\df{\bfk}.\end{aligned}$$ In Fourier space, the potential vorticity fields $q_\ga$ and streamfunction fields $\psi_\ga$ are related by $$\hat q _\ga (\bfk,t) = \sum_{\gb} L_{\ga\gb} (\nrm{\bfk}) \hat\psi_\ga (\bfk,t). \label{eq:QpsiRel}$$ Here, the sum runs over all layers, in this case $\gb=1,2$, and $L_{\ga\gb} (k)$ is a wavenumber matrix defined as $$L_{\ga\gb} (k) = \mattwo{-k^2-k_R^2/2}{+k_R^2/2}{+k_R^2/2}{-k^2-k_R^2/2}.
\label{eq:TheMatrixL}$$ In real space, the same relation between the potential vorticity $q_\ga$ and the streamfunction $\psi_\ga$ can be written in terms of a corresponding differential operator $\ccL_{\ga\gb}$, as follows: $$q_\ga (\bfx,t) = \sum_{\gb} \ccL_{\ga\gb} \psi_\ga (\bfx,t).$$ It is easy to see that the matrix $L_{\ga\gb} (k)$ is non-singular, for $k>0$, and can therefore be inverted. The inverse matrix $L_{\ga\gb}^{-1} (k)$ defines a corresponding inverse integrodifferential operator $\ccL_{\ga\gb}^{-1}$. Note that in Eq.  we have neglected the $\gb$ contribution to the Coriolis term $f$, since, for the case of our planet, the impact of the $\gb$-effect on the Nastrom-Gage energy spectrum is negligible. We have also neglected the latitude dependence of $f$, on the premise that we are interested in the ensemble average of the energy spectrum restricted on a thin strip of the Earth’s surface that is oriented parallel to the equator. These approximations cause the Coriolis term $f$ to 