Peregrine's system revisited

In 1967, D. H. Peregrine proposed a Boussinesq-type model for long waves in shallow waters of varying depth Peregrine (J Fluid Mech 27:815-827, 1967, [70]). This prominent paper turned a new leaf in coastal hydrodynamics along with contributions by Serre (La Houille Blanche 8:374-388, 1953, [72]) and Green and Naghdi (J Fluid Mech 78:237-246, 1976, [47]) and many others since then. Several modern Boussinesq-type systems stem from these pioneering works. In the present work, we revise the long wave model traditionally referred to as the Peregrine system. Namely, we propose a modification of the governing equations, which is asymptotically similar to the initial model for weakly nonlinear waves, while preserving an additional symmetry of the complete water wave problem. This modification procedure is called the invariantization. We show that the improved system has well-conditioned dispersive terms in the swash zone, hence allowing for efficient and stable run-up computations.