Number and Energy Spectra in One and Two Dimensions

The ordinary power spectra $\vert f_k\vert^2$ and $\vert f_k'\vert^2$ are output in the same way regardless of the number of dimensions, but the spectra $n_k$ and $\rho_k$ have well defined normalizations, so in one and two dimensions they are normalized so as to try to approximate the values that they would have in three dimensions. Let $f_{3k}$ be the Fourier transform of a three dimensional field $f(x)$, and let $f_{1k}$ and $f_{2k}$ be the Fourier transforms of the field taken on one and two dimensional slices respectively. Assuming that isotropy holds on average we show in section 6.3.5 that

$$\vert f_{1k}\vert^2 \approx {dx^4 \over 2 \pi L^2} k^2 \vert f_{3k}\vert^2$$ (5.36)

and

$$\vert f_{2k}\vert^2 \approx {dx^2 \over \pi L} \vert k\vert \vert f_{3k}\vert^2.$$ (5.37)

Recall that the wave vector $\vec{k}$ is given by

$$\vec{k} = {2 \pi \over L} \vec{i}$$ (5.38)

where $\vec{i}$ is the position on the grid in Fourier space (i.e. a triplet of integers from $-N/2$ to $N/2$). Thus

$$\vert f_{1k}\vert^2 \approx {2 \pi dx^4 \over L^4} i^2 \vert f_{3k}\vert^2 = {2 \pi \over N^4} i^2 \vert f_{3k}\vert^2,$$ (5.39)

$$\vert f_{2k}\vert^2 \approx {2 dx^2 \over L^2} \vert i\vert ... ..._{3k}\vert^2 = {2 \over N^2} \vert i\vert \vert f_{3k}\vert^2.$$ (5.40)

The Fourier transform $f_k$ that is actually calculated in one and two dimensional simulations will correspond to $f_{1k}$ and $f_{2k}$. The definitions of $n_k$ and $\rho_k$, however, are given in terms of $f_{3k}$. Thus equations [5.33] and [5.34] are multiplied by prefactors equal to ${N^4 \over 2 \pi i^2}$ and ${N^2 \over 2 \vert i\vert}$ in one and two dimensions respectively.