Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-24T02:57:57.424Z Has data issue: false hasContentIssue false
  • English
  • Français

Available potential energy and buoyancy variance in horizontal convection

Published online by Cambridge University Press:  15 June 2009

KRAIG B. WINTERS  and
WILLIAM R. YOUNG
KRAIG B. WINTERS*
Affiliation:
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0209, USA Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0209, USA
WILLIAM R. YOUNG
Affiliation:
Scripps Institution of Oceanography, University of California San Diego, La Jolla, CA 92093-0209, USA
*
†Email address for correspondence: kraig@coast.ucsd.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the mechanical energy budget for horizontal Boussinesq convection and show that there are two distinct energy pathways connecting the mechanical energy (i.e. kinetic, available potential and background potential energies) to the internal energy reservoir and the external energy source. To obtain bounds on the magnitudes of the energy transfer rates around each cycle, we first show that the volume-averaged dissipation rate of buoyancy variance χ ≡ κ 〈|∇b|2〉, where b is the buoyancy, is bounded from above by 4.57h−1κ2/3ν−1/3b7/3max. Here h is the depth of the container, κ the molecular diffusion, ν the kinematic viscosity and bmax the maximum buoyancy difference that exists on the surface. The bound on χ is used to estimate the generation rate of available potential energy Ea and the rate at which Ea is irreversibly converted to background potential energy via diapycnal fluxes, both of which are shown to vanish at least as fast as κ1/3 in the limit κ → 0 at fixed Prandtl number Pr = ν/κ. As a thought experiment, consider a hypothetical ocean insulated at all boundaries except at the upper surface, where the buoyancy is prescribed. The bounds on the energy transfer rates in the mechanical energy budget imply that buoyancy forcing alone is insufficient by at least three orders of magnitude to maintain observed oceanic dissipation rates and that additional energy sources such as winds, tides and perhaps bioturbation are necessary to sustain observed levels of turbulent dissipation in the world's oceans.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 629 , 25 June 2009 , pp. 221 - 230
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Balmforth, N. J. & Young, W. R. 2003. Diffusion-limited scalar cascades. J. Fluid Mech. 482, 91–100.CrossRefGoogle Scholar
Bjerknes, V. 1916. Über thermodynamische Maschinen, die unter Mitwirkung der Schwerkraft arbeiten. Abh. Akad. Wissensch. Leipzig 35 (1), 1–33.Google Scholar
Coman, M. A., Griffiths, R. W. & Hughes, G. O. 2006 Sandström's experiments revisited. J. Mar. Res. 64, 783–796.CrossRefGoogle Scholar
Defant, A. 1961 Physical Oceanography, vol. 1. MacMillan.Google Scholar
Dewar, W. K., Bingham, R. J., Iverson, R. L., Nowacek, D. P., St. Laurent, L. C. & Wiebe, P. H. 2006 Does the marine biosphere mix the ocean? J. Mar. Res. 64, 541–561.CrossRefGoogle Scholar
Howard, L. N. 1972 Bounds on flow quantities. Annu. Rev. Fluid Mech. 17, 473–494.CrossRefGoogle Scholar
Hughes, G. O. & Griffiths, R. W. 2008 Horizontal convection Annu. Rev. Fluid Mech. 40, 185–208.CrossRefGoogle Scholar
Hughes, G. O., Hogg, A. M. & Griffiths, R. W. 2008 Available potential energy and irreversible mixing in the meridional overturning circulation J. Phys. Oceanogr. Submitted.CrossRefGoogle Scholar
Jeffreys, H. 1925 On fluid motions produced by differences of temperature and humidity. Quart. J. R. Meteorol. Soc. 51, 347–356.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, 181209.CrossRefGoogle Scholar
Munk, W. H. & Wunsch, C. 1998 Abyssal recipes. Part 2. Energetics of tidal and wind mixing. Deep-Sea Res. 45, 1977–2010.CrossRefGoogle Scholar
Nycander, J., Nilsson, J., Döös, K. & Brostr, öm 2007 Thermodynamic analysis of ocean circulation. J. Phys. Oceanogr. 37, 2038–2052.CrossRefGoogle Scholar
Osborn, T. R. & Cox, C. S. 1972 Oceanic fine structure. Geophys. Fluid Dyn. 3, 321–345.CrossRefGoogle Scholar
Paparella, F. & Young, W. R. 2002 Horizontal convection is non-turbulent. J. Fluid Mech. 466, 205–214.CrossRefGoogle Scholar
Rossby, T. 1965 On thermal convection driven by non-uniform heating from below: an experimental study. Deep-Sea Res. 12, 9–16.Google Scholar
Sandström, J. W. 1908 Dynamische Versuche mit Meerwasser. Ann. Hydrodyn. Mar. Meteorol. 36, 6–23.Google Scholar
Siggia, E. D. 1994 High Rayleigh number convection. Annu. Rev. Fluid Mech. 26, 137–168.CrossRefGoogle Scholar
Stern, M. E. 1975 Ocean Circulation Physics. International Geophysics Series, vol. 19. Academic.Google Scholar
Wang, W. & Huang, R. X. 2005. An experimental study on thermal convection driven by horizontal differential heating. J. Fluid Mech 540, 4973.CrossRefGoogle Scholar
Winters, K. B. & D'Asaro, E. A. 1996. Diascalar flux and the rate of fluid mixing. J. Fluid Mech. 317, 179–193.CrossRefGoogle Scholar
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, 115–128.CrossRefGoogle Scholar