Possible Explanation of the Atmospheric Kinetic and Potential Energy Spectra

We hypothesize that the observed wave number spectra of kinetic and potential energy in the atmosphere can be explained by assuming that there are two related cascade processes emanating from the same large-scale energy source, a downscale cascade of potential enstrophy, giving rise to the $k^{-3}$ spectrum at synoptic scales and a downscale energy cascade giving rise to the $k^{-5/3}$ spectrum at mesoscales. The amount of energy which is going into the downscale energy cascade is determined by the rate of system rotation, with negligible energy going downscale in the limit of very fast rotation. We present a set of simulations of a system with strong rotation and stratification, supporting these hypotheses and showing good agreement with observations.

Figure 1

Atmospheric spectra of kinetic energy of the zonal and meridional wind components and potential energy measured by means of the potential temperature. The spectra of meridional wind and potential temperature are shifted one and two decades to the right, respectively. Reproduced from Nastrom & Gage (1985).

Figure 2

(a) Energy flux as a function of wave number $k$ normalized by the energy injection rate. The magnitude of the flux is increasing with increasing Rossby number. From bottom to top: $\mathrm{Ro}=[0,0.025,0.05,0.075,0.1,0.2]$, (b) energy spectrum for different Rossby numbers with the same colors as in (a). The $k^{-3}$ (dashed) and $k^{-5/3}$ slopes (dotted) are indicated.

Figure 3

Third-order structure function $D$, for different Rossby numbers, with the same colors as in Fig. 2. (a) Negative range normalized by the kinetic energy dissipation rate, (b) positive range normalized by the enstrophy injection rate. The theoretically predicted slopes are indicated.