The role of boundary conditions in the simulation of rotating, stratified turbulence

In this paper we use the CASL method to explore the role of boundary conditions in determining the long-time behaviour of rotating, stratified, quasi-geostrophic turbulence. We End that initially two-dimensional (sufficiently tall) columns of potential vorticity (PV) break down through three-dimensional instability to give a fully three-dimensional dow consisting of ellipsoidal structures. This is the case both for rigid-lid (isothermal) vertical boundary conditions and for vertically periodic boundaries. However, the rigid boundary case gives rise to semi-ellipsoids at both the top and bottom boundaries, and, for sufficient domain depths, preferred depths for the formation of ellipsoids in the interior. By contrast, in the vertically periodic case, the distribution of ellipsoids is homogeneous in depth. The role of the horizontal boundaries is indirect, but still significant. In all cases doubly periodic horizontal boundary conditions are imposed. We consider a range of initial conditions where in each case equal numbers of two-dimensional columns of positive and negative vorticity are used, taking up a fixed, but relatively small fraction of the domain (approximately 5%). Thus when there is only a small number of vortices, they have larger radius. When the initial number of vortices is small enough (i.e., when the radius is not small compared with the horizontal domain width), at long time there is a two-dimensionalisation giving rise to a single column of positive PV and a single column of negative PV, as has been observed in some previous simulations. We find the same phenomenon for both vertically periodic and rigid lid boundary conditions, but it occurs over a broader range of initial conditions in the vertically periodic case. However, in all cases fully three-dimensional final states are regained when the number of vortices is increased while keeping the fraction of the domain occupied by vortices fixed, i.e., when the vortex radius/domain width ratio is sufficiently small.