Hostname: page-component-5b777bbd6c-cp4x8 Total loading time: 0 Render date: 2025-06-22T11:44:38.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear and nonlinear dynamics in radiatively forced cold water

Published online by Cambridge University Press:  13 May 2025

Ruiqi Huang Open the ORCID record for Ruiqi Huang [Opens in a new window] ,
Zhen Ouyang  and
Zijing Ding Open the ORCID record for Zijing Ding [Opens in a new window]
Ruiqi Huang
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, PR China
Zhen Ouyang
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, PR China
Zijing Ding*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, PR China Institute of Mechanics, Chinese Academy of Sciences, Beijing, PR China
*
Corresponding author: Zijing Ding, z.ding@hit.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates the linear and nonlinear dynamics of two-dimensional penetrative convection subjected to radiative volumetric thermal forcing, focusing on ice-covered freshwater systems. Linear stability analysis reveals how critical wavenumbers $k_c$ and Rayleigh numbers $Ra_c$ are influenced by the attenuation lengths and incoming heat flux. In this configuration, the system easily becomes unstable with a small $Ra_c$, which is two decades smaller than that of the classical Rayleigh–Bénard convection problem, with typically $O(10)$. Weakly nonlinear analysis figures out that this configuration is supercritical, contrasting with the subcritical case by Veronis (Astrophys. J., vol. 137, 1963, 641–663). Numerical bifurcation solutions are performed from the critical points and over several decades, up to $Ra \sim O(10^6)$. This paper found that the system exhibits multiple steady solutions, and under certain specific conditions, a staircase temperature profile emerges. Meanwhile, we further discuss the influence of incoming heat flux and the Prandtl number $Pr$ on the primary bifurcation. Direct numerical simulations are also carried out, showing that heat is transported more efficiently via unsteady convection.

JFM classification

Convection: Buoyancy-driven instability Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1011 , 25 May 2025 , A24
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Álvarez-Regueiro, Eva, V., Javier, P., Fernández-Seara, J., Fernández, J. & Lugo, L. 2019 Experimental convection heat transfer analysis of a nano-enhanced industrial coolant. Nanomaterials 9 (2), 267.CrossRefGoogle ScholarPubMed
Barker, A.J., Dempsey, A.M. & Lithwick, Y. 2014 Theory and simulations of rotating convection. Astrophys. J. 791 (1), 13.CrossRefGoogle Scholar
Barnes, D.F. & Hobbie, J.E. 1960 Rate of melting at the bottom of floating ice. Geological survey research: short papers in the geological sciences. Geol. Surv. Prof. Paper 400, B392–B394.Google Scholar
Bengtsson, L. 1996 Mixing in ice-covered lakes. In The First International Lake Ladoga Symposium (eds. Heikki, H. S., Viljanen, M., Slepukhina, T. & Murthy, R.), pp. 91–97, Springer Netherlands.CrossRefGoogle Scholar
Bergé, P. & Dubois, M. 1984 Rayleigh-Bénard convection. Contemp. Phys. 25 (6), 535–582.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh-Bénard convection. Annu. Rev. Fluid Mech. 32 (1), 709–778.CrossRefGoogle Scholar
Bouffard, D. et al. 2019 Under-ice convection dynamics in a boreal lake. Inland Waters 9 (2), 142–161.CrossRefGoogle Scholar
