| Abstract: |
| We outline the role of vorticity in the Euler-like system for weakly viscous water waves, specifically with respect to the mass conservation of this system.
The derivation of a viscous Euler system was performed by Ruvinksky et al. in 1991 and Longuet-Higgins in 1992. In 2008, Dias, Dyachenko and Zakharov adapted this system to express the viscosity contribution in terms of the surface elevation $\eta$. This Euler-like system has been widely used to perform higher order spectral method-based simulations, or as a starting point to derive Nonlinear Schr\odinger (NLS) like propagation equations.
However, in the above cases small vorticity terms (of higher order in wave steepness $\eps$ and viscosity $\nu$) that arise in the derivation when moving from Navier Stokes to the Euler-like system are neglected. In our findings, omitting these terms is not always justified, namely when the thickness of the boundary layer where vorticity is active becomes smaller than the wave amplitude. Moreover, their omission breaks the conservation of mass. We derive an expression for these `lost` vorticity terms in terms of the velocity potential $\phi$. |
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