Time-dependent fluxes across double-diffusive interfaces

Abstract

A simple, time-dependent model is proposed to predict the heat and salt fluxes transferred between a hot, salty fluid layer and an overlying colder, fresher fluid layer when the overall density of the lower layer is greater than that of the upper layer. The interface separating the two layers, known as a ‘diffusive’ interface, consists of a purely diffusing core sandwiched between vigorously convecting boundary layers. This type of interface occurs whenever the density of superposed fluid layers depends on two diffusing agents with different diffusivities such that the faster diffusing agent causes a statically unstable density variation while the slower diffusing agent causes a statically stable variation. Early experimental and theoretical studies sought a single relationship between the heat flux $F_T$ across a diffusive interface and the buoyancy ratio $R_\rho\,{=}\,\beta\uDelta S/\alpha\uDelta T$, where $\uDelta T$ and $\uDelta S$ are the differences in temperature and salinity between the well-mixed fluid layers, while $\alpha$ and $\beta$ are the coefficients of thermal and solutal expansion respectively. The model presented here supports more recent experimental findings that the relationship is time-dependent and therefore that results depend on the initial conditions and any forcing applied to the mixed regions. It is predicted that the evolution of the thickness of the core is determined principally by the mismatch between the transport rates of the slower-diffusing species (salt) diffusively from the core and convectively into the mixed regions. The introduction of time-dependence brings together the data from many previous experimental studies for both heat–salt and salt–sugar systems over a wide range of $R_\rho$, and both links and puts into context two previous theories of diffusive interfaces.