The variability of the atmospheric power spectrum

The kinetic energy spectrum of the atmosphere is a measure of how much energy is at each length scale - It is essentially the spatial Fourier transform of the kinetic energy field. Figure 1 shows the spectrum calculated by Nastrom and Gage (1985) from aircraft data. Motions on the largest length scales contain most of the kinetic energy. Figure 1 is on a log scale, so the gradients of the lines correspond to a power law relationship between spectral density of kinetic energy and wavelength. The spectral gradient is significant due to its link with predictability. Lorenz (1969) showed that, under certain statistical assumptions, the 2D vorticity equations -a good approximation to atmospheric motion on synoptic scales- will have an intrinsically finite range of predictability if the energy of the system has a -3 or shallower spectral gradient. Note that taken with figure 1, that re-ally does imply we will never be able to predict the weather more than 2-3 weeks in advance.
A controversial question.
The origin of the -3 range of figure 1 was convincingly explained by Charney (1971) as resulting from a large scale quasi-2D flow that is well-mixed via baroclinic in-stability. A similarly convincing explanation for the -5/3 range has not been provided so far. For some intuition, 2D turbulence will display an upscale energy cascade from the energy injection scale that has a -5/3 power law and a downscale enstrophy cascade of -3. 3D turbu-lence will have no transfer of energy upscale, instead it will just have a -5/3 downscale energy cascade (see Val-lis 2017, ch 14).
Figure 1: The result of thousands of aircraft measurements, this fig-ure shows the mysterious -5/3 spectral gradient at the mesoscales. [Nastrom 1985]
What could be causing the unexpectedly energetic mesoscale? In 1979, Gage suggested that it was the re-sult of 2D upscale energy transfer from small scale mix-ing such as storms. In opposition to this explanation, VanZandt (1982) pointed out a downscale motion of en-ergy associated with gravity waves could produce the observation. Further complicating the issue, the spec-trum is variable. Orographic forcing was shown to have a large effect with mesoscale energy over mountains up to 10 times larger than over ocean (Nastrom et al 1987). Similarly, precipitation has been seen to energize the mesoscales in models, (Selz et al 2019) for example. It has even be suggested that there may be no universal dynamical mechanism that governs the observed spec-trum, the lack of compelling theory simply being due to the statistical assumptions not applying to the highly complex real atmosphere (Selz et al 2019). The answer is likely to be a combination of the above with which cause dominates depending on latitude (Cho et al 1999), and possibly altitude.
In the last decade computing power has continued to grow. ECMWF analysis data can now see the -5/3 spec-trum and we are approaching a period where our ques-tions about the cause can be answered in a more direct manner. To provide evidence for the origin of the -5/3, a thorough categorization of the power spectrum in analysis is
therefore proposed. A tantalizing question is whether observed variation in the power spectrum correlates with changes in the state-dependent predictability. If this was found to be the case, forecasters could tell in advance how many ensemble members would be needed to sufficiently explore the state-space.