On The Quasi-streamfunction Formalism for Waves and Vorticity

TL;DR

This paper extends the quasi-streamfunction formalism to include vertical vorticity, providing a systematic framework for studying waves with vorticity over arbitrary bathymetry, with applications to coastal dynamics.

Contribution

It develops a modified Psi-formalism that incorporates vertical vorticity and reformulates the problem with boundary and interior equations, connecting to Hamiltonian and Lagrangian formulations.

Findings

Formalism accommodates vertical vorticity in wave analysis.

Asymptotic Hamiltonian matches previous potential flow results.

Lagrangian expansion aligns with Zakharov's formalism for wave scattering.

Abstract

The quasi-streamfunction (Psi) formalism proposed by Kim et. al. (J.W. Kim, K.J. Bai, R.C. Ertekin, W.C. Webster, J. Eng. Math. 40, 17 (2001)) provides a natural framework for systematically studying zero-vorticity waves over arbitrary bathymetry. The modified Psi-formalism developed here discards the original constraints of zero-vorticity by allowing for vertical vorticity which is the case of most interest for coastal dynamics. The problem is reformulated in terms of two dynamical equations on the boundary supplemented by one equation that represents a kinematic constraint in the interior of the domain. In this framework, the kinematic constraint can be solved to express Psi in terms of canonically-conjugated variables. The formalism is demonstrated for horizontally homogeneous flows over mild topography, where asymptotic formulations for the Hamiltonian and Lagrangian functions are derived based on the Helmholz-Hodge decomposition. For potential flows, the asymptotic form of the Hamiltonian is identical to previous results. The Lagrangian function is also expressed as an expansion in terms of the surface height and its time derivative and compared with Zakharov's formalism where agreement is found for one-dimensional wave scattering.