• Received : 2015.09.14
  • Accepted : 2015.10.15
  • Published : 2016.02.29


The aim of this study is to describe the physical processes taking place in the solar photosphere. Based on 3D hydrodynamic simulations including a detailed radiation transfer scheme, we investigate thermodynamic structures and radiation fields in solar surface convection. As a starting model, the initial stratification in the outer envelope calculated using the solar calibrations in the context of the standard stellar theory. When the numerical fluid becomes thermally relaxed, the thermodynamic structure of the steady-state turbulent flow was explicitly collected. Particularly, a non-grey radiative transfer incorporating the opacity distribution function was considered in our calculations. In addition, we evaluate the classical approximations that are usually adopted in the onedimensional stellar structure models. We numerically reconfirm that radiation fields are well represented by the asymptotic characteristics of the Eddington approximation (the diffusion limit and the streaming limit). However, this classical approximation underestimates radiation energy in the shallow layers near the surface, which implies that a reliable treatment of the non-grey line opacities is crucial for the accurate description of the photospheric convection phenomenon.


Sun:photosphere;atmospheres;granulation;numerical:hydrodynamics;radiative transfer


  1. Asplund, M., Grevesse, N., & Sauval, A. J. 2005, The Solar Chemical Composition, ASPC, 336, 25
  2. Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, The Chemical Composition of the Sun, ARA&A, 47, 481
  3. Bach, K., & Kim, Y.-C. 2012, Hydrodynamical Comparison Test of Solar Models, Astron. Nachr., 333, 934
  4. Bahcall, J. N., & Loeb, A. 1990, Element Diffusion in Stellar Interiors, ApJ, 360, 267
  5. Baran, O. A., & Stodilka, M. I. 2014, Specifics of the Solar Photospheric Convection at Granulation, Mesogranulation, and Supergranulation Scales, Sol. Phys., 30, 173
  6. Basu, S., & Antia, H. M. 2008, Helioseismology and Solar Abundances, Phy. Rep., 457, 217
  7. Böhm-Vitense, E. 1958, Über Die Wasserstoffkonvektionszone in Sternen Verschiedener Effektivtemperaturen und Leuchtkräfte. Mit 5 Textabbildungen, Z. Astrophys., 46, 108
  8. Cannon, C. J. 1973, Angular Quadrature Perturbations in Radiative Transfer Theory, J.QSRT, 13, 627
  9. Cannon, C. J. 1973, Frequency-Quadrature Perturbations in Radiative-Transfer Theory, ApJ, 185, 621
  10. Castelli, F., Gratton, R. G., & Kurucz, R. L. 1997, Notes on the convection in the ATLAS9 model atmospheres, A&A, 318, 841
  11. Chan, K. L., & Wolff, C. L. 1982, ADI on Staggered Mesh - A Method for the Calculation of Compressible Convection, J. Comp. Phys., 47, 109
  12. Chan, K. L., & Sofia, S. 1987, Validity Tests of the Mixing-Length Theory of Deep Convection, Science, 235, 465
  13. Deardorff, J. W. 1970, A Numerical Study of Three-dimensional Turbulent Channel Flow at Large Reynolds Numbers, J. Fluid Mechanics, 41, 453
  14. Dravins, D. 1987, Stellar Granulation II: Stellar Photospheric Line Asymmetries, A&A, 172, 211
  15. Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005, Low- Temperature Opacities, ApJ, 623, 585
  16. Freytag, B., Ludwig, H.-G., & Steffen, M. 1996, Hydrodynamical Models of Stellar Convection. The Role of Overshoot in DA White Dwarfs, A-Type Stars, and the Sun, A&A, 313, 497
  17. Grevesse, N., & Sauval, A. J. 1998, Standard Solar Composition, SSRv, 85, 161
  18. Hathaway, D. H., Teil, T., Norton, A. A., & Kitiashvili, I. 2015, The Sun’s Photospheric Convection Spectrum, ApJ, 811, 105
  19. Hubeny, I. 2003, Stellar Atmosphere Modeling, ASPC, 288, 17
  20. Iglesias, C. A., & Rogers, F. J. 1996, Updated Opal Opacities, ApJ, 464, 943
