On the pseudomomentum and generalized Stokes drift in a spectrum of rotational waves

Two Lagrangian-mean measures crucial to the accurate estimation of mean particle velocities in wavy or turbulent shear flows are considered. The measures are the pseudomomentum and generalized Stokes drift and of particular interest is their expression in terms of quantities directly measurable by fixed instruments. To proceed, the measures are first calculated for broad spectra of progressive symmetric rotational wave pairs of small amplitude. Both discrete and continuous spectra are considered and the waves may grow or decay. The expressions are then cast into a form composed of quantities that are measurable in a fixed reference frame, such as the surface slope spectrum of surface gravity waves or space-time velocity correlations in the interior of wavy shear flows. Finally, an example is given in which the measures are calculated for a plane channel flow subject to a broad spectrum of discrete progressive waves, specifically a numerical simulation of turbulent channel flow. It is seen that while the streamwise component of pseudomomentum is everywhere negative in the flow, the generalized Stokes drift changes sign, giving rise to an enhanced mass transport close to the boundary and a reduction in transport some distance from it. The sign change occurs 12.5 viscous units from the wall, near the centre of a 15 viscous units thick highly sheared layer of Stokes drift