Hostname: page-component-7b9c58cd5d-wdhn8 Total loading time: 0 Render date: 2025-03-13T16:17:09.768Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of fingering convection. Part 1 Small-scale fluxes and large-scale instabilities

Published online by Cambridge University Press:  19 April 2011

A. TRAXLER ,
S. STELLMACH ,
P. GARAUD ,
T. RADKO  and
N. BRUMMELL
A. TRAXLER*
Affiliation:
Applied Mathematics and Statistics, Baskin School of Engineering, University of California, Santa Cruz, CA 96064, USA
S. STELLMACH
Affiliation:
Applied Mathematics and Statistics, Baskin School of Engineering, University of California, Santa Cruz, CA 96064, USA Institut für Geophysik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany Institute of Geophysics and Planetary Physics, University of California, Santa Cruz, CA 96064, USA
P. GARAUD
Affiliation:
Applied Mathematics and Statistics, Baskin School of Engineering, University of California, Santa Cruz, CA 96064, USA
T. RADKO
Affiliation:
Department of Oceanography, Naval Postgraduate School, Monterey, CA 93943, USA
N. BRUMMELL
Affiliation:
Applied Mathematics and Statistics, Baskin School of Engineering, University of California, Santa Cruz, CA 96064, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Double-diffusive instabilities are often invoked to explain enhanced transport in stably stratified fluids. The most-studied natural manifestation of this process, fingering convection, commonly occurs in the ocean's thermocline and typically increases diapycnal mixing by 2 orders of magnitude over molecular diffusion. Fingering convection is also often associated with structures on much larger scales, such as thermohaline intrusions, gravity waves and thermohaline staircases. In this paper, we present an exhaustive study of the phenomenon from small to large scales. We perform the first three-dimensional simulations of the process at realistic values of the heat and salt diffusivities and provide accurate estimates of the induced turbulent transport. Our results are consistent with oceanic field measurements of diapycnal mixing in fingering regions. We then develop a generalized mean-field theory to study the stability of fingering systems to large-scale perturbations using our calculated turbulent fluxes to parameterize small-scale transport. The theory recovers the intrusive instability, the collective instability and the γ-instability as limiting cases. We find that the fastest growing large-scale mode depends sensitively on the ratio of the background gradients of temperature and salinity (the density ratio). While only intrusive modes exist at high density ratios, the collective and γ instabilities dominate the system at the low density ratios where staircases are typically observed. We conclude by discussing our findings in the context of staircase-formation theory.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 530 - 553
Copyright
Copyright © Cambridge University Press 2011

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

Baines, P. G. & Gill, A. E. 1969 On thermohaline convection with linear gradients. J. Fluid Mech. 37, 289–306.CrossRefGoogle Scholar
Borue, V. & Orszag, S. A. 1996 Turbulent convection with constant temperature gradient. In APS, Division of Fluid Dynamics Meeting, 24–26 November 1996, abstract no. CG.01. pp. 1.Google Scholar
Calzavarini, E., Doering, C. R., Gibbon, J. D., Lohse, D., Tanabe, A. & Toschi, F. 2006 Exponentially growing solutions in homogeneous Rayleigh-Bénard convection. Phys. Rev. E 73 (3), 035301–035304.Google Scholar
Canuto, C. G., Hussaini, M. Y., Quarteroni, A. M. & Zang, T. A. 2007 Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Springer.CrossRefGoogle Scholar
