Amplitude equilibration of sugar-salt fingers

Abstract

The mechanism by which amplifying salt fingers in an unbounded uniform $T/S$ gradient are equilibrated is determined, starting with a time-dependent asymptotic field equation for $( {R\tau })^{ - 1} \,{-}\, 1 \,{=}\, \varepsilon \,{\to}\, 0$, where $R \,{>}\, 1$ is the $T/S$ density ratio and $\tau \,{=}\, K_S / K_T\,{<}\,1$ is the molecular diffusivity ratio. A mode truncation of that equation yields an ODE which shows that the fastest growing finger mode transfers energy to two ‘slave’ modes with relatively small vertical scale; the finger mode thereby attains a statistically steady amplitude. The results for $\tau \,{=}\, 1 / 3$ are compared with spectral solutions of the non-truncated equations in two and three dimensions; the predicted fluxes are testable in sugar ($S$), salt ($T$) laboratory experiments.