Double-diffusive salt-particle systems: The role of settling-driven collective instabilities on the growth of γ-instabilities

High Prandtl number, double-diffusive systems such as salt and particles in water can be, in theory and in the absence of settling, unstable to the γ-instability but stable to collective instabilities. We show through numerical simulations that such systems can remain in a fingering regime for much longer times than suggested by the growth rate of γ-modes. We extend previous work on generalized mean-field theory of double-diffusive systems to incorporate settling and find a generalized expression for the growth rate of instabilities. We find the collective instability condition in the presence of settling, and recover known results in the limit of purely vertical γ-modes in a unifying framework. We estimate fluxes using small-scale numerical simulations and show that there is a critical settling velocity beyond which collective instabilities grow from the fingering regime. Visualization of the energy spectrum of the system reveals the transfer of energy from the inclined-wave driven collective instability to the γ-modes. We propose that the generalized mean-field theory with settling can fully characterize the stability of such systems.