Salt-fingering convection in the small diffusivity ratio limit

Doubly diffusive processes play a fundamental role in many physical phenomena of substantial geophysical and astrophysical importance. Despite much research basic questions, e.g. how primary and secondary instabilities saturate and how mean fields are generated, remain unresolved. In the salt-finger regime, we use a systematic asymptotic procedure to reduce the primitive equations to a prognostic equation for the evolution of the salinity field coupled to a novel diagnostic relation between the salinity and streamfunction. The reduced model preserves both the linear and nonlinear properties of the original system including primary and secondary instabilities and the establishment of a statistically steady saturated state. We identify the flux spectrum as a good diagnostic for the presence of salt fingers: in the horizontal spectrum, we find two distinctive scaling laws whose ranges are separated by the optimal wavenumber, while a Gaussian distribution is discovered for the vertical spectrum.