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Number and Energy Spectra in One and Two Dimensions
The ordinary power spectra
and
are output in the same way regardless of the number of dimensions, but the spectra and 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 be the Fourier transform of a three dimensional field , and let and 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

(5.36) 
and

(5.37) 
Recall that the wave vector is given by

(5.38) 
where is the position on the grid in Fourier space (i.e. a triplet of integers from to ). Thus

(5.39) 

(5.40) 
The Fourier transform that is actually calculated in one and two dimensional simulations will correspond to and . The definitions of and , however, are given in terms of . Thus equations [5.33] and [5.34] are multiplied by prefactors equal to
and
in one and two dimensions respectively.
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