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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
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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

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References

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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