Grids

The spectral transform (the Spherical Harmonic Transform) in SpeedyWeather.jl supports any ring-based equi-longitude grid. Several grids are already implemented but other can be added. The following pages will describe an overview of these grids and but let's start but how they can be used

using SpeedyWeather
spectral_grid = SpectralGrid(Grid = FullGaussianGrid)

SpectralGrid:
├ Spectral:   T31 LowerTriangularMatrix{Complex{Float32}}, radius = 6.371e6 m
├ Grid:       48-ring FullGaussianGrid{Float32}, 4608 grid points
├ Resolution: 333km (average)
└ Vertical:   8-level SigmaCoordinates

The life of every SpeedyWeather.jl simulation starts with a SpectralGrid object which defines the resolution in spectral and in grid-point space. The generator SpectralGrid() can take as a keyword argument Grid which can be any of the grids described below. The resolution of the grid, however, is not directly chosen, but determined from the spectral resolution trunc and the dealiasing factor. More in SpectralGrid and Matching spectral and grid resolution.

Ring-based equi-longitude grids

SpeedyWeather.jl's spectral transform supports all ring-based equi-longitude grids. These grids have their grid points located on rings with constant latitude and on these rings the points are equi-spaced in longitude. There is technically no constrain on the spacing of the latitude rings, but the Legendre transform requires a quadrature to map those to spectral space and back. Common choices for latitudes are the Gaussian latitudes which use the Gaussian quadrature, or equi-angle latitudes (i.e. just regular latitudes but excluding the poles) that use the Clenshaw-Curtis quadrature. The longitudes have to be equi-spaced on every ring, which is necessary for the fast Fourier transform, as one would otherwise need to use a non-uniform Fourier transform. In SpeedyWeather.jl the first grid point on any ring can have a longitudinal offset though, for example by spacing 4 points around the globe at 45˚E, 135˚E, 225˚E, and 315˚E. In this case the offset is 45˚E as the first point is not at 0˚E.

Implemented grids

All grids in SpeedyWeather.jl are a subtype of AbstractGrid, i.e. <: AbstractGrid. We further distinguish between full, and reduced grids. Full grids have the same number of longitude points on every latitude ring (i.e. points converge towards the poles) and reduced grids reduce the number of points towards the poles to have them more evenly spread out across the globe. More evenly does not necessarily mean that a grid is equal-area, meaning that every grid cell covers exactly the same area (although the shape changes).

Currently the following full grids <: AbstractFullGrid are implemented

• FullGaussianGrid, a full grid with Gaussian latitudes
• FullClenshawGrid, a full grid with equi-angle latitudes

and additionally we have FullHEALPixGrid and FullOctaHEALPixGrid which are the full grid equivalents to the HEALPix grid and the OctaHEALPix grid discussed below. Full grid equivalent means that they have the same latitude rings, but no reduction in the number of points per ring towards the poles and no longitude offset. Other implemented reduced grids are

• OctahedralGaussianGrid, a reduced grid with Gaussian latitudes based on an octahedron
• OctahedralClenshawGrid, similar but based on equi-angle latitudes
• HEALPixGrid, an equal-area grid based on a dodecahedron with 12 faces
• OctaHEALPixGrid, an equal-area grid from the class of HEALPix grids but based on an octahedron.

An overview of these grids is visualised here, and a more detailed description follows below.

Visualised are each grid's grid points (white dots) and grid faces (white lines). All grids shown have 16 latitude rings on one hemisphere, Equator included. The total number of grid points is denoted in the top left of every subplot. The sphere is shaded with grey, orange and turquoise regions to denote the hemispheres in a and b, the 8 octahedral faces c, d, f and the 12 dodecahedral faces (or base pixels) in e. Coastlines are added for orientation.

Grid resolution

All grids use the same resolution parameter nlat_half, i.e. the number of rings on one hemisphere, Equator included. The Gaussian grids (full and reduced) do not have a ring on the equator, so their total number of rings nlat is always even and twice nlat_half. Clenshaw-Curtis grids and the HEALPix grids have a ring on the equator such their total number of rings is always odd and one less than the Gaussian grids at the same nlat_half.

