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Horizontal convection dynamics: insights from transient adjustment

Published online by Cambridge University Press:  11 June 2013

Ross W. Griffiths ,
Graham O. Hughes  and
Bishakhdatta Gayen
Ross W. Griffiths*
Affiliation:
Research School of Earth Sciences, Australian National University, Canberra, ACT 0200, Australia
Graham O. Hughes
Affiliation:
Research School of Earth Sciences, Australian National University, Canberra, ACT 0200, Australia
Bishakhdatta Gayen
Affiliation:
Research School of Earth Sciences, Australian National University, Canberra, ACT 0200, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The dynamics of horizontal convection are revealed by examining transient adjustment toward thermal equilibrium. We restrict attention to high Rayleigh numbers (of $O(1{0}^{12} )$) and a Prandtl number ${\sim }5$ that characterize many practical applications, and consider responses to small changes in the thermal boundary conditions, using laboratory experiments, three-dimensional direct numerical simulations (DNS) and simple theoretical models. The experiments and the mechanical energy budget from the DNS demonstrate that unsteady forcing can produce flow dramatically more active than horizontal convection under steady forcing. The physical mechanisms at work are indicated by the time scales of approach to the new equilibrium, and we show that these can range over two orders of magnitude depending on the imposed change in boundary conditions. Changes that lead to a net destabilizing buoyancy flux give rapid adjustments: for applied heat flux conditions the whole of the circulation is controlled by conduction through the stable portion of the boundary layer, whereas for applied temperature difference the circulation is controlled by small-scale convection within the unstable part of the boundary layer. The experiments, DNS and models are in close agreement and show that the time scale under applied temperatures is as small as 0.01 vertical diffusion time scales, a factor of four smaller than for imposed flux. Both cases give adjustments too rapid for diffusion in the interior to play a significant role, at least through 99 % of the adjustment, and we conclude that diffusion through the full depth is not significant in setting the equilibrium state. Boundary changes leading to a net stabilizing buoyancy flux give a very different response, causing the convection to quickly form a shallow circulation cell, followed eventually by a return to full-depth overturning through a combination of penetrative convection and conduction. The time scale again varies by orders of magnitude, depending on the boundary conditions and the location of the imposed boundary perturbation.

JFM classification

Convection: Convection Convection: Convection in cavities Geophysical and Geological Flows: Ocean circulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 726 , 10 July 2013 , pp. 559 - 595
Copyright
©2013 Cambridge University Press 

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References

Borowski, D., Gerdes, R. & Olbers, D. 2002 Thermohaline and wind forcing of a circumpolar channel with blocked geostrophic contours. J. Phys. Oceanogr. 32, 2520–2539.CrossRefGoogle Scholar
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford University Press.Google Scholar
Chiu-Webster, S., Hinch, E. J. & Lister, J. R. 2008 Very viscous horizontal convection. J. Fluid Mech. 611, 395–426.CrossRefGoogle Scholar
Coman, M. A., Griffiths, R. W. & Hughes, G. O. 2010 The sensitivity of convection from a horizontal boundary to the distribution of heating. J. Fluid Mech. 647, 71–90.CrossRefGoogle Scholar
Ganopolski, A. & Rahmstorf, S. 2001 Rapid changes of glacial climate simulated in a coupled climate model. Nature 409, 153–158.CrossRefGoogle Scholar
Gayen, B., Griffiths, R. W., Hughes, G. O. & Saenz, J. A. 2013 Energetics of horizontal convection. J. Fluid Mech. 716, R10.CrossRefGoogle Scholar
Griffiths, R. W., Maher, N. & Hughes, G. O. 2011 Horizontal convection under oscillatory buoyancy forcing. J. Mar. Res. 69, 523–543.CrossRefGoogle Scholar
Hogg, A. McC. 2010 An antarctic circumpolar current driven by surface buoyancy forcing. Geophys. Res. Lett. 37, L23601.CrossRefGoogle Scholar
Hughes, G. O. & Griffiths, R. W. 2006 A simple convective model of the global overturning circulation, including effects of entrainment into sinking regions. Ocean Model. 12, 46–79.CrossRefGoogle Scholar
Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection. Annu. Rev. Fluid Mech. 40, 185–208.CrossRefGoogle Scholar
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, 251–276.CrossRefGoogle Scholar
Hughes, G. O., Hogg, A. McC. & Griffiths, R. W. 2009 Available potential energy and irreversible mixing in the meridional overturning circulation. J. Phys. Oceanogr. 39, 3130–3146.CrossRefGoogle Scholar
Marshall, J. & Speer, K. 2012 Closure of the meridional overturning circulation through southern ocean upwelling. Nat. Geosci. 5, 171–180.CrossRefGoogle Scholar
Mori, A. & Niino, H. 2002 Time evolution of nonlinear horizontal convection: its flow regimes and self-similar solutions. J. Atmos. Sci. 59, 1841–1856.2.0.CO;2>CrossRefGoogle Scholar
Mullarney, J. C., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181–209.CrossRefGoogle Scholar
Munk, W. H. 1966 Abyssal recipes. Deep-Sea Res. 13, 707–730.Google Scholar
Pierce, D. W. & Rhines, P. B. 1996 Convective building of a pycnocline: laboratory experiments. J. Phys. Oceanogr. 26, 176–190.2.0.CO;2>CrossRefGoogle Scholar
Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 9–16.Google Scholar
Rossby, H. T. 1998 Numerical experiments with a fluid non-uniformly heated from below. Tellus 50, 242–257.CrossRefGoogle Scholar
Saenz, J., Hogg, A. McC., 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.CrossRefGoogle Scholar
Scotti, A. & White, B. 2011 Is horizontal convection really non-turbulent? Geophys. Res. Lett. 38, L21609.CrossRefGoogle Scholar
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, 495–507.CrossRefGoogle Scholar
Stewart, K. D., Hughes, G. O. & Griffiths, R. W. 2011 When do marginal seas and topographic sills modify the ocean density structure? J. Geophys. Res. 116, C08021.Google Scholar
Stewart, K., 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, 1907–1922.CrossRefGoogle Scholar
Stommel, H. 1962 On the smallness of sinking regions in the ocean. Proc. Natl Acad. Sci. Washington 48, 766–772.CrossRefGoogle ScholarPubMed
Wang, W. & Huang, R. X. 2005 An experimental study on thermal convection driven by horizontal differential heating. J. Fluid Mech. 540, 49–73.CrossRefGoogle Scholar
Whitehead, J. A. & Wang, W. 2008 A laboratory model of vertical ocean circulation driven by mixing. J. Phys. Oceanogr. 38, 1091–1106.CrossRefGoogle Scholar
Wunsch, C. & Ferrari, R. 2004 Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314.CrossRefGoogle Scholar