  21. Kim, Y. -C., Fox, P. A., Sofia, S., & Demarque, P. 1995, Modeling of Shallow and Inefficient Convection in the Outer Layers of the Sun Using Realistic Physics, ApJ, 442, 422
  22. Kim, Y. -C., Fox, P. A., Demarque, P., & Sofia, S. 1996, Modeling Convection in the Outer Layers of the Sun: A Comparison with Predictions of theMixing-Length Approximation, ApJ, 461, 499
  23. Kurucz, H. L. 1995, Laboratory and Astronomical High Resolution Spectra, ASPC, 81, 17
  24. Kurucz, R. L. 1996, Model Atmospheres and Spectrum Synthesis, ASPC, 108, 160
  25. Ludwig, H. G., Freytag, B., & Steffen, M. 1999, A Calibration of theMixing-Length for Solar-Type Stars Based on Hydrodynamical Simulations, A&A, 346, 111
  26. Mihalas, D. 1978, Stellar Atmospheres 2nd edn, Freeman and Co. (San Francisco: Freeman and Co.)
  27. Nordlund, A. 1982 Numerical Simulations of the Solar Granulation. I - Basic Equations and Methods, A&A, 107, 1
  28. Nordlund, A., & Dravins, D. 1990, Stellar Granulation. III - Hydrodynamic Model Atmospheres, A&A, 228, 155
  29. Peaceman, D. W., & Rachford, Jr. H. H. 1955, The Numerical Solution of Parabolic and Elliptic Differential Equations, J. Soc. Ind. Appl. Math, 3, 28
  30. Robinson, F. J., Demarque, P., Li, L. H., Sofia, S., Kim, Y.-C., Chan, K. L., & Guenther, D. B. 2003, Three-Dimensional Convection Simulations of the Outer Layers of the Sun Using Realistic Physics, MNRAS, 340, 923
  31. Robinson, F. J., Demarque, P., Li, L. H., Sofia, S., Kim, Y.-C., Chan, K. L., & Guenther, D. B. 2004, Three-Dimensional Simulations of the Upper Radiation-Convection Transition Layer in Subgiant Stars, MNRAS, 347, 1208U
  32. Robinson, F. J., Demarque, P., Guenther, D. B. Kim, Y.-C., & Chan, K. L. 2005, Simulating the Outer Layers of Procyon A : A Comparison with the Sun, MNRAS, 362, 1031
  33. Rogers, F. J., Swenson, F. J., & Iglesias, C. A. 1996, OPAL Equation-of-State Tables for Astrophysical Applications, ApJ, 456, 902
  34. Sbordone, L., Bonifacio, P., Castelli, F., & Kurucz, R. L. 2004, ATLAS and SYNTHE under Linux, MSAIS, 5, 93
  35. Smagorinsky, J. 1963, General Circulation Experiments with the Primitive Equations I. The Basic Experiment, Monthly Weather Rev., 91, 99<0099:GCEWTP>2.3.CO;2
  36. Spiegel, E. A. 1957, The Smoothing of Temperature Fluctuations by Radiative Transfer, ApJ, 126, 202
  37. Steffen, M., Ludwig, H.-G., & Kruess, A. 1989, A Numerical Simulation Study of Solar Granular Convection in Cells of Different Horizontal Dimension, A&A, 213, 371
  38. Stein, R. F., & Nordlund, Å. 1989, Topology of Convection Beneath the Solar Surface, ApJL, 342, 95
  39. Tanner, J. D., Basu, S., & Demarque, P. 2014, The Effect of Metallicity-dependent T-tau Relations on Calibrated Stellar Models, ApJL, 785, 13
  40. Thoul, A. A., Bahcall, J. N., & Loeb, A. 1994, Element Diffusion in the Solar Interior, ApJ, 421, 828
  41. Trampedach, R., & Stein, R. F. 2011, The Mass Mixing Length in Convective Stellar Envelopes, ApJ, 731, 78
  42. Trampedach, R., Stein, R. F., Christensen-Dalsgaard, J., Nordlund, Å., & Asplund, M. 2014, Improvements to Stellar Structure Models, Based on a Grid of 3D Convection Simulations - I. T(τ) Relations, MNRAS, 442, 805
  43. Trampedach, R., Stein, R. F., Christensen-Dalsgaard, J., Nordlund, Å., & Asplund, M. 2014, Improvements to Stellar Structure Models, Based on a Grid of 3D Convection Simulations - II. Calibrating theMixing-length Formulation, MNRAS, 445, 4366
  44. Unno, W., & Spiegel, E. A. 1966, The Eddington Approximation in the Radiative Heat Equation, PASJ, 18, 85
  45. Vögler, A., Bruls, J. H. M. J., & Schüssler, M. 2004, Approximations for Non-grey Radiative Transfer in Numerical Simulations of the Solar Photosphere, A&A, 421, 741