# Mathematics Calendar

Thursday, January 12, 2017 - 14:00

Weierstraß-Institut

Mohrenstr. 39, 10117 Berlin, Erhard-Schmidt-Hörsaal, Erdgeschoss

Seminar Numerische Mathematik

In the last decade, some weather and climate modeling centers started to develop atmospheric models that reside on tessellations of the icosahedron. The resulting hexagonal or triangular meshes are essential for two reasons: first, the numerical difficulties arising from CFL restrictions near the poles are avoided, and second, the lower boundary (land structure and ocean) is represented by approximately equally sized areas.

C-staggered meshes are common in atmospheric modeling because they allow for good wave propagation properties. However, this staggering, which positions the mass points at the grid box centers and the velocity points at the grid box edges, generates other problems for tessellations of the icosahedron, which will be discussed in the talk.

The first problem is the overspecification of velocity components in comparison to the mass components. This problem can only be solved for hexagonal C-grid meshes by defining discretization procedures which guarantee the linear dependency of velocity components during time stepping. This applies to the Coriolis term and the momentum diffusion term. Such methods are understood by the community since the work of Thuburn (2008).

The second problem is the grid deformation in the vicinity of the 12 pentagon grid boxes. It will be discussed that this deformation is responsible for non-convergence of some measures which are needed for the evaluation of the friction tensor. The formulation of the friction in dependency on vorticity and divergence instead on strain and shear deformations avoids this problem. However, a direct numerical integration by parts which delivers the frictional heating is then no longer possible. Results with different approaches for the friction term will be presented.

C-staggered meshes are common in atmospheric modeling because they allow for good wave propagation properties. However, this staggering, which positions the mass points at the grid box centers and the velocity points at the grid box edges, generates other problems for tessellations of the icosahedron, which will be discussed in the talk.

The first problem is the overspecification of velocity components in comparison to the mass components. This problem can only be solved for hexagonal C-grid meshes by defining discretization procedures which guarantee the linear dependency of velocity components during time stepping. This applies to the Coriolis term and the momentum diffusion term. Such methods are understood by the community since the work of Thuburn (2008).

The second problem is the grid deformation in the vicinity of the 12 pentagon grid boxes. It will be discussed that this deformation is responsible for non-convergence of some measures which are needed for the evaluation of the friction tensor. The formulation of the friction in dependency on vorticity and divergence instead on strain and shear deformations avoids this problem. However, a direct numerical integration by parts which delivers the frictional heating is then no longer possible. Results with different approaches for the friction term will be presented.

submitted by lawrenz (marion.lawrenz@wias-berlin.de, 030 20372566)