Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T14:17:11.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Applicability and failure of the flux-gradient laws in double-diffusive convection

Published online by Cambridge University Press:  30 May 2014

Timour Radko
Timour Radko*
Affiliation:
Department of Oceanography, Naval Postgraduate School, Monterey, CA 93943, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Double-diffusive flux-gradient laws are commonly used to describe the development of large-scale structures driven by salt fingers – thermohaline staircases, collective instability waves and intrusions. The flux-gradient model assumes that the vertical transport is uniquely determined by the local background temperature and salinity gradients. While flux-gradient laws adequately capture mixing characteristics on scales that greatly exceed those of primary double-diffusive instabilities, their accuracy rapidly deteriorates when the scale separation between primary and secondary instabilities is reduced. This study examines conditions for the breakdown of the flux-gradient laws using a combination of analytical arguments and direct numerical simulations. The applicability (failure) of the flux-gradient laws at large (small) scales is illustrated through the example of layering instability, which results in the spontaneous formation of thermohaline staircases from uniform temperature and salinity gradients. Our inquiry is focused on the properties of the ‘point-of-failure’ scale ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H_{pof}$) at which the vertical transport becomes significantly affected by the non-uniformity of the background stratification. It is hypothesized that $H_{pof} $ can control some key characteristics of secondary double-diffusive phenomena, such as the thickness of high-gradient interfaces in thermohaline staircases. A more general parametrization of the vertical transport – the flux-gradient-aberrancy law – is proposed, which includes the selective damping of relatively short wavelengths that are inadequately represented by the flux-gradient models. The new formulation is free from the unphysical behaviour of the flux-gradient laws at small scales (e.g. the ultraviolet catastrophe) and can be readily implemented in theoretical and large-scale numerical models of double-diffusive convection.

JFM classification

Convection: Convection Convection: Double diffusive convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 750 , 10 July 2014 , pp. 33 - 72
Copyright
© Cambridge University Press 2014. This is a work of the U.S. Government and is not subject to copyright protection in the United States. 

