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Integrating the local two-layer quasi-geostrophic energy conservation law over a horizontal domain and using the specified vanishing boundary flux givesPeriodic boundaries, or suitable fixed streamfunction boundary data eliminating the displayed energy flux, provide examples. The gradient terms are the two layer kinetic energies; the last term is available potential energy, equal to $g^{\prime }{\chi }^2/2$ under the interface-displacement relation.

The baroclinic energy ratio and deformation scale for unequal layer depths uses a decomposition uses the depth-weighted barotropic streamfunction $\Psi =(H_1{\psi }_1+H_2{\psi }_2)/(H_1+H_2)$ and $d=2\tilde{\psi }$. Define $H_r=H_1{H}_2/(H_1+H_2)$. ThenFor variations on horizontal scale $l$, this givesFor comparable layer depths $H_r$ is of order either $H_i$, reproducing the requested scale $l^2{f}_0^2/(g^{\prime }{H}_i)$. If one layer is much thinner, its depth controls this ratio; a depth-independent arithmetic barotropic average would leave unwanted cross terms in the energy decomposition.

Baroclinic potential energy dominates at scales much larger than the two-layer internal deformation radius; baroclinic kinetic energy dominates at much smaller scales. They are comparable near $l\sim R_d$. The independent barotropic kinetic energy has no interface-displacement partner, so this scale comparison refers specifically to the baroclinic component.

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.