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Fluids 2017, 2(1), 6; doi:10.3390/fluids2010006

A Reduced Model for Salt-Finger Convection in the Small Diffusivity Ratio Limit

1
Department of Physics, University of California, Berkeley, CA 94720, USA
2
Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309, USA
*
Author to whom correspondence should be addressed.
Received: 27 December 2016 / Accepted: 23 January 2017 / Published: 30 January 2017
(This article belongs to the Collection Geophysical Fluid Dynamics)
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Abstract

A simple model of nonlinear salt-finger convection in two dimensions is derived and studied. The model is valid in the limit of a small solute to heat diffusivity ratio and a large density ratio, which is relevant to both oceanographic and astrophysical applications. Two limits distinguished by the magnitude of the Schmidt number are found. For order one Schmidt numbers, appropriate for astrophysical applications, a modified Rayleigh–Bénard system with large-scale damping due to a stabilizing temperature is obtained. For large Schmidt numbers, appropriate for the oceanic setting, the model combines a prognostic equation for the solute field and a diagnostic equation for inertia-free momentum dynamics. Two distinct saturation regimes are identified for the second model: the weakly driven regime is characterized by a large-scale flow associated with a balance between advection and linear instability, while the strongly-driven regime produces multiscale structures, resulting in a balance between energy input through linear instability and energy transfer between scales. For both regimes, we analytically predict and numerically confirm the dependence of the kinetic energy and salinity fluxes on the ratio between solutal and thermal Rayleigh numbers. The spectra and probability density functions are also computed. View Full-Text
Keywords: salt-finger convection; reduced model; asymptotic expansion; turbulence salt-finger convection; reduced model; asymptotic expansion; turbulence
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Xie, J.-H.; Miquel, B.; Julien, K.; Knobloch, E. A Reduced Model for Salt-Finger Convection in the Small Diffusivity Ratio Limit. Fluids 2017, 2, 6.

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