 Internal-wave momentum deposition

A slowly established upward plane internal gravity wave has $\overline{u^{\prime }{w}^{\prime }}=-m|a{|}^2/(2k)$. The mean horizontal conservation of momentum equation gives ${\overline{u}}_t=-{\partial }_z\overline{u^{\prime }{w}^{\prime }}$. Using envelope transport and an initially vanishing mean gives $\overline{u}=-m|a{|}^2/(2k{c}_{gz})$. Its energy density per unit mass is $\mathcal{E}=(k^2+m^2)|a{|}^2/(2k^2)$, so $\overline{u}=(k/\omega )\mathcal{E}$. This is the mean flow left behind the advancing wave envelope, rather than a claim that an established uniform wavetrain has nonzero local stress divergence.