Abstract. A numerical model of propagation of internal gravity waves in a stratified medium is applied to the problem of tsunami wave run-up onto a shore. In the model, the ocean and the atmosphere is considered as a united continuum whose the density varies with height with a saltus at a water–air boundary. Correct conditions of join at a water–air interlayer are automatically ensured because the solution is searched for as a generalised one. The density stratification in the ocean and in the atmosphere is supposed to be an exponential one, but in the ocean, a scale of stratification of density is large and the density varies slightly. The wave running to a shore is taken as a long solitary wave. The wave evolution is simulated with consideration of time-varying vertical wave structure. Inshore, the wave breaks down, and intensive turbulent mixing develops in water thickness. The effect of breakdown depends on shape of the bottom. If slope of the bottom is small, and inshore the depth grows slowly with distance from a shore, then mixing happens only in the upper stratum of the fluid, thanks to formation of a dead region near the bottom. If the bottom slope inshore is significant, then the depth of fluid mixing is dipped up to 50 metres. The developed model shows the depth of mixing effects strongly depends on shape of a bottom, and the model may be useful for investigation of influences of strong gales and hurricanes on coastline and beaches and investigation of dependence of stability of coastline and beaches on bottom shape.