Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-16T09:20:19.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of fingering convection. Part 2 The formation of thermohaline staircases

Published online by Cambridge University Press:  04 May 2011

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

Abstract

Regions of the ocean's thermocline unstable to salt fingering are often observed to host thermohaline staircases, stacks of deep well-mixed convective layers separated by thin stably stratified interfaces. Decades after their discovery, however, their origin remains controversial. In this paper we use three-dimensional direct numerical simulations to shed light on the problem. We study the evolution of an analogous double-diffusive system, starting from an initial statistically homogeneous fingering state, and find that it spontaneously transforms into a layered state. By analysing our results in the light of the mean-field theory developed in Part 1 (Traxler et al., J. Fluid Mech. doi:10.1017/jfm.2011.98, 2011), a clear picture of the sequence of events resulting in the staircase formation emerges. A collective instability of homogeneous fingering convection first excites a field of gravity waves, with a well-defined vertical wavelength. However, the waves saturate early through regular but localized breaking events and are not directly responsible for the formation of the staircase. Meanwhile, slower-growing, horizontally invariant but vertically quasi-periodic γ-modes are also excited and grow according to the γ-instability mechanism. Our results suggest that the nonlinear interaction between these various mean-field modes of instability leads to the selection of one particular γ-mode as the staircase progenitor. Upon reaching a critical amplitude, this progenitor overturns into a fully formed staircase. We conclude by extending the results of our simulations to real oceanic parameter values and find that the progenitor γ-mode is expected to grow on a time scale of a few hours and leads to the formation of a thermohaline staircase in about one day with an initial spacing in the order of 1–2 m.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 554 - 571
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

