Computing Dimensional Spectra from Gridded Data and Compensating for Discretization Errors

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 we show can be virtually eliminated though a simple multiplicative correction. 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. 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.