  In this work we discuss an approximate model for the propagation of deep
irrotational water waves, specifically the model obtained by keeping only
quadratic nonlinearities in the water waves system under the
Zakharov/Craig-Sulem formulation. We argue that the initial-value problem
associated with this system is most likely ill-posed in finite regularity
spaces, and that it explains the observation of spurious amplification of
high-wavenumber modes in numerical simulations that were reported in the
literature. This hypothesis has already been proposed by Ambrose, Bona, and
Nicholls [4] but we identify a different instability mechanism. On the basis of
this analysis, we show that the system can be "rectified". Indeed, by
introducing appropriate regularizing operators, we can restore the
well-posedness without sacrificing other desirable features such as a canonical
Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient
numerical integration. This provides a first rigorous justification for the
common practice of applying filters in high-order spectral methods for the
numerical approximation of surface gravity waves. While our study is restricted
to a quadratic model, we believe it can be generalized to any order and paves
the way towards the rigorous justification of a robust and efficient strategy
to approximate water waves with arbitrary accuracy. Our study is supported by
detailed and reproducible numerical simulations.
