A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

Abstract

The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type ("soliton" equations), but with vorticity. The familiar problem of propagation governed by the Korteweg-de Vries (KdV) equation is introduced, but allowing for an arbitrary distribution of vorticity. The effect of the constant vorticity on the solitary wave is described. The corresponding problem for the Nonlinear Schrödinger (NLS) equation is briefly mentioned but not explored here. The problem of two-way propagation (admitting head-on collisions), as described by the Boussinesq equation, is examined next. This leads to a new equation: the Boussinesq-type equation valid for constant vorticity. However, this cannot be transformed into an integrable Boussinesq equation (as is possible for the corresponding KdV and NLS equations). The solitary-wave solution for this new equation is presented. A description of the Camassa-Holm equation for water waves, with constant vorticity, with its solitary-wave solution, is described. Finally, we outline the problem of propagation of small-amplitude, large-radius ring waves over a flow with vorticity (representing a background flow in one direction). Some properties of this flow, for constant vorticity, are described.

Nonlinear water waves in shallow water in the presence of constant vorticity: A Whitham approach

“…Using an asymptotic expansion method Choi [3] derived a Green-Naghdi system for long gravity waves in uniform shear flows (constant vorticity) and for weakly nonlinear waves he deduced from this system a Boussinesq-type equation and a KdV equation. The derivation of a Boussinesq-type equation and a Camassa-Holm equation with constant vorticity was carried out by Johnson [8]. Very recently, Richard and Gavrilyuk [11] derived a dispersive shallow water model which is a generalisation of the classical Green-Naghdi model to the case of shear flows.…”

Ring-type multisoliton dynamics in shallow water

“…and using a multiple scale method [6,17,25], one can reduce the fluid Eqs. (1)- (5) to the cylindrical (or concentric) Kortewegde Vries equation (cKdVE), viz.,…”