Stagnation Points Beneath Rotational Solitary Waves in Gravity-Capillary Flows

M. V. Flamarion


Stagnation points beneath solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A Korteweg-de Vries equation that incorporates the surface tension and the vorticity effects is obtained asymptotically from the full Euler equations. The velocity field in the bulk fluid is approximated which allow us to compute stagnation points in the solitary wave moving frame. We show that stagnation points bellow the crest of elevation solitary waves exist for large values of the vorticity and Bond numbers less than a critical value that depends on the vorticity. Remarkably, the positions of these stagnation points do not depend on the surface tension. Besides, we show that when there are two stagnation points located at the bottom of the channel, they are pulled towards the horizontal coordinate of the solitary wave crest as the Bond number increases until its critical value.


Gravity-capillary waves; Euler equations; KdV equation; stagnation points

Full Text:



Stokes GG. On the theory of oscillatory waves. Trans Cambridge Phil Soc. 1847; 8:441-455.

Ursell F. Mass transport in gravity waves. Proc Cambridge Phil Soc. 1953; 40:145-150.

Constantin A, Villari G Particle trajectories in linear water waves. J Math Fluid Mech. 2008; 10:1336-1344.

Constantin A, Strauss W Pressure beneath a Stokes wave. Comm Pure Appl Math. 2010; 63:533-557.

Nachbin A, Ribeiro-Jr R A boundary integral method formulation for particle trajectories in Stokes Waves. DCDS-A. 2014; 34(8):3135-3153.

Borluk H, Kalisch H Particle dynamics in the kdv approximation. Wave Mo- tion. 2012; 49:691-709.

Alfatih A, Kalisch H Reconstruction of the pressure in long-wave models with constant vorticity. Eur J Mech B-Fluid. 2013; 37:187-194.

Gagnon L Qualitative description of the particle trajectories for n-solitons so- lution of the korteweg-de vries equation. Discrete Contin Dyn Syst. 2017; 37:1489-1507.

Guan, X Particle trajectories under interactions between solitary waves and a linear shear current. Theor App Mech Lett. 2020;10:125-131.

Khorsand, Z Particle trajectories in the Serre equations. Appl Math Comput. 2020;230:35-42.

Curtis C, Carter J, Kalisch H Particle paths in nonlinear schrödinger models in the presence of linear shear currents. J. Fluid Mech. 2018;855:322-350.

Carter J, Curtis C, Kalisch H Particle trajectories in nonlinear Schrödinger models. Water Waves. 2020;2:31-57.

Teles Da Silva AF, Peregrine DH Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 1988;195:281-302.

Ribeiro-Jr R, Milewski PA, Nachbin A Flow structure beneath rotational water waves with stagnation points. J. Fluid Mech. 2017;812:792-814.

Flamarion MV, Nachbin A, Ribeiro-Jr R Time-dependent Kelvin cat-eye struc- ture due to current-topography interaction. J. Fluid Mech. 2020;889:A11.

Johnson RS On the nonlinear critical layer below a nonlinear unsteady surface wave. J. Fluid Mech. 1986;167:327-351.

Martin CI Equatorial wind waves with capillary effects and stagnation points. Nonlinear Anal-Theor. 2014;96:1-8.

Hur V, Wheeler M Exact free surfaces in constant vorticity flows. J. Fluid Mech. 2020;896:R1.

Shoji M, Okamoto H Stationary water waves on rotational flows of two vortical layers. Jpn J Ind Appl Math. 2021;38:79-103.

G. B. Whitham (1974) Linear and Nonlinear Waves, Wiley.


Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM


  • There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)


Indexed in:




Desenvolvido por:

Logomarca da Lepidus Tecnologia