of spectra in terms of narrow spherical shells in Fourier space (see Eq. ). It is also superior in that one can generalize the definition of spectra to non-homogeneous flows by removing the spatial integral in Eq. , thereby obtaining location-dependent spectra.

It is useful to be able to rewrite the above spectra in terms of a streamfunction spectrum $C_{\ga\gb}(k)=\innerf{\psi_\ga}{\psi_\gb}{k}$. From the bilinear property of the bracket, it follows that the energy spectrum $E(k)$ reads: $$\begin{aligned}
E(k) &= -\sum_{\ga}  \innerf{\psi_\ga}{q_\ga}{k} = -\sum_{\ga} \innerf{\psi_\ga}{\sum_{\gb} \ccL_{\ab} \psi_\gb}{k} = -\sum_{\ab} L_{\ab}(k) \innerf{\psi_\ga}{\psi_\gb}{k} \\
&= -\sum_{\ab} L_{\ab}(k) C_{\ab}(k). \label{eq:Ek}\end{aligned}$$ Likewise, the potential enstrophy spectrum $G(k)$ reads: $$\begin{aligned}
G (k) &= \sum_{\ga} \innerf{q_\ga}{q_\ga}{k} = \sum_{\ga} \innerf{\sum_{\gb} \ccL_{\ab} \psi_\gb}{\sum_{\gc} \ccL_{\ac} \psi_\gc}{k} \\ 
&= \sum_{\ab} L_{\ab} (k)\innerf{\psi_\gb}{\sum_{\gc} \ccL_{\ac} \psi_\gc}{k} =  \sum_{\abc} L_{\ab} (k) L_{\ac}(k) \innerf{\psi_\gb}{\psi_\gc}{k}\\
&= \sum_{\abc} L_{\ab} (k) L_{\ac}(k) C_{\bc}(k). \label{eq:Gk}\end{aligned}$$ Stated in this way, these expressions easily generalize to multiple-layer quasi-geostrophic models simply by using a different matrix $L_{\ab} (k)$ with more rows and columns.

Forcing Spectrum in general
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Now let us turn our attention to the forcing spectra of the two-layer model. We begin by writing the potential vorticity equations in the following more abstract form: $$\pderiv{q_\ga}{t}+J(\psi_\ga, q_\ga) = \sum_\gb \cD_{\ga\gb} \psi_\gb + 