Viscous attenuation of an internal-wave beam

When kinematic viscosity and mass diffusivity are equal to $ϵ$, a monochromatic internal-wave beam with wavenumber $k$ and ray angle $\theta$ has leading stream-function amplitudeThe cubic dependence on $k$ makes slope-focused, short internal waves dissipate especially rapidly.

In the prescribed along-beam scalar model $U=A(\xi )\cos \omega t$ with $A=U_0{e}^{-\lambda \xi }$, the first-order fluid displacement is $A\sin \omega t/\omega$. Its leading sampled-velocity correction is $-\lambda A^2\sin (2\omega t)/(2\omega )$, whose period average vanishes. This result concerns the supplied component model and does not determine an Eulerian mean streaming flow.