Multiscale Analyses for the Shallow Water Equations

Abstract This paper explores several asymptotic limit regimes for shallow water flows over multiscale topography. Depending on the length and time scales consid-ered and on the characteristic water depth and height of topography, a variety of mathematically quite different asymptotic limit systems emerges. Specifically, we recover the classical “lake equations ” for balanced flow without gravity waves in the single time, single space scale limit (Greenspan, Cambridge Univ. Press, (1968)), discuss a weakly nonlinear and a strongly nonlinear multi-scale version of these wave-free equations involving short-range topography, and we re-derive the equa-tions for long-wave shallow water waves passing over short-range topography by Le Maı̂tre et al., JCP (2001)