Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

James N. Steer,Alistair G. L. Borthwick,Mark L. McAllister,Ton S. van den Bremer

doi:10.3390/fluids4020105

James N. Steer, Alistair G. L. Borthwick + Show 2 more

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https://doi.org/10.3390/fluids4020105

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Abstract

The coupled nonlinear Schrödinger equation (CNLSE) is a wave envelope evolution equation applicable to two crossing, narrow-banded wave systems. Modulational instability (MI), a feature of the nonlinear Schrödinger wave equation, is characterized (to first order) by an exponential growth of sideband components and the formation of distinct wave pulses, often containing extreme waves. Linear stability analysis of the CNLSE shows the effect of crossing angle, θ , on MI, and reveals instabilities between 0 ∘ < θ < 35 ∘ , 46 ∘ < θ < 143 ∘ , and 145 ∘ < θ < 180 ∘ . Herein, the modulational stability of crossing wavetrains seeded with symmetrical sidebands is determined experimentally from tests in a circular wave basin. Experiments were carried out at 12 crossing angles between 0 ∘ ≤ θ ≤ 88 ∘ , and strong unidirectional sideband growth was observed. This growth reduced significantly at angles beyond θ ≈ 20 ∘ , reaching complete stability at θ = 30–40 ∘ . We find satisfactory agreement between numerical predictions (using a time-marching CNLSE solver) and experimental measurements for all crossing angles.

Highlights

Crossing seas, in which waves travel in multiple directions, have been identified as an important challenge to offshore operations, linked to an increased probability of extreme waves [1,2]
We contribute to the understanding of extreme waves in crossing seas by reporting on an experimental study of modulational instability in waves crossing at angles between 0◦ ≤ θ ≤ 88◦
In doing so we identify the region over which the finite-crest effect does not play a role and we can exclusively examine modulational instability

Summary

Introduction

In which waves travel in multiple directions, have been identified as an important challenge to offshore operations, linked to an increased probability of extreme waves [1,2]. We contribute to the understanding of extreme waves in crossing seas by reporting on an experimental study of modulational instability in waves crossing at angles between 0◦ ≤ θ ≤ 88◦. For long-crested or unidirectional seas, it is well established that weakly nonlinear regular wavetrains in sufficiently deep water rapidly evolve into pulses of wave groups through modulational instability (MI) [7,8]. Breather waves are characterized by a sudden increase in amplitude of initially regular waves to either three or five times their initial value [11,12], and provide close approximations to rogue waves in long-crested seas. Breather waves are sensitive to initial conditions, which must be specified precisely for the waves

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