Marginally unstable salt fingers: Limits to growth
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The theory of long, thin salt fingers is extended to allow for a finite length. The temporal evolution of a marginally unstable, single mode disturbance is then examined as it interacts with and modifies the basic state, which is constrained to be at rest in hydrostatic equilibrium. After a few e-folding periods of initial, exponential growth, salt fingers will have modified the horizontally averaged salinity field to the extent that it is no longer unstable. For the cases considered, a stable steady-state is achieved between heat and salt transport by salt finger mode and diffusive adjustment of the horizontally averaged field. The nature of the approach to the steady state is studied numerically and is oscillatory for long fingers with large initial growth and monotonic for short fingers with small initial growth rates. It is suggested that long fingers will be unstable leading to small, local instabilities and a resulting layered structure. Further, salt fingers with large growth rates are more likely to be transient in nature rather than dynamically steady.