 Solution

Let $d={\psi }_1-{\psi }_2$ and $C=f_0^2/g^{\prime }=H_1{F}_1=H_2{F}_2$. Multiply each two-layer quasi-geostrophic potential vorticity equation by $H_i{\psi }_i$ and sum. The time-derivative terms areThe interface terms combine to $-Cd{d}_t$, since the two layers share the same depth-weighted coupling $H_i{F}_i$. The planetary term $\beta y$ has no time derivative. For the nonlinear terms, ${\nabla }_h\cdot {\mathbf{u}}_i=0$ and ${\mathbf{u}}_i\cdot {\nabla }_h{\psi }_i=0$ implyConsequently the local two-layer quasi-geostrophic energy conservation law isThe flux expression is one convenient form obtained directly from the requested multiplication; divergence-free modifications would represent the same local balance. $E$ is physical energy per horizontal area divided by the common reference density. Multiplying both $E$ and ${\mathbf{F}}_E$ by that density restores the dimensional physical-energy convention.