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Turbulent horizontal convection under spatially periodic forcing: a regime governed by interior inertia

  • Madelaine G. Rosevear (a1), Bishakhdatta Gayen (a1) and Ross W. Griffiths (a1)

Abstract

Differential heating applied at a single horizontal boundary forces ‘horizontal convection’, even when there is no net heat flux through the boundary. However, almost all studies of horizontal convection have been limited to a special class of problem in which temperature or heat flux differences were applied in only one direction and over the horizontal length of a box (the Rossby problem; Rossby, Deep-Sea Res., vol. 12, 1965, pp. 9–16). These conditions strongly constrain the flow. Here we report laboratory experiments and direct numerical simulations (DNS) extending the results of Griffiths & Gayen (Phys. Rev. Lett., vol. 115, 2015, 204301) for horizontal convection forced by boundary conditions imposed in a two-dimensional periodic array at a horizontal boundary. The experiments use saline and freshwater fluxes at a permeable base with the imposed boundary salinity having a horizontal length scale one quarter of the width of the box. The flow reaches a state in which the net boundary buoyancy flux vanishes and the bulk of the fluid shows an inertial range of turbulence length scales. A regime transition is seen for increasing water depth, from an array of individual coherent plumes on the forcing scale to convection dominated by emergent larger scales of overturning. The DNS explore the analogous thermally forced case with sinusoidal boundary temperature of wavenumber $n=4$ , and are used to examine the Rayleigh number ( $Ra$ ) dependence for shallow- and deep-water cases. For shallow water the flow transitions with increasing $Ra$ from laminar to turbulent boundary layer regimes that are familiar from the Rossby problem and which have normalised heat transport scaling as $Nu\sim Ra^{1/5}$ and $Nu\sim (Ra\,Pr)^{1/5}$ , with $Nu$ the Nusselt number and $Pr$ the Prandtl number, in this case maintaining a stable array of coherent turbulent plumes. For deep-water and large $Ra$ the laminar scaling transitions to $Nu\sim (Ra\,Pr)^{1/4}$ , with the scales of turbulence extending to the dimensions of the box. The $1/4$ power law regime is explained in terms of the momentum of symmetric, inviscid large scales of motion in the interior coupled to diffusive loss of heat through stabilised parts of the boundary layer. The turbulence production is predominantly by shear instability rather than convection, with viscous dissipation distributed throughout the bulk of the fluid. These conditions are not seen in the highly asymmetric flow in the Rossby problem even at Rayleigh numbers up to six orders of magnitude greater than the transition found here. The new inertial interior regime has the rate of supply of available potential energy, and its removal by mixing of density, increasing as $Ra^{5/4}$ , which is faster than $Ra^{6/5}$ in the Rossby problem. Irreversible mixing is confined close to the forcing boundary and is very much larger than the viscous dissipation, which is proportional to $Ra$ .

