Evolution of Spectral Distributions in Deep-Water Constant Vorticity Flows
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Evolution of Spectral Distributions in Deep-Water Constant Vorticity Flows Christopher W. Curtis1
· Mackensie Murphy1
Received: 27 September 2019 / Accepted: 30 March 2020 © Springer Nature Switzerland AG 2020
Abstract A central question in sea-state modeling is the role that various physical effects have on the evolution of the statistical properties of random sea states. This becomes a critical issue when one is concerned with the likelihood of rare events such as rogue, or freak, waves which can have significant destructive potential on deep sea ships and other offshore structures. In this paper then, using a recently derived higher-order model of deep water nonlinear waves, we examine the impact of constant vorticity currents on the statistical properties of nonlinearly evolving random sea states. As we show, these currents can both decrease and increase the kurtosis of the affiliated distributions of the sea states, thereby diminishing or enhancing the likelihood of rare events. We likewise numerically study the relationship between the kurtosis and a non-dimensional parameter, the Benjamin–Feir Index, which has proven to be a useful measure of when rare events are likely in oceanographic application. Keywords Water waves · Modulational instability · Shear currents · Dysthe equation · Random sea states
1 Introduction There is now a wide range of literature which shows that nonlinear instabilities, in particular the modulational instability (MI), are responsible for significant modifications to the statistical properties of water waves; see [1–4] among others. As explained in [5], MI is especially important due to the relative rapidity with which it acts, being orders of magnitude faster than say four-wave resonant interactions which form the traditional backbone of our understanding of how nonlinearity drives changes in statistics of random-wave fields [6,7]. A detailed set of theoretical and experimental results [8–11] have also shown that MI could play a significant role in unidirectional deepwater-rogue wave formation among other oceanographic phenomena. It is important
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Christopher W. Curtis [email protected] San Diego State University, San Diego, CA, USA
C. W. Curtis, M. Murphy
to note though that there are several possible mechanisms for rogue-wave generation; see [12], for example, which rules out MI as a mechanism altogether in certain oceanic conditions. However, much of the understanding around MI relies on highly idealized assumptions which cannot be expected to hold in natural settings. In particular, most analytic understandings of MI rely on looking at the stability of perturbations to carrier waves at essentially a fixed wave number. To address this shortcoming, in now seminal papers, [13] and [14] analytically studied families of perturbations of wave packets with narrow, but non-zero, spectral width around a central wave number. Using the Nonlinear Schrödinger equation (NLSE), an analytic criterion determining when MI is either manifested or suppressed depending on this spectral wid
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