drop out of the nonlinear Jacobian terms altogether.

Let us now introduce the following notation. Consider any arbitrary abstract scalar fields $a(\bfx)$ and $b(\bfx)$, which can be snapshots in time of either the streamfunction fields $\psi_\ga (\bfx,t)$ or the potential vorticity fields $q_\ga (\bfx,t)$ for a given level $\ga$. Let $a^{<k}(\bfx)$ and $b^{<k}(\bfx)$ be the fields obtained from $a(\bfx)$ and $b(\bfx)$ by setting to zero, in Fourier space, the components corresponding to wavenumbers whose norm is greater than $k$. Formally, $a^{<k}(\bfx)$ is defined as $$\begin{aligned}
a^{<k}(\bfx) &= \int_{\bbR^2} \df{\bfx_0} \int_{\bbR^2} \df{\bfk_0}\;  \frac{H(k-\nrm{\bfk_0})}{4\pi^2} \exp (i\bfk_0\cdot (\bfx-\bfx_0)) a(\bfx_0) \\
&\equiv \int_{\bbR^2} \df{\bfx_0} \; P(k|\bfx-\bfx_0) a(\bfx_0),\end{aligned}$$ with $H(x)$ the Heaviside function, defined as the integral of a delta function: $$\begin{aligned}
H(x) &= \int_0^x \gd (\tau) \; d\tau = \casethree{1}{\text{if } x\in (0,+\infty)}{1/2}{\text{if } x=0}{0}{\text{if } x\in (-\infty,0)}.\end{aligned}$$ Here $P(k|\bfx-\bfx_0)$ is the corresponding low-pass filter kernel. Obviously, $b^{<k}(\bfx)$ is defined similarly. We now use the two filtered fields $a^{<k}(\bfx)$ and $b^{<k}(\bfx)$ to define the bracket $\innerf{a}{b}{k}$ as: $$\begin{aligned}
\innerf{a}{b}{k} &= \dD{k} \int_{\bbR^2} \df{\bfx}\;\avg{a^{<k}(\bfx) b^{<k}(\bfx)} \label{eq:BracketDef} \\
&= \frac{1}{2} \int_{A\in \SO{2}} \df{\Omega (A)}\; \avg{[\hat a^{\ast}(kA\bfe) \hat b(kA\bfe) + \hat a(kA\bfe) \hat b^{\ast}(kA\bfe)]}. \label{eq:BracketFourier} \end{aligned}$$ Here, $\hat a (\bfk)$ and $\hat b (\bfk)$ are the Fourier 