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 inverse spectral transform (spectral space to grid point space), this is accomplished in two computational steps. Firstly, an inverse Legendre transform is performed in the latitudinal direction,

$$X_{m,k}(\phi_j) = \sum_{n=|m|}^{N} X_{m,n,k} P_{m,n}(\sin(\phi_j)).$$

Then, an inverse Fourier transform is performed in the longitudinal direction,

$$X_k(\lambda_i, \phi_j) = \sum_{n=-N}^{N} X_{m,k}(\phi_j) e^{im\lambda_i}.$$

Parallelizing a spectral transform

Basic usage of ecTrans