Effect of vorticity on the generation of rogue waves due to dispersive focusing

Abstract

The kinematic and dynamic of steep two-dimensional focusing wave trains on a shearing flow in deep water are investigated analytically and numerically. In the absence of waves, the vorticity due to the vertical gradient of the horizontal current velocity is assumed constant. A linear kinematic model based on the spatio-temporal evolution of the frequency is derived, predicting the focusing distance and time of a chirped wave packet in the presence of constant vorticity. Furthermore, a linear model, based on a Fourier integral, is used to describe the evolution of the free surface on shearing current. To compute the fully nonlinear evolution of the wave group in the presence of vorticity, a new numerical model, based on a BIEM approach, is developed. On the basis of these different approaches, the role of constant vorticity on rogue wave occurrence is analysed. Two main resultsare obtained: (1) the linear behaviour expected in the presence of constant vorticity is significantly different from what is commonly expected in the presence of constant current and (2) the nonlinear effects are found to be of significant influence in the case at hand

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

“…The wavelength of the following waves is also modified by vortical effects, it is shortened in the presence of negative vorticity and stretched for positive vorticity. Within the framework of deep water, Touboul and Kharif [14] using a Boundary Integral Element Method found similar results. An investigation on the breaking time of dispersive waves in the presence of constant vorticity has been carried out within the framework of the Whitham equation and generalised Whitham equation, respectively.…”

“…Substituting this expression into equation (11) gives (19) is fully nonlinear and describes the spatio-temporal evolution of hyperbolic water waves in shallow water in the presence of constant vorticity. This equation is equivalent to the system of equations (11) and (14) for waves moving rightwards.…”

Focusing Wave Group Propagating in Finite Depth in the Presence of Surface Current and Vorticity

“…Among them, one can cite Tsao [33], Dalrymple [8], Brevik [3], Simmen [24], Simmen and Saffman [25] Teles da Silva and Peregrine [26], Kishida and Sobey [19], Pak and Chow [22], Constantin [7], etc. Thus, Touboul and Kharif [32] extended their previous study [28] to this more realistic case of water waves propagating in the presence of vorticity.…”

“…where σ min and σ max are the intrinsic, Doppler shifted, frequencies, respectively given by σ min (ω min − k min U 0 ) √ gk min , and σ max (ω max − k max U 0 ) √ gk max . In a recent work, Touboul and Kharif [32] investigated the evolution of a chirped wave packet in the presence of a horizontally constant current presenting linear variations with respect to depth, so that nizar.abcha@unicaen.fr…”

Evolution of Spectral Distributions in Deep-Water Constant Vorticity Flows

“…Therefore, while a relatively idealized case of shear, the interplay between nonlinearity and constant vorticity can manifest in markedly different physics. Building then on the work in [4] and [17], and complementing the results in [18], we numerically explore the statistical properties of random-nonlinear-wave fields moving over deep water and linear shear profiles. Moving beyond just examining the properties of the NLSE with vorticity, we look at the statistical properties of solutions to a higher order model, the vor-Dysthe equation (VDE) derived in [17].…”