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 discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax-Friedrichs scheme. A third-order total variation diminishing Runge-Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

a b s t r a c tA class of new benchmark deformational flow test cases for the two-dimensional horizontal linear transport problems on the sphere is proposed. The scalar field follows complex trajectories and undergoes severe deformation during the simulation; however, the flow reverses its course at half-time and the scalar field returns to its initial position and shape. This process makes the exact solution available at the end of the simulation, and facilitates assessment of the accuracy of the underlying transport scheme. A procedure to eliminate possible cancellations of errors when the flow reverses is proposed.The test suite consists of four cases. Three are based on non-divergent flow fields and one on a divergent flow. The initial conditions are prescribed in terms of regular latitude-longitude spherical coordinates and are easy to implement. The divergent flow is specifically aimed for conservative global transport schemes to test for conservation, consistency and monotonicity (or positivity) of limiters/filters in a challenging flow environment. In the context of semi-Lagrangian schemes, the time-varying flow fields can be used to test trajectory algorithms where the exact trajectories do not follow great-circle arcs. The characteristics of the test cases are demonstrated with two different transport schemes.

It is the purpose of this paper to provide a comprehensive documentation of the new NCAR (National Center for Atmospheric Research) version of the spectral element (SE) dynamical core as part of the Community Earth System Model (CESM2.0) release. This version differs from previous releases of the SE dynamical core in several ways. Most notably the hybrid sigma vertical coordinate is based on dry air mass, the condensates are dynamically active in the thermodynamic and momentum equations (also referred to as condensate loading), and the continuous equations of motion conserve a more comprehensive total energy that includes condensates. Not related to the vertical coordinate change, the hyperviscosity operators and the vertical remapping algorithms have been modified. The code base has been significantly reduced, sped up, and cleaned up as part of integrating SE as a dynamical core in the CAM (Community Atmosphere Model) repository rather than importing the SE dynamical core from High‐Order Methods Modeling environment as an external code.

A conservative transport scheme based on the discontinuous Galerkin (DG) method has been developed for the cubed sphere. Two different central projection methods, equidistant and equiangular, are employed for mapping between the inscribed cube and the sphere. These mappings divide the spherical surface into six identical subdomains, and the resulting grid is free from singularities. Two standard advection tests, solid-body rotation and deformational flow, were performed to evaluate the DG scheme. Time integration relies on a third-order total variation diminishing (TVD) Runge–Kutta scheme without a limiter. The numerical solutions are accurate and neither exhibit shocks nor discontinuities at cube-face edges and vertices. The numerical results are either comparable or better than a standard spectral element method. In particular, it was found that the standard relative error metrics are significantly smaller for the equiangular as opposed to the equidistant projection.

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A new two-dimensional advection test on the surface of the sphere is proposed. The test combines a solid-body rotation and a deformational flow field to form moving vortices over the surface of the sphere. The resulting time-dependent deforming vortex centers are located on diametrically opposite sides of the sphere and move along a predetermined great circle trajectory. The horizontal wind field is deformational and nondivergent, and the analytic solution is known at any time. During one revolution around the sphere the initially smooth transported scalar develops strong gradients. Such an approach is therefore more challenging than existing advection test cases on the sphere. To demonstrate the effectiveness and versatility of the proposed test, three different advection schemes are employed, such as a discontinuous Galerkin method on a cubed-sphere mesh, a classical semi-Lagrangian method, and a finite-volume algorithm with adaptive mesh refinement (AMR) on a regular latitude-longitude grid. The numerical results are compared with the analytic solution for different flow orientation angles on the sphere.

The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.