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Third-order momentum advection on the quasi-hexagonal C-grid on the sphere
  • Almut Gassmann
Almut Gassmann
TRR 181

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The 3rd-order upstream advection scheme for scalars on the Voronoi C-grid, once introduced by Skamarock and Gassmann (2011), is applied to horizontal momentum advection. A prerequisite is that the 2nd-order momentum advection is available in advection form for a trivariate coordinate system, so that the higher order terms can be formulated as an add-on. Three key ingredients for a successful application are (i) the determination of the advecting velocity, (ii) the determination of directional Laplacians of wind components and (iii) the determination of the upstream direction. The scheme is tested in two settings, a shallow water framework on the regular hexagonal mesh and the baroclinic wave test on the sphere, where the mesh is slightly deformed. In both cases, the trailing ripples and waves known to represent dispersion errors are impressively reduced. If they are not removed, they can lead to spurious excitation of gravity waves or wavy vorticity patterns. After upscale error growth, they can no longer be identified as a result of numerical errors. The effects of the 3rd-order upstream add-on and a Smagorinsky diffusion are compared. The Smagorinsky model reduces the amplitude of the mentioned waves, but does not erase them. With regard to the dissipation properties, the Smagorinsky diffusion is in accordance with the 2nd law of thermodynamics and dissipation is locally only positive. In contrast, dissipation can be locally negative in runs with the 3rd-order upstream add-on. Therefore, physical and numerical requirements cannot be fulfilled simultaneously.