Bouillaut, V., Lepot, S., Aumaître, S. & Gallet, B. 2019 Transition to the ultimate regime in a radiatively driven convection experiment. J. Fluid Mech. 861, R5.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Chang, Y. & Scotti, A. 2023 Characteristic scales during the onset of radiatively driven convection: linear analysis and simulations. J. Fluid Mech. 973, A14.CrossRefGoogle Scholar
Chen, C.-T.A. & Millero, F.J. 1986 Thermodynamic properties for natural waters covering only the limnological range 1. Limnol. Oceanogr. 31 (3), 657–662.CrossRefGoogle Scholar
Christopher, T.W., Le Bars, M. & Smith, S.S.G.L. 2023 Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux. J. Fluid Mech. 961, A1.CrossRefGoogle Scholar
Cortés, A. & Macintyre, S. 2020 Mixing processes in small arctic lakes during spring. Limnol. Oceanogr. 65 (2), 260–288.CrossRefGoogle Scholar
Creyssels, M. 2020 Model for classical and ultimate regimes of radiatively driven turbulent convection. J. Fluid Mech. 900, A39.CrossRefGoogle Scholar
Cross, M.C. & Hohenberg, P.C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.CrossRefGoogle Scholar
Davaille, A. & Jaupart, C. 1994 Onset of thermal convection in fluids with temperature-dependent viscosity: application to the oceanic mantle. J. Geophys. Res.: Solid Earth 99 (B10), 19853–19866.CrossRefGoogle Scholar
Ding, Z., Huang, R. & Ouyang, Z. 2024 Scaling laws behind penetrative turbulence: history and perspectives. Adv. Atmos. Sci. 41, 1881–1900.CrossRefGoogle Scholar
Ding, Z. & Ouyang, Z. 2023 Penetrative convection: heat transport with marginal stability assumption. J. Fluid Mech. 960, A26.CrossRefGoogle Scholar
Ding, Z. & Wu, J. 2021 Coherent heat transport in two-dimensional penetrative Rayleigh-Bénard convection. J. Fluid Mech. 920, A48.CrossRefGoogle Scholar
Efremova, T.V. & Pal’shin, N.I. 2011 Ice phenomena terms on the water bodies of northwestern russia. Russian Meteorol. Hydrol 36 (8), 559–565.CrossRefGoogle Scholar
Farmer, D.M. 1975 Penetrative convection in the absence of mean shear. Q. J. R. Meteorol. Soc. 101 (430), 869–891.CrossRefGoogle Scholar
Gebhart, B. & Mollendorf, J.C. 1977 A new density relation for pure and saline water. Deep Sea Res. 24 (9), 831–848.CrossRefGoogle Scholar
Goluskin, D. 2016 Internally Heated Convection and Rayleigh-Bénard Convection. Springer.CrossRefGoogle Scholar
Goluskin, D. & van der Poel, E. P. 2016 Penetrative internally heated convection in two and three dimensions. J. Fluid Mech. 791, R6.CrossRefGoogle Scholar
Goluskin, D. & Spiegel, E.A. 2012 Convection driven by internal heating. Phys. Lett. A 377 (1-2), 83–92.CrossRefGoogle Scholar
Gotzinger, G. 1909 Studien über das eis des lunzer unter-und obersees. Internationale Revue der Gesamten Hydrobiologie Und Hydrographie 2 (3), 386–396.CrossRefGoogle Scholar
Grace, A.P., Fogal, A. & Stastna, M. 2023 Restratification in late winter lakes induced by cabbeling. Geophys. Res. Lett. 50 (14), e2023GL103402.CrossRefGoogle Scholar
Grace, A.P., Stastna, M., Lamb, K. & Scott, K. Andrea 2022 Numerical simulations of the three-dimensionalization of a shear flow in radiatively forced cold water below the density maximum. Phys. Rev. Fluids 7 (2), 023501.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16 (12), 4462–4472.CrossRefGoogle Scholar
Hampton, S.E. et al. 2017 Ecology under lake ice. Ecol. Lett. 20 (1), 98–111.CrossRefGoogle ScholarPubMed
