Self-localized solitons of the nonlinear wave blocking problem

https://doi.org/10.1016/j.dynatmoce.2020.101189Get rights and content

Highlights

  • A numerical framework is introduced for analyzing the nonlinear soliton blocking.

  • A spectral renormalization method is proposed for finding solitons of the Smith’s NLSE.

  • Dynamics and stabilities of the nonlinearly blocked solitons are analyzed for various current velocity profiles.

Abstract

In this paper, we propose a numerical framework to study the shapes, dynamics, and stabilities of the self-localized solutions of the nonlinear wave blocking problem. To our best knowledge, blocking of the solitons in the marine environment has not been studied before in the existing literature. With this motivation, we use the nonlinear Schrödinger equation (NLSE) derived by Smith as a model for the nonlinear wave blocking. We propose a spectral renormalization method (SRM) to find the self-localized solitons of this model. We show that for constant, linearly varying or sinusoidal current gradient, i.e. dUdx, the self-localized solitons of the Smith’s NLSE do exist. Additionally, we propose a spectral scheme with a 4th order Runge–Kutta time integrator to study the temporal dynamics and stabilities of such solitons. We observe that self-localized solitons are stable for the cases of constant or linearly varying current gradient however, they are unstable for sinusoidal current gradient, at least for the selected parameters. We comment on our findings and discuss the importance and applicability of the proposed approach.

Introduction

The phenomenon of wave blocking can be described as the stopping of the propagating waves by opposing ocean currents and circulations. The celerity of waves propagating into the opposing oceanic currents and circulations reduces and if the opposing current is strong enough, then they may be blocked, that is the group velocity of the wavefield becomes zero. This phenomenon is observed in nature. One of the most common places of their observation is tidal inlets where tidal currents are strong enough to block the propagating waves. Due to the sharp increase in wave steepness during wave blocking processes, the wave fields tend to exhibit higher waves which may eventually become rogue waves and may break. Therefore, wave blocking may result in catastrophic navigational hazards in the marine environment. The boats have been known to capsize during crossing inlets, where wave blocking phenomena are observed. Additionally, due to blocking of the waves by Agulhas current, the South Africa off coast is known as one of the most dangerous seas of the earth (Kharif and Pelinovsky, 2003).

With these motivations, scientists have spent considerable effort to study the wave blocking problem using alternative approaches. Spectral modeling of current-induced wave blocking is given in Ris and Holthuijsen (1996). Reflection of short gravity waves on a non-uniform current is studied in Smith (1975). The blockage of gravity and capillary waves by longer waves and currents are analyzed in Shyu and Phillips (1990). Reflection of oblique waves by currents is analyzed analytically and numerically in Shyu and Tung (1999). Using the wave focusing and hydraulic jump formation approach discussed in Chin, 1979, Chin, 1980a, Chin, 1980b, the kinematic barrier for gravity waves on variable current is discussed in Chin (1981). Monochromatic and random wave breaking at blocking points is studied in Chawla and Kirby (2002). A model for blocking of periodic waves is proposed in Suastika (2004) and Suastika and Battjes (2009). Additionally, some of the experimental studies on wave blocking can be seen in Lai et al. (1989) and Pokazayev and Rozenberg (1983) and Chawla and Kirby (1998). While the majority of these studies analyze wave blocking phenomena within the framework of linear theory, some studies utilizing the nonlinear theory also exist (Smith, 1976, Peregrine, 1976, Chawla and Kirby, 2005, Bayındır, 2016a).

However, to our best knowledge, the problem of soliton blocking in the marine environment has not been investigated in the existing literature. With this motivation, in this paper, we propose and follow a methodology for analyzing the nonlinear soliton blocking in the marine environment. We use an extended version of the nonlinear Schrödinger equation (NLSE), first derived in Smith (1976) as a nonlinear wave–current interaction model. This equation is known as Smith’s NLSE. We develop a spectral renormalization method (SRM) for the numerical solutions of Smith’s NLSE which can be implemented for arbitrary current gradient forms, i.e. dUdx. The proposed SRM can be used to find the self-localized solitons starting from an arbitrary wave profile. We show that self-localized solutions of the nonlinear wave blocking problem can be found using the approach proposed in this paper. Additionally, we perform the stability analysis for the solitons blocked by the oceanic current. The stability analysis includes an assessment of the Vakhitov-Kolokolov slope condition and the time stepping of the solitons found by SRM. We implement a Fourier spectral scheme with 4th order Runge–Kutta time integrator to analyze the temporal dynamics and stabilities of the self-localized solitons, for the time stepping part of the stability problem. We show that self-localized solitons are stable for the cases of the constant or linearly varying current gradient, at least for selected parameters. However, our results also indicate that they are unstable for sinusoidal current gradient, dUdx, at least for the parameters considered. We discuss the uses, applicability, and limitations of the proposed approach and comment on our findings.

Section snippets

Review of the nonlinear Schrödinger equation for wave blocking

Various models have been proposed to study the wave blocking phenomena. In the majority of these studies, wave blocking is studied within the frame of the linear theory. However, there are some studies which utilize nonlinear theories with this aim (Smith, 1976, Peregrine, 1976, Chawla and Kirby, 2005, Bayındır, 2016a). Smith (1976) derived the formula iψtω8k2ψxxωk22|ψ|2ψ+k|dUdx|xψ=0to study the nonlinear wave field under the effect of wave blocking. This formula is valid in the vicinity of

Self-localized solitons for constant current velocity gradient, dUdx= constant case

In this section, we consider the case of constant current velocity gradient and select dUdx=0.1s1 in our simulations. The initial condition for this case is selected to be a Gaussian in the form of exp((xx0)2), where x0 is taken as 0. We define the convergence as the normalized change of α to be less than 10−7 in our iterations. In Fig. 1, we plot the self-localized single soliton power as a function of soliton eigenvalue, which is obtained by using the SRM summarized above for different

Conclusion and future work

In this paper, we have proposed a numerical framework to study self-localized solutions of Smith’s NLSE derived for modeling the nonlinear wave blocking phenomena. The method we have proposed is a spectral renormalization method for Smith’s NLSE and we have shown that self-localized solitons of this model equation can be found for (i) constant current velocity gradient, (ii) linearly varying current velocity gradient, and (iii) sinusoidally varying current velocity gradient cases by our

CRediT authorship contribution statement

Cihan Bayındır: Conceptualization, Methodology, Software, Validation, Writing - original draft, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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  • Cited by (0)

    1

    Associate Professor at İstanbul Technical University.

    2

    Adjunct Professor at Boğaziçi University.

    3

    International Collaboration Board Member at CERN.

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