transformFromSpatialDomainWithFg

transforms from the spatial domain (z,:,:) to the spectral domain (j,:,:) using the geostrophic F-modes

---

Declaration

 u_bar = transformFromSpatialDomainWithFg(u)

Parameters

• u variable with dimensions \((z,:,:)\)

Returns

• u_bar variable with dimensions \((j,:,:)\)

Discussion

This is the component of the discrete transformation \(D\) that transforms \((x,y) \mapsto (k,l)\) with a discrete Fourier transform, followed by a projection onto the F-modes. Mathematically we write,

\[\tilde{f}_{klj} = \mathcal{F} \cdot \mathcal{DFT}_y \cdot \mathcal{DFT}_x \left[ f(x,y,z) \right]\]

The \(F\) mode projection is applicable to dynamical variables \(u\), \(v\), \(p\).

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.