spectra for the case of a generalized multi-layer quasi-geostrophic model, and discuss the random Gaussian forcing model. These results are applied to the two-layer quasi-geostrophic model itself in section 4. Conclusions and discussion are given in section 5 and technical matters are discussed in the appendices.

The two-layer model equations
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The two-layer model is defined by two vorticity-streamfunction equations and a temperature equation which read $$\begin{aligned}
\pderiv{\gz_1}{t} &+ J(\gy_1, \gz_1+f) = -\frac{2f}{h} \gw + d_1, \label{eq:RelativeVortOne} \\
\pderiv{\gz_2}{t} &+ J(\gy_2, \gz_2+f) = +\frac{2f}{h} \gw + d_2 + e_2, \label{eq:RelativeVortTwo}  \\
\pderiv{T}{t} &+ J\left( \frac{\gy_1+\gy_2}{2}, T\right)  = -\frac{N^2}{f}\gw + Q_0. \label{eq:Temperature} \end{aligned}$$ We see that the temperature $T$ is advected by the average streamfunction $(\gy_1+\gy_2)/2$. Here $\gy_1$ and $\gy_2$ are the streamfunctions of the top and bottom layers; $\gw$ is the vertical velocity; $\gz_1 = \del^2 \gy_1$ and $\gz_2 = \del^2 \gy_2$ are the relative vorticities, and $d_1$, $d_2$, and $e_2$ are the dissipation terms given by $$\begin{aligned}
d_1 &= (-1)^{\gk+1}\nu \del^{2\gk} \gz_1 =  (-1)^{\gk+1}\nu \del^{2\gk+2} \gy_1 ,\\
d_2 &= (-1)^{\gk+1}\nu \del^{2\gk} \gz_2 =  (-1)^{\gk+1}\nu \del^{2\gk+2} \gy_2, \\
e_2 &= -\nu_E \gz_2 = -\nu_E \del^2 \gy_2.\end{aligned}$$ The terms $d_1$ and $d_2$ represent momentum dissipation of relative vorticity and $e_2$ represents 