Trough instabilities in Boussinesq formulations for water waves

Modern Boussinesq-type formulations for water waves typically incorporate fairly accurate linear dispersion relations and similar accuracy in nonlinear properties. This has extended their application range to higher values of $kh$ ( $k$ being wavenumber and $h$ the water depth) and has allowed for a better representation of nonlinear irregular waves with a fairly large span of short waves and long waves. Unfortunately, we have often experienced a number of ‘mysterious’ breakdowns or blowups, which have perplexed us for some time. A closer inspection has revealed that short-period noise can typically evolve in the deep troughs of wave trains in cases having relatively high spatial resolution. It appears that these potential ‘trough instabilities’ have not previously been discussed in the literature. In the present work, we analyse this problem in connection with the fourth- and fifth-order Pade formulations by Agnon et al.  ( J. Fluid Mech. , vol. 399, 1999, pp. 319–333) the one-step Pade and the two-step Taylor–Pade formulations by Madsen et al.  ( J. Fluid Mech. , vol. 462, 2002, pp. 1–30) and the multi-layer formulations by Liu et al.  ( J. Fluid Mech. , vol. 842, 2018, pp. 323–353). For completeness, we also analyse the popular, but older, formulations by Nwogu ( ASCE J. Waterway Port Coastal Ocean Engng , vol. 119, 1993, pp. 618–638) and Wei et al. ( J. Fluid Mech. , vol. 294, 1995, pp. 71–92). We generally conclude that trough instabilities may occur in any Boussinesq-type formulation incorporating nonlinear dispersive terms. This excludes most of the classical Boussinesq formulations, but includes all of the so-called ‘fully nonlinear’ formulations. Our instability analyses are successfully verified and confirmed by making simple numerical simulations of the same formulations implemented in one dimension on a horizontal bottom. Furthermore, a remedy is proposed and tested on the one-step and two-step formulations by Madsen et al.  ( J. Fluid Mech. , vol. 462, 2002, pp. 1–30). This demonstrates that the trough instabilities can be moved or removed by a relatively simple reformulation of the governing Boussinesq equations. Finally, we discuss the option of an implicit Taylor formulation combined with exact linear dispersion, which is the starting point for the explicit perturbation formulation by Dommermuth and Yue ( J. Fluid Mech ., vol. 184, 1987, pp. 267–288), i.e. the popular higher-order-spectral formulations. In this case, we find no sign of trough instabilities.