 Baroclinic energy ratio and deformation scale

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.