f_\gc}{k} \\
 &= \sum_\gc L_{\ac}^{-1}(k) \innerf{f_\gc}{\psi_\gb}{k} + \sum_\gc L_{\bc}^{-1}(k) \innerf{\psi_\ga}{f_\gc}{k} \\
 &= \sum_\gc [ L_{\ac}^{-1}(k) \phi_{\cb} (k) + L_{\bc}^{-1}(k) \phi_{\ca} (k)].\end{aligned}$$ Here, we have replaced the time derivatives in Eq.  with the forcing term from Eq. . We now note that governing equations for the energy spectrum $E(k)$ and the potential enstrophy spectrum $G(k)$ can be obtained by applying the operators indicated by equations and to the governing equation for the streamfunction spectrum $C_{\ab}(k)$. These equations are analogous to Eq.  and they take the form: $$\begin{aligned}
&\pderiv{E(k)}{t} + T_E (k) = D_E (k) + F_E (k),\\
&\pderiv{G(k)}{t} + T_G (k) = D_G (k) + F_G (k).\end{aligned}$$ Here, the terms above have analogous definitions. The next step is to write the forcing spectrum $F_E (k)$ for the energy and $F_G (k)$ for the potential enstrophy in terms of $F_{\ab}(k)$. Using the operator indicated by Eq. , the energy forcing spectrum $F_E (k)$ reads: $$\begin{aligned}
F_E (k) &= -\sum_{\ab} L_{\ab}(k) F_{\ab}(k) = -\sum_{\abc} L_{\ab}(k) [ L_{\ac}^{-1}(k) \phi_{\cb} (k) + L_{\bc}^{-1}(k) \phi_{\ca} (k)] \\
&= -\sum_{\bc} \left[  \left(  \sum_{\ga} L_{\ba}(k) L_{\ac}^{-1}(k) \right) \phi_{\cb} (k) \right] - \sum_{\ac}  \left[  \left(  \sum_{\gb} L_{\ab}(k) L_{\bc}^{-1}(k) \right)  \phi_{\ca} (k) \right] \\
&= -\sum_{\bc} \gd_{\bc} \phi_{\cb} (k) -  