# Computation at a coordinate singularity

## Abstract

Coordinate singularities are sometimes encountered in computational problems. An impor-tant example involves global atmospheric models used for climate and weather prediction. Classical spherical coordinates can be used to parameterize the manifold – that is, generate a grid for the computational spherical shell domain. This particular parameterization offers significant benefits such as orthogonality and exact representation of curvature and con-nection (Christoffel) coefficients. But it also exhibits two polar singularities and at or near these points typical continuity/integral constraints on dependent fields and their deriva-tives are generally inadequate and lead to poor model performance and erroneous results. Other parameterizations have been developed that eliminate polar singularities, but prob-lems of weaker singularities and enhanced grid noise compared to spherical coordinates (away from the poles) persist. In this study reparameterization invariance of geometric ob-jects (scalars, vectors and the forms generated by their covariant derivatives) is utilized to generate asymptotic forms for dependent fields of interest valid in the neighborhood of a pole. The central concept is that such objects cannot be altered by the metric structure of a parameterization.The new boundary conditions enforce symmetries that are required for transformations of geometric objects. They are implemented in an implicit polar filter of a structured grid,more »

- Authors:

- Teraflux Corporation, Boca Raton FL (United States).

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Laboratory, Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC).

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1463654

- DOE Contract Number:
- AC02-05CH11231

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 361; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Prusa, Joseph M.
```*Computation at a coordinate singularity*. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2018.01.044.

```
Prusa, Joseph M.
```*Computation at a coordinate singularity*. United States. doi:10.1016/j.jcp.2018.01.044.

```
Prusa, Joseph M. Tue .
"Computation at a coordinate singularity". United States. doi:10.1016/j.jcp.2018.01.044.
```

```
@article{osti_1463654,
```

title = {Computation at a coordinate singularity},

author = {Prusa, Joseph M.},

abstractNote = {Coordinate singularities are sometimes encountered in computational problems. An impor-tant example involves global atmospheric models used for climate and weather prediction. Classical spherical coordinates can be used to parameterize the manifold – that is, generate a grid for the computational spherical shell domain. This particular parameterization offers significant benefits such as orthogonality and exact representation of curvature and con-nection (Christoffel) coefficients. But it also exhibits two polar singularities and at or near these points typical continuity/integral constraints on dependent fields and their deriva-tives are generally inadequate and lead to poor model performance and erroneous results. Other parameterizations have been developed that eliminate polar singularities, but prob-lems of weaker singularities and enhanced grid noise compared to spherical coordinates (away from the poles) persist. In this study reparameterization invariance of geometric ob-jects (scalars, vectors and the forms generated by their covariant derivatives) is utilized to generate asymptotic forms for dependent fields of interest valid in the neighborhood of a pole. The central concept is that such objects cannot be altered by the metric structure of a parameterization.The new boundary conditions enforce symmetries that are required for transformations of geometric objects. They are implemented in an implicit polar filter of a structured grid, nonhydrostatic global atmospheric model that is simulating idealized Held–Suarez flows. Aseriesof test simulations using different configurations of the asymptotic boundary con-ditions are made, along with control simulations that use the default model numerics with no absorber, at three different grid sizes. Typically the test simulations are ~ 20% faster in wall clock time than the control—resulting from a decrease in noise at the poles in all cases. In the control simulations adverse numerical effects from the polar singularity are observed to increase with grid resolution. In contrast, test simulations demonstrate robust polar behavior independent of grid resolution.},

doi = {10.1016/j.jcp.2018.01.044},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = C,

volume = 361,

place = {United States},

year = {2018},

month = {5}

}