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.
Highlights
Doubly-diffusive systems in which two components with different diffusivities contribute to buoyancy in opposite ways arise frequently in geophysics and astrophysics [1,2,3]: for example, heat and salt contribute to the stratification in the oceanic setting, while heat and chemical composition contribute in stellar astrophysics and sugar and salt in laboratory experiments
We mention that the two-dimensional model we study is expected to capture the dynamics of the three-dimensional system for O(1) Prandtl numbers and sufficiently large density ratios [26], but not for low Prandtl numbers for which simulations of the two-dimensional and three-dimensional primitive equations lead to qualitatively different predictions [27]
In view of its simplicity and its strong relevance to oceanic flows, we focus in this paper on the inertia-free salt convection (IFSC) model (20) and leave the modified Rayleigh–Bénard convection (mRBC) model (18) for future study
Summary
Doubly-diffusive systems in which two components with different diffusivities contribute to buoyancy in opposite ways arise frequently in geophysics and astrophysics [1,2,3]: for example, heat and salt contribute to the stratification in the oceanic setting, while heat and chemical composition contribute in stellar astrophysics and sugar and salt in laboratory experiments. In the oceanic parameter regime with fast momentum diffusion, Sc 1, we derive a reduced model with a prognostic-diagnostic form, which we refer to as the inertia-free salt convection model This model is the main topic of this paper. We employ a doubly-periodic setting as appropriate for oceanic and astrophysical applications with distant boundaries, in contrast to earlier studies of salt-finger convection in vertically-bounded domains, e.g., [21,22,23,24,25] In this formulation, elevator modes consisting of vertically-invariant upward and downward moving fingers are exact nonlinear solutions, and we believe that analogous modes are crucial even in vertically-bounded domains because of their efficiency in extracting energy from the salinity field. Dependence of the results on the domain size and the other an analytical approximation to one of the saturation regimes, complete the paper
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