Pressure induced breather overturning on deep water: Exact solution
Introduction
There is currently a considerable interest in understanding the mechanisms underlying the formation and breaking of strongly nonlinear waves such as solitons, breathers and rogue waves. This problem was initially considered in hydrodynamics and was later extended to other fields of physics [1], [2], [3], [4], [5], [6], [7].
Overturning of surface gravity waves on water is one of the brightest examples of their nonlinear behavior. Two types of breaking waves: spilling and plunging breakers may be distinguished [8], [9]. The spilling breaker gets steeper slowly. As its steepness amounts to some critical value, the water at the crest begins to slide (“spill”) down the wave front, reducing the wave height. So the wave steepness doesn't exceed this critical number. The plunging breaker becomes steep very quickly. Its final steepness is greater than of the spilling breaker. The top of the plunging wave gets far ahead of the bottom of the wave, thus forming an arc profile. Mathematically this corresponds to appearance of a vertical tangent of a free surface profile, which is the basic difference between the two wave types. We will be interested in the plunging breakers.
This phenomenon has been well studied in the shallow water approximation [10], [11]. The deep water situation is less clear. Full details of the wave profile, velocity and pressure fields in the overturning waves were obtained from the comprehensive numerical computations [12], [13], [14]. But there are some analytical results too. Using the semi-Lagrangian approach proposed by John [15], Longuet-Higgins [16], [17] and New [18] developed simple analytical models of the underside or the loop of a plunging breaking wave. Longuet-Higgins suggested that a solution for a rotating hyperbola falling under gravity could represent the motion of the jet of fluid ejected from the top of the breaking wave. New found that the free surface underneath the jet is described well by an ellipse, and obtained unsteady solutions for a zero-gravity flow around an ellipse. Greenhow [19] showed that both the jet of fluid from the top of a breaking wave and the ellipse model describing a loop are, for large times, complementary solutions of the same free surface equation. That, in turn, suggested solutions which combined both the jet and the loop and enabled obtaining a much more complete model of the entire overturning region not too far from the wave crest.
All those theoretical studies were performed for potential flows. The pressure on the free boundary was considered to be constant. But most waves in the ocean occur under storm conditions, where wind action must be taken into account [20]. Kharif et al. considered the wind action as a linear equation relating the pressure and the steepness of the wave profile [21]; i.e., the wind action was simulated by a nonuniform pressure distribution.
We present an exact solution of 2D hydrodynamical equations that describes overturning of a breather on deep water. The solution is written in Lagrangian variables and belongs to the class of Ptolemaic flows [22], [23]. The fluid particles rotate in circles of different radii and there is no drift current. The pressure on the free surface is nonuniform. The breather dynamics and pressure are studied. Unlike other models, the analyzed wave motion is vortical. The vorticity is maximal in the neighborhood of the overturning slope.
Section snippets
Ptolemaic flows
The equations of 2D hydrodynamics for waves on the surface of an incompressible inviscid fluid in Lagrangian coordinates are written in the form [22], [23]: where X, Y are Cartesian coordinates and a, b are Lagrangian coordinates of fluid particles, D denotes Jacobian, t is time, ρ is fluid density, p is pressure, g is acceleration of gravity, the subscripts mean differentiation by the corresponding variable, and
Breather solution
Let us consider the following solution where β is a real value, , ω is the frequency of oscillations. The functions F, G have a pole at the point which is situated outside the flow region. The solution is dimensionless. Values of W, χ are normalized using the lengthscale α, the parameter β is normalized to . The expression (10) describes the standing wave structure localized in space and “breathing” in time. We shall call it a breather. The properties of
Overturning condition and mechanism
The overturning regime is realized in a definite region of β values. The overturning of the breather is connected with the existence of a vertical tangent to the breather profile. This condition can be written as follows Since a Ptolemaic flow is periodic, the wave overturning is also repeated periodically. The conditions (12) are equivalent to the inequality for some values of the Lagrange coordinate a. In a particular case when the function G is linear, the
Conclusion
The solution describing a standing breather on deep water was obtained in the class of Ptolemaic flows (6). This breather is created and evolves due to nonuniform pressure acting on the surface of the liquid. The effect of breather overturning and the impact of liquid vorticity on this process were studied in detail.
In the mathematical analysis of free boundary dynamics, we chose the functions G and F in (6) in the simplest possible form so that the conditions of breather existence could be
Acknowledgements
The authors have the pleasure to thank Prof. E.M. Gromov for numerous stimulating discussions.
References (28)
- et al.
Phys. Lett. A
(2013) - et al.
Phys. Lett. A
(2013) - et al.
Phys. Lett. A
(2013) - et al.
Phys. Lett. A
(2014) - et al.
Phys. Lett. A
(2014) - et al.
Phys. Lett. A
(2014) - et al.
Phys. Lett. A
(2013) - et al.
Annu. Rev. Fluid Mech.
(1993) - et al.
J. Fluid Mech.
(2007) Annu. Rev. Fluid Mech.
(1983)