Related: Effective grid resolution and Available horizontal resolutions.

Matching spectral and grid resolution

A given spectral resolution can be matched to a variety of grid resolutions. A cubic grid, for example, combines a spectral truncation $T$ with a grid resolution $N$ (=nlat_half) such that $T + 1 = N$. Using T31 and an O32 is therefore often abbreviated as Tco31 meaning that the spherical harmonics are truncated at $l_{max}=31$ in combination with N=32, i.e. 64 latitude rings in total on an octahedral Gaussian grid. In SpeedyWeather.jl the choice of the order of truncation is controlled with the dealiasing parameter in the SpectralGrid construction.

Let J be the total number of rings. Then we have

• $T \approx J$ for linear truncation, i.e. dealiasing = 1
• $\frac{3}{2}T \approx J$ for quadratic truncation, i.e. dealiasing = 2
• $2T \approx J$ for cubic truncation, , i.e. dealiasing = 3

and in general $\frac{m+1}{2}T \approx J$ for m-th order truncation. So the higher the truncation order the more grid points are used in combination with the same spectral resolution. A higher truncation order therefore makes all grid-point calculations more expensive, but can represent products of terms on the grid (which will have higher wavenumber components) to a higher accuracy as more grid points are available within a given wavelength. Using a sufficiently high truncation is therefore one way to avoid aliasing. A quick overview of how the grid resolution changes when dealiasing is passed onto SpectralGrid on the FullGaussianGrid

You will obtain this information every time you create a SpectralGrid(; Grid, trunc, dealiasing).

Full Gaussian grid

(called FullGaussianGrid)

The full Gaussian grid is a grid that uses regularly spaced longitudes which points that do not reduce in number towards the poles. That means for every latitude $\theta$ the longitudes $\phi$ are

\[\phi_i = \frac{2\pi (i-1)}{N_\phi}\]

with $i = 1, ..., N_\phi$ the in-ring index (1-based, counting from 0˚ eastward) and $N_\phi$ the number of longitudinal points on the grid. The first longitude is therefore 0˚, meaning that there is no longitudinal offset on this grid. There are always twice as many points in zonal direction as there are in meridional, i.e. $N_\phi = 2N_\theta$. The latitudes, however, are not regular, but chosen from the $j$-th zero crossing $z_j(l)$ of the $l$-th Legendre polynomial. For $\theta$ in latitudes

\[\sin(\theta_j) = z_j(l) \]

As it can be easy to mix up latitudes, colatitudes and as the Legendre polynomials are defined in $[0, 1]$ an overview of the first Gaussian latitudes (approximated for $l>2$ for brevity)

Only even Legendre polynomials are used, such that there is always an even number of latitudes, with no latitude on the Equator. As you can already see from this short table, the Gaussian latitudes do not nest, i.e. different resolutions through different $l$ do not share latitudes. The latitudes are also only approximately evenly spaced. Due to the Gaussian latitudes, a spectral transform with a full Gaussian grid is exact as long as the truncation is at least quadratic, see Matching spectral and grid resolution. Exactness here means that only rounding errors occur in the transform, meaning that the transform error is very small compared to other errors in a simulation. This property arises from that property of the Gauss-Legendre quadrature, which is used in the Spherical Harmonic Transform with a full Gaussian grid.

On the full Gaussian grid there are in total $N_\phi N_\theta$ grid points, which are squeezed towards the poles, making the grid area smaller and smaller following a cosine. But no points are on the poles as $z=-1$ or $1$ is never a zero crossing of the Legendre polynomials.

Octahedral Gaussian grid

(called OctahedralGaussianGrid)

The octahedral Gaussian grid is a reduced grid, i.e. the number of longitudinal points reduces towards the poles. It still uses the Gaussian latitudes from the full Gaussian grid so the exactness property of the spherical harmonic transform also holds for this grid. However, the longitudes $\phi_i$ with $i = 1, ..., 16+4j$ on the $j$-th latitude ring (starting with 1 around the north pole), $j=1, ..., \tfrac{N_\theta}{2}$, are now, on the northern hemisphere,

\[\phi_i = \frac{2\pi (i-1)}{16 + 4j}.\]

We start with 20 points, evenly spaced, starting at 0˚E, around the first latitude ring below the north pole. The next ring has 24 points, then 28, and so on till reaching the Equator (which is not a ring). For the southern hemisphere all points are mirrored around the Equator. For more details see Malardel, 2016^[M16].

