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Abstract We examine a two-dimensional deep-water surface gravity wave packet generated by a pressure disturbance in the Lagrangian reference frame. The pressure disturbance has the form of a narrow-banded weakly nonlinear deep-water wave packet. During forcing, the vorticity equation implies that the momentum resides entirely in the near-surface Lagrangian-mean flow, which in this context is often called the “Stokes drift”. After the forcing turns off, the wave packet propagates away from the forcing region, carrying with it most of the energy imparted by the forcing. These waves together with their induced long wave response have no momentum in a depth integrated sense, in agreement with the classical results of Longuet-Higgins and Stewart (Deep Sea Research and Oceanographic Abstracts 11, 592−562) and McIntyre (Journal of Fluid Mechanics 106, 331−347). The total flow associated with the propagating packet has no net momentum. In contrast with the finite-depth scenario discussed by McIntyre (Journal of Fluid Mechanics 106, 331−347), however, momentum imparted to the fluid during forcing resides in a dipolar structure that persists in the forcing region—rather than being carried away by shallow-water waves. We conclude by examining waves propagating from deep to shallow water and show that wave packets, which initially have no momentum, may have non-zero momentum in finite-depth water through reflected and trapped long waves. This explains how deep water waves acquire momentum as they approach shore. The artificial form of the parameterized forcing from the wind facilitates the thought experiments considered in this paper, as opposed to striving to model more realistic wind forcing scenarios. 

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.