| Periodic 2D surface water waves propagating steadily on a rotational current have been
studied by many authors (see [1] and references therein). Although the recent important
theoretical developments have confirmed that periodic waves can exist over flows with
arbitrary vorticity, their stability and their nonlinear evolution have not been much studied
extensively so far. In fact, even in the rather simple case of uniform vorticity (linear
shear), few papers have been published on the effect of a vertical shear current on the
side-band instability of a uniform wave train over finite depth. In most of these
studies [2-5], asymptotic expansions and multiple scales method have been used to
obtain envelope evolution equations, which allow eventually to formulate a condition
of (linear) instability to long modulational perturbations. It is noted here that this
instability is often referred in the literature as the Benjamin-Feir or modulational
instability.
In the present study, we consider the linear stability of finite amplitude two-dimensional,
periodic water waves propagating steadily on the free surface of a fluid with constant vorticity
and finite depth. First, the steadily propagating surface waves are computed with steepness up
to very close to the highest, using a Fourier series expansions and a collocation
method, which constitutes a simple extension of Fenton’s method [6] to the cases
with a linear shear current. Then, the linear stability of these permanent waves to
infinitesimal 2D perturbations is developed from the fully nonlinear equations in the
framework of normal modes analysis. This linear stability analysis is an extension of
[7] to the case of waves in the presence of a linear shear current and permits the
determination of the dominant instability as a function of depth and vorticity for a given
steepness. The numerical results are used to assess the accuracy of the vor-NLS
equation derived in [5] for the characteristics of modulational instabilities due to
resonant four-wave interactions, as well as to study the influence of vorticity and
nonlinearity on the characteristics of linear instabilities due to resonant five-wave
and six-wave interactions. Depending on the dimensionless depth, superharmonic
instabilities due to five-wave interactions can become dominant with increasing positive
vorticiy.
Acknowledgments: This work was supported by the Direction Générale de l’Armement
and funded by the ANR project n∘. ANR-13-ASTR-0007.
References
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vorticity beneath a surface wave train, Eur. J. Mech. B/Fluids, 2011, 30, 12–16.
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[3] M. Oikawa, K. Chow, D. J. Benney, The propagation of nonlinear wave
packets in a shear flow with a free surface, Stud. Appl. Math., 1987, 76, 69–92.
[4] A. I Baumstein, Modulation of gravity waves with shear in water, Stud. Appl.
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[5] R. Thomas, C. Kharif, M. Manna, A nonlinear Schrödinger equation for
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[7] M. Francius, C. Kharif, Three–dimensional instabilities of periodic gravity
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