f_\ga.$$ Here, $\cD_{\ga\gb}$ is a matrix operator that accounts for all the dissipation terms and $f_\ga$ is the forcing term acting on the $\ga$-layer. Using this abstract formulation will shorten the calculations below considerably. For the case of thermal forcing, the forcing terms take the form $f_1 = \gf$ and $f_2 = -\gf$. Multiplying both sides with the inverse operator $\ccL_{\ga\gb}^{-1}$ gives the following governing equation for the streamfunctions: $$\pderiv{\psi_\ga}{t}+ \sum_\gb \ccL_{\ab}^{-1} J(\psi_\gb, q_\gb) = \sum_{\bc} \ccL_{\ab}^{-1} \cD_{\bc} \psi_\gc + \sum_\gb \ccL_{\ab}^{-1} f_\gb. \label{eq:GovEqStreamFunc}$$ Now, let us define a streamfunction-forcing spectrum $\phi_{\ab}(k) = \innerf{f_\ga}{\psi_\gb}{k}$ and recall our definition of the streamfunction spectrum $C_{\ga\gb}(k)=\innerf{\psi_\ga}{\psi_\gb}{k}$. Differentiating $C_{\ab}(k)$ with respect to time gives $$\pderiv{C_{\ab}(k)}{t} = \innerf{\pderiv{\psi_\ga}{t}}{\psi_\gb}{k} + \innerf{\psi_\ga}{\pderiv{\psi_\gb}{t}}{k}, \label{eq:TimeDerCab}$$ and we may write a governing equation for $C_{\ab}(k)$ in the form $$\pderiv{C_{\ab}(k)}{t} + T_{\ab}(k) = D_{\ab}(k) + F_{\ab}(k). \label{eq:GovEqCab}$$ Here, $T_{\ab}(k)$ is the contribution from the nonlinear Jacobian term, $D_{\ab}(k)$ is the contribution from the dissipation term, and $F_{\ab}(k)$ is the contribution from the forcing term. Our interest here is in the forcing contribution $F_{\ab}(k)$. It is easy to see that $F_{\ab}(k)$ can be written in terms of the streamfunction-forcing spectrum $\phi_{\ab}(k)$ as follows: $$\begin{aligned}
F_{\ab}(k) &= \innerf{\sum_\gc \ccL_{\ac}^{-1} f_\gc}{\psi_\gb}{k} + \innerf{\psi_\ga}{\sum_\gc \ccL_{\bc}^{-1} 