The mass-conservative cell-integrated semi-Lagrangian advection scheme on the sphere

• Published in: Monthly weather review 2002-03, Vol.130 (3), p.649-667
• Main Authors: NAIR, Ramachandran D, MACHENHAUER, Bennert
• Format: Article
• Language: English
• Subjects: Earth, ocean, space; Exact sciences and technology; External geophysics; Geophysics. Techniques, methods, instrumentation and models
• ISSN: 0027-0644; 1520-0493
• DOI: 10.1175/1520-0493(2002)130<0649:tmccis>2.0.co;2

A mass-conservative cell-integrated semi-Lagrangian (CISL) scheme is presented and tested for 2D transport on the sphere. The total mass is conserved exactly and the mass of each individual grid cell is conserved in general. The scheme is based on a general scheme developed by Machenhauer and Olk that has increased cost effectiveness without loss of accuracy, compared to the CISL scheme of Rancic. A regular latitude-longitude grid is used on the sphere and upstream trajectories from the corner points of the regular grid cells (the Eulerian cells) define the corner points of the departure cells. The sides in these so-called Lagrangian cells are generally defined as straight lines in a ( lambda , mu ) plane, where lambda is the longitude and mu is the sine of the latitude. The mass distribution within each Eulerian grid cell is defined by quasi-biparabolic functions, which are used to integrate analytically the mass in each Lagrangian computational cell. The auxiliary computational cells are polygons with each side parallel to the coordinate axis. Also, the computational cells have the same area as the Lagrangian cells they approximate. They were introduced in order to simplify the analytical integrals of mass. Near the poles, the east and west sides of certain Lagrangian cells cannot be approximated by straight lines in the ( lambda , mu ) plane, and are instead represented by straight lines in polar tangent plane coordinates. Each of the latitudinal belts of Lagrangian cells in the polar caps are split up into several latitudinal belts of subcells, which can be approximated by computational cells as in the case of cells closer to the equator. One latitudinal belt in each hemisphere, which encloses the Eulerian pole (singular belt), is treated in a special way. First the total mass in the singular belt is determined and then it is redistributed among the cells in the belt using weights determined by a traditional SL scheme at the midpoints of the cells. By this procedure the total mass is still conserved while the conservation is only approximately maintained for the individual cells in the singular belt. These special treatments in the polar caps fit well into the general structure of the code and can be implemented with minor modifications in the code used for the rest of the sphere. Compared to two other conservative advection schemes implemented on the sphere the CISL scheme used here was found to be competitive in terms of accuracy for the same resolut