  Bifurcation analysis of confined salt-finger convection using single-mode
equations obtained from a severely truncated Fourier expansion in the
horizontal is performed. Strongly nonlinear staircase-like solutions having,
respectively, one (S1), two (S2) and three (S3) regions of mixed salinity in
the vertical direction are computed using numerical continuation, and their
stability properties are determined. Near onset, the one-layer S1 solution is
stable and corresponds to maximum salinity transport among the three solutions.
The S2 and S3 solutions are unstable but exert an influence on the statistics
observed in direct numerical simulations (DNS) in larger two-dimensional (2D)
domains. Secondary bifurcations of S1 lead either to tilted-finger (TF1) or to
traveling wave (TW1) solutions, both accompanied by the spontaneous generation
of large-scale shear, a process favored for lower density ratios and Prandtl
numbers (Pr). These states at low Pr are associated, respectively, with
two-layer and three-layer staircase-like salinity profiles in the mean. States
breaking reflection symmetry in the midplane are also computed. In 2D and for
low Pr the DNS results favor direction-reversing tilted fingers (RTF)
resembling the pulsating wave state observed in other systems. Two-layer and
three-layer mean salinity profiles corresponding to RTF and TW1 are observed in
2D DNS averaged over time. The single-mode solutions close to the high
wavenumber onset are in an excellent agreement with 2D DNS in small horizontal
domains and compare well with 3D DNS.
