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

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 m...

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Published inFluids (Basel) Vol. 2; no. 1; p. 6
Main Authors Xie, Jin-Han, Miquel, Benjamin, Julien, Keith, Knobloch, Edgar
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.03.2017
Subjects
Astrophysics
asymptotic expansion
Convection
Damping
Density ratio
Dependence
Diagnostic systems
Diffusivity
Energy transfer
Experiments
Gravitational waves
Heat
Kinetic energy
Mathematical models
Probability density functions
reduced model
Salinity
salt-finger convection
Schmidt number
Stability analysis
Studies
turbulence
Two dimensional models
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Summary: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.
ISSN:2311-5521
2311-5521
DOI:10.3390/fluids2010006