Refinement (AMR) techniques provide an attractive framework for
atmospheric flows since they allow an improved resolution in limited
regions without requiring a fine grid resolution throughout the entire
model domain. The model regions at high resolution are kept at a
minimum and can be individually tailored towards the research problem
associated with atmospheric model simulations.
A solution-adaptive grid is a virtual necessity for resolving a problem
with different length scales. In order to avoid under-resolving
high-gradient regions in the problem, or conversely, over-resolving
low-gradient regions at the expense of more critical regions, solution
adaptation is a powerful tool saving several orders of magnitude in
computing resources for many problems. Climate and weather models, or
generally speaking computational fluid dynamics (CFD) codes, are among
the many applications that are characterized by multiscale
phenomena and their resulting interactions.
For instance, large-scale weather systems such as midlatitude cyclones
drive small-scale frontal zones, thunderstorms or rain events. These
small-scale features may then influence the larger scale if, as an
example, evaporation processes and turbulence at the surface trigger
sensible and latent heat fluxes. But although today's atmospheric
general circulation models (GCMs), and in particular weather prediction
codes, are already capable of uniformly resolving horizontal scales of
order 10-20 km (e.g. the model IFS of the European Centre for
Medium-Range Weather Forecasts), the atmospheric motions of interest
span many more scales than those captured in a fixed resolution model
run. The widely varying spatial and temporal scales, in addition to the
nonlinearity of the dynamical system, raise an interesting and
challenging modeling problem. Solving such a problem more efficiently
and accurately requires variable resolution.
The adaptive FV dynamical core has been
run in two configurations: the full 3D hydrostatic dynamical core on
the sphere and the corresponding 2D shallow water model that has been
extracted out of the 3D version (Jablonowski 2004
, Jablonowski et al. 2004
, Jablonowski et al. 2006
, St-Cyr et al. 2008
, Jablonowski et al. 2009
). In general, the shallow water system
can be considered a 1-level version of the 3D dynamical core. This
shallow water setup serves as an
ideal testbed for the horizontal discretization and the 2D
adaptive-mesh strategy. It further allows the efficient and quick
interpolation routines at fine-coarse grid interfaces.
Both static and dynamic adaptation strategies have been tested. Static
adaptations can be used to vary the resolution in pre-defined regions
of interest. This includes static refinements near mountain ranges or
static coarsenings in the longitudinal direction for the implementation
of a so-called reduced grid in polar regions. Dynamic adaptations are
based on flow characteristics and guided by refinement criteria that
detect user-defined features of interest during a simulation. In
particular, flow-based refinement criteria, such as vorticity or
gradient indicators, have been tested. Refinements and coarsenings then
occur according to pre-defined threshold values.
An example of an adaptive passive advection test on the sphere is shown
below. The figure shows the initial conditions for the shallow water
standard test case 1 with a 90o
rotation angle (see Williamson et a. 1992
for the test
specifications). Here the adapted blocks track a cosine bell as it is
transported once around the sphere. Note that each self-similar blocks
contains 9x6 grid points in lon x lat direction so that the finest grid
resolution in this example corresponds to a 0.625o
grid. The maximum number of refinement levels is set to 3.
The adaptation criterion is based on a simple threshold assessment. A
block is refined as soon as the height of the cosine bell exceeds a
user-determined threshold value at (at least) one grid point within the
block. On the other hand, a block gets coarsened if the height of the
cosine bell in the block no longer meets the criterion. The
corresponding mpeg movie (4.1 MB)
shows that the cosine bell is successfully captured as indicated by the
overlaid block distribution. The movie shows a 12-day simulation. After 12-days
the tracer distribution returns to its initial position which then serves
as the reference solution. There are no visible distortions of the
height field as the cosine bell approaches, passes over and leaves the
poles. The increased resolution clearly helps preserve the shape and
The cosine bell represents a rather smooth tracer
distribution. An alternative tracer field is shown below and also in this 12-day simulation (4.4 MB)
with a rotation angle of 30o
The slotted cylinder is initialized with a constant value and is set to zero outside the inner domain. The movie shows that the sharp edges are tracked successfully by a gradient-based refinement criterion.
An example of a dynamically adapted nonlinear flow field is presented in
the next figure. It shows an idealized flow over a single mountain at
model day 10 (test case 5, Williamson
et a. 1992
). Here the adaptations are guided by the absolute value
of the geopotential height gradient. The refined regions pick out the
strong gradient regimes that are associated with the evolving wave
train behind the mountain. Other refinement criteria are also feasible for this mountain-induced wave response.
For example, the following
15-day mpeg movie (5 MB)
shows the evolution of the geopotential height field that is dynamically
tracked by a relative vorticity refinement criterion.
It detects the evolving lee-side wave reliably and highlights slightly different
refinement regions in comparison to the gradient criterion shown below.
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