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Sudakov I. A., Vakulenko S. A., Sukacheva T. G. New type of bifurcations in the modified Rayleigh–Benard convection problem. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 2, pp. 145-162. DOI: 10.18500/0869-6632-2013-21-2-145-162

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532.516, 517.957

New type of bifurcations in the modified Rayleigh–Benard convection problem

Sudakov Ivan Alekseevich, University of Utah
Vakulenko Sergej Avgustovich, Institute of Problems of Mechanical Engineering of Russian Academy of Science
Sukacheva Tamara Gennadevna, Yaroslav-the-Wise Novgorod State University

The original Rayleigh–Benard convection is a standard example of the system where bifurcations occur with changing of a control parameter. In this paper we consider the modified Rayleigh–Benard convection problem including radiative effects as well as gas sources on a surface. Such formulation leads to the identification of new type of bifurcations in the problem besides the well-known Benard cells. This problem is very important for mathematics of climate because it proves the occurrence of the climate system tipping point related to greenhouse gas emission into the atmosphere.

  1. Thompson JMT, Sieber J. Predicting climate tipping as a noisy bifurcation: A review. Int J. Bif. Chaos. 2011;21(2):399–423. DOI:10.1142/S0218127411028519.
  2. Monin AS, Shishkov YuA. “Climate as a problem of physics. Phys. Usp. 2000;43(4):381–406. DOI: 10.3367/UFNr.0170.200004d.0419.
  3. Lorenz EN. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. 1963;20(2):130–141. DOI:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
  4. Chulichkov AI. Mathematical models of nonlinear dynamics. Moscow: Fizmatlit; 2000. 294 p. (In Russian).
  5. Tucker W. The Lorenz attractor exists. C. R. Acad. Sci. Paris. 1999;328(12):1197–1202. DOI:10.1016/S0764-4442(99)80439-X.
  6. Goody RM. The influence of radiative transfer on cellular convection. J. Fluid Mech. 1956;1(4):424–435. DOI:10.1017/S0022112056000263.
  7. Larson VE. The effects of thermal radiation on dry convective instability. Dynamics of Atmospheres and Oceans. 2001;34(1):45–71. DOI:10.1016/S0377-0265(01)00060-4.
  8. Goody RM. Atmospheric Radiation. I. Theoretical Basis. New York: Oxford University Press; 1964. 436 p.
  9. Goody RM. Corrigendum. J. Fluid Mech. 1956;1(6):670.
  10. Goody RM, Yung YL. Atmospheric Radiation: Theoretical Basis, 2nd Edition. New York: Oxford University Press; 1989. 536 p.
  11. Gille J, Goody RM. Convection in a radiating gas. J. Fluid Mech. 1964;20(1):47–79. DOI: 10.1017/S002211206400101X.
  12. Goody RM. Principles of Atmospheric Physics and Chemistry. New York: Oxford University Press; 1995. 324 p.
  13. Spiegel EA. The convective instability of a radiating fluid layer. Astrophys. J.V. 1960;132:716–728.
  14. Spiegel EA, Veronis G. On the Boussinesq approximation for a compressible fluid. Astrophys. J. 1960;131:442–447.
  15. Larson VE. Stability properties of and scaling laws for a dry radiative-convective atmosphere. Q. J. R. Meteorol. Soc. 2000;126(562):145–171. DOI:10.1002/qj.49712656208.
  16. Murgai MP, Khosla PK. A study of the combined effect of thermal radiative transfer and a magnetic field on the gravitational convection of an ionized fluid. J. Fluid Mech. 1962;14:433–451. DOI:10.1017/S0022112062001342.
  17. Narasimha R, Vasudeva Murthy AS. The energy balance in the Ramdas layer. Bound. Layer Meteorol. 1995;76(4):307–321. DOI:10.1007/BF00709236.
  18. Vasudeva Murthy AS, Srinivasan J, Narasimha R. A theory of the lifted temperature minimum on calm clear nights. Phil. Trans. R. Soc. London A. 1993;344(1671):183–206. DOI:10.1098/rsta.1993.0087.
  19. Sudakov I, Vakulenko S. Bifurcations of the climate system and greenhouse gas emissions. Philos. Trans. A Math. Phys. Eng. Sci. 2013;371(1991):20110473. DOI: 10.1098/rsta.2011.0473.
  20. Bledoui F, Soufani A. The onset of Rayleigh–Benard instability in molecular radiating gases. Phys. Fluids A. 1997;9:3858–3872.
  21. Getling AV. On the scales of convection flows in a horizontal layer with radiative energy transfer. Atmos. Oceanic Phys. 1980;16(5):363–365.
  22. Veronis G. Penetrative convection. Astrophys. J. 1963;137(2):641–663.
  23. Vincenti WG, Traugott SC. The coupling of radiative transfer and gas motion. Annu. Rev. Fluid Mech. 1971;3:89–116. DOI:10.1146/ANNUREV.FL.03.010171.000513.
  24. Polezhaev VI, Yaremchuk VP. Numerical Modeling of Unsteady Two-Dimensional Convection in a Finite-Length Horizontal Layer Heated from Below. Fluid Dynamics. 2001;36(4):556-565.
  25. Joseph DD. Nonlinear stability of the Boussinesq equations by the method of energy. Arch. Ration. Mech. Anal. 1965;22:163–184. DOI:10.1007/BF00266474.
  26. Sudakov IA. Dynamics of permafrost lakes and climate change. St. Petersburg Polytechnic University Journal: Physics and Mathematics. 2011;2:74–79.
  27. Henry D. Geometric theory of semilinear parabolic equations. Moscow: Mir; 1984. 376 p. (In Russian).
  28. Drazin PG, Reid WH. Hydrodynamic Stability. New York: Cambridge University Press; 1981. 525 p.
  29. Besov OV, Ilyin VP, Nikolsky SM. Integral representations of functions and embedding theorems. Moscow: Nauka; 1975. 480 p. (In Russian).
  30. Ladyzhenskaya OA. Mathematical questions of dynamics of viscous incompressible fluid. Moscow: Nauka; 1970. 288 p. (In Russian).
  31. Stein EM. Singular integrals and differentiability properties of function. Moscow: Mir; 1973. 342 p. (In Russian).
  32. Friedman A. Equations with partial parabolic derivatives. Moscow: Mir; 1968. 427 p. (In Russian).
  33. Sorokin VS. On stationary movements of liquid heated from below.Journal of Applied Mathematics and Mechanics. 1954;18(2):197–204.
  34. Gershuni HZ, Zhukhovitsky EM. Convective stability of incompressible fluid. Moscow: Nauka; 1972. 392 p. (In Russian).
  35. Gershuni HZ, Zhukhovitsky EM. Convective stability. Fluid Dynamics. 1978;11:66–154.
  36. Straughan B. The Energy Method, Stability, and Nonlinear Convection. New York: Springer; 1992. 242 p.
  37. Barnsley MF, Demko S. Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A. 1978;399(1817):243–275. DOI:10.1098/rspa.1985.0057.
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