Hamiltonian formulation of the stochastic surface wave problem

TL;DR

This paper develops a stochastic Hamiltonian framework for water wave problems, incorporating uncertainty and noise, and connects it with classical models through systematic approximations of the Dirichlet-Neumann operator.

Contribution

It introduces a novel stochastic Hamiltonian formulation for water waves based on the location uncertainty framework, extending classical theory to include noise.

Findings

Derivation of stochastic Hamiltonian structure under small noise.

Explicit appearance of Dirichlet-Neumann operator in the energy functional.

Systematic approximation methods for simplified stochastic wave models.

Abstract

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 different simplified wave models including noise in a natural way.