A sustainability balance between heat transport and dynamical complexity in Rayleigh–Benard convection
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In seventeen direct numerical simulations of two-dimensional Rayleigh–Bénard convection (Pr = 0.71, Ra = 1.3 × 10⁴–2.6 × 10⁵, n = 1–5 convective wavelengths at the critical wavelength), we report that the dominant phase-space contraction rate |λ₂| satisfies n|λ₂| ≈ Nu, where n is the number of convective wavelengths and Nu is the Nusselt number. The ratio n|λ₂|/Nu is approximately Ra-independent for n = 3 and n = 4, bracketing unity, while it departs systematically for other values of n. These departures are consistent with the forced geometry of the domain, where integer n and fixed roll width at the critical wavelength prevent the balance from being satisfied exactly. The leading Lyapunov exponent λ₁ is consistent with zero in all cases within the uncertainty of these integrations. The Nusselt number is independent of n: the balance constrains the Lyapunov spectrum given the transport, not the reverse. We hypothesize that the dominant contraction mode is spatially localized to one convective wavelength; this is testable but not yet verified.
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- Dataset: 10.5281/zenodo.19351317 (DOI)
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2026-03-31
References
- W. V. R. Malkus, "The heat transport and spectrum of thermal turbulence," Proc. R. Soc. Lond. A 225, 196–212 (1954).
- 2S. Grossmann and D. Lohse, "Scaling in thermal convection: a unifying theory," J. Fluid Mech. 407, 27–56 (2000).
- 3J. J. Niemela, L. Skrbek, K. R. Sreenivasan, and R. J. Donnelly, "Turbulent convection at very high Rayleigh numbers," Nature (London) 404, 837–840 (2000).
- 4K. P. Iyer, J. D. Scheel, J. Schumacher, and K. R. Sreenivasan, "Classical 1/3 scaling of convection holds up to Ra = 1015," Proc. Natl. Acad. Sci. U.S.A. 117, 7594–7598 (2020).
- 7M. R. Paul, M. C. Cross, P. F. Fischer, and H. S. Greenside, "Pattern formation and dynamics in Rayleigh–B´enard convection: numerical simulations of experimentally realistic geometries," Phys. Rev. Lett. 87, 154501 (2001).
- 8L. Sirovich and A. E. Deane, "A computational study of Rayleigh–B´enard convection. Part 2. Dimension considerations," J. Fluid Mech. 222, 251–265 (1991).
- 10C. R. Doering and P. Constantin, "Variational bounds on energy dissipation in incompressible flows. III. Convection," Phys. Rev. E 53, 5957–5981 (1996).
- 11G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, "Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them," Meccanica 15, 9–20 (1980).
- 12D. R. Hewitt, J. A. Neufeld, and J. R. Lister, "Ultimate regime of high Rayleigh number convection in a porous medium," Phys. Rev. Lett. 108, 224503 (2012).
- 15C. R. Guttridge, "Data for: An empirical relationship between the Lyapunov spectrum and heat transport in Rayleigh–Bénard convection" (Zen odo, 2026). https://doi.org/10.5281/zenodo.19351318
- 6J. L. Kaplan and J. A. Yorke, "Chaotic behavior of multidimen sional difference equations," in Functional Differential Equations and Approximation of Fixed Points, edited by H.-O. Peitgen and H.-O. Walther, Lecture Notes in Mathematics Vol. 730 (Springer, Berlin, 1979), pp. 204–227.
- 10E. P. van der Poel, R. J. A. M. Stevens, K. Sugiyama, and D. Lohse, "Flow states in two-dimensional Rayleigh–B´enard con vection as a function of aspect-ratio and Rayleigh number," Phys. Fluids 24, 085104 (2012).
- I. Prigogine, "Time, structure and fluctuations" (Nobel Lecture, 1977).