...
SPHEREPACK requires gridded data on either a Gaussian grid or an equal angle lat/lon “cap” “cap” grid which includes points at the poles. It relies on the FFT for much of the transforms, and for a combination of associated Legendre transforms and FFTs. For efficiency the grid should have dimensions N x 2N (Gaussian), or (N+1) x 2N (equal angle cap) with N the product of powers of small primes. The most common lat/lon data is an equal angle offset grid, which avoids pole points (see. CAM-FV Grid Overview ). Data on this offset grid would have to first be interpolated to a cap grid before computing spherical harmonic transforms.
...
To compute spectral from native grid EAM output, we will first interpolate to a (N+1)x2N lat/lon cap grid. The KE spectra turns out to be sensitive to both the resolution of the lat/lon grid, and the algorithm used for interpolation. To determine sufficient resolution and the best algorithm, we use examine this sensitivity, we look at spectra below from an NE=256 (13km) Aqua planet simulation. This grid has 3072 points at degrees of freedom along the Equator, meaning it can potentially resolve spherical harmonics up to degree (or wavenumber) 1536 (corresponding to 2dx).
To look at the impacts of the interpolation algorithm, we compute the spectra from a single snapshot using bilinear, integrated bilinear and TR’s highorder (integrated native SE shapefunctionsshape functions), and two different resolutions. We plot E(k)/E_0(k), where we normalize all spectra by E_0, the spectra compute using the high order algorithm on an oversampled the highest resolution 3073x6144 lat/lon cap grid.
...
Computing E(k) at lower resolution or with certain algorithms introduces a roll off (reduced energy) and high frequencies. Since the behavior of the spectra, especially the rolloff starting around wave number 200 is of most interest, this should be avoided.
Using “PG2” output (cyan curve below) is not recommended - this is most likely due to the downscalings downscaling of the velocity when transforming from the dynamics to the PG2 physics grid, removing energy from small scales.
“intbilin” algorithm (green and brown) is not recommended (brow and green curves below). This algorithm is very nice for analysis, but does remove energy from small scales. “bilin” is a little better (orange, blue)
Best results are obtained by the highorder algorithm (red and straight line reference).
...
We thus recommend spectra be computed by first transforming to a lat/lon grid using TR’s high-order algorithm.
We next look at the effects of interpolation grid resolution: Again plotting E(k)/E_0(k). Observations:
The HO algorithm is quite good enough (no signficant significant rolloff at wave number 200) at a resolution of 1537x3072 with slight rolloff visable visible at 1025x2048.
The bilin algorithm requires more than twice as much resolution to match the results of highorder.
...
Note that in the actual spectra plot, the differences are quite small, with the lower resolution spectra matching very well at low wave numbers, and only slightly below at high wave numbers (zoomed in version on right). Only the yellow curve (1025x2048 high order) has visable visible roll off different that the highest resolution result.
...
The spectra above from single snaphots have a lot of noise which can be removed by time averaging. We next analyze how much averaging is necessary from an aqua planet NE256 simulation with 1 month of daily snapshots. The legend indicates how many snapshots (spaced 1 day apart) were used when averaging E(k). With 28 snapshots the data is still a touch noisy at the lower frequencies, so we recommend at least 60 snapshots. Need to test the minimum spacing: 60 snapshots from hourly data is probably not as good as 60 snapshots from daily data. In the plot below, each curves are all shifted downward so they would not be plotted on top of each other.
...