Abstract
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.