Abstract
Spectra are often computed from gridded data to determine the horizontal-scale dependence of quantities such as kinetic energy, vertical velocity, or perturbation potential temperature. This paper discusses several important considerations for the practical computation of such spectra. To ensure that the sum of the spectral energy densities in wavenumber space matches the sum of the energies in the physical domain (the discrete Parseval relation), the constant coefficient multiplying the spectral energy density must properly account for the way the discrete Fourier transform pair is normalized. The normalization factor appropriate of many older FORTRAN-based fast Fourier transforms (FFTs) differs from that in Matlab and Python’s numpy.fft, and as a consequence, the correct scaling factor for the kinetic energy (KE) spectral density differs between one-dimensional FFTs computed using these two approaches by a factor equal to the square of the number of physical grid points. A common algorithm used to compute two-dimensional spectra as a function of the total-wavenumber magnitude sums the contributions from all pairs of x- and y-component wavenumbers whose vector magnitude lies with a series of bins. This approach introduces systematic short-wavelength noise, which can be largely eliminated though a simple multiplicative correction. One- and two-dimensional spectra will differ by a constant if computed for flows in which the KE spectral density decreases as a function of the wavenumber to some negative power. This constant is evaluated and the extension of theoretical results to numerically computed FFTs is examined.
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