Charbonnel, C. & Zahn, J. P. 2007 Thermohaline mixing: a physical mechanism governing the photospheric composition of low-mass giants. Astron. Astrophys. 467 (1), L-15–L-18.CrossRefGoogle Scholar
Dietze, H., Oschlies, A. & Kähler, P. 2004 Internal-wave-induced and double-diffusive nutrient fluxes to the nutrient-consuming surface layer in the oligotrophic subtropical North Atlantic. Ocean Dyn. 54 (1), 1–7.CrossRefGoogle Scholar
Garaud, P. 2011 What happened to the other Mohicans? The case for a primordial origin to the planet-metallicity connection. Astrophys. J. Lett. 728 (2), L30.Google Scholar
Gargett, A. E. & Holloway, G. 1992 Sensitivity of the gfdl ocean model to different diffusivities for heat and salt. J. Phys. Oceanogr. 22 (10), 1158–1177.Google Scholar
Glessmer, M. S., Oschlies, A. & Yool, A. 2008 Simulated impact of double-diffusive mixing on physical and biogeochemical upper-ocean properties. J. Geophys. Res. 113, C08029.Google Scholar
Holyer, J. Y. 1981 On the collective instability of salt fingers. J. Fluid Mech. 110, 195–207.CrossRefGoogle Scholar
Inoue, R., Kunze, E., St. Laurent, L., Schmitt, R. W. & Toole, J. M. 2008 Evaluating salt-fingering theories. J. Mar. Res. 66, 413–440.Google Scholar
Kluikov, Y. Y. U. & Karlin, L. N. 1995 A model of the ocean thermocline stepwise stratification caused by double diffusion. In Double-Diffusive Convection, Geophysical Monograph, vol. 94, pp. 287–292. American Geophysical Union.Google Scholar
Krishnamurti, R. 2003 Double-diffusive transport in laboratory thermohaline staircases. J. Fluid Mech. 483, 287–314.Google Scholar
Kunze, E. 1987 Limits on growing, finite-length salt fingers: a richardson number constraint. J. Mar. Res. 45, 533–556.Google Scholar
Kunze, E. 2003 A review of oceanic salt-fingering theory. Prog. Oceanogr. 56 (3–4), 399–417.CrossRefGoogle Scholar
McDougall, T. J. 1985 a Double-diffusive interleaving. Part I. Linear stability analysis. J. Phys. Oceanogr. 15 (11), 1532–1541.2.0.CO;2>CrossRefGoogle Scholar
McDougall, T. J. 1985 b Double-diffusive interleaving. Part II. Finite amplitude, steady state interleaving. J. Phys. Oceanogr. 15 (11), 1542–1556.Google Scholar
Merryfield, W. J., Holloway, G. & Gargett, A. E. 1999 A global ocean model with double-diffusive mixing. J. Phys. Oceanogr. 29 (6), 1124–1142.2.0.CO;2>CrossRefGoogle Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Viscous Flow. Springer.Google Scholar
Radko, T. 2003 A mechanism for layer formation in a double-diffusive fluid. J. Fluid Mech. 497, 365–380.Google 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
Ruddick, B. & Kerr, O. 2003 Oceanic thermohaline intrusions: theory. Prog. Oceanogr. 56 (3–4), 483–497.Google Scholar
Ruddick, B. & Richards, K. 2003 Oceanic thermohaline intrusions: observations. Prog. Oceanogr. 56 (3–4), 499–527.CrossRefGoogle Scholar
Ruddick, B. R. & Turner, J. S. 1979 The vertical length scale of double-diffusive intrusions. Deep Sea Res. Part A. Oceanogr. Res. Papers 26 (8), 903–904, IN1–IN3, 905–913.CrossRefGoogle Scholar
Ruddick, B. R., Phillips, O. M. & Turner, J. S. 1999 A laboratory and quantitative model of finite-amplitude thermohaline intrusions. Dyn. Atmos. Oceans 30 (2-4), 71–99.Google Scholar
Schmitt, R. W. 1979 The growth rate of super-critical salt fingers. Deep-Sea Res. 26A, 23–40.CrossRefGoogle Scholar
Schmitt, R. W. 1981 Form of the temperature-salinity relationship in the central water: evidence for double-diffusive mixing. J. Phys. Oceanogr. 11 (7), 1015–1026.Google Scholar
Schmitt, R. W. 1983 The characteristics of salt fingers in a variety of fluid systems, including stellar interiors, liquid metals, oceans, and magmas. Phys. Fluids 26, 2373.CrossRefGoogle Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26 (1), 255–285.Google 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 (5722), 685.CrossRefGoogle ScholarPubMed
