Two-layer quasi-geostrophic energy conservation ($E=\frac{1}{2}[{\sum }_i{H}_i|\nabla {\psi }_i{|}^2+(f_0^2/g\prime )({\psi }_1-{\psi }_2{)}^2]$)

For inviscid unforced two-layer QG dynamics, $H_1{F}_1=H_2{F}_2=f_0^2/g\prime$ makes the coupling symmetric in the layer-depth inner product. Multiplication by $H_i{\psi }_i$ gives the positive energy $E=\frac{1}{2}[{\sum }_i{H}_i|\nabla {\psi }_i{|}^2+(f_0^2/g\prime )({\psi }_1-{\psi }_2{)}^2]$. It is conserved after integration when the lateral energy flux vanishes. The interface term is available potential energy; the gradient terms are kinetic energy.

Depth-weighted decomposition gives reduced depth $H_r=H_1{H}_2/(H_1+H_2)$ and internal radius $R_d^2=g\prime H_r/f_0^2$. On scale $l$, the ratio of interface available potential energy to baroclinic kinetic energy is of order $(l/R_d{)}^2$. Comparable depths recover $l^2{f}_0^2/(g\prime H_i)$; for strongly unequal depths the thinner layer sets the scale.