Design

The spectral transform

ecTrans transforms a batch of meteorological fields from a grid point space representation $X_k(\lambda_i, \phi_j)$, where $\lambda_i$ is the $i^{\text{th}}$ longitude, $\phi_j$ is the $j^{\text{th}}$ latitude, and $k$ is the index which ranges over the batch of fields, to a spectral space representation $X_{m,n,k}$, where $m$ is the zonal wavenumber, and $n$ is the total wavenumber. This constitutes a direct spectral transform. ecTrans can also carry out the inverse spectral transform.

Beginning with the direct spectral transform (grid point space to spectral space), this is accomplished in two computational steps. Firstly, a Fourier transform is performed in the longitudinal direction at each latitude and for each field, independently,

$$X_{m,k}(\phi_j) = \sum_{i=0}^{N_j} X_k(\lambda_i, \phi_j) \exp{\left(-I2\pi\frac{m}{N_j}i\right)}.$$

Here, $N_j$ is the number of longitudinal points at latitude $j$ and $I$ is the imaginary identity. The input meteorological fields, $X_k(\lambda_i, \phi_j)$, are always entirely real. Thus, although a full Fourier transform would be evaluated for $(m = 0, N_j - 1)$, terms $(m = N_j / 2 + 1, N_j - 1)$ are simply the conjugate of terms $(m = 1, N_j / 2)$ so we needn't calculate them. This is referred to as an "RFFT" algorithm.

The second step is the Legendre transform which is performed in the latitudinal direction. This is performed by Gaussian quadrature with Gaussian weights $w_j$,

$$X_{m,n,k} = \sum_{j=0}^{N_\text{lat}} X_{m,n}(\phi_j) w_j P_{m,n}(\sin(\phi_j)),$$

where $P_{m,n} (\sin(\phi_j))$ is the associated Legendre polynomial of zonal wavenumber $m$ and total wavenumber $n$ which is evaluated at the point $\sin(\phi_j)$. The Legendre transform is implemented as a matrix-matrix multiplication, collapsing over the latitudinal dimension with the fields and total wavenumber dimensions "free". This operation is performed independently at each zonal wavenumber $m$. Notably, the fact that $P_{m,n}$s is symmetric when $n - m$ is even and is antisymmetric when $n - m$ is odd is exploited to halve the required number of floating- point operations. The Legendre transform is split into two transforms over the Northern hemisphere: one for each of the even and odd polynomials.

Parallelizing a spectral transform

Basic usage of ecTrans