as it is in the solar environment itself. The composite seismic signature of acoustic radiation accumulated over preceding hours drowns out the signature of newly emitted “acoustic pings.” This problem could be alleviated in simulations by periodically damping the accumulated acoustic radiation – if this can be done benignly, i.e. in such a way that the onset transient of the damping (and its subsequent termination) does not emit its own acoustic noise. We introduce a way of doing this based upon a study of the butterfly effect in compressible radiative MHD simulations of convection that excites p-modes. This gives us an encouraging preview of what further development of this utility offers for an understanding of the character of p-mode generation in convective atmospheres.

This is a preview of subscription content, log in to check access.

Figure 1
Figure 2
Figure 3

Notes

  1. 1.

    The simulations do not associate geodetic directionality, neither terrestrial nor solar, with the computational grid. We arbitrarily designate the grid boundary toward the top of the page “north” and the rightward “east,” reserving “upward” and “downward” for reference with respect to gravity.

References

  1. Ahmad, Q.R., Allen, R.C., Andersen, T.C., Anglin, J.D., Bühler, G., Barton, J.C., et al.: 2001, Measurement of the rate of \(\nu _{e} + d \rightarrow p + p + e^{-}\) interactions produced by 8B solar neutrinos at the sudbury neutrino observatory. Phys. Rev. Lett.87(7), 071301. DOI . ADS .

  2. Alvarado-Gómez, J.D., Buitrago-Casas, J.C., Martínez-Oliveros, J.C., Lindsey, C., Hudson, H.S., Calvo-Mozo, B.: 2012, Magneto-acoustic energetics study of the seismically active flare of 15 February 2011. Solar Phys.280(2), 335. DOI . ADS .

  3. Braun, D.C., Lindsey, C.: 2001, Seismic imaging of the far hemisphere of the Sun. Astrophys. J. Lett.560(2), L189. DOI . ADS .

  4. Brown, T.M., Bogdan, T.J., Lites, B.W., Thomas, J.H.: 1992, Localized sources of propagating acoustic waves in the solar photosphere. Astrophys. J. Lett.394, L65. DOI . ADS .

  5. Christensen-Dalsgaard, J., Proffitt, C.R., Thompson, M.J.: 1993, Effects of diffusion on solar models and their oscillation frequencies. Astrophys. J. Lett.403, L75. DOI . ADS .

  6. Donea, A.-C., Braun, D.C., Lindsey, C.: 1999, Seismic images of a solar flare. Astrophys. J. Lett.513(2), L143. DOI . ADS .

  7. Evans, J.W., Michard, R., Servajean, R.: 1963, Observational study of macroscopic inhomogeneities in the solar atmosphere. V. Statistical study of the time variations of the solar inhomogeneities. Ann. Astrophys.26, 368. ADS .

  8. Evans, J.W., Main, P., Michard, R., Servajean, R.: 1962, Correlations in the time variations of macroscopic inhomogeneities in the solar atmosphere. Astrophys. J. Lett.136(2), L682. DOI . ADS .

  9. Goldreich, P., Kumar, P.: 1990, Wave generation by turbulent convection. Astrophys. J.363, 694. DOI . ADS .

  10. Kumar, P.: 1993, Solar oscillations with frequencies above the acoustic cutoff frequency. In: Brown, T.M. (ed.) GONG 1992. Seismic Investigation of the Sun and Stars 42, Astron. Soc. Pacific, San Francisco, 15. ADS .

  11. Kumar, P.: 1994, Properties of acoustic sources in the Sun. Astrophys. J.428, 827. DOI . ADS .

  12. Kumar, P., Lu, E.: 1991, The location of the source of high-frequency solar acoustic oscillations. Astrophys. J. Lett.375, L35. DOI . ADS .

  13. Leighton, R.B., Noyes, R.W., Simon, G.W.: 1962, Velocity fields in the solar atmosphere. I. Preliminary report. Astrophys. J.135, 474. DOI . ADS .

  14. Lighthill, M.J.: 1952, On sound generated aerodynamically. I. General theory. Proc. Roy. Soc. London Ser. A, Math. Phys. Sci.211, 564. DOI . ADS .

  15. Lindsey, C., Braun, D.C.: 1997, Helioseismic holography. Astrophys. J.485(2), 895. DOI . ADS .

  16. Lindsey, C., Braun, D.C.: 2000, Seismic images of the far side of the Sun. Science287(5459), 1799. DOI . ADS .

  17. Lindsey, C., Braun, D.C.: 2017, Seismic imaging of the Sun’s far hemisphere and its applications in space weather forecasting. Space Weather15, 761. DOI .

  18. Lindsey, C., Donea, A.-C.: 2013, Statistics of local seismic emission from the solar granulation. In: Cally, P., Erdélyi, R., Norton, A. (eds.) Eclipse on the Coral Sea: Cycle 24 Ascending (GONG 2012, LWS/SDO-5, and SOHO 27), J. Phys.CS-440, 012044. DOI . ADS .

