Variants of HEALPix grids for global climate modelling

Various grids are being used to discretize the sphere for general circulation models of ocean and atmosphere, coupled to sea ice and land models. Depending on the numerics (or the network architecture for machine learning-based models) some grids are better suited and more widely used but no single grid meets all requirements. The HEALPix grid was originally invented for applications in cosmology and designed for spectral transform efficiency, hierarchical nested ordering and equal-area grid cells. Here, we present several variants: The OctaHEALPix grid, and the octaminimal Gaussian and Clenshaw-Curtis grids. The OctaHEALPix grid inherits all the properties of the original HEALPix grid but based on a single square matrix instead of 12 used for the base pixels (faces) of the HEALPix grid. This eliminates singularities for the 8 corner cells with 7 instead of 8 neighbouring cells. The nested hierarchy of the OctaHEALPix grid is therefore a quadtree from the coarsest to the finest zoom level. Data on the OctaHEALPix grid can be arranged as a single square matrix, representing an equal-area map projection of the sphere, allowing for interpolation-free visualisation and data storage. However, the HEALPix grids do not provide an exact quadrature in the Legendre transform, such that transform errors are higher than with Gaussian, or equi-angle latitude-based grids using the Clenshaw-Curtis quadrature. The inexact transform with the HEALPix grids does not pose any problems in simulations where other sources of error dominate. We present the grid-flexible spectral transform implemented in the atmospheric circulation model SpeedyWeather.jl that simultaneously supports all ring-based, equi-longitude grids both on CPU (including multithreading) and GPU. The OctaHEALPix grid is also favourable for machine learning-based models like diffusion models based on the UNet architecture which only require custom boundary conditions. The other HEALPix variant we present is the octaminimal Gaussian grid, which imposes Gaussian latitudes on the OctaHEALPix grid which reduces transform errors while preserving more of the HEALPix’s equal-area and hierarchy properties. Similarly, the octaminimal Clenshaw-Curtis grid uses regular latitudes with the Clenshaw-Curtis quadrature. Simulations based on these grids are presented for coupled climate simulations and implications for hybrid numerical and machine learning-based models are discussed.