  Double-diffusive instabilities are often invoked to explain enhanced
transport in stably-stratified fluids. The most-studied natural manifestation
of this process, fingering convection, commonly occurs in the ocean's
thermocline and typically increases diapycnal mixing by two orders of magnitude
over molecular diffusion. Fingering convection is also often associated with
structures on much larger scales, such as thermohaline intrusions, gravity
waves and thermohaline staircases. In this paper, we present an exhaustive
study of the phenomenon from small to large scales. We perform the first
three-dimensional simulations of the process at realistic values of the heat
and salt diffusivities and provide accurate estimates of the induced turbulent
transport. Our results are consistent with oceanic field measurements of
diapycnal mixing in fingering regions. We then develop a generalized mean-field
theory to study the stability of fingering systems to large-scale
perturbations, using our calculated turbulent fluxes to parameterize
small-scale transport. The theory recovers the intrusive instability, the
collective instability, and the gamma-instability as limiting cases. We find
that the fastest-growing large-scale mode depends sensitively on the ratio of
the background gradients of temperature and salinity (the density ratio). While
only intrusive modes exist at high density ratios, the collective and
gamma-instabilities dominate the system at the low density ratios where
staircases are typically observed. We conclude by discussing our findings in
the context of staircase formation theory.
