Re-analysis of the critical nucleation work in the case of diffusive interface

The formation and evolution of a diffusive interface in a stable salt-stratified layer cooled from above is studied in a two-dimensional geometry by direct numerical simulation. For a typical example with realistic parameters, the evolution of the flow is computed up to the moment where three layers can be distinguished. Focus is on the development of the first mixed layer. The convective velocity scaling as proposed by Hunt (1984) and previously proposed expressions for the interfacial heat flux (Huppert 1971; Fernando 1989a) are shown to correspond well with the results of the simulation. The evolution of the first layer can be well described by an entrainment relation based on a local balance between kinetic and potential energy with mixing efficiency γ. The new entrainment relation is shown to fit the numerical results well and an interpretation of γ in terms of the overall energy balances of the flow is given.Previously, two rival mechanisms have been proposed that determine the final thickness of the first layer (Turner 1968; Fernando 1987). One of the distinguishing features of both mechanisms is whether a transition in entrainment regime – as the first layer develops – is a necessary condition for the mixed layer to stop growing. Another is the presence of a buoyancy jump over the interface before substantial convection in the second layer occurs. From the numerical results, we find a significant buoyancy jump even before the thermal boundary layer ahead of the first layer becomes unstable. Moreover, the convective activity in the second layer is too small to be able to stop the growth of the first layer. We therefore favour the view proposed by Fernando (1987) that a transition in entrainment regime determines the thickness of the first layer. Following this, a new one-dimensional model of layer formation is proposed. Important expressions within this model are verified using the results of the numerical simulation. The model contains two constants which are determined from the numerical results. The results of the new model fit experimental results quite well and the parameter dependence of the thickness of the first layer is not sensitive to the values of the two constants.

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Simulations of a double-diffusive interface in the diffusive convection regime

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.399 ◽

2012 ◽

Vol 711 ◽

pp. 411-436 ◽

Cited By ~ 28

Author(s):

J. R. Carpenter ◽

T. Sommer ◽

A. Wüest

Keyword(s):

Boundary Layer ◽

Boundary Layers ◽

Molecular Diffusion ◽

Layer Structure ◽

Previous Theory ◽

Convection Regime ◽

Graphic Interface ◽

Diffusive Convection ◽

Diffusive Interface ◽

Double Diffusive

AbstractThree-dimensional direct numerical simulations are performed that give us an in-depth account of the evolution and structure of the double-diffusive interface. We examine the diffusive convection regime, which, in the oceanographically relevant case, consists of relatively cold fresh water above warm salty water. A ‘double-boundary-layer’ structure is found in all of the simulations, in which the temperature ($T$) interface has a greater thickness than the salinity ($S$) interface. Therefore, thin gravitationally unstable boundary layers are maintained at the edges of the diffusive interface. The $TS$-interface thickness ratio is found to scale with the diffusivity ratio in a consistent manner once the shear across the boundary layers is accounted for. The turbulence present in the mixed layers is not able to penetrate the stable stratification of the interface core, and the $TS$-fluxes through the core are given by their molecular diffusion values. Interface growth in time is found to be determined by molecular diffusion of the $S$-interface, in agreement with a previous theory. The stability of the boundary layers is also considered, where we find boundary layer Rayleigh numbers that are an order of magnitude lower than previously assumed.

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