Copyright

Corresponding author

Email address for correspondence: Ross.Griffiths@anu.edu.au

References

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Baines, W. D. & Turner, J. S. 1969 Turbulent buoyant convection from a source in a confined region. J. Fluid Mech. 37, 5180.
Barkan, R., Winters, K. & Llewellyn Smith, S. 2013 Rotating horizontal convection. J. Fluid Mech. 723, 556586.
Chiu-Webster, S., Hinch, E. J. & Lister, J. R. 2008 Very viscous horizontal convection. J. Fluid Mech. 611, 395426.
Gayen, B., Griffiths, R. W. & Hughes, G. O. 2014 Stability transitions and turbulence in horizontal convection. J. Fluid Mech. 751, 111.
Gayen, B., Griffiths, R. W., Hughes, G. O. & Saenz, J. A. 2013a Energetics of horizontal convection. J. Fluid Mech. 716, R10.
Gayen, B., Hughes, G. O. & Griffiths, R. W. 2013b Completing the energy budget and pathways for Rayleigh–Bénard convection. Phys. Rev. Lett. 111, 124301.
Griffiths, R. W. & Gayen, B. 2015 Turbulent convection insights from small-scale thermal forcing with zero net heat flux at a horizontal boundary. Phys. Rev. Lett. 115, 204301.
Griffiths, R. W., Hughes, G. O. & Gayen, B. 2013 Horizontal convection dynamics: insights from transient adjustment. J. Fluid Mech. 726, 559595.
Hignett, P., Ibbetson, A. & Killworth, P. D. 1981 On rotating thermal convection driven by non-uniform heating from below. J. Fluid Mech. 109, 161187.
Hughes, G. O., Gayen, B. & Griffiths, R. W. 2013 Available potential energy in Rayleigh–Bénard convection. J. Fluid Mech. 729, R3.
Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection. Annu. Rev. Fluid Mech. 40, 185208.
Hughes, G. O., Griffiths, R. W., Mullarney, J. C. & Peterson, W. H. 2007 A theoretical model for horizontal convection at high Rayleigh number. J. Fluid Mech. 581, 251276.
Hughes, G. O., Hogg, A. M. & Griffiths, R. W. 2009 Available potential energy and irreversible mixing in the meridional overturning circulation. J. Phys. Oceanogr. 39, 31303146.
Mullarney, J. C., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181209.
Nokes, R.2014 Streams, version 2.03: system theory and design. Tech. Rep. Department of Civil and Natural Resources Engineering, University of Canterbury, New Zealand.
Paparella, F. & Young, W. R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205214.
Park, Y. G. & Whitehead, J. A. 1999 Rotating convection driven by differential bottom heating. J. Phys. Oceanogr. 29, 12081220.
Pierce, D. W. & Rhines, P. B. 1996 Convective building of a pycnocline: laboratory experiments. J. Phys. Oceanogr. 26, 176190.
Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 916.
Rossby, H. T. 1998 Numerical experiments with a fluid non-uniformly heated from below. Tellus 50, 242257.
Saenz, J. A., Hogg, A. M., Hughes, G. O. & Griffiths, R. W. 2012 Mechanical power input from buoyancy and wind to the circulation in an ocean model. Geophys. Res. Lett. 39, L13605.
Scotti, A. & White, B. 2011 Is horizontal convection really ‘non-turbulent’? Geophys. Res. Lett. 38, L21609.
Sheard, G. J., Hussama, W. K. & Tsai, T. 2016 Linear stability and energetics of rotating radial horizontal convection. J. Fluid Mech. 795, 135.
Sheard, G. J. & King, M. P. 2011 Horizontal convection: effect of aspect ratio on Rayleigh number scaling and stability. Appl. Math. Model. 35, 16471655.
Shishkina, O., Grossmann, S. & Lohse, D. 2016 Prandtl number dependence of heat transport in horizontal convection. Geophys. Res. Lett. 116, 024302.
Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495507.
Stewart, K. D., Hughes, G. O. & Griffiths, R. W. 2012 The role of turbulent mixing in an overturning circulation maintained by surface buoyancy forcing. J. Phys. Oceanogr. 42, 19071922.
Stommel, H. 1962 On the smallness of sinking regions in the ocean. Proc. Natl Acad. Sci. USA 48, 766772.
Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy. J. Fluid Mech. 638, 339382.
Vreugdenhil, C., Gayen, B. & Griffiths, R. W. 2016 Mixing and dissipation in a geostrophic buoyancy-driven circulation. J. Geophys. Res. Oceans 121, 60766091.
Vreugdenhil, C., Griffiths, R. W. & Gayen, B. 2017 Geostrophic and chimney regimes in rotating horizontal convection with imposed heat flux. J. Fluid Mech. 823, 5799.
Whitehead, J. A. & Wang, W. 2008 A laboratory model of vertical ocean circulation driven by mixing. J. Phys. Oceanogr. 38, 10911106.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.
Winton, M. 1995 Why is the deep sinking narrow? J. Phys. Oceanogr. 25, 9971005.
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