Abstract
The role of the Lagrangian mean flow, or drift, in modulating the geometry, kinematics
and dynamics of rotational and irrotational deep-water surface gravity waves is examined.
A general theory for permanent progressive waves on an arbitrary vertically sheared
steady Lagrangian mean flow is derived in the Lagrangian reference frame and mapped
to the Eulerian frame. A Lagrangian viewpoint offers tremendous flexibility due to the
particle labelling freedom and allows us to reveal how key physical wave behaviour
arises from a kinematic constraint on the vorticity of the fluid, inter alia the nonlinear
correction to the phase speed of irrotational finite amplitude waves, the free surface
geometry and velocity in the Eulerian frame, and the connection between the Lagrangian
drift and the Benjamin–Feir instability. To complement and illustrate our theory, a small
laboratory experiment demonstrates how a specially tailored sheared mean flow can almost
completely attenuate the Benjamin–Feir instability, in qualitative agreement with the
theory. The application of these results to problems in remote sensing and ocean wave
modelling is discussed. We provide an answer to a long-standing question: remote sensing
techniques based on observing current-induced shifts in the wave dispersion will measure
the Lagrangian, not the Eulerian, mean current.
and dynamics of rotational and irrotational deep-water surface gravity waves is examined.
A general theory for permanent progressive waves on an arbitrary vertically sheared
steady Lagrangian mean flow is derived in the Lagrangian reference frame and mapped
to the Eulerian frame. A Lagrangian viewpoint offers tremendous flexibility due to the
particle labelling freedom and allows us to reveal how key physical wave behaviour
arises from a kinematic constraint on the vorticity of the fluid, inter alia the nonlinear
correction to the phase speed of irrotational finite amplitude waves, the free surface
geometry and velocity in the Eulerian frame, and the connection between the Lagrangian
drift and the Benjamin–Feir instability. To complement and illustrate our theory, a small
laboratory experiment demonstrates how a specially tailored sheared mean flow can almost
completely attenuate the Benjamin–Feir instability, in qualitative agreement with the
theory. The application of these results to problems in remote sensing and ocean wave
modelling is discussed. We provide an answer to a long-standing question: remote sensing
techniques based on observing current-induced shifts in the wave dispersion will measure
the Lagrangian, not the Eulerian, mean current.