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

Balmforth, N. J., Llewellyn Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329–358.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2002 Stratified Kolmogorov flow. J. Fluid Mech. 450, 131–167.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2005 Stratified Kolmogorov flow. Part 2. J. Fluid Mech. 528, 23–42.CrossRefGoogle Scholar
Bryden, H. L., Schroeder, K., Sparnocchia, S., Borghini, M. & Vetrano, A. 2014 Thermohaline staircases in the western Mediterranean Sea. J. Mar. Res. (in press).CrossRefGoogle Scholar
Gama, S., Vergassola, M. & Frisch, U. 1994 Negative eddy viscosity in isotropically forced two-dimensional flow – linear and nonlinear dynamics. J. Fluid Mech. 260, 95–126.CrossRefGoogle Scholar
Gargett, A. E. & Schmitt, R. W. 1982 Observations of salt fingers in the central waters of the eastern North Pacific. J. Geophys. Res. 87, 8017–8092.CrossRefGoogle Scholar
Holyer, J. Y. 1981 On the collective instability of salt fingers. J. Fluid Mech. 110, 195–207.CrossRefGoogle Scholar
Holyer, J. Y. 1984 The stability of long, steady, two-dimensional salt fingers. J. Fluid Mech. 147, 169–185.CrossRefGoogle Scholar
Holyer, J. Y. 1985 The stability of long steady three-dimensional salt fingers to long wavelength perturbations. J. Fluid Mech. 156, 495–503.CrossRefGoogle Scholar
Kelley, D. E., Fernando, H. J. S., Gargett, A. E., Tanny, J. & Ozsoy, E. 2003 The diffusive regime of double-diffusive convection. Prog. Oceanogr. 56, 461–481.CrossRefGoogle Scholar
Kimura, S. & Smyth, W. D. 2007 Direct numerical simulation of salt sheets and turbulence in a double-diffusive shear layer. Geophys. Res. Lett. 34, L21610.CrossRefGoogle Scholar
Krishnamurti, R. 2003 Double-diffusive transport in laboratory thermohaline staircases. J. Fluid Mech. 483, 287–314.CrossRefGoogle Scholar
Kunze, E. 1987 Limits on growing, finite length salt fingers: a Richardson number constraint. J. Mar. Res. 45, 533–556.CrossRefGoogle Scholar
Kunze, E. 2003 A review of salt fingering theory. Prog. Oceanogr. 56, 399–417.CrossRefGoogle Scholar
Legras, B. & Villone, B. 2009 Large-scale instability of a generalized turbulent Kolmogorov flow. Nonlinear Process. Geophys. 16, 569–577.CrossRefGoogle Scholar
Linden, P. F. 1974 Salt fingers in a steady shear flow. Geophys. Fluid Dyn. 6, 1–27.CrossRefGoogle Scholar
Manfroi, A. & Young, W. 1999 Slow evolution of zonal jets on the beta plane. J. Atmos. Sci. 56, 784–800.2.0.CO;2>CrossRefGoogle Scholar
Manfroi, A. & Young, W. 2002 Stability of beta-plane Kolmogorov flow. Physica D 162, 208–232.CrossRefGoogle Scholar
Mei, C. C. & Vernescu, M. 2010 Homogenization Methods for Multiscale Mechanics. p. 330. World Scientific.CrossRefGoogle Scholar
Merryfield, W. J. 2000 Origin of thermohaline staircases. J. Phys. Oceanogr. 30, 1046–1068.2.0.CO;2>CrossRefGoogle Scholar
Meshalkin, L. & Sinai, Y. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid. J. Appl. Math. Mech. 25, 1700–1705.CrossRefGoogle Scholar
Mirouh, G. M., Garaud, P., Stellmach, S., Traxler, A. L. & Wood, T. S. 2012 A new model for mixing by double-diffusive convection (semi-convection). I. The conditions for layer formation. Astrophys. J. 750, 61.CrossRefGoogle Scholar
Mueller, R. D., Smyth, W. D. & Ruddick, B. 2007 Shear and convective turbulence in a model of thermohaline intrusions. J. Phys. Oceanogr. 37, 2534–2549.CrossRefGoogle Scholar
Novikov, A. & Papanicolau, G. 2001 Eddy viscosity of cellular flows. J. Fluid Mech. 446, 173–198.CrossRefGoogle Scholar
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid is it unstable? Deep-Sea Res. 19, 79–81.Google Scholar
Posmentier, E. S. 1977 The generation of salinity fine structure by vertical diffusion. J. Phys. Oceanogr. 7, 298–300.2.0.CO;2>CrossRefGoogle Scholar
Radko, T. 2003 A mechanism for layer formation in a double-diffusive fluid. J. Fluid Mech. 497, 365–380.CrossRefGoogle Scholar
