 Solution

Write $Z=\tilde{z}$ and introduce an arbitrary constant streamfunction scale ${\Psi }_0$, with $\hat{\psi }={\Psi }_0\tilde{\psi }$. The displacement jump condition acquires only a constant prefactor $2{\Psi }_0/\Delta U$, which can be divided out:For the second of the jump conditions for stratified inviscid shear flow, the two derivative terms have common scale $\Delta U{\Psi }_0/h$. Dividing by it givesThe term proportional to the reference density has zero jump by the first condition. ThusNo physical reference-density force has been discarded inside either layer; its continuous interface contribution has canceled. The layer equation becomes ${\tilde{\psi }}_{ZZ}={\alpha }^2\tilde{\psi }$. This explains why only the density anomaly appears in the nondimensional jump condition.