Hussein, A.M., Bakar, R.A. & Kadirgama, K. 2014 Study of forced convection nanofluid heat transfer in the automotive cooling system. Case Stud. Therm. Engng 2, 50–61.CrossRefGoogle Scholar
Jansen, J. et al. 2021 Winter limnology: how do hydrodynamics and biogeochemistry shape ecosystems under ice? J. Geophys. Res.: Biogeosci. 126 (6), e2020JG006237.CrossRefGoogle Scholar
Joseph, D.D. 1976 Stability of Fluid Motions II. Springer.CrossRefGoogle Scholar
Kazemi, S., Ostilla-Mónico, R. & Goluskin, D. 2022 Transition between boundary-limited scaling and mixing-length scaling of turbulent transport in internally heated convection. Phys. Rev. Lett. 129 (2), 024501.CrossRefGoogle ScholarPubMed
Kippenhahn, R. & Weigert, A. 1990 Stellar Structure and Evolution. Springer.CrossRefGoogle Scholar
Kirillin, G. et al. 2012 Physics of seasonally ice-covered lakes: a review. Aquat. Sci. 74, 659–682.CrossRefGoogle Scholar
Kirillin, G. & Terzhevik, A. 2011 Thermal instability in freshwater lakes under ice: effect of salt gradients or solar radiation? Cold Reg. Sci. Technol. 65 (2), 184–190.CrossRefGoogle Scholar
Lecoanet, D., Bars, L., Michael, B., Keaton, J., Vasil, G.M., Brown, B.P., Quataert, E. & Oishi, J.S. 2015 Numerical simulations of internal wave generation by convection in water. Phys. Rev. E 91 (6), 063016.CrossRefGoogle ScholarPubMed
Lepot, S., Aumaître, S. & Gallet, B. 2018 Radiative heating achieves the ultimate regime of thermal convection. Proc. Natl Acad. Sci. USA 115 (36), 8937–8941.CrossRefGoogle ScholarPubMed
Leppäranta, M., Terzhevik, A. & Shirasawa, K. 2010 Solar radiation and ice melting in lake vendyurskoe, Russian karelia. Hydrol. Res. 41 (1), 50–62.CrossRefGoogle Scholar
Malkus, W.V.R. 1960 Considerations on localized velocity fields in stellar atmospheres. In Aerodynamic Phenomena in Stellar Atmospheres IAU Symposium 12 (ed. R.N. Thomas), Zanichelli, p. 346.Google Scholar
Matthews, P.C. & Heaney, S.I. 1987 Solar heating and its influence on mixing in ice-covered lakes. Freshwater Biol. 18 (1), 135–149.CrossRefGoogle Scholar
McCutchan, A.L. & Johnson, B.A. 2022 Laboratory experiments on ice melting: a need for understanding dynamics at the ice-water interface. J. Mar. Sci. Engng 10 (8), 1008.CrossRefGoogle Scholar
McDougall, T. J., Jackett, D.R., Wright, D.G. & Feistel, R. 2003 Accurate and computationally efficient algorithms for potential temperature and density of seawater. J. Atmos. Ocean. Tech. 20 (5), 730–741.2.0.CO;2>CrossRefGoogle Scholar
Mironov, D., Terzhevik, A., Kirillin, G., Jonas, T., Malm, J. & Farmer, D. 2002 Radiatively driven convection in ice-covered lakes: observations, scaling, and a mixed layer model. J. Geophys. Res.: Oceans 107 (C4), 7–1.CrossRefGoogle Scholar
Moore, D.R. & Weiss, N. 1973 Nonlinear penetrative convection. J. Fluid Mech. 61 (3), 553–581.CrossRefGoogle Scholar
Musman, S. 1968 Penetrative convection. J. Fluid Mech. 31 (2), 343–360.CrossRefGoogle Scholar
Olsthoorn, J., Tedford, E.W. & Lawrence, G.A. 2021 The cooling box problem: convection with a quadratic equation of state. J. Fluid Mech. 918, A6.CrossRefGoogle Scholar
Plumley, M. & Julien, K. 2019 Scaling laws in rayleigh-bénard convection. Earth Space Sci. 6 (9), 1580–1592.CrossRefGoogle Scholar
Ponce, G.H.S.F., Alves, M., Miranda, J.C.C., Maciel Filho, R. & Wolf Maciel, M.R. 2015 Using an internally heat-integrated distillation column for ethanol–water separation for fuel applications. Chem. Engng Res. Des. 95, 55–63.CrossRefGoogle Scholar
