The viscous lee wave problem and its implications for ocean modelling

- Ocean models employ elevated horizontal viscosity and diffusivity.
- Linear lee wave theory for such viscous 'model fluids' is derived.
- The theory is applied to wave-resolving models of lee waves with O(1) m2/s viscosity.
- Waves in such models will dissipate the majority of their energy within a few hundred metres of the boundary.
- Caution is necessary in comparing model results to ocean observations.

Ocean circulation models employ 'turbulent' viscosity and diffusivity to represent unresolved sub-gridscale processes such as breaking internal waves. Computational power has now advanced sufficiently to permit regional ocean circulation models to be run at sufficiently high (100 m-1 km) horizontal resolution to resolve a significant part of the internal wave spectrum. Here we develop theory for boundary generated internal waves in such models, and in particular, where the waves dissipate their energy. We focus specifically on the steady lee wave problem where stationary waves are generated by a large-scale flow acting across ocean bottom topography. We generalise the energy flux expressions of [Bell, T., 1975. Topographically generated internal waves in the open ocean. J. Geophys. Res. 80, 320-327] to include the effect of arbitrary viscosity and diffusivity. Applying these results for realistic parameter choices we show that in the present generation of models with O(1) m2s−1 horizontal viscosity/diffusivity boundary-generated waves will inevitably dissipate the majority of their energy within a few hundred metres of the boundary. This dissipation is a direct consequence of the artificially high viscosity/diffusivity, which is not always physically justified in numerical models. Hence, caution is necessary in comparing model results to ocean observations. Our theory further predicts that O(10−2) m2s−1 horizontal and O(10−4) m2s−1 vertical viscosity/diffusivity is required to achieve a qualitatively inviscid representation of internal wave dynamics in ocean models.