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Stressed horizontal convection

Published online by Cambridge University Press:  05 January 2012

J. Hazewinkel ,
F. Paparella  and
W. R. Young
J. Hazewinkel
Affiliation:
Scripps Institution of Oceanography, La Jolla CA 92093-0213, USA
F. Paparella*
Affiliation:
Department of Mathematics, University of Salento, Lecce 73100, Italy
W. R. Young
Affiliation:
Scripps Institution of Oceanography, La Jolla CA 92093-0213, USA
*
†Email address for correspondence: [email protected]
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Abstract

We consider the problem of a Boussinesq fluid forced by applying both non-uniform temperature and stress at the top surface. On the other boundaries the conditions are thermally insulating and either no-slip or stress-free. The interesting case is when the direction of the steady applied surface stress opposes the sense of the buoyancy driven flow. We obtain two-dimensional numerical solutions showing a regime in which there is an upper cell with thermally indirect circulation (buoyant fluid is pushed downwards by the applied stress and heavy fluid is elevated), and a second deep cell with thermally direct circulation. In this two-cell regime the driving mechanisms are competitive in the sense that neither dominates the flow. A scaling argument shows that this balance requires that surface stress vary as the horizontal Rayleigh number to the three-fifths power.

JFM classification

Convection: Buoyant boundary layers Geophysical and Geological Flows: Ocean circulation Geophysical and Geological Flows: Ocean processes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 692 , 10 February 2012 , pp. 317 - 331
Copyright
Copyright © Cambridge University Press 2012

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References

1.Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1), 119–143.CrossRefGoogle Scholar
2.Beardsley, R. C. & Festa, J. F. 1972 A numerical model of convection driven by a surface stress and non-uniform horizontal heating. J. Phys. Oceanogr. 2 (4), 444–455.2.0.CO;2>CrossRefGoogle Scholar
3.Briggs, W. 1987 A Multigrid Tutorial. SIAM.Google Scholar
4.Cessi, P. 2007 Regimes of thermohaline scaling: the interaction of wind stress and surface buoyancy. J. Phys. Oceanogr. 37, 2009–2021.CrossRefGoogle Scholar
5.Chiu-Webster, S., Hinch, E. J. & Lister, J. R. 2008 Very viscous horizontal convection. J. Fluid Mech. 611, 395–426.CrossRefGoogle Scholar
6.Coman, M. A., Griffiths, R. W. & Hughes, G. O. 2006 Sandström’s experiments revisited. J. Mar. Res. 64 (14), 783–796.CrossRefGoogle Scholar
7.Defant, A. 1961 Physical Oceanography: vol. I. Macmillan.Google Scholar
8.Dewar, W. K., Bingham, R. J., Iverson, R. L., Nowacek, D. P., Laurent, L. C. S. & Wiebe, P. H. 2006 Does the marine biosphere mix the ocean? J. Mar. Res. 64 (21), 541–561.CrossRefGoogle Scholar
9.DuFort, E. C. & Frankel, S. P. 1953 Stability conditions in the numerical treatment of parabolic differential equations. Math. Tables Other Aids Comput. 7 (43), 135–152.Google Scholar
10.Gayen, B. & Sarkar, S. 2010 Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett. 104, 218502.CrossRefGoogle ScholarPubMed
11.Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection. Annu. Rev. Fluid Mech. 40, 185–208.CrossRefGoogle Scholar
12.Hughes, G., Griffith, R., Mullarney, J. & Peterson, W. 2007 A theoretical model for horizontal convection at high Rayleigh number. J. Fluid Mech. 581, 251–276.CrossRefGoogle Scholar
13.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, 3130–3146.CrossRefGoogle Scholar
14.Ilicak, M. & Vallis, G. K. 2011 Simulations and scaling of horizontal convection. Tellus (submitted).CrossRefGoogle Scholar
15.Kuhlbrodt, T. 2008 On Sandström’s inferences from his tank experiments: a hundred years later. Tellus A 60 (5), 819–836.CrossRefGoogle Scholar
16.McIntyre, M. E. 2009 On spontaneous imbalance and ocean turbulence: generalizations of the Paparella–Young epsilon theorem. In Turbulence in the Atmosphere and Oceans (ed. Dritschel, D. G. ). Proc. Intl. IUTAM/Newton Workshop held 8–12 December 2008 , pp. 3–15. Springer.Google Scholar
17.Mullarney, J., Griffiths, R. W. & Hughes, G. O. 2004 Convection driven by differential heating at a horizontal boundary. J. Fluid Mech. 516, 181–209.CrossRefGoogle Scholar
18.Munk, W. & Wunsch, C. I. 1998 Abyssal recipes. Part II. Energetics of tidal and wind mixing. Deep Sea Res. I 45 (12), 1977–2010.CrossRefGoogle Scholar
19.Nycander, J. 2010 Horizontal convection with a nonlinear equation of state: generalization of a theorem of Paparella and Young. Tellus A 62 (2), 134–137.CrossRefGoogle Scholar
20.Paparella, F. & Young, W. R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205–214.CrossRefGoogle Scholar
21.Rossby, H. T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep Sea Res. Oceanogr. Abst. 12 (1), 9–10 IN9–IN14, 11–16.CrossRefGoogle Scholar
22.Rossby, H. T. 1998 Numerical experiments with a fluid heated non-uniformly from below. Tellus A 50 (2), 242–257.CrossRefGoogle Scholar
23.Sandström, J. W. 1908 Dynamische Versuche mit Meerwasser. Ann. Hydrogr. Mar. Met. 36, 6–23.Google Scholar
24.Scotti, A. & White, B. 2011 Is horizontal convection really non-turbulent? Geophys. Res. Lett. (accepted).CrossRefGoogle Scholar
25.Siggers, J. H., Kerswell, R. R. & Balmforth, N. J. 2004 Bounds on horizontal convection. J. Fluid Mech. 517, 55–70.CrossRefGoogle Scholar
26.Stern, M. E. 1975 Ocean Circulation Physics. Academic Press.Google Scholar
27.Tailleux, R. 2009 On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models, and the ocean heat engine controversy. J. Fluid Mech. 638, 339–382.CrossRefGoogle Scholar
28.Tailleux, R. & Rouleau, L. 2010 The effect of mechanical stirring on horizontal convection. Tellus A 62 (2), 138–153.CrossRefGoogle Scholar
29.Talley, L. D. 2003 Shallow, intermediate and deep overturning components of the global heat budget. J. Phys. Oceanogr. 33, 530–560.2.0.CO;2>CrossRefGoogle Scholar
30.Thomas, L. 2005 Destruction of potential vorticity by fronts. J. Phys. Oceanogr. 35, 2457–2466.CrossRefGoogle Scholar
31.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
32.Whitehead, J. A. & Wang, W. 2008 A laboratory model of vertical ocean circulation driven by mixing. J. Phys. Oceanogr. 39, 1091–1106.CrossRefGoogle Scholar
33.Winters, K. B. & Young, W. R. 2009 Available potential energy and buoyancy variance in horizontal convection. J. Fluid Mech. 629, 221–230.CrossRefGoogle Scholar