Abstract
The modulational instability of two-dimensional nonlinear traveling-wave solutions of the Whitham equation in the presence of constant vorticity is considered. It is shown that vorticity has a significant effect on the growth rate of the perturbations and on the range of unstable wavenumbers. Waves with kh greater than a critical value, where k is the wavenumber of the solution and h is the fluid depth, are modulationally unstable. This critical value decreases as the vorticity increases. Additionally, it is found that waves with large enough amplitude are always unstable, regardless of wavelength, fluid depth, and strength of vorticity. Furthermore, these new results are in qualitative agreement with those obtained by considering fully nonlinear solutions of the water-wave equations.
Highlights
It is well known that small-amplitude, two-dimensional, periodic wave trains are stable with respect to the modulational instability when the dispersive parameter kh, where k is the wavenumber and h is the mean fluid depth, is less than the critical value 1.363
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
The Whitham equation is an extension of the KdV equation which takes into account the full range of dispersion
Summary
It is well known that small-amplitude, two-dimensional, periodic wave trains are stable with respect to the modulational instability when the dispersive parameter kh, where k is the wavenumber and h is the mean fluid depth, is less than the critical value 1.363. 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.
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