transformToSpatialDomainWithG

transforms from the spectral domain (k,l,j) to the spatial domain (x,y,z) using the G-modes

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Declaration

 w = transformToSpatialDomainWithG( w_bar )

Parameters

• u_bar variable with dimensions \((k,l,j)\)

Discussion

This is the component of the inverse discrete transformation \(D^{-1}\) that projects from the vertical modes $G$, followed by a transformation \((k,l) \mapsto (x,y)\) with a discrete Fourier transform. Mathematically we write,

\[g(x,y,z) = \mathcal{DFT}_x^{-1} \left[\mathcal{DFT}_y^{-1} \left[ \mathcal{G}^{-1} \left[ \tilde{f}^{klj} \right] \right] \right]\]

The \(G\) mode projection is applicable to dynamical variables \(w\), \(\eta\).

As noted in Early, et al. (2021), the vertical transforms $\mathcal{F}$ and $\mathcal{G}$ require a matrix multiplication and thus have a computational cost of,

\[N_z^2 N_x N_y\]

while the horizontal transforms use an FFT algorithm and therefore scale as,

\[\frac{5}{2} N_z N_x N_y \log_2 N_x N_y - N_z N_x N_y\]

Assuming that \(N_x = N_y\), the total computational cost of the horizontal and vertical transforms are approximately equal when \(10 log_2 N_x = N_z\) , or \(13 log_2 N_x = N_z\) for the hydrostatic case. This means that for a horizontal resolution of \(256^2\) the horizontal transformations dominate the computation time until approximately \(80-100\) vertical modes are used.