Bois, P. A. & Kubicki, A. 2002 Double diffusive aspects of the convection in moist-saturated air. In Continuum Thermomechanics, Solid Mechanics and its Applications, vol. 76, pp. 29–42. 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).CrossRefGoogle Scholar
Fer, I., Nandi, P., Holbrook, W. S., Schmitt, R. W. & Páramo, P. 2010 Seismic imaging of a thermohaline staircase in the western tropical North Atlantic. Ocean Sci. 6, 621–631.CrossRefGoogle Scholar
Guillot, T. 1999 Interiors of giant planets inside and outside the solar system. Science 286 (5437), 72.CrossRefGoogle ScholarPubMed
Huppert, H. E. 1971 On the stability of a series of double-diffusive layers. Deep-Sea Res. 18 (10), 1005–1021.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.CrossRefGoogle Scholar
Krishnamurti, R. 2003 Double-diffusive transport in laboratory thermohaline staircases. J. Fluid Mech. 483, 287–314.CrossRefGoogle Scholar
Krishnamurti, R. 2009 Heat, salt and momentum transport in a laboratory thermohaline staircase. J. Fluid Mech. 638, 491–506.CrossRefGoogle Scholar
Marmorino, G. O., Brown, W. K. & Morris, W. D. 1987 Two-dimensional temperature structure in the C-SALT thermohaline staircase. Deep Sea Res. Part A. Oceanogr. Res. Paper 34 (10), 1667–1676.CrossRefGoogle Scholar
Merceret, F. J. 1977 A possible manifestation of double diffusive convection in the atmosphere. Boundary-Layer Meteorol. 11 (1), 121–123.CrossRefGoogle Scholar
Merryfield, W. J. 2000 Origin of thermohaline staircases. J. Phys. Oceanogr. 30 (5), 1046–1068.2.0.CO;2>CrossRefGoogle Scholar
Özgökmen, T., Esenkov, O. & Olson, D. 1998 A numerical study of layer formation due to fingers in double-diffusive convection in a vertically-bounded domain. J. Mar. Res. 56 (2), 463–487.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. 2007 Mechanics of merging events for a series of layers in a stratified turbulent fluid. J. Fluid Mech. 577, 251–273.CrossRefGoogle Scholar
Rosenblum, E., Garaud, P., Traxler, A. & Stellmach, S. 2011 Turbulent mixing and layer formation in double-diffusive convection: 3D numerical simulations and theory. Astrophys. J. 731, 66.CrossRefGoogle Scholar
Ruddick, B. R. 1985 Momentum transport in thermohaline staircases. J. Geophys. Res. 90 (C1), 895–902.CrossRefGoogle Scholar
Ruddick, B. & Gargett, A. E. 2003 Oceanic double-diffusion: introduction. Prog. Oceanogr. 56, 381–393.CrossRefGoogle 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), 1015–1026.2.0.CO;2>CrossRefGoogle Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26 (1), 255–285.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 (5722), 685.CrossRefGoogle ScholarPubMed
Schmitt, R. W., Perkins, H., Boyd, J. D. & Stalcup, M. C. 1987 C-SALT: an investigation of the thermohaline staircase in the western tropical North Atlantic. Deep-Sea Res. Part A. Oceanogr. Res. Paper 34 (10), 1655–1665.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.CrossRefGoogle 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).CrossRefGoogle Scholar
Stern, M.E. 1960 The salt fountain and thermohaline convection. Tellus 12 (2), 172–175.CrossRefGoogle Scholar
Stern, M. E. 1967 Lateral mixing of water masses. Deep-Sea Res. 14 (1), 747–753.Google Scholar
Stern, M. E. 1969 Collective instability of salt fingers. J. Fluid Mech. 35.CrossRefGoogle Scholar
Stern, M. E., Radko, T. & Simeonov, J. 2001 Salt fingers in an unbounded thermocline. J. Mar. Res. 59 (3), 355–390.CrossRefGoogle Scholar
Stern, M. E. & Turner, J. S. 1969 Salt fingers and convecting layers. Deep-Sea Res. 16 (1), 97–511.Google Scholar
Tait, R. I. & Howe, M. R. 1971 Thermohaline staircase. Nature 231 (5299), 178–179.CrossRefGoogle ScholarPubMed
Tait, S. & Jaupart, C. 1989 Compositional convection in viscous melts. Nature 338 (6216), 571–574.CrossRefGoogle Scholar
Toole, J. & Georgi, D. 1981 On the dynamics of double diffusively driven intrusions. Prog. Oceanogr. 10, 123–145.CrossRefGoogle 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
Traxler, A., Stellmach, S., Garaud, P., Radko, T. & Brummell, N. 2011 Dynamics of fingering convection. Part 1 Small-scale fluxes and large-scale instabilities. J. Fluid Mech. doi:10.1017/jfm.2011.98.CrossRefGoogle Scholar
Vauclair, S. 2004 Metallic fingers and metallicity excess in Exoplanets' host stars: the accretion hypothesis revisited. Astrophys. J. 605 (2), 874–879.CrossRefGoogle Scholar
Veronis, G. 2007 Updated estimate of double diffusive fluxes in the C-SALT region. Deep-Sea Res. I 54.Google Scholar
Walsh, D. & Ruddick, B. 1995 Double-diffusive interleaving: the influence of nonconstant diffusivities. J. Phys. Oceanogr. 25, 348–358.2.0.CO;2>CrossRefGoogle Scholar

Stellmach et al. supplementary movie

Temperature perturbation from direct numerical simulation of fingering convection at Prandtl number 7, diffusivity ratio 1/3, and background density ratio 1.1, in a domain of size 335d x 335d x 536d. The dynamics of the system are divided into three distinct phases, as discussed in section 3 of the paper. In Phase I, initial perturbations are amplified by the fingering instability, grow and eventually saturate into a state of vigorous fingering convection. Gravity waves then rapidly emerge, grow and saturate (Phase II), later followed by a sharp transition to a layered state (Phase III). A higher-resolution movie is available on request.

Download Stellmach et al. supplementary movie(Video)
Video 65.7 MB

Stellmach et al. supplementary movie

Temperature perturbation from direct numerical simulation of fingering convection at Prandtl number 7, diffusivity ratio 1/3, and background density ratio 1.1, in a domain of size 335d x 335d x 536d. The dynamics of the system are divided into three distinct phases, as discussed in section 3 of the paper. In Phase I, initial perturbations are amplified by the fingering instability, grow and eventually saturate into a state of vigorous fingering convection. Gravity waves then rapidly emerge, grow and saturate (Phase II), later followed by a sharp transition to a layered state (Phase III). A higher-resolution movie is available on request.

Download Stellmach et al. supplementary movie(Video)
Video 10.5 MB