Practical Considerations for Computing Dimensional Spectra from Gridded Data

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.

Extension of the subgrid-scale gradient model for compressible magnetohydrodynamics turbulent instabilities

“…For further technical considerations about normalizations, caveats (e.g. the systematic noise introduced by the conversion to radial coordinates in the Fourier space) and possible corrective factors, we refer to a recent dedicated paper 83 . Note that the calculation of spectra for simulations with a large number of points (N 1000 3 ) require a large computational memory.…”

The scale dependence of initial‐condition sensitivities in simulations of convective systems over the southeastern United States

“…At a given height and time, denoting the zonal and meridional velocities of the m th ensemble member, including the unperturbed control member, as u m and v m , respectively, and denoting the two‐dimensional discrete Fourier transform of a function ϕ as and its complex conjugate as , the total (or background) two‐dimensional KE spectral density is ((Durran et al. , ), equation 24) where k h is the magnitude of the horizontal wavenumber, n x and n y are the number of grid points in the zonal and meridional directions, respectively, and Δ x and Δ y are the horizontal grid spacing in the zonal and meridional directions. The spectral density at each individual horizontal wavenumber ( k x , k y ) is added to the bin for which , where Δ k = 2 π /1,500 km−1 is the smallest resolved wavenumber corresponding to a full‐domain wave.…”

“…Spectra are also scaled by a corrective factor ((Durran et al. , ), equation 29) to reduce systematic noise introduced by binning. As noted in Section 2, spectra are calculated on the 1500 × 1500 km 2 square analysis subdomains highlighted in blue in Figure after detrending the data following the procedure in Errico ().…”

The Influence of Gravity Waves on the Slope of the Kinetic Energy Spectrum in Simulations of Idealized Midlatitude Cyclones

“…Data are plotted as a function of the nondimensional discrete horizontal wavenumberk [ (L y /2p)k, and smoothed to reduce the noise introduced by 2D binning using the technique in Durran et al (2017).…”

“…Using convection-permitting horizontal grid spacings of 1 km, Durran and Weyn (2016) showed a k 25/3 horizontal KE spectrum can be generated in an initially quiescent horizontally uniform environment by idealized convective systems. Sun et al (2017) analyzed the spectral KE budget in simulations similar to those in Durran and Weyn (2016) and showed that buoyancy forces and vertical energy fluxes play an important role in regulating the KE spectrum across a wide range of scales, again suggesting the k 25/3 slope is not exclusively generated through an inertial cascade.…”