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Arrested development of the Rayleigh-Taylor instability in the cabbeling regime

Marek Stastna*

Andrew P. Grace

  • *Contact author: mmstastn@uwaterloo.ca
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Phys. Rev. Fluids 11, 084505 – Published 31 August, 2026

DOI: https://doi.org/10.1103/6m6k-tkjy

Abstract

The fresh water equation of state exhibits a fundamentally important nonlinearity; the temperature at which it attains its maximum density occurs at roughly 4∘C as opposed to at 0∘C. The form of the equation of state in this regime is quadratic and has the well-known implication that two fluid parcels may mix and produce a child parcel with a higher density; a process known as cabbeling. In this work we report on direct numerical simulations of the Rayleigh-Taylor instability in the cabbeling regime. We demonstrate that the mature, three-dimensional state of the instability leads to cabbeling, which, in turn, serves to sustain an instability that would otherwise saturate outside of the cabbeling regime. We present a new quantitative measure of strong cabbeling potential and use it, along with the mechanical energy, to demonstrate that the Rayleigh-Taylor instability is arrested in two ways that are unique to the cabbeling regime. Our results have implications for radiatively driven, under-ice flows in late winter during which ice prevents mechanical driving by the wind. While our focus is on the freshwater setting, we extend the derivation of the strong cabbeling potential to heat-salt systems, thereby extending its usefulness to the broader community.

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References (37)