Note that starting with 20 grid points on the first ring is a choice that ECMWF made with their grid for accuracy reasons. An octahedral Gaussian grid can also be defined starting with fewer grid points on the first ring. However, in SpeedyWeather.jl we follow ECMWF's definition.

The grid cells of an octahedral Gaussian grid are not exactly equal area, but are usually within a factor of two. This largely solves the efficiency problem of having too many grid points near the poles for computational, memory and data storage reasons.

Full Clenshaw-Curtis grid

(called FullClenshawGrid)

The full Clenshaw-Curtis grid is a regular longitude-latitude grid, but a specific one: The colatitudes $\theta_j$, and the longitudes $\phi_i$ are

\[\theta_j = \frac{j}{N_\theta + 1}\pi, \quad \phi_i = \frac{2\pi (i-1)}{N_\phi}\]

with $i$ the in-ring zonal index $i = 1, ..., N_\phi$ and $j = 1, ... , N_\theta$ the ring index starting with 1 around the north pole. There is no grid point on the poles, but in contrast to the Gaussian grids there is a ring on the Equator. The longitudes are shared with the full Gaussian grid. Being a full grid, also the full Clenshaw-Curtis grid suffers from too many grid points around the poles, this is addressed with the octahedral Clenshaw-Curtis grid.

The full Clenshaw-Curtis grid gets its name from the Clenshaw-Curtis quadrature that is used in the Legendre transform (see Spherical Harmonic Transform). This quadrature relies on evenly spaced latitudes, which also means that this grid nests, see Hotta and Ujiie^[HU18]. More importantly for our application, the Clenshaw-Curtis grids (including the octahedral described below) allow for an exact transform with cubic truncation (see Matching spectral and grid resolution). Recall that the Gaussian latitudes allow for an exact transform with quadratic truncation, so the Clenshaw-Curtis grids require more grid points for the same spectral resolution to be exact. But compared to other errors during a simulation this error may be masked anyway.

Octahedral Clenshaw-Curtis grid

(called OctahedralClenshawGrid)

In the same as we constructed the octahedral Gaussian grid from the full Gaussian grid, the octahedral Clenshaw-Curtis grid can be constructed from the full Clenshaw-Curtis grid. It therefore shares the latitudes with the full grid, but the longitudes with the octahedral Gaussian grid.

\[\theta_j = \frac{j}{N_\theta + 1}\pi, \quad \phi_i = \frac{2\pi (i-1)}{16 + 4j}.\]

Notation as before, but note that the definition for $\phi_i$ only holds for the northern hemisphere, Equator included. The southern hemisphere is mirrored. The octahedral Clenshaw-Curtis grid inherits the exactness properties from the full Clenshaw-Curtis grid, but as it is a reduced grid, it is more efficient in terms of computational aspects and memory than the full grid. Hotta and Ujiie^[HU18] describe this grid in more detail.

HEALPix grid

(called HEALPixGrid)

Technically, HEALPix grids are a class of grids that tessalate the sphere into faces that are often called basepixels. For each member of this class there are $N_\varphi$ basepixels in zonal direction and $N_\theta$ basepixels in meridional direction. For $N_\varphi = 4$ and $N_\theta = 3$ we obtain the classical HEALPix grid with $N_\varphi N_\theta = 12$ basepixels shown above in Implemented grids. Each basepixel has a quadratic number of grid points in them. There's an equatorial zone where the number of zonal grid points is constant (always $2N$, so 32 at $N=16$) and there are polar caps above and below the equatorial zone with the border at $\cos(\theta) = 2/3$ ($\theta$ in colatitudes).

