An Evaluation of Eulerian and Semi-Lagrangian Advection Schemes in Simulations of Rotating, Stratified Flows in the Laboratory. Part I: Axisymmetric Flow

Abstract

A series of numerical simulations of steady, thermally stratified flow of a Boussinesq, incompressible fluid in a rotating, cylindrical fluid annulus were carried out over ranges of spatial resolution, grid stretch, and rotation rate. A range of different numerical advection schemes were used for the representation of heat transport, including a conventional conservative second-order Eulerian scheme and three different variants of a semi-Lagrangian scheme used either for temperature advection alone, or for both thermal and momentum advection. The resulting simulations were compared both with each other, and with high precision measurements of velocity, temperature, and total heat transport in the laboratory. The performance of the semi-Lagrangian scheme was found to be quite strongly sensitive to the spatial interpolation algorithm. A basic tensor cubic scheme generally produced good simulations of steady 2D and 3D flows, although the somewhat more accurate tensor quintic scheme (which is, however, also significantly more expensive) appeared to offer some detectable improvements in accuracy and performance in some cases. A split cubic scheme (which is computationally cheaper but formally less accurate) gave generally poor results in practice and is not recommended. In all cases considered, both the fully Eulerian and most forms of the semi-Lagrangian schemes gave good quantitative agreement with the laboratory measurements when extrapolated to very high resolution. Some significant systematic errors in the simulated heat transport and zonal momentum were found with all schemes, however, when run at moderate (though by no means very low) resolution. The semi-Lagrangian schemes had a tendency to overestimate heat transport relative to the laboratory measurements compared with the Eulerian schemes, but the latter tended to overestimate zonal momentum relative to the laboratory flows compared with the fully semi-Lagrangian simulations.