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In this work we discuss an approximate model for the propagation of deepirrotational water waves, specifically the model obtained by keeping onlyquadratic nonlinearities in the water waves system under theZakharov/Craig-Sulem formulation. We argue that the initial-value problemassociated with this system is most likely ill-posed in finite regularityspaces, and that it explains the observation of spurious amplification ofhigh-wavenumber modes in numerical simulations that were reported in theliterature. This hypothesis has already been proposed by Ambrose, Bona, andNicholls [4] but we identify a different instability mechanism. On the basis ofthis analysis, we show that the system can be "rectified". Indeed, byintroducing appropriate regularizing operators, we can restore thewell-posedness without sacrificing other desirable features such as a canonicalHamiltonian structure, cubic accuracy as an asymptotic model, and efficientnumerical integration. This provides a first rigorous justification for thecommon practice of applying filters in high-order spectral methods for thenumerical approximation of surface gravity waves. While our study is restrictedto a quadratic model, we believe it can be generalized to any order and pavesthe way towards the rigorous justification of a robust and efficient strategyto approximate water waves with arbitrary accuracy. Our study is supported bydetailed and reproducible numerical simulations.

Rectification of a deep water model for surface gravity waves