Nonlinear water waves in shallow water in the presence of constant vorticity: A Whitham approach

Abstract

Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface elevation and the horizontal velocity are derived. Using Riemann invariants of these equations, that are obtained analytically, a closed-form nonlinear evolution equation for the surface elevation is derived. A dispersive term is added to this equation using the exact linear dispersion relation. With this new single first-order partial differential equation, vorticity effects on undular bores are studied. Within the framework of weakly nonlinear waves, a KdV-type equation and a Whitham equation with constant vorticity are derived from this new model and the effect of vorticity on solitary waves and periodic waves is considered. Futhermore, within the framework of the new model and the Whitham equation a study of the effect of vorticity on the breaking time of dispersive waves and hyperbolic waves as well is carried out.

Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

“…values of the wave steepness, c 0 increases as increases. This feature was observed by Kharif and Abid [9] within the framework of a fully nonlinear, generalized vor-Whitham equation, whereas the profiles shown in Fig. 1 are weakly nonlinear.…”

“…To compute these solutions, we use the numerical method of Ehrnström and Kalisch [7]. Details of the numerical method and its validation are found in Kharif and Abid [9]. However, herein we add a supplementary equation that fixes the wave amplitude when following solutions using c 0 as the continuation parameter.…”

“…Ehrnström and Kalisch [7,8] proved rigorously that the Whitham equation admits small and large amplitude, periodic traveling-wave solutions and numerically computed traveling-wave solutions with a variety of amplitudes including those close to the highest wave. Later on, Kharif and Abid [9] computed steadily propagating periodic waves in the presence of constant vorticity. The method of computing these solutions was developed by Ehrnström and Kalisch [8] and is also found in Sanford et al [10] and Kharif and Abid [9].…”

“…Later on, Kharif and Abid [9] computed steadily propagating periodic waves in the presence of constant vorticity. The method of computing these solutions was developed by Ehrnström and Kalisch [8] and is also found in Sanford et al [10] and Kharif and Abid [9]. Sanford et al [10] studied the Whitham equation and found that two-dimensional, periodic wave trains with kh = 1 are stable when the wave steepness, ak, is less than approximately 0.142 and are unstable when the wave steepness is larger than this threshold.…”

“…Later on, Hur and Johnson [14] incorporated in the Whitham equation the effect of constant vorticity which modifies the threshold value of the dispersive parameter. Following Whitham [3], Kharif et al [15] and Kharif and Abid [9] proposed a new model derived from the Euler equations for fully nonlinear water waves propagating on a vertically sheared current of constant vorticity in shallow water that satisfies the unidirectional linear dispersion relation. From this model they derived, within the framework of weakly nonlinear waves, a generalization of the Whitham equation which they named the vor-Whitham equation.…”