A mass-conservative, positive-definite, and efficient Eulerian advection scheme in spherical geometry and on a nonuniform grid system

The flux-form advection scheme of Bott is modified for the spherical coordinates, combined with the expanded-polar-zone (EPZ) technique to improve the overall performance of the advection calculations. With the EPZ technique, this Eulerian scheme has comparable efficiency as semi-Lagrangian methods for advection of nonreactive tracers on a sphere but with somewhat better overall numerical accuracy. The conservation of global tracer mass and the positive definiteness of the algorithm are achieved to machine precision. For the test problem of solid body rotations on a sphere, this scheme shows small numerical diffusion, almost undetectable phase errors, and very little artificial deformation of the test shape even far cross-polar transport. In comparison with some semi-Lagrangian schemes and other high-order Eulerian methods, it shows very competitive performance. Numerical tests also indicate that, without any modifications, it performs just as well on slightly nonuniform Gaussian grid as on uniform grid. For the vertical advection, a fourth-order and two second-order versions of this scheme formulated on a nonuniform grid system have also been derived. The performance of these versions is tested with a nonuniform sigma grid system by using ideal one-dimensional test problems. This accurate numerical scheme is recommended for models where resolving the sharp vertical gradients of atmospheric trace species such as water vapor is important.