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

Oliver Bühler
1998 Journal of the Atmospheric Sciences  
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
more » ... 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. 1 The time for shocks to form is inversely proportional to the wave amplitude and is generally very short. For instance, an SSW gravity wave with a moderate amplitude of relative depth disturbance a ϭ 0.1 will break after just one wavelength of propagation; see also (38).
doi:10.1175/1520-0469(1998)055<2884:aswmtp>2.0.co;2 fatcat:zsgmjn7phjc3xmqetgrmgqarau