Spatially adaptive Fibonacci
grids
R. James Purser
IMSG at NOAA/NCEP
Camp Springs, Maryland, USA.
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When we develop a finite difference numerical prediction
model for the spherical domain we usually try to find the
best grid before we begin.
The latitude-longitude has well-known problems around
the poles where the resolution is not only inhomogeneous
but is seriously anisotropic also.
The grids based on symmetrical polyhedral mappings,
such as the icosahedral triangular grid, or the various
spherical cubic grids are attractive, assuming we seek
a uniform resolution and have ways to overcome the
numerical problems at the vertex singularities. (Overset
variants of polyhedral grids are possible ways to overcome
these problems. (See Purser and Rancic poster)
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What if we don’t want uniform
resolution?
Traditionally, when we want a global simulation to have
enhanced resolution, the solution is to embed a regional
model, or possibly several models, nested within the
larger global domain. This choice requires interpolating
and blending the solution from the coarser grid to the finer,
and vice versa if two-way nesting is involved.
One can obtain a SINGLE region of enhanced resolution in
some traditional global models, for example, by applying
the F. Schmidt (or Mobius) transformations, or by changing
the spacing of the two sets of grid lines (Canadian model).
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Multiple regional enhancement in a
single global model?
It would seem impossible to achieve multiple regions of
enhanced resolution in any conventional global model.
However, there is an alternative form of grid, the
FIBONACCI GRID
proposed by Swinbank and Purser (QJ, 132, 1769—1793),
which seems less constrained by the rules the other grids
must obey.
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A very uniform global gridding
is certainly possible with the Fibonacci
grid construction, as Swinbank and
Purser Showed.
But it is also possible to engineer it so that prespecified
regions are given higher than average resolution for
a grid that covers less than the whole sphere.
For example, the following slides show a grid with three
distinct regions of high resolution.
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One elongated,
Two circular,
Regions of
Higher
Resolution.
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Intermediate
region
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Inner
region
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The construction begins with an orthogonal, or at least
an almost orthogonal, ‘skeleton grid’, which is generated
in these examples by a one-sided integration from some
point or line. The resolution is predefined as a variable
density function, and the Jacobian of the mapping
from ordinary space to the space of these curvilinear
coordinates is made to conform to the prescribed
density.
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Skelton grid constructed from a small circular center
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Dynamical adaptivity
It should be possible in many circumstances to
anticipate where the higher resolution is required.
In that case, the grid can be made to evolve in time.
We might start with a uniform grid and enhance the
resolution is a moving region for a while, then relax
the resolution to its former state:
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Or we may choose to follow existing moving systems
with a grid enhancement:
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Global adaptive grids
A single global Fibonacci grid has two polar singularities
Even if it were possible to construct an adaptive variant
Of this ‘polar’ grid, there would still be a need to deal
With the poles by special numerics (as was done in
The Swinbank and Purser study).
An alternative might be to generate a ‘Yin-Yang’ pair of
Overlapping adaptive Fibonacci grids, where the problem
Of singularities is replaced by interpolation/merging (again!)
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Numerical considerations
and questions
Most models assume a single time step and all grid points
march forward in step; it would be simpler if this could
also be done for the adaptive grids. This probably means
that, for efficient modeling, one would need to use
essentially fully-implicit methods to guarantee stability.
The Fibonacci grid is inherently NOT staggered. Methods
would need to be developed that overcome the tendency
of nonlinear computational instability.
Are there Arakawa-type differencing schemes that would
apply?
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Conclusion
There exist grids, based upon the Fibonacci spiral construction,
that allow, in principle, mutiple regions of enhanced resolution.
While a fully global version of this form of grid as a single unity
does not seem possible with the existing method of construction,
a Yin-Yang variant of it does seem quite feasible, with two large
essentially rectangular regions and a single continuous
ribbon of overlap linking them.
There remain many numerical challenges to overcome before
this grid can become part of a reliable numerical prediction
framework.
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