1 Spatially adaptive Fibonacci grids R. James Purser IMSG at NOAA/NCEP Camp Springs, Maryland, USA.

2 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)

3 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).

4 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.

5 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.

6 One elongated, Two circular, Regions of Higher Resolution.

7 Intermediate region

8 Inner region

9 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.

10 Skelton grid constructed from a small circular center

11 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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13 Or we may choose to follow existing moving systems with a grid enhancement:

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15 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!)

16 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?

17 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.