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Transition to time-dependent flow in highly viscous horizontal convection

Lennart Ramme* and Ulrich Hansen

  • Institut für Geophysik, WWU Münster, Corrensstraße 24, 48149 Münster, Germany

  • *lennart.ramme@uni-muenster.de
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Phys. Rev. Fluids 4, 093501 – Published 10 September, 2019

DOI: https://doi.org/10.1103/PhysRevFluids.4.093501

Abstract

Horizontal convection is studied numerically in two dimensions for the case of an infinite Prandtl number and shear-free boundary conditions on all sides. Three different temperature distributions are applied at the top boundary. Steady states become unstable, giving way to unsteady flow with increasing Rayleigh number Ra. The transition to time-varying flow depends on the heating configuration. For Ra≥5×109 simulations show a time-dependent flow behavior for a linearly varying temperature at the top, whereas other temperature distributions show this transition at even lower Rayleigh numbers. The scaling laws of heat flow and circulation strength are investigated and the results indicate that both are changed by the upcoming time-dependent flow. While in the steady boundary layer flow regime the heat flux scales as Ra1/5, the exponent increases to a value of approximately 0.25 in the unsteady regime. A similar transition is observed for the circulation strength. The observed scaling laws are in good agreement with established theories in both the steady and unsteady regimes. After the transition, the flow shows a periodic oscillation of the boundary layer circulation, which turns into a more irregular flow behavior as the Rayleigh number is increased further. We conclude that, even for an infinite Prandtl number, horizontal convection exhibits unsteady flow at sufficiently high Rayleigh numbers and that this flow resembles the turbulent circulation at low Prandtl numbers.

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References (31)

  1. M. E. Stern, Ocean Circulation Physics (Academic, New York, 1975).
  2. G. O. Hughes and R. W. Griffiths, Horizontal convection, Annu. Rev. Fluid Mech. 40, 185 (2008).
  3. P.-Y. Passaggia, A. Scotti, and B. White, Transition and turbulence in horizontal convection: Linear stability analysis, J. Fluid Mech. 821, 31 (2017).
  4. H. Stommel, On the smallness of sinking regions in the ocean, Proc. Natl. Acad. Sci. USA 48, 766 (1962).
  5. H. T. Rossby, On thermal convection driven by non-uniform heating from below: An experimental study, Deep Sea Research and Oceanographic Abstracts 12, 9 (1965).
  6. H. T. Rossby, Numerical experiments with a fluid heated non-uniformly from below, Tellus A 50, 242 (1998).
  7. H. Jeffreys, On fluid motions produced by differences of temperature and humidity, Q. J. R. Meteorol. Soc. 51, 347 (1925).
  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. W. Wang and R. X. Huang, An experimental study on thermal circulation driven by horizontal differential heating, J. Fluid Mech. 540, 49 (2005).
  10. 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).
  11. O. Shishkina, S. Grossmann, and D. Lohse, Heat and momentum transport scalings in horizontal convection, Geophys. Res. Lett. 43, 1219 (2016).
  12. O. Shishkina and S. Wagner, Prandtl-Number Dependence of Heat Transport in Laminar Horizontal Convection, Phys. Rev. Lett. 116, 024302 (2016).
  13. S. Chiu-Webster, E. J. Hinch, and J. R. Lister, Very viscous horizontal convection, J. Fluid Mech. 611, 395 (2008).
  14. F. Paparella and W. R. Young, Horizontal convection is non-turbulent, J. Fluid Mech. 466, 205 (2002).
  15. A. Scotti and B. White, Is horizontal convection really non-turbulent? Geophys. Res. Lett. 38, L21609 (2011).
  16. K. D. Stewart, G. O. Hughes, and R. W. Griffiths, When do marginal seas and topographic sills modify the ocean density structure? J. Geophys. Res. 116, C08021 (2011).
  17. B. Gayen, R. W. Griffiths, G. O. Hughes, and J. A. Saenz, Energetics of horizontal convection, J. Fluid Mech. 716, R10 (2013).
  18. R. W. Griffiths, G. O. Hughes, and B. Gayen, Horizontal convection dynamics: Insights from transient adjustment, J. Fluid Mech. 726, 559 (2013).
  19. T. Tsai, W. K. Hussam, A. Fouras, and G. J. Sheard, The origin of instability in enclosed horizontally driven convection, Int. J. Heat Mass Transfer 94, 509 (2016).
  20. B. Gayen, R. W. Griffiths, and G. O. Hughes, Stability transitions and turbulence in horizontal convection, J. Fluid Mech. 751, 698 (2014).
  21. S. Grossmann and D. Lohse, Scaling in thermal convection: A unifying theory, J. Fluid Mech. 407, 27 (2000).
  22. J. H. Siggers, R. R. Kerswell, and N. J. Balmforth, Bounds on horizontal convection, J. Fluid Mech. 517, 55 (2004).
  23. G. J. Sheard and M. P. King, Horizontal convection: Effect of aspect ratio on Rayleigh number scaling and stability, Appl. Math. Model. 35, 1647 (2011).
  24. R. X. Huang, Mixing and energetics of the oceanic thermohaline circulation, J. Phys. Oceanogr. 29, 727 (1999).
  25. J. T. Houghton, The Physics of Atmospheres (Cambridge University Press, Cambridge, 2002).
  26. H. J. J. Gramberg, P. D. Howell, and J. R. Ockendon, Convection by a horizontal thermal gradient, J. Fluid Mech. 586, 41 (2007).
  27. D. Zhao, Global tomographic images of mantle plumes and subducting slabs: Insight into deep earth dynamics, Phys. Earth Planet. Inter. 146, 3 (2004).
  28. M. A. Coman, R. W. Griffiths, and G. O. Hughes, The sensitivity of convection from a horizontal boundary to the distribution of heating, J. Fluid Mech. 647, 71 (2010).
  29. U. Hansen and A. Ebel, Experiments with a numerical model related to mantle convection: Boundary layer behaviour of small-and large scale flows, Phys. Earth Planet. Inter. 36, 374 (1984).
  30. U. Hansen and A. Ebel, Time-dependent thermal convection—A possible explanation for a multiscale flow in the earth's mantle, Geophys. J. Int. 94, 181 (1988).
  31. B. Blankenbach et al., A benchmark comparison for mantle convection codes, Geophys. J. Int. 98, 23 (1989).

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Phys. Rev. Fluids 4, 093501– Published 10 September, 2019
  • Received 17 March 2019
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