varies from about 0.3U for q=0.1 to about 0.5U for q=0.3. This difference between our numerical results and the theory of [6] arises from the fact that our model includes the unsteady nonlinear evolution of the wake layer and its vorticity field.

At incipient breaking conditions, an instability appears to develop at the toe of the wave as discussed in the previous section. For a given value of q, we define a range for the wave amplitude parameter Hmax in the following manner. The lower limit of the range corresponds to the maximum initial amplitude of the gravity wave for which no instability develops at the toe of the wave during the simulation, indicating no breaking. The upper limit of the range corresponds to the minimum initial wave amplitude for which a strong instability originates at the toe of the wave, indicating a definite breaking condition. These numerically determined boundaries are plotted in Figure 9 and are in good agreement with the experimental data.

6 Conclusion

The experiments and numerical simulations reported herein indicate that, as proposed by Banner and Phillips [6], the maximum amplitudes of steady nonbreaking water waves are reduced in the presence of a surface drift layer that flows in the direction of wave propagation, and the reduction increases with increasing drift velocity. Our results, though, indicate that these maximum heights are much smaller than the ones predicted by [6] whose breaking criterion is based on the assumption of zero fluid velocity at the wave crest in the frame of reference moving with the crest. For example, for the highest surface drift velocity used herein, q=0.27, our incipient breaking height is about 60% of the one predicted by Banner and Phillips [6] and about 33% of the Stokes’ wave limiting height. Our incipient breaking heights are independent of the wake momentum thickness over the range from θ=0.145 cm to 0.210 cm (λ/θ=286 to 197, where is the wavelength from linear theory in calm water). The numerical simulations show that breaking is associated with the formation of a bulge on the forward face of the crest of the wave and a point of high upward curvature (called the toe) at the leading edge of this bulge. For large enough wave steepness, it is observed that vorticity fluctuations increase dramatically just under the surface in this region of the flow. The incipient breaking amplitudes based on whether or not this local growth of vorticity fluctuations occurs are in good agreement with the experimental data.

Through the experimental measurements of the surface drift velocity without the wave and the incipient breaking wave heights and theoretical analysis like that of Banner and Phillips (1974), it was found that the crest fluid velocity at the incipient breaking condition in the reference frame of the crest is 50% of the wave phase speed, U, relative to the water at infinite depth. Our numerical simulations, though, indicate that the fluid particle velocity at the crest for incipient breaking conditions varies from 0.3U for lower drift velocities to 0.5U for higher velocities. The difference between these values of the crest velocity and those from the experimental data/theoretical analysis mentioned above is assumed to be due to the inclusion of the unsteady nonlinear evolution of the wake layer and its vorticity field in the numerical calculations.

It is concluded, therefore, that the incipient wave height is lower than the one predicted by Banner and Phillips [6] due to the vorticity action in the toe region, which renders the use of a zero-crest-velocity criterion impractical to characterize incipient breaking conditions.


The support of the Office of Naval Research under contract N000149610368, Program Officer Dr. Edwin P.Rood, is gratefully acknowledged.


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

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

[3] Duncan, J.H. “The Breaking and Non-Breaking Resistance of a Two-Dimensional Hydrofoil” J Fluid Mech., Vol. 126, 1983, pp. 507–520

[4] Parker, F. “The Royal Navy’s Boxer,” Proceedings of the U.S. Naval Institute, Vol. 120, 3, 1994, Cover.

[5] Stokes, G.G. “On the Theory of Oscillatory Waves,” Trans. Camb. Phil. Soc., Vol. 8, 1847, pp. 441.

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

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