Displacement pseudomomentum of an internal gravity wave ($P=\overline{{\xi }_{x}u+{\zeta }_{x}w}=-\omega K^2|\hat{\sigma }{|}^2/(2k{N}^4)$)

The displayed displacement convention is the negative of the usual positive-frequency wave pseudomomentum $k\overline{E}/\omega$ for a rightgoing internal gravity wave. Here energy per unit reference mass density is $\overline{E}=|\hat{\sigma }{|}^2/(2N^2)$. From the buoyancy polarization of a plane internal gravity wave, horizontal averaging gives $\overline{{\zeta }_{x}p}=km|\hat{\sigma }{|}^2/(2N^2{K}^2)=c_{gz}P$, where $p$ is kinematic pressure. The sign convention changes both density and flux and leaves their propagation at the group velocity unchanged.

With weak kinematic viscosity $\nu$, zero buoyancy diffusion and locally unchanged plane internal gravity wave polarization, the displacement pseudomomentum of an internal gravity wave obeys $P_t+{\partial }_z(c_{gz}P)=-\nu K^{2}P$. For constant wavenumbers and a stationary wave envelope, $P\propto |\hat{\sigma }{|}^2$ proves the displayed amplitude decay. Decay is along the group velocity, so $c_{gz}<0$ describes a source above the receiving level. In a slowly varying shear flow, use wave action and the changing intrinsic frequency rather than treating $P/|\hat{\sigma }{|}^2$ as constant.