Reconstruction of the pressure in long-wave models with constant vorticity

The effect of constant background vorticity on the pressure beneath steady long gravity waves at the surface of a fluid is investigated. Using an asymptotic expansion for the streamfunction, we derive a model equation and a formula for the pressure in a flow with constant vorticity. The model equation was previously found by Benjamin (1962), [3], and is given in terms of the vorticity omega(0), and three parameters Q, R and S representing the volume flux, total head and momentum flux, respectively. The focus of this work is on the reconstruction of the pressure from solutions of the model equation and the behavior of the surface wave profiles and the pressure distribution as the strength of the vorticity changes. In particular, it is shown that for strong enough vorticity, the maximum pressure is no longer located under the wave crest, and the fluid pressure near the surface can be below atmospheric pressure. (C) 2012 Elsevier Masson SAS. All rights reserved.