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.
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