When <kinematic viscosity> and <mass diffusivity> are equal to $\epsilon$, a monochromatic internal-wave beam with <wavenumber> $k$ and ray angle $\theta$ has leading stream-function amplitude
$$
A(\zeta)=A(0)\exp\left(-\frac{\epsilon k^3\zeta}{N\cos\theta}\right)
=A(0)e^{-k\zeta/\operatorname{Re}},
\qquad
\operatorname{Re}=\frac{N\cos\theta}{\epsilon k^2}.
$$
The cubic dependence on $k$ makes slope-focused, short internal waves dissipate especially rapidly.


 Viscous attenuation of an internal-wave beam

ID: viscous-attenuation-of-an-internal-wave-beam