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1. Stokes waves with constant vorticity: I. Numerical computation

   https://doi.org/10.1111/sapm.12250  

   Dyachenko, Sergey A.; Hur, Vera Mikyoung (February 2019, Studies in Applied Mathematics)

   Abstract Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk. 

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2. The decay of a dipolar vortex in a weakly dispersive environment

   https://doi.org/10.1017/jfm.2021.298  

   Johnson, Edward R.; Crowe, Matthew N. (June 2021, Journal of Fluid Mechanics)

   null (Ed.)

   A simple model is presented for the evolution of a dipolar vortex propagating horizontally in a vertical-slice model of a weakly stratified inviscid atmosphere, following the model of Flierl & Haines ( Phys. Fluids , vol. 6, 1994, pp. 3487–3497) for a modon on the $${\rm beta}$$ -plane. The dipole is assumed to evolve to remain within the family of Lamb–Chaplygin dipoles but with varying radius and speed. The dipole loses energy and impulse through internal wave radiation. It is argued, and verified against numerical solutions of the full equations, that an appropriately defined centre vorticity for the dipole is closely conserved throughout the flow evolution. Combining conservation of centre vorticity with the requirement that the dipole energy loss balances the work done on the fluid by internal wave radiation gives a model that captures much of the observed dipole decay. Similar results are noted for a cylindrical dipole propagating along the axis of a rotating fluid when the dipole axis is perpendicular to the axis of rotation and for a spherical vortex propagating horizontally in a weakly stratified fluid. The model extends to fluids of small viscosity and so provides an estimate for the relative importance of wave drag and dissipation in dipole decay. 

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3. Almost extreme waves

   https://doi.org/10.1017/jfm.2022.1047  

   Dyachenko, Sergey A.; Hur, Vera Mikyoung; Silantyev, Denis A. (January 2023, Journal of Fluid Mechanics)

   Numerically computed with high accuracy are periodic travelling waves at the free surface of a two-dimensional, infinitely deep, and constant vorticity flow of an incompressible inviscid fluid, under gravity, without the effects of surface tension. Of particular interest is the angle the fluid surface of an almost extreme wave makes with the horizontal. Numerically found are the following. (i) There is a boundary layer where the angle rises sharply from $$0^\circ$$ at the crest to a local maximum, which converges to $$30.3787\ldots ^\circ$$ , independently of the vorticity, as the amplitude increases towards that of the extreme wave, which displays a corner at the crest with a $$30^\circ$$ angle. (ii) There is an outer region where the angle descends to $$0^\circ$$ at the trough for negative vorticity, while it rises to a maximum, greater than $$30^\circ$$ , and then falls sharply to $$0^\circ$$ at the trough for large positive vorticity. (iii) There is a transition region where the angle oscillates about $$30^\circ$$ , resembling the Gibbs phenomenon. Numerical evidence suggests that the amplitude and frequency of the oscillations become independent of the vorticity as the wave profile approaches the extreme form. 

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4. Boundary layer formulations in orthogonal curvilinear coordinates for flow over wind-generated surface waves

   https://doi.org/10.1017/jfm.2020.32  

   Yousefi, K; Veron, F. (January 2020, Journal of fluid mechanics)

   null (Ed.)

   The development of the governing equations for fluid flow in a surface-following coordinate system is essential to investigate the fluid flow near an interface deformed by propagating waves. In this paper, the governing equations of fluid flow, including conservation of mass, momentum and energy balance, are derived in an orthogonal curvilinear coordinate system relevant to surface water waves. All equations are further decomposed to extract mean, wave-induced and turbulent components. The complete transformed equations include explicit extra geometric terms. For example, turbulent stress and production terms include the effects of coordinate curvature on the structure of fluid flow. Furthermore, the governing equations of motion were further simplified by considering the flow over periodic quasi-linear surface waves wherein the wavelength of the disturbance is large compared to the wave amplitude. The quasi-linear analysis is employed to express the boundary layer equations in the orthogonal wave-following curvilinear coordinates with the corresponding decomposed equations for the mean, wave and turbulent fields. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocity–vorticity formulations. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over wind-generated surface waves. Experimental results are presented in a companion paper. 

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5. Two-way wave–vortex interactions in a Lagrangian-mean shallow water model

   https://doi.org/10.1017/jfm.2022.889  

   Maitland-Davies, Cai; Bühler, Oliver (January 2023, Journal of Fluid Mechanics)

   We derive and investigate numerically a reduced model for wave–vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere–ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of ‘greedy’ waves, i.e. in the long run the wave field always extracts energy from the mean flow. 

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