  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
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 which
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
and applications for potential new ways of inferring drift are discussed.
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 disturbance transfer to the wave field, recognizing the source
of the mean Lagrangian drift as the net momentum required to generate an
irrotational surface wave by any conservative force.