Following Górski, 2004^[G04], the $z=cos(\theta)$ colatitude of the $j$-th ring in the north polar cap, $j=1, ..., N_{side}$ with $2N_{side} = N$ is

\[z = 1 - \frac{j^2}{3N_{side}^2}\]

and on that ring, the longitude $\phi$ of the $i$-th point ($i$ is the in-ring-index) is at

\[\phi = \frac{\pi}{2j}(i-\tfrac{1}{2})\]

The in-ring index $i$ goes from $i=1, ..., 4$ for the first (i.e. northern-most) ring, $i=1, ..., 8$ for the second ring and $i = 1, ..., 4j$ for the $j$-th ring in the northern polar cap.

In the north equatorial belt $j=N_{side}, ..., 2N_{side}$ this changes to

\[z = \frac{4}{3} - \frac{2j}{3N_{side}}\]

and the longitudes change to ($i$ is always $i = 1, ..., 4N_{side}$ in the equatorial belt meaning the number of longitude points is constant here)

\[\phi = \frac{\pi}{2N_{side}}(i - \frac{s}{2}), \quad s = (j - N_{side} + 1) \mod 2\]

The modulo function comes in as there is an alternating longitudinal offset from the prime meridian (see Implemented grids). For the southern hemisphere the grid point locations can be obtained by mirror symmetry.

Grid cell boundaries

The cell boundaries are obtained by setting $i = k + 1/2$ or $i = k + 1/2 + j$ (half indices) into the equations above, such that $z(\phi, k)$, a function for the cosine of colatitude $z$ of index $k$ and the longitude $\phi$ is obtained. These are then (northern polar cap)

\[z = 1 - \frac{k^2}{3N_{side}^2}\left(\frac{\pi}{2\phi_t}\right)^2, \quad z = 1 - \frac{k^2}{3N_{side}^2}\left(\frac{\pi}{2\phi_t - \pi}\right)^2\]

with $\phi_t = \phi \mod \tfrac{\pi}{2}$ and in the equatorial belt

\[z = \frac{2}{3}-\frac{4k}{3N_{side}} \pm \frac{8\phi}{3\pi}\]

OctaHEALPix grid

(called OctaHEALPixGrid)

While the classic HEALPix grid is based on a dodecahedron, other choices for $N_\varphi$ and $N_\theta$ in the class of HEALPix grids will change the number of faces there are in zonal/meridional direction. With $N_\varphi = 4$ and $N_\theta = 1$ we obtain a HEALPix grid that is based on an octahedron, which has the convenient property that there are twice as many longitude points around the equator than there are latitude rings between the poles. This is a desirable for truncation as this matches the distances too, $2\pi$ around the Equator versus $\pi$ between the poles. $N_\varphi = 6, N_\theta = 2$ or $N_\varphi = 8, N_\theta = 3$ are other possible choices for this, but also more complicated. See Górski, 2004^[G04] for further examples and visualizations of these grids.

We call the $N_\varphi = 4, N_\theta = 1$ HEALPix grid the OctaHEALPix grid, which combines the equal-area property of the HEALPix grids with the octahedron that's also used in the OctahedralGaussianGrid or the OctahedralClenshawGrid. As $N_\theta = 1$ there is no equatorial belt which simplifies the grid. The latitude of the $j$-th isolatitude ring on the OctaHEALPixGrid is defined by

\[z = 1 - \frac{j^2}{N^2},\]

with $j=1, ..., N$, and similarly for the southern hemisphere by symmetry. On this grid $N_{side} = N$ where $N$= nlat_half, the number of latitude rings on one hemisphere, Equator included, because each of the 4 basepixels spans from pole to pole and covers a quarter of the sphere. The longitudes with in-ring- index $i = 1, ..., 4j$ are

\[\phi = \frac{\pi}{2j}(i - \tfrac{1}{2})\]

and again, the southern hemisphere grid points are obtained by symmetry.

Grid cell boundaries

Similar to the grid cell boundaries for the HEALPix grid, the OctaHEALPix grid's boundaries are

\[z = 1 - \frac{k^2}{N^2}\left(\frac{\pi}{2\phi_t}\right)^2, \quad z = 1 - \frac{k^2}{N^2}\left(\frac{\pi}{2\phi_t - \pi}\right)^2\]

The $3N_{side}^2$ in the denominator of the HEALPix grid came simply $N^2$ for the OctaHEALPix grid and there's no separate equation for the equatorial belt (which doesn't exist in the OctaHEALPix grid).

References