Energy of tsunami waves generated by bottom motion

In the vast literature on tsunami research, few articles have been devoted to energy issues. A theoretical investigation on the energy of waves generated by bottom motion is performed here. We start with the full incompressible Euler equations in the presence of a free surface and derive both dispersive and non-dispersive shallow-water equations with an energy equation. It is shown that dispersive effects only appear at higher order in the energy budget. Then we solve the Cauchy-Poisson problem of tsunami generation for the linearized water wave equations. Exchanges between potential and kinetic energies are clearly revealed.

💡 Research Summary

The paper addresses a gap in tsunami research concerning the energetics of waves generated by sudden bottom motion. Starting from the full three‑dimensional incompressible Euler equations with a free surface, the authors perform a systematic nondimensional scaling that introduces two small parameters: ε, measuring non‑linearity, and δ = h₀/λ, measuring shallowness. Under the shallow‑water assumption (δ ≪ 1) they derive both the classical non‑dispersive Saint‑Venant system and its energy balance. The mass conservation equation ∂η/∂t + ∇·(h u)=0 and the momentum equation ∂u/∂t + g∇η = 0 are multiplied by appropriate variables and integrated to obtain an exact energy equation ∂E/∂t + ∇·F = 0, where E = ½ h|u|² + ½ gη² is the sum of kinetic and potential energy densities and F is the corresponding energy flux. This shows that, even in the simplest shallow‑water model, total mechanical energy is rigorously conserved.

Next, the authors extend the analysis to include weak dispersion by retaining O(δ²) terms, leading to a Boussinesq‑type system. The dispersive contributions appear only as higher‑order corrections in the energy budget; they do not create or destroy energy but merely redistribute it spatially, affecting wave shape and phase speed. This result clarifies a common misconception that dispersion might act as an energy sink.

The core of the paper is the solution of the Cauchy‑Poisson problem for a prescribed initial bottom displacement ζ(x). Linearizing the water‑wave equations (∂²η/∂t² = g∇²η) and applying a Fourier transform yields the modal solution

 η̂(k,t) = ζ̂(k) ·