Nigel Wood, Andy White and Andrew Staniforth
The Met Office uses a single Unified Model (the MetUM) for all operational weather
forecasts and climate simulations. Its single dynamical core must therefore be
capable of simulating spatial scales from around one kilometre to planetary
scales, and time scales from minutes to centuries and beyond. Another key aspect
of the MetUM is its avoidance of certain conventional approximations - made by other
operational centres - to the underpinning continuous equation set: the MetUM is a
deep-atmosphere model in that it does not make the shallow-atmosphere approximation
and retains all elements of the Coriolis force; and it uses the fully compressible and
nonhydrostatic equations.
Two particular challenges for the next version of the dynamical core for the
MetUM are: to permit the relaxation of the main remaining approximation of the
equation set, that is the assumption of spherical geopotential surfaces; and to
retain the deep-atmosphere, nonhydrostatic formulation whilst allowing the
numerical scheme to be straightforwardly switchable to shallow-atmosphere and/or
hydrostatic versions. Achieving both of these aims will permit careful study of
any scientific impact and benefits achieved by relaxing the conventional
approximations.
Use of the semi-Lagrangian scheme for advection allows the above challenges to be
met in a relatively straightforward manner. The particular aspect of the
semi-Lagrangian advection scheme that is of interest here, in the context
of global atmospheric modelling, is its applicability to the advection of a
vector field in curvilinear geometries without the need to explicitly discretise
the curvature terms. This is achieved by application of a particular matrix to
rotate vector components in the departure point frame of reference to components
in the arrival point frame of reference.
This approach has previously been used for discretising the momentum equations in
a shallow-atmosphere model. It is straightforward to relax the shallowness to
allow for a truly spherical coordinate system. But, additionally, this coordinate
system can be generalised within the matrix framework, to a particular spheroidal
coordinate system that captures 2/3rds of the ellipticity of the Earth's geoid. It
will also be shown how the matrix approach can be applied to the evaluation of the
departure points vector. The result is a consistent numerical handling of both the
momentum vector and the departure point position vector that can be switched from
spheroidal to spherical, to shallow-atmosphere to Cartesian geometry.
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