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Spatially adaptive Fibonacci grids

R. James Purser

IMSG at NOAA/NCEPCamp 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 thebest grid before we begin.

The latitude-longitude has well-known problems aroundthe poles where the resolution is not only inhomogeneousbut is seriously anisotropic also.

The grids based on symmetrical polyhedral mappings,such as the icosahedral triangular grid, or the variousspherical cubic grids are attractive, assuming we seeka uniform resolution and have ways to overcome thenumerical problems at the vertex singularities. (Oversetvariants of polyhedral grids are possible ways to overcomethese 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 haveenhanced resolution, the solution is to embed a regionalmodel, or possibly several models, nested within the larger global domain. This choice requires interpolatingand 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 insome traditional global models, for example, by applyingthe F. Schmidt (or Mobius) transformations, or by changingthe 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 ofenhanced 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 gridsmust obey.

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A very uniform global griddingis certainly possible with the Fibonaccigrid construction, as Swinbank and Purser Showed.

But it is also possible to engineer it so that prespecifiedregions are given higher than average resolution fora grid that covers less than the whole sphere.

For example, the following slides show a grid with threedistinct regions of high resolution.

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One elongated,Two circular,Regions ofHigherResolution.

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Intermediate region

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Innerregion

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The construction begins with an orthogonal, or at least an almost orthogonal, ‘skeleton grid’, which is generatedin these examples by a one-sided integration from some point or line. The resolution is predefined as a variabledensity function, and the Jacobian of the mapping from ordinary space to the space of these curvilinearcoordinates is made to conform to the prescribeddensity.

10Skelton grid constructed from a small circular center

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Dynamical adaptivity

It should be possible in many circumstances toanticipate 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 theresolution is a moving region for a while, then relaxthe resolution to its former state:

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Or we may choose to follow existing moving systemswith 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 variantOf this ‘polar’ grid, there would still be a need to deal With the poles by special numerics (as was done inThe Swinbank and Purser study).

An alternative might be to generate a ‘Yin-Yang’ pair ofOverlapping adaptive Fibonacci grids, where the problemOf singularities is replaced by interpolation/merging (again!)

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Numerical considerationsand questions

Most models assume a single time step and all grid pointsmarch forward in step; it would be simpler if this couldalso be done for the adaptive grids. This probably meansthat, for efficient modeling, one would need to use essentially fully-implicit methods to guarantee stability.

The Fibonacci grid is inherently NOT staggered. Methodswould need to be developed that overcome the tendencyof nonlinear computational instability.

Are there Arakawa-type differencing schemes that wouldapply?

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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 unitydoes not seem possible with the existing method of construction,a Yin-Yang variant of it does seem quite feasible, with two largeessentially rectangular regions and a single continuous ribbon of overlap linking them.

There remain many numerical challenges to overcome beforethis grid can become part of a reliable numerical predictionframework.