Hamiltonian formulation of the stochastic surface wave problem

Evgueni Dinvay (1, 2) , Etienne Mémin (1, 2)

1 ODYSSEY - Océan Dynamique Observations Analyse
2 IRMAR - Institut de Recherche Mathématique de Rennes

We devise a stochastic Hamiltonian formulation of the water wave problem. This stochastic representation is built within the framework of the modelling under location uncertainty. Starting from restriction to the free surface of the general stochastic fluid motion equations, we show how one can naturally deduce Hamiltonian structure under a small noise assumption. Moreover, as in the classical water wave theory, the non-local Dirichlet-Neumann operator appears explicitly in the energy functional. This, in particular, allows us, in the same way as in deterministic setting, to conduct systematic approximations of the Dirichlet-Neumann operator and to infer di↵erent simplified wave models including noise in a natural way.

Domaines: Physique mathématique [math-ph]

Evgueni Dinvay, Etienne Mémin. Hamiltonian formulation of the stochastic surface wave problem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2022, pp.1-26. ⟨10.1098/rspa.2022.0050⟩. ⟨hal-03777888⟩