  1. H.-J. Kull, Theory of the Rayleigh–Taylor instability, Phys. Rep. 206, 197 (1991).
  2. P. Kundu, Fluid Mechanics (Elsevier, Amsterdam, 2016).
  3. H. N. Ulloa, A. Wüest, and D. Bouffard, Mechanical energy budget and mixing efficiency for a radiatively heated ice-covered waterbody, J. Fluid Mech. 852, R1 (2018).
  4. M. J. Andrews and S. B. Dalziel, Small Atwood number Rayleigh–Taylor experiments, Philos. Trans. R. Soc. A 368, 1663 (2010).
  5. T. J. McDougall, Double-diffusive convection with a nonlinear equation of state: Part II. Laboratory experiments and their interpretation, Prog. Oceanogr. 10, 91 (1981).
  6. T. T. Clark, A numerical study of the statistics of a two-dimensional Rayleigh–Taylor mixing layer, Phys. Fluids 15, 2413 (2003).
  7. 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).
  8. A. Sengupta, H. N. Ulloa, and B. Joshi, Multi-layer Rayleigh-Taylor instability: Consequences for naturally occurring stratified mixing layers, Phys. Fluids 35, 102110 (2023).
  9. B. Yang, J. Young, L. Brown, and M. Wells, High-frequency observations of temperature and dissolved oxygen reveal under-ice convection in a large lake, Geophys. Res. Lett. 44, 12 (2017).
  10. I. Yilmaz, Analysis of Rayleigh-Taylor instability at high atwood numbers using fully implicit, non-dissipative, energy-conserving large eddy simulation algorithm, Phys. Fluids 32, 054101 (2020).
  11. 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).
  12. T. D. Foster and E. C. Carmack, Temperature and salinity structure in the Weddell Sea, J. Phys. Oceanogr. 6, 36 (1976).
  13. T. J. McDougall, Thermobaricity, cabbeling, and water-mass conversion, J. Geophys. Res.: Oceans 92, 5448 (1987).
  14. C. J. Shakespeare and L. N. Thomas, A new mechanism for mode water formation involving cabbeling and frontogenetic strain at thermohaline fronts. Part II: Numerical simulations, J. Phys. Oceanogr. 47, 1755 (2017).
  15. A. P. Grace, M. Stastna, K. G. Lamb, and K. A. Scott, Asymmetries in gravity currents attributed to the nonlinear equation of state, J. Fluid Mech. 915, A18 (2021).
  16. N. Castro-Folker, A. P. Grace, and M. Stastna, Three-dimensional structure of cold-water gravity currents, Phys. Rev. Fluids 8, 113901 (2023).
  17. K. Bhavsar, M. Stastna, and N. Castro-Folker, Effects of weak buoyancy on shear instabilities in cold water, Phys. Rev. Fluids 11, 023904 (2026).
  18. E. C. Carmack, C. B. Gray, C. H. Pharo, and R. J. Daley, Importance of lake-river interaction on seasonal patterns in the general circulation of Kamloops Lake, British Columbia, Limnol. Oceanogr. 24, 634 (1979).
  19. 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).
  20. C.-C. P. Caulfield and W. R. Peltier, The anatomy of the mixing transition in homogeneous and stratified free shear layers, J. Fluid Mech. 413, 1 (2000).
  21. K. B. Winters, H. N. Ulloa, A. Wüest, and D. Bouffard, Energetics of radiatively heated ice-covered lakes, Geophys. Res. Lett. 46, 8913 (2019).
  22. R. Tailleux, Negative available potential energy dissipation as the fundamental criterion for double diffusive instabilities, J. Fluid Mech. 994, A5 (2024).
  23. R. Tailleux, On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy, J. Fluid Mech. 638, 339 (2009).
  24. R. Tailleux, Local available energetics of multicomponent compressible stratified fluids, J. Fluid Mech. 842, R1 (2018).
  25. J. Nycander, Energy conversion, mixing energy, and neutral surfaces with a nonlinear equation of state, J. Phys. Oceanogr. 41, 28 (2011).
  26. C. J. Subich, K. G. Lamb, and M. Stastna, Simulation of the Navier-Stokes equations in three dimensions with a spectral collocation method, Int. J. Numer. Methods Fluids 73, 103 (2013).
  27. A. P. Grace, M. Stastna, K. G. Lamb, and K. A. Scott, Gravity currents in the cabbeling regime, Phys. Rev. Fluids 8, 014502 (2023).
  28. T. J. McDougall, D. R. Jackett, D. G. Wright, and R. Feistel, Accurate and computationally efficient algorithms for potential temperature and density of seawater, J. Atmos. Ocean. Technol. 20, 730 (2003).
  29. A. P. Grace, A. Fogal, and M. Stastna, Restratification in late winter lakes induced by cabbeling, Geophys. Res. Lett. 50, e2023GL103402 (2023).
  30. A. Cortés and S. MacIntyre, Mixing processes in small Arctic lakes during spring, Limnol. Oceanogr. 65, 260 (2020).
  31. J. Olsthoorn, E. W. Tedford, and G. A. Lawrence, Salt-fingering in seasonally ice-covered lakes, Geophys. Res. Lett. 49, e2022GL097935 (2022).
  32. T. Hanson, M. Stastna, and A. Coutino, Stratified shear instability in the cabbeling regime, Phys. Rev. Fluids 6, 084802 (2021).
  33. J. Leyrer, H. Ulloa, J. Ortega, and J. Letelier, Onset of cabbeling instabilities in superconfined two-fluid systems, Phys. Fluids 36, 126613 (2024).
  34. J. Letelier, H. Ulloa, J. Leyrer, and J. Ortega, Scaling CO2-brine mixing in permeable media via analogue models, J. Fluid Mech. 962, A8 (2023).
  35. E. Carmack and S. Vagle, Thermobaric processes both drive and constrain seasonal ventilation in deep Great Slave Lake, Canada, J. Geophys. Res.: Earth Surf. 126, e2021JF006288 (2021).
  36. J. I. Bisits, J. D. Zika, and T. Sohail, Cabbeling as a catalyst and driver of turbulent mixing, J. Fluid Mech. 1011, A17 (2025).
  37. D. J. Allum, A. P. Grace, and M. Stastna, Two-dimensional simulations of flow in ice-covered lakes with horizontal variations in surface albedo, Phys. Rev. Fluids 7, 103501 (2022).

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Phys. Rev. Fluids 11, 084505– Published 31 August, 2026
  • Received 6 November 2025
  • Accepted 15 July 2026
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