Spatially localised doubly diffusive convection in an axisymmetric spherical shell

Doubly diffusive convection describes the fluid motion driven by the competition of temperature and salinity gradients diffusing at different rates. While the convective motions driven by these gradients usually occupy the entire domain, parameter regions exist where the convection is spatially localised. Although well-studied in planar geometries, spatially localised doubly diffusive convection has never been investigated in a spherical shell, a geometry of relevance to astrophysics. In this paper, numerical simulation is used to compute spatially localised solutions of doubly diffusive convection in an axisymmetric spherical shell. Several families of spatially localised solutions, named using variants of the word convecton, are found and their bifurcation diagram computed. The various convectons are distinguished by their symmetry and by whether they are localised at the poles or at the equator. We find that because the convection rolls that develop in the spherical shell are not straight but curve around the inner sphere, their strength varies with latitude, and the system is spatially modulated. As a result spatially periodic states can no longer form and, much like in a planar system with non-standard boundary conditions, localised states are forced to arise via imperfect bifurcations. While the direct relevance is to doubly diffusive convection, parallels drawn with the Swift Hohenberg equation suggest a wide applicability to other pattern forming systems in similar geometries.