Double-diffusive convection concerns the study of fluids in which there are gradients of two or more properties with different molecular diffusivities. In the case of the ocean, salt and heat are the two opposing effects.
The review article focusses on the situation where warm salty fluid lies below cold fluid. The warm fluid tends to rise and the salty fluid, being heavier than fresh fluid, tends to fall, so that if we begin with the liquid at rest, then increasing the temperature difference will eventually destabilize the rest state and yield either an oscillatory motion or a steady convective motion. In particular, we address the oscillatory onset of instability (the ``diffusive-convection'' problem) and examine a family of patterns that may arise. These are fully three-dimensional patterns that are doubly periodic on a hexagonal lattice with sufficient symmetry that the Hopf equivariant theorem can be applied. From the literature, those solutions are: (i) Standing rolls, (ii) Standing hexagons, (iii) Standing regular triangles, (iv) Standing patchwork quilt, (v) Travelling rolls, (vi) Travelling patchwork quilt (1), which is always unstable, (vii) Travelling patchwork quilt (2), (viii) Oscillating triangles, (ix) Wavy rolls (1), (x) Twisted patchwork quilt, (xi) Wavy rolls (2). The conditions for stability of each solution with respect to perturbations with the double periodicity is investigated for our particular physical system.
At the American Geophysical Union's 1993 Chapman Conference on Double Diffusive Convection, Ray Schmitt (funded by the Ocean, Atmosphere and Space Modeling and Prediction Division, ONR) of Woods Hole Oceanographic Institution gave an invited talk on thermohaline convection in the ocean. One open problem which he described was the observation by T. Osborn (``Observations of the salt fountain'', Atmosphere-Ocean 29(2), 340-356 (1991)) of salt fingers off the coast of San Diego which appeared to have a triangular or hexagonal structure. At the same conference, Yuriko Renardy gave a talk on pattern selection for the time-periodic onset in double diffusive convection, where group theoretic results that take advantage of the symmetries of the hexagonal lattice were applied. Yuriko and Ray began to consider why triangular or hexagonal salt fingers would be observed at the onset of steady convection in a layer of fluid sandwiched in the ocean. In order to have this theoretically, it is known that there must be a strong up-down asymmetry. This is affirmed by the observations, in which Osborne reports on a strong downflow accompanied by a weak upwelling. In discussing this problem with Melvin Stern at the Geophysical Fluid Dynamics Summer School at Woods Hole in 1994, we concluded that two effects which must be examined are surface evaporation and surface warming, due to solar radiation. The surface evaporation would increase the salinity in a sharp surface boundary layer, and provide a destabilizing and nonlinear base salinity profile. Surface warming would provide a nonlinear base temperature profile. The mathematical modeling and numerical treatment for the linearized stability analysis by Yuriko Renardy led to the conclusion that surface warming, but not evaporation, would likely be responsible for the asymmetry. The collaboration continued with a visit by Y. Renardy to Woods Hole Oceanographic Institution for the GFD Summer School in 1995, and both Schmitt and Renardy reported on their progress at the Society for Industrial and Applied Mathematics Annual Meeting in Charlotte, NC, during October 23-27, 1995.
In the final paper, we investigate the effect of non-Boussinesq effects, such as the dependence of fluid properties on temperature and salinity and nonuniform temperature and salinity gradients in the base state, and we evaluate a variety of such factors to determine whether they would favor downward fingers (as observed) or upward fingers. A nonlinear temperature profile is very evident in the data of Osborn, but we find the effect on pattern selection to be weak. Factors which are more likely to be important include a nonlinear salinity profile, the temperature dependence of viscosity, and the dependence of the density on temperature and salinity.