  19. Lorenz, E.N.: 1969, Atmospheric predictability as revealed by naturally occurring analogues. J. Atmos. Sci.26, 636. DOI . ADS .

  20. Noyes, R.W., Leighton, R.B.: 1963, Velocity fields in the solar atmosphere. II. The oscillatory field. Astrophys. J.138, 631. DOI . ADS .

  21. Rast, M.P.: 1997, Photospheric downflows: How deep, how coherent, how important? In: Pijpers, F.P., Christensen-Dalsgaard, J., Rosenthal, C.S. (eds.) SCORe’96: Solar Convection and Oscillations and Their Relationship, Astrophys. Space Sci. Lib.225, Kluwer, Dordrecht, 135. DOI . ADS .

  22. Rast, M.P.: 1999a, The thermal starting plume as an acoustic source. Astrophys. J. Lett.524, 462. DOI . ADS .

  23. Rast, M.P.: 1999b, Thermal starting plumes, solar granulation, and the excitation of solar acoustic oscillations. In: Rimmele, T.R., Balasubramaniam, K.S., Radick, R.R. (eds.) High Resolution Solar Phyics: Theory, Observations and TechniquesCS-183, Astron. Soc. Pacific, San Francisco, 443. ADS .

  24. Stein, R.F., Nordlund, A.: 1998a, Convection and p-modes. In: Korzennik, S.G., Wilson, A. (eds.) Structure and Dynamics of the Interior of the Sun and Sun-Like Stars SOHO 6/GONG 98 WorkshopSP-418, ESA, Noordwijk, 693. ADS .

  25. Stein, R.F., Nordlund, A.: 1998b, Simulations of solar granulation. I. General properties. Astrophys. J.499, 914. DOI . ADS .

  26. Stein, R.F., Nordlund, A.: 2001, Solar oscillations and convection. II. Excitation of radial oscillations. Astrophys. J.546, 585. DOI . ADS .

Download references

Acknowledgments

We very much appreciate the insight of the anonymous reviewer, especially for their recognition of alternatives to our primary initial interpretation of the butterfly results. We equally appreciate the insight of Robert Stein, Mark Rast, and Douglas Braun. We would like to acknowledge high-performance computing support from Cheyenne (doi:10.5065/D6RX99Hx) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation.

Author information

Correspondence to Charles Lindsey.

Ethics declarations

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The spatial domain of the simulations in this study was a square slab whose horizontal dimensions were \(98~\mbox{Mm}\times 98~\mbox{Mm}\) (\(768\times 768\) pixels), with a depth of 18.4 Mm (384 pixels), noting that the horizontal separation between neighboring pixels, 128 km, is 8/3 the vertical pixel separation of 48 km. The simulation domain is evolved with a cadence of 0.5 seconds, but it is saved only every 22.5 seconds.

The disturbance injected into the butterfly version of the simulation was a pseudo-random perturbation in the rms range of \({\pm}\,3~\mbox{m}\,\mbox{s}^{-1}\) applied independently to the vertical velocity of each pixel extending downward from the base of the photosphere to a depth of 1 Mm. This disturbance was applied in a single 3D frame of the butterfly version of the simulation, the computation proceeding from that moment with no further intervention from the authors.

In standard convective simulations run in the past, the bottom boundary has been, like the top, an acoustic reflector hence the strong accumulation of p-modes discussed in Sections 14, and 5, and the sharp resonances these show in \(\ell \)\(\nu \) diagrams computed from standard compressible simulations of convection (Stein and Nordlund, 2001). For the simulations run in this study, we replaced the reflecting lower boundary by one that efficiently absorbs low-\(\ell \) modes that penetrate to it. This is devised by a boundary condition that prescribes a pressure perturbation \([\delta p]\) at the lower boundary that resists vertical motion by the rule

$$ \delta p = -c\rho v_{z}, $$
(2)

a simple adaptation of the standard practice in classical electronics of terminating a transmission line with a resistor that matches its characteristic impedance to suppress a reflection back to the source that would occur in one that was terminated by a short or left simply open.

By this contrivance, the back-reflection of the pulse that delivers the butterfly perturbation to the granulation passes through the bottom boundary and disappears, leaving \(\Delta ^{\mathrm{lep}}v_{z}\) in the purely acoustic region undisturbed until the first acoustic disturbances from the growing butterfly-difference sources arrive therein. This innovation, then, represents an early step in the development of the acoustic mop anticipated in Section 5 and reiterated in item vi) of the Summary (Section 7).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lindsey, C., Rempel, M. Using the Butterfly Effect to Probe How the Sun Generates Acoustic Noise. Sol Phys 295, 26 (2020). https://doi.org/10.1007/s11207-020-1580-x

Download citation

Keywords

  • Convection
  • Helioseismology
  • \(p\)-Modes