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KE spectra is useful for understanding and tuning the dissipation mechanisms in the model. For a global model, it can be computed via a vector spherical harmonic transform of the instantaneous velocity field, or two spherical harmonic transforms of the instantaneous vorticity and divergence scalars on a spherical surface. For all the data below, we interpolate to 250mb (I think because this is the most energetic part of the atmosphere - check this?). Spherical harmonics can be thought of as polynomials in Cartesian coordinates (x,y,z) restricted to the sphere. For each degree k there will be 2k+1 polynomials (spherical harmonics) of total degree k, or 2k+1 spherical harmonics of degree n. To compute the KE spectra, E(k), we sum the coefficient squared over all degree n k spherical harmonics.
The atmosphere is in close to geostrophic balance, resulting in a kinetic energy spectrum that scales like E(k) ~ k^-3 as k ranges from 3000km to 800km. But at 600km (wave number 134), the atmosphere transitions into a k^−5/3 scaling that holds down to a few km. The transition, or breakdown of the geostrophic balance, is still not fully explained. The −5/3 scaling suggests a transition to three-dimensional isotropic turbulence, but this explanation can be quickly ruled out. This transition was first observed by Nastrom and Gage, who suggested a combination of the known enstrophy generation from baroclinic instability with an unknown small scale energy source. Later investigators have speculated on where this source could come from: there are pure dynamical explanations and explanations which require convection and moist physics (such as thunderstorms).
Example KE spectra. Plotted is the “compensated” spectra, E(k)*k^5/3, to better illustrate the possible transition to ^-5/3. E(k) was computed from many flow snapshots and then E(k) is averaged over all the results to get the nice curves shown below. Unfortunately, one cannot compute E(k) from time averaged flow fields.
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In CAM4 at 1/8 degree (14km) resolution the model had a nice Nastrom-Gage transition. This has slowly deteriorated, perhaps due to increased dissipation being added to the model as it evolved. It may also have been that CAM4 had this transition for the wrong reasons. It will be interesting to see the specta for higher resolutions now that we are able to run at 6km and 3km resolutions.
Computing KE Spectra
Interpolate to the correct lat/lon grid.
Computing the KE spectra requires performing spherical harmonic transforms. This can be done via the long standing SPHEREPACK Fortran package ( https://www2.cisl.ucar.edu/resources/legacy/spherepack ), which has an easy to use interface in NCL (and pyNGL?).
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