transforms of $a(\bfx)$ and $b(\bfx)$, $\SO{2}$ is the set of all non-reflecting rotation matrices in two dimensions, $d\Omega (A)$ is the measure of a spherical integral, $\bfe$ is a two-dimensional unit vector, and $\avg{\cdot}$ represents taking an ensemble average. The star superscript represents taking the complex conjugate. Also note that Eq.  is the definition of the bracket, and Eq.  is a consequence of the definition.

It is easy to see that the bracket is both symmetric and bilinear, in that it satisfies: $$\begin{aligned}
&\innerf{a}{b}{k} = \innerf{b}{a}{k},\\
&\innerf{a}{b+c}{k} = \innerf{a}{b}{k} + \innerf{a}{c}{k}, \\
&\innerf{a+b}{c}{k} = \innerf{a}{c}{k} + \innerf{b}{c}{k}.\end{aligned}$$ Moreover, every $(\ab)$-component of the operator $\ccL_{\ga\gb}$ is self-adjoint with respect to the bracket, which gives $$\innerf{\ccL_{\ab} a}{b}{k}  = \innerf{a}{\ccL_{\ab} b}{k} = L_{\ab} (k)\innerf{a}{b}{k},$$ and the same property is also satisfied by every component of the inverse operator $\ccL_{\ga\gb}^{-1}$: $$\innerf{\ccL_{\ab}^{-1} a}{b}{k}  = \innerf{a}{\ccL_{\ab}^{-1} b}{k} = L_{\ab}^{-1} (k)\innerf{a}{b}{k}.$$

Using the bracket, we now define the energy spectrum $E(k)= - \innerf{\psi_1}{q_1}{k} - \innerf{\psi_2}{q_2}{k}$ and the potential enstrophy spectra $G_1(k)= \innerf{q_1}{q_1}{k}$ and $G_2(k)= \innerf{q_2}{q_2}{k}$ for each layer. We also define $G (k)=G_1(k)+G_2(k)$ as the total potential enstrophy spectrum. This method of defining spectra was previously used by @book:Frisch:1995, and it is equivalent to the standard definition 