The Mass-Conservative Cell-Integrated Semi-Lagrangian Advection Scheme on the Sphere

Abstract

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 Rančić. 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 (,) plane, where is the longitude and 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 (,) 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 resolution. In addition the CISL scheme has the advantage over these schemes that it is applicable for Courant numbers larger than one. In plane geometry the scheme of Rančić had an overhead factor of 2.5 in CPU time compared to a traditional bicubic semi-Lagrangian scheme. This factor is reduced to 1.1 for the Machenhauer and Olk scheme on the plane while on the sphere the factor is found to be 1.28 for the present scheme. This overhead seems to be a reasonable price to pay for increased accuracy and exact mass conservation.

A mass‐conservative version of the semi‐implicit semi‐Lagrangian HIRLAM

“…Regarding phase errors (not shown) the CISL-HIRLAM performs slightly better than HIRLAM at both resolutions. When using the cascade scheme of instead of the fully twodimensional CISL scheme of Nair and Machenhauer (2002), the accuracy in terms of the l 2 difference is not altered (Figure 3(b)). The importance of the consistent Lagrangian discretization of the energy conversion term introduced in section 3 is demonstrated on Figure 3(c).…”

“…The two-dimensional integral in (25) is computed by a CISL scheme. Here we use the scheme of Nair and Machenhauer (2002) or alternately the cascade scheme of , both using a parabolic subgrid-scale representation. When the mass transported into the arrival column from all NLEV model layers (Figure 1) have been computed, the cell-averaged pressure at the bottom of each Lagrangian cell p k+1/2 n+1 expl can be determined by the assumption of hydrostatic balance.…”

“…A first step toward a locally mass-conservative semi-Lagrangian semi-implicit model has been taken with the development of the cell-integrated semi-Lagrangian (CISL) or, equivalently, finite-volume semi-Lagrangian advection scheme (e.g. Nair and Machenhauer, 2002). For a recent review of finite-volume methods used in meteorology, see Machenhauer et al (2008) and the detailed stability analysis given in Lauritzen (2007 A semi-implicit time-stepping of the fast waves requires that the term involving the flow divergence in the continuity equation must be handled in a semi-implicit manner.…”

“…The cells move with vertical walls and the horizontal extension is a polygon. The polygon is as in the two-dimensional CISL scheme of Nair and Machenhauer (2002), but the general idea applies to all CISL schemes. p k n * is the integral mean value of the pressure-layer thickness over the irregular departure cell area δA n k , and δp k n+1 is the mean value of the pressure-layer thickness over the regular arrival cell area A.…”

An explicit and conservative remapping strategy for semi‐Lagrangian advection

“…The flow field is deformational and an analytic solution is available (see Nair et al (1999a);Nair & Machenhauer (2002) for details).…”