Radko, T. 2005 What determines the thickness of layers in a thermohaline staircase? J. Fluid Mech. 523, 79–98.CrossRefGoogle Scholar
Radko, T. 2008 The double-diffusive modon. J. Fluid Mech. 609, 59–85.CrossRefGoogle Scholar
Radko, T. 2011 Mechanics of thermohaline interleaving: beyond the empirical flux laws. J. Fluid Mech. 675, 117–140.CrossRefGoogle Scholar
Radko, T. 2013 Double-Diffusive Convection. p. 344. Cambridge University Press.CrossRefGoogle Scholar
Radko, T., Bulters, A., Flanagan, J. & Campin, J.-M. 2014 Double-diffusive recipes. Part 1: Large-scale dynamics of thermohaline staircases. J. Phys. Oceanogr. 44, 1269–1284.CrossRefGoogle Scholar
Radko, T. & Smith, D. P. 2012 Equilibrium transport in double-diffusive convection. J. Fluid Mech. 692, 5–27.CrossRefGoogle Scholar
Radko, T. & Stern, M. E. 2011 Finescale instabilities of the double-diffusive shear flow. J. Phys. Oceanogr. 41, 571–585.CrossRefGoogle Scholar
Ruddick, B. R., McDougall, T. J. & Turner, J. S. 1989 The formation of layers in a uniformly stirred density gradient. Deep-Sea Res. 36, 597–609.CrossRefGoogle Scholar
Ruddick, B. 1997 Differential fluxes of heat and salt: implications for circulation and ecosystem modelling. Oceanography 10, 122–127.CrossRefGoogle Scholar
Ruddick, B.2014 On the scale of layers formed in strongly stratified turbulent mixing. Unpublished manuscript.Google Scholar
Ruddick, B. & Kerr, O. 2003 Oceanic thermohaline intrusions: theory. Prog. Oceanogr. 56, 483–497.CrossRefGoogle Scholar
Schmitt, R. W. 1979a The growth rate of supercritical salt fingers. Deep-Sea Res. 26A, 23–44.CrossRefGoogle Scholar
Schmitt, R. W. 1979b Flux measurements on salt fingers at an interface. J. Mar. Res. 37, 419–436.Google Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285.CrossRefGoogle Scholar
Schmitt, R. W. 2003 Observational and laboratory insights into salt finger convection. Prog. Oceanogr. 56, 419–433.CrossRefGoogle Scholar
Schmitt, R. W., Ledwell, J. R., Montgomery, E. T., Polzin, K. L. & Toole, J. M. 2005 Enhanced diapycnal mixing by salt fingers in the thermocline of the tropical Atlantic. Science 308, 685–688.CrossRefGoogle ScholarPubMed
Shen, C. Y. & Schmitt, R. W. 1995 The wavenumber spectrum of salt fingers. In Double-Diffusive Convection (ed. Brandt, A. & Fernando, H.), AGU Geophysical Monograph, vol. 94, pp. 305–312. American Geophysical Union.Google Scholar
Sivashinsky, G. 1985 Weak turbulence in periodic flows. Physica D 17, 243–255.CrossRefGoogle Scholar
Smyth, W. D. & Ruddick, B. 2010 Effects of ambient turbulence on interleaving at a baroclinic front. J. Phys. Oceanogr. 40, 685–712.CrossRefGoogle Scholar
Stellmach, S., Traxler, A., Garaud, P., Brummell, N. & Radko, T. 2011 Dynamics of fingering convection. II: The formation of thermohaline staircases. J. Fluid Mech. 677, 554–571.CrossRefGoogle Scholar
Stern, M. E. 1960 The ‘salt-fountain’ and thermohaline convection. Tellus 12, 172–175.CrossRefGoogle Scholar
Stern, M. E. 1967 Lateral mixing of water masses. Deep-Sea Res. 14, 747–753.Google Scholar
Stern, M. E., Radko, T. & Simeonov, J. 2001 3D salt fingers in an unbounded thermocline with application to the Central Ocean. J. Mar. Res. 59, 355–390.CrossRefGoogle Scholar
Stern, M. E. & Simeonov, J. 2002 Internal wave overturns produced by salt fingers. J. Phys. Oceanogr. 32, 3638–3656.2.0.CO;2>CrossRefGoogle Scholar
Traxler, A., Stellmach, S., Garaud, P., Radko, T. & Brummel, N. 2011 Dynamics of fingering convection. I: Small-scale fluxes and large-scale instabilities. J. Fluid Mech. 677, 530–553.CrossRefGoogle Scholar
Turner, J. S. 1967 Salt fingers across a density interface. Deep-Sea Res. 14, 599–611.Google Scholar
Walsh, D. & Ruddick, B. R. 2000 Double-diffusive interleaving in the presence of turbulence: the effect of a non-constant flux ratio. J. Phys. Oceanogr. 30, 2231–2245.2.0.CO;2>CrossRefGoogle Scholar