Abstract
Modern Boussinesqtype 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 shortperiod 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 fifthorder Padé formulations by Agnon et al. (J. Fluid Mech., vol. 399, 1999, pp. 319333) the onestep Padé and the twostep TaylorPadé formulations by Madsen et al. (J. Fluid Mech., vol. 462, 2002, pp. 130) and the multilayer formulations by Liu et al. (J. Fluid Mech., vol. 842, 2018, pp. 323353). For completeness, we also analyse the popular, but older, formulations by Nwogu (ASCE J. Waterway Port Coastal Ocean Engng, vol. 119, 1993, pp. 618638) and Wei et al. (J. Fluid Mech., vol. 294, 1995, pp. 7192). We generally conclude that trough instabilities may occur in any Boussinesqtype formulation incorporating nonlinear dispersive terms. This excludes most of the classical Boussinesq formulations, but includes all of the socalled '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 onestep and twostep formulations by Madsen et al. (J. Fluid Mech., vol. 462, 2002, pp. 130). 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. 267288), i.e. the popular higherorderspectral formulations. In this case, we find no sign of trough instabilities.
Original language  English 

Article number  A38 
Journal  Journal of Fluid Mechanics 
Volume  889 
Number of pages  25 
ISSN  00221120 
DOIs  
Publication status  Published  2020 
Keywords
 Surface gravity waves
 Computational methods
 Nonlinear instability
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Research featured in "Focus on Fluids" article by Journal of Fluid Mechanics
Madsen, P. A. (Recipient) & Fuhrman, D. R. (Recipient), 10 Jul 2020
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