NOTES AND CORRESPONDENCE A Shallow-Water Model that Prevents Nonlinear Steepening of Gravity Waves

The shallow-water model is used as a testbench for understanding many fundamental dynamical problems (e.g. certain wave–mean interaction problems). One sometimes wants to allow large-amplitude gravity waves to propagate significant distances in such models without forming shocks. This paper presents a simple, and apparently unique, modification of the standard shallow-water model that prevents gravity wave shock formation, but which, at the same time, introduces only minimal changes in other aspects of the behavior. For instance, the presented modification is nondissipative as well as nondispersive, and it preserves the linear structure of the shallow-water equations as well as the nonlinear functional form and material invariance of shallow-water potential vorticity. The modification is derived theoretically and has been tested numerically in several ways in one and two dimensions.