The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

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.

How Currents Trigger Extreme Sea Waves. The Roles of Stokes Drift, Eulerian Return Flow, and a Background Flow in the Open Ocean

“…Similarly, using the second‐order approximation for $$\mathcal{L}$$ instead and with the neglected nonlinear terms in the round brackets, negligible y dependence, and vanishing Eulerian return flow, Equation leads to the equation in a similar structure as that derived by Pizzo et al. (2023, their Equation 3.16) for long‐crested waves in a Lagrangian frame.…”

Bragg scattering of nonlinear surface waves by sinusoidal sandbars

“…2012; Francius & Kharif 2017; Dhar & Kirby 2023), wave kinematic and dynamic properties (Pizzo et al. 2023), and the geometry of the fluid particle trajectories (Wang, Guan & Vanden-Broeck 2020), we included this term in the present study.…”

“…The first-order problem is composed of waves propagating both forwards and backwards, as Mei (1985) demonstrated, and an additional non-propagative mode, B, which represents the wave-induced mean flow and is generated in the process of nonlinear wave modulations in slow spatial and time scales (Thomas et al 2012). Considering the significant influence of term B on stability of wave modulation (Thomas et al 2012;Francius & Kharif 2017;Dhar & Kirby 2023), wave kinematic and dynamic properties (Pizzo et al 2023), and the geometry of the fluid particle trajectories (Wang, Guan & Vanden-Broeck 2020), we included this term in the present study.…”

Statistics of weakly nonlinear waves on currents with strong vertical shear

“…A positive vorticity, which corresponds to a following current -i.e. U (z) > 0 and U (z) < 0 with U (z) the current oriented along the wave propagation direction, z the vertical coordinate, and a prime denotes the derivative -can remove the modulational instability altogether, demonstrated experimentally by Steer et al [40] and Pizzo et al [41] (the definition of positive/negative shear in ref. [40] is different from ours due to a different choice of the coordinate system).…”

“…Francius and Kharif [42] have extended [34] to two-dimensional Stokes waves where new quartet and quintet instabilities have been discovered arising from the presence of a uniform vorticity, while Abrashkin and Pelinovsky [43] derived a nonlinear Schrödinger equation for arbitrary, weak vertical shear in a Lagrangian framework, generalized in ref. [41].…”