Semi-convection in the ocean and in stars: A multi-scale analysis

Fluid stratified by gravitation can be subject to a number of instabilities which eventually lead to a flow that causes enhanced mixing and transport of heat. The special case where a destabilizing temperature gradient counteracts the action of a stabilizing gradient in molecular weight is of interest to astrophysics (inside stars and giant planets) and geophysics (lakes, oceans) as well as to some engineering applications. The detailed dynamics of such a system depend on the molecular diffusivities of heat, momentum, and solute as well as system parameters including the ratio of the two gradients to each other. Further important properties are the formation and merging of well-defined layers in the fluid which cannot be derived from linear stability analysis. Moreover, the physical processes operate on a vast range of length and time scales. This has made the case of semi-convection, where a mean temperature gradient destabilizes the stratification while at the same time the mean molecular gradient tends to stabilize it, a challenge to physical modelling and to numerical hydrodynamical simulation. During the MetStröm project the simulation codes ANTARES and MITgcm have been extended such that they can be used for the simulations of such flows. We present a comparison of effective diffusivities derived from direct numerical simulations. For both stars and the oceanic regimes, the Nusselt numbers (scaled diffusivities) follow similar relationships. Semi-convection quickly becomes inefficient, because the formation of layers limits vertical mixing. In contrast to the complementary saltfingering, these layers tend to damp instabilities so that effective diffusivities of salinity (concentration) are up to two orders of magnitudes smaller than in the former case