Shen, C. Y. 1995 Equilibrium salt-fingering convection. Physics of Fluids 7 (4), 706–717.CrossRefGoogle Scholar
Simeonov, J. & Stern, M. 2004 Double-diffusive intrusions on a finite-width thermohaline front. J. Phys. Oceanogr. 34 (7), 1723–1740.2.0.CO;2>CrossRefGoogle Scholar
Simeonov, J. & Stern, M. E. 2007 Equilibration of two-dimensional double-diffusive intrusions. J. Phys. Oceanogr. 37 (3), 625–643.CrossRefGoogle Scholar
Smyth, W. D. & Ruddick, B. 2010 Effects of ambient turbulence on interleaving at a baroclinic front. J. Phys. Oceanogr. 40 (4), 685–712.Google Scholar
St. Laurent, L. & Schmitt, R. W. 1999 The contribution of salt fingers to vertical mixing in the north atlantic tracer release experiment. J. Phys. Oceanogr. 29 (7), 1404–1424.2.0.CO;2>CrossRefGoogle Scholar
Stancliffe, R. J., Glebbeek, E., Izzard, R. G. & Pols, O. R. 2007 Carbon-enhanced metal-poor stars and thermohaline mixing. Astron. Astrophys. 464, L57–L60.Google Scholar
Stellmach, S. & Hansen, U. 2008 An efficient spectral method for the simulation of dynamos in Cartesian geometry and its implementation on massively parallel computers. Geochem. Geophys. Geosyst. 9 (5), Q05003.CrossRefGoogle Scholar
Stellmach, S., Traxler, A., Garaud, P., Brummell, N. & Radko, T. 2011 Dynamics of fingering convection. Part 2. The formation of thermohaline staircases. J. Fluid Mech. doi:10.1017/jfm.2011.99.CrossRefGoogle Scholar
Stern, M. E. 1960 The ‘salt fountain’ and thermohaline convection. Tellus 12 (2), 172–175.Google Scholar
Stern, M. E. 1969 Collective instability of salt fingers. J. Fluid Mech. 35, 209–218.CrossRefGoogle Scholar
Stern, M. E. & Simeonov, J. A. 2002 Internal wave overturns produced by salt fingers. J. Phys. Oceanogr. 32 (12), 3638–3656.2.0.CO;2>CrossRefGoogle Scholar
Stern, M. E. & Simeonov, J. 2005 The secondary instability of salt fingers. J. Fluid Mech. 533 (−1), 361–380.CrossRefGoogle Scholar
Stern, M. E. & Turner, J. S. 1969 Salt fingers and convecting layers. Deep-Sea Res. 16 (1), 97–511.Google Scholar
Stern, M. E., Radko, T. & Simeonov, J. 2001 Salt fingers in an unbounded thermocline. J. Mar. Res. 59 (3), 355–390.CrossRefGoogle Scholar
Tait, R. I. & Howe, M. R. 1968 Some observations of thermohaline stratification in the deep ocean. Deep-Sea Res. 15, 275–280.Google Scholar
Tait, R. I. & Howe, M. R. 1971 Thermohaline staircase. Nature 231 (5299), 178–179.Google Scholar
Toole, J. & Georgi, D. 1981 On the dynamics of double diffusively driven intrusions. Prog. Oceanogr. 10, 123–145.Google Scholar
Traxler, A., Garaud, P. & Stellmach, S. 2011 Numerically determined transport laws for fingering (‘thermohaline’) convection in astrophysics. Astrophys. J. Lett. 728 (2), L29.CrossRefGoogle Scholar
Vauclair, S. 2004 Metallic fingers and metallicity excess in exoplanets' host stars: the accretion hypothesis revisited. Astrophys. J. 605 (2), 874–879.Google Scholar
Veronis, G. 2007 Updated estimate of double diffusive fluxes in the C-SALT region. Deep-Sea Res. I 54, 831–833.Google Scholar
Walsh, D. & Ruddick, B. 1995 Double-diffusive interleaving: the influence of nonconstant diffusivities. J. Phys. Oceanogr. 25 (3), 348–358.2.0.CO;2>CrossRefGoogle Scholar
Walsh, D. & Ruddick, B. 2000 Double-diffusive interleaving in the presence of turbulence: the effect of a nonconstant flux ratio. J. Phys. Oceanogr. 30 (9), 2231–2245.Google Scholar
You, Y. 2002 A global ocean climatological atlas of the Turner angle: implications for double-diffusion and water-mass structure. Deep-Sea Res. 49 (11), 2075–2093.CrossRefGoogle Scholar
Zhang, J., Schmitt, R. W. & Huang, R. X. 1998 Sensitivity of the gfdl modular ocean model to parameterization of double-diffusive processes. J. Phys. Oceanogr. 28 (4), 589–605.Google Scholar