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Spatially adaptive Fibonacci grids

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Spatially adaptive Fibonacci grids

R. James Purser

IMSG at NOAA/NCEP

Camp Springs, Maryland, USA.

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)

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).

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.

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.

One elongated,

Two circular,

Regions of

Higher

Resolution.

Intermediate

region

Inner

region

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.

Skelton grid constructed from a small circular center

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:

Or we may choose to follow existing moving systems

with a grid enhancement:

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!)

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?

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.