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distinction between these regions, shows that the surface discontinuous point does not serve as a source of vorticity, but rather as point from which the vorticity separates into a shear layer.

Case 2, the lower Reynolds and Froude number case, shows that the capillary curvature provides a negligible contribution of the gravity term to the vorticity flux, and that the contribution is dominated by the deceleration term as the flow passes through the capillaries. The net flux of vorticity into the flow is shown to be one order of magnitude smaller than that seen in case 1, which would also serve as an indication of the separating shear layer in case 2. Also, the vorticity seen beneath the capillaries is due to the free surface curvature, and while remaining at the free surface, does not flux into the flow from the free surface. The vorticity is thus confined to a region within a thickness of the order of the capillary amplitude.


This work is supported by URI research grant number N00014–92-J-1618 by the office of Naval Research. We gratefully acknowledge the insightful discussions with Dr. Longuet-Higgins, Dr. Doug Dommermuth, and Dr. Edwin Rood that lead to our insight into this topic.


1. Banner, M.L. & Peregrine, D.H., “Wave Breaking in Deep Water,” Annu. Rev. fluid Mech., Vol. 25, 1993, pp. 373–397

2. Banner, M.L. & Phillips, O.M., “On the Incipient Breaking of Small Scale Waves,” J. Fluid Mech., Vol. 65, 1974, pp. 647–656.

3. Tulin, M.P. & Cointe, R., “A Theory of Spilling Breakers,” Proc. 16th Symp. Naval Hydrodynamics, Berkeley, pp. 93–105. National Academy Press, Washington D.C., 1986

4. Peregrine, D.H. & Svendson, I.A., “Spilling breakers, bores and hydraulic jumps,” Proc. 16th Coastal Engng. Conf: ASCE, Hamburg, Germany, 1978, pp. 540–550.

5. Battjes, J.A., & Sakai, T., “Velocity Field in a Steady Breaker,” J. Fluid Mech., Vol. 111, 1981, pp. 421–437.

6. Cointe, R. & Tulin, M., “A Theory of Steady Breakers,” J. Fluid Mech., Vol. 276, 1994, pp. 1–20.

7. Duncan, J.H. “An Experimental Investigation of Breaking Waves Produced by a Towed Hydrofoil,” Proc. R. Soc. Lond. A, Vol. 377, 1981, pp. 331–348.

8. Duncan, J.H., “The Breaking and Non-breaking Wave Resistance of Two-Dimensional Hydrofoil,” J. Fluid Mech., Vol. 126, 1983, pp. 507–520.

9. Duncan, J.H. & Philomin, V., “The Formation of Spilling Breaking Water Waves,” Phys. Fluids 6, Vol. 8, 1994, pp. 2558–2560.

10. Lin, J.C. & Rockwell, D., “Instantaneous Structure of a Breaking Wave,” Phys. Fluids, Vol. 6, 1994, pp. 2877–2879.

11. Lin, J.C. & Rockwell, D., “Evolution of a Quasi-Steady Breaking Wave,” J. Fluid Mech., Vol. 302 , 1995, pp. 29–44.

12. Longuet-Higgins, M.S., “Capillary Rollers and Bores.” J. Fluid Mech., Vol. 240, 1992, pp. 659–679.

13. Hornung, H.G.; Willert C.E. & Turner, S., “The Flow Field Downstream of a Hydraulic Jump,” J. Fluid Mech., Vol. 287, 1995, pp. 299–316.

14. Willert, C.E.; & Gharib, M., “Digital Particle Image Velocimetry,” Exp. Fluids, Vol. 10, 1991, pp. 181–193.

15. Cox, C.S., “Measurements of Slopes of High-Frequency Wind Waves,” J. Marine Res., Vol. 16, 1958, pp. 199–225.

16. Longuet-Higgins, M.S., “The Generation of Capillary Waves by Steep Gravity Waves,” J. Fluid Mech., Vol. 16, 1963, pp. 138–159.

17. Rood, E.P., “Interpreting Vortex Interactions with a Free Surface,” Trans. ASME I: J. Fluid Engng., Vol. 116, No. 1, 1994a, pp. 91–94.

18. Rood, E.P., “Free Surface Vorticity,” chapter 17 in Fluid Vortices, S.Green (ed.), Kluwer Academic Publishing, Norwell, MA 1993, in review.

19. Gharib, M. & Weigand, A., “Experimental Studies of Vortex Disconnection and Connection at a Free Surface,” Submitted to J. Fluid Mech., 1995

20. Longuet-Higgins, M.S., “Shear Instability in Spilling Breakers,” Proc. R. Soc. Lond. A, Vol. 446, 1994, pp. 399–409.

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