h Q_0$, and $f_2 = (1/4)k_R^2 h Q_0$. The derivation is shown in Appendix \[app:VorticityStreamfunctionEquations\]. Although the argument is well-known folklore, we want to note mainly that: (a) the dissipation terms have the same form in the relative vorticity equations as they do in the potential vorticity equations; (b) the thermal forcing term $Q$ appears on both top and bottom potential vorticity equations with opposite signs. Consequently, both layers are forced anti-symmetrically by the same forcing term, except with opposite signs.

It is also well-known that the two-layer model, in the absence of forcing and dissipation, conserves the total energy E given by $$E(t) = - \int_{\bbR^2} [\psi_1 (\bfx,t) q_1 (\bfx,t)  + \psi_2 (\bfx,t)  q_2 (\bfx,t) ] \;\df{\bfx},$$ and the potential enstrophies $G_1$ and $G_2$ for each layer given by: $$\begin{aligned}
G_1 (t) &= \int_{\bbR^2} q_1^2  (\bfx,t)  \;\df{\bfx},\\
G_2 (t) &= \int_{\bbR^2} q_2^2  (\bfx,t)  \;\df{\bfx}.\end{aligned}$$

To properly define all the relevant spectra associated with these conserved quantities, consider first the Fourier expansions of the streamfunctions fields $\psi_\ga (\bfx,t)$ and potential vorticity fields $q_\ga (\bfx,t)$ ($\ga = 1,2$): $$\begin{aligned}
\psi_\ga (\bfx,t) &= \int_{\bbR^2} \hat\psi_\ga (\bfk,t) \exp (i\bfk\cdot\bfx) \; \df{\bfk}, \\
q_\ga (\bfx,t) &= \int_{\bbR^2} \hat q _\ga (\bfk,t) \exp (i\bfk\cdot\bfx) \; 