\sum_{\ac} \gd_{\ac}  \phi_{\ca} (k) = -2 \sum_{\ga} \phi_{\ga\ga} (k). \label{eq:FE}\end{aligned}$$ A similar calculation gives the potential enstrophy forcing spectrum $F_G (k)$. We use the operator indicated by Eq. , and we find that $F_G (k)$ reads: $$\begin{aligned}
F_G (k)  &= \sum_{\abc} L_{\ab} (k) L_{\ac}(k) F_{\bc}(k) \\
 &= \sum_{\abc} L_{\ab} (k) L_{\ac}(k) \sum_\gd  \left[ L_{\bd}^{-1}(k) \phi_{\dc} (k) + L_{\cd}^{-1}(k) \phi_{\db} (k) \right]\\
 &= \sum_{\acd}  L_{\ac}(k) \left[ \sum_\gb L_{\ab} (k) L_{\bd}^{-1}(k) \right] \phi_{\dc} (k) + \sum_{\abd} L_{\ab} (k) \left[ \sum_\gc  L_{\ac}(k) L_{\cd}^{-1}(k) \right] \phi_{\db} (k) \\
 &= \sum_{\acd} \gd_{\ad} L_{\ac}(k) \phi_{\dc} (k) + \sum_{\abd} L_{\ab} (k) \gd_{\ad} \phi_{\db} (k) 
 = 2\sum_{\ab}L_{\ab} (k)  \phi_{\ab} (k). \label{eq:FG}\end{aligned}$$ Note that the $F_E (k)$ calculation is dependent on the symmetry assumption $L_{\ab} (k) = L_{\ba}(k)$, which multi-layer quasi-geostrophic models do satisfy. On the contrary, the $F_G (k)$ calculation is not dependent on this symmetry assumption.

Because of the dependence of the streamfunction-forcing spectrum $\phi_{\ga\gb} (k)$ on the streamfunction $\gy_\ga$, it is not possible to predict the forcing spectra $F_E (k)$ and $F_G (k)$ solely from the statistical properties of the forcing term $f_\ga$. The sole exception is the case where $f_\ga$ is a random Gaussian field that is delta-correlated in time such that $$\avg{f_\ga (\bfx_1, t_1) 