Powers, S.M. & Hampton, S.E. 2016 Winter limnology as a new frontier. Limnol. Oceanogr. Bull. 25 (4), 103–108.CrossRefGoogle Scholar
Ramón, C.L., Ulloa, H.N., Doda, T., Winters, K.B. & Bouffard, D. 2021 Bathymetry and latitude modify lake warming under ice. Hydrol. Earth Syst. Sci. 25 (4), 1813–1825.CrossRefGoogle Scholar
Ricard, Y. 2007 Physics of mantle convection. Treatise Geophys. 7, 31–87.CrossRefGoogle Scholar
Schmid, P.J. & Henningson, D.S. 2002 Stability and Transition in Shear Flows. Springer.Google Scholar
Schubert, G. 2015 Treatise On Geophysics. Elsevier.Google Scholar
Shen, Y., Zikanov, O. 2016 Thermal convection in a liquid metal battery. Theor. Comput. Fluid Dyn. 30 (4), 275–294.CrossRefGoogle Scholar
Smirnov, S., Smirnovsky, A., Zdorovennova, G., Zdorovennov, R., Efremova, T., Palshin, N. & Bogdanov, S. 2023 Numerical simulation of radiatively driven convection in a small ice-covered lake with a lateral pressure gradient. Water 15 (22), 3953.CrossRefGoogle Scholar
Smirnov, S., Smirnovsky, A., Zdorovennova, G., Zdorovennov, R., Palshin, N., Novikova, I., Terzhevik, A. & Bogdanov, S. 2022 Water temperature evolution driven by solar radiation in an ice-covered lake: a numerical study and observational data. Water 14 (24), 4078.CrossRefGoogle Scholar
Toppaladoddi, S. & Wettlaufer, J.S. 2018 Penetrative convection at high Rayleigh numbers. Phys. Rev. Fluids 3 (4), 043501.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral methods in MATLAB. SIAM.CrossRefGoogle Scholar
Ulloa, H.N., Winters, K.B., Wüest, A. & Bouffard, D. 2019 Differential heating drives downslope flows that accelerate mixed-layer warming in ice-covered waters. Geophys. Res. Lett. 46 (23), 13872–13882.CrossRefGoogle Scholar
Ulloa, H.N., Wüest, A. & Bouffard, D. 2018 Mechanical energy budget and mixing efficiency for a radiatively heated ice-covered waterbody. J. Fluid Mech. 852, R1.CrossRefGoogle Scholar
Verescagin, G. 1925 A selection of works from lake baikal expedition. Dokl. Akad. Nauk SSSR 12, 161–164.Google Scholar
Veronis, G. 1963 Penetrative convection. Astrophys. J. 137, 641–663.CrossRefGoogle Scholar
Wang, Q., Zhou, Q., Wan, Z.-H. & Sun, D.-J. 2019 Penetrative turbulent Rayleigh–Bénard convection in two and three dimensions. J. Fluid Mech. 870, 718–734.CrossRefGoogle Scholar
Wang, Z., Calzavarini, E., Sun, C. & Toschi, F. 2021 How the growth of ice depends on the fluid dynamics underneath. Proc. Natl Acad. Sci. 118 (10), e2012870118.CrossRefGoogle ScholarPubMed
Winters, K.B., Ulloa, H.N., Wüest, A. & Bouffard, D. 2019 Energetics of radiatively heated ice-covered lakes. Geophys. Res. Lett. 46 (15), 8913–8925.CrossRefGoogle Scholar
Yang, B., Wells, M.G., Li, J. & Young, J. 2020 Mixing, stratification, and plankton under lake-ice during winter in a large lake: implications for spring dissolved oxygen levels. Limnol. Oceanogr. 65 (11), 2713–2729.CrossRefGoogle Scholar
Yang, B., Young, J., Brown, L. & Wells, M. 2017 High-frequency observations of temperature and dissolved oxygen reveal under-ice convection in a large lake. Geophys. Res. Lett. 44 (24), 12–218.CrossRefGoogle Scholar
Zdorovennova, G. et al. 2021 Dissolved oxygen in a shallow ice-covered lake in winter: effect of changes in light, thermal and ice regimes, Water 13 (17), 2435.CrossRefGoogle Scholar