Griffiths et al. supplementary movie

Advection of passive dye tracer for the case of an applied temperature difference of 23.2˚C and a warming boundary perturbation applied at the heating end (a repeat of Run 14, table 1, and shown in figure 1). The clip shows the initial equilibrium flow and the more vigorous flow immediately after the boundary perturbation. The movie speed is 20 times actual speed in the laboratory and the clip shows a total of 14min of flow. The whole length of the box is shown. The clock at bottom reads zero until (at 9s from the beginning of the 42s clip) the heating temperature on the right hand half of the base is increased (by 3.9˚C) and the dye release is changed from red to blue. The flow has barely begun its long adjustment to the new boundary conditions by the end of this clip (where the clock reads 11min:17s). Once adjusted, this flow appeared indistinguishable from the initial equilibrated flow.

Download Griffiths et al. supplementary movie(Video)
Video 3.6 MB

Griffiths et al. supplementary movie

Advection of passive dye tracer for the case of an applied heat flux and a cooling boundary perturbation applied at the heating end (a repeat of Run 6, table 1, and shown in figure 5). The clip shows 3min of the initial equilibrium flow followed the early shutdown of deep convection to form a shallow circulation cell immediately after the applied heat flux was decreased by 10%. The movie speed is 20 times actual speed in the laboratory and approximately 60% of the box length is shown. The clock at bottom reset to zero (at 9s from the beginning of the 42s clip). A 7min interval has been removed from the middle of the clip (6min:30s to 13min:40s after the boundary perturbation), during which a few Potassium Permanganate crystals were dropped into the box and settled on the base to give up the purple tracer. After a long adjustment to the new boundary conditions the full-depth convection was re-established and the flow appeared indistinguishable from the initial equilibrium state.

Download Griffiths et al. supplementary movie(Video)
Video 4.5 MB

Griffiths et al. supplementary movie

Advection of passive dye tracer for the case of an applied heat flux and a cooling boundary perturbation in the temperature Tc applied at the cooling end (a repeat of Run 9, table 1, and shown in figure 6). The movie speed is 20 times actual speed in the laboratory and begins at the time the boundary temperature perturbation was applied. The clock at bottom was reset to zero at this time and the boundary proceeded to take 1 to 2 min to reach its new temperature. Approximately 60% of the box length is shown. The clip shows five separate periods: t=0 to 11min:48s, t = 54min:20s to 59min:45s, t = 4hrs:30min to 4hrs:44min, t = 9hrs:6min to 9hrs:12min, and t = 20hrs:58min to 21hrs:4min. New dye releases provided tracer for each time segment. In the final segment the flow is fully adjusted to the new boundary conditions and is indistinguishable from the initial equilibrium state.

Download Griffiths et al. supplementary movie(Video)
Video 15.1 MB

Griffiths et al. supplementary movie

A DNS solution (Run DNSb, table 1) showing the horizontal velocity field u(x, z) on the spanwise (y) mid-plane of the box for an increase in the temperature of the heating half of the base . The solution begins with the thermally equilibrated solution under the initial boundary conditions. The temperature of the heated (left hand) half of the base is then changed increased instantaneously. Immediately after this change the flow is far from equilibrium and velocities are very much larger than in equilibrium states. As the flow evolves toward equilibrium under the new boundary conditions, the velocities slowly decay back toward a solution close to the initial flow. Note that the velocity colour scale in the movie is held constant, whereas a varying colour scale is used in the instantaneous fields shown in figure 12.

Download Griffiths et al. supplementary movie(Video)
Video 3.1 MB