Recurrence in truncated Boussinesq models for nonlinear waves in shallow water

Abstract

The rapid spatial recurrence of weakly nonlinear, weakly dispersive, progressive shallow‐water waves is examined with numerical simulations using a discretized and truncated (i.e., finite number of allowed frequency modes) form of the Boussinesq equations. Laboratory observations of sandbar formation under recurring, mechanically generated monochromatic waves with small Ursell number have motivated others to suggest that recurrence in naturally occurring random waves contributes to the establishment of periodic longshore sandbars on beaches. The present study primarily examines recurrence in wave fields with Ursell number O(1) and characterizes the sensitivity of recurrence to initial spectral shape and number of allowed frequency modes. It is shown that rapid spatial recurrence is not an inherent property of discretized and truncated Boussinesq systems for evolution distances of 10–50 wavelengths. When a small number of Fourier modes are used to represent an initially monochromatic wave field with significant nonlinearity (the Ursell number is O(1)), there is a trend toward recurrence of initial modal amplitudes, consistent with the known periodic solutions for a primary wave and its harmonic. However, for 32 modes or more, numerical simulations indicate only a few cycles of a damped recurrence, followed by disordered evolution of the Fourier amplitudes. For initial conditions similar to ocean field measurements of frequency‐sorted swell with Ursell number O(1) and many (>300) modes in the numerical model, the Fourier coefficients of the wave field do not recur rapidly. Thus in these cases the predictions of many rapid recurrence cycles by few‐mode models is an artifact of severe truncation. On the other hand, even with many allowed modes, pronounced recurrence is predicted when the Ursell number is small and the initial wave field is monochromatic. In this case, few‐ and many‐mode solutions are similar.