Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

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considering only the initial stages of modulational instability, before breaking takes place. 48 Although extensively studied both theoretically and experimentally in one dimension, the 49 applicability of the 1D+1 NLSE to the open ocean is limited by the equation's unidirectionality. 50 In the open ocean, waves may be created from multiple sources, interact, and cross at an angle. 51 Additionally, in fetch-limited seas it has been observed that spectral components above and below 52 the peak frequency become bimodal with energy naturally spreading symmetrically to angles above 53 and below that of the peak frequency direction [22,23]. As derived for deep-water by Onorato et al. 54 [24] from the 2D+1 Zakharov equation [25], the coupled nonlinear Schrödinger equation (CNLSE) 55 is a system of nonlinear wave equations describing the interaction of two narrow-banded weakly 56 nonlinear wave systems propagating at an angle (see also [26]). This deep-water CNLSE has since been 57 extended to finite depth by Kundu [29,30]. Within this paper, crossing angle, θ is the angle at which waves 66 propagate to the x-axis, i.e. when two waves cross at ±θ the angle of bisection is 2θ. Along with the 67 general investigation into plane wave stability, rogue wave solutions to the CNLSE are known to exist 68 and have been classified and, through numerical computations, compared to their 1D+1 analogue, the 69 Peregrine breather [31].  In addition to possible MI, changes to the second-order bound waves occur when waves cross.

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The wave-averaged free surface, represented spectrally by second-order difference waves, is the local 80 mean surface elevation formed by temporal averaging over the rapidly varying waves that make 81 up the slowly varying group. Whereas a set-down of the wave-averaged free surface is expected in 82 the absence of crossing, packets are accompanied by a set-up for sufficiently large crossing angles. In this paper, we report on regular wave experiments with seeded sidebands for two crossing  This paper is laid out as follows. First, §2 reviews the theoretical background, followed by an  , is a narrow-banded wave equation describing the evolution of coupled, complex wave envelopes A and B. Both wave envelopes propagate on an associated carrier wave whose properties define the CNLSE coefficients and thus (along with the initial conditions) the envelope evolution. Scaled for water waves, and under the assumption of identical but symmetrical carrier waves (about the x-axis) with distinct amplitude envelopes, the CNLSE is given, in a Cartesian coordinate system (x, y, t), by [24], where carrier properties: frequency, ω 0 ; x-axis wavenumber, k; y-axis wavenumber, l; and absolute wavenumber, k 0 = √ k 2 + l 2 , define the group velocities C x and C y along their respective axes, the linear coefficients α, β, and γ are given by, and the nonlinear coefficients ξ and ζ by, The carrier frequency ω 0 and absolute wavenumber k 0 are related through the deep-water dispersion relation, ω 0 = k 0 g, with g denoting the gravitational constant.
In the special case of envelopes propagating along the x-axis, a Galilean transformation into the group reference frame reduces the CNLSE to [24], where X = x − C x t. From the wave packet amplitudes, the (linear) free surface elevation is reconstructed by reintroducing the carrier waves through,

Linear stability analysis
108 Linear stability analysis of the CNLSE reveals many properties of the equation and, using a seeded carrier solution, allows prediction of the initial sideband growth rate. Identical plane waves are admitted as solutions to (6-7) and we therefore add perturbations of infinitesimal amplitude and phase to obtain (see also [24]), where a 0 and b 0 are carrier amplitudes, and δ a , δ b , δφ a , and δφ b are small perturbations in amplitude and phase. In this linear stability analysis, the assumed form of the sideband solutions a δ and b δ is, where a δ,0 and b δ,0 are the initial sideband amplitudes, K is the perturbation wavenumber, and Ω is the perturbation frequency. The relationship between K and Ω is found through linear stability analysis as [24], where it is apparent that Ω may take either real or imaginary values. Following substitution of this 109 relationship into (10), either oscillatory (when Ω ∈ Re) or exponential (when Ω ∈ Im) behaviour can 110 be expected from the sidebands.  in water waves, energy may be lost to wave breaking resulting in a nonconservative system but we 144 note that FPU recurrence is a long-term behaviour, and strong MI is required to observe it in the space 145 available in most experimental facilities. simple recurrence. Complex recurrence is expected when K lies less than (or at) half-way through 148 the instability region (0 < K ≤ K c /2), and primary sidebands themselves act as unstable carriers, 149 continually spawning new sidebands. When K lies more than half way to the stability boundary 150 (K c /2 < K < K c ) new sidebands will lie in the stable region, and simple recurrence is observed.  The experimental campaign was split into two parts. Part I aimed to quantify the effect of finite-length crests in the facility in the absence of seeded sidebands, which is a manifestation of the inability of a finite number of wavemakers encircling a finite-size circular basin to create perfectly long-crested waves spanning the entire basin diameter. This finite-crest effect needed to be quantified in order to estimate the length over which components travelling with different directions would interact. Part II aimed to measure the growth of frequency sidebands about carrier waves travelling at crossing angles ±θ. Crossing carrier and sideband waves only interact fully in regions of total crest overlap, and so the extent that these regions cover the chosen wave gauge locations is defined by the carrier crest length and angle. Experiments 1a-d (Part I) were therefore designed to determine the effective sideband evolution region in the basin at each angle. In these experiments, a single unseeded carrier wave was propagated at the angles given in Table 1 (Part I).
For Part I, the amplitude profiles of experiments 1a-d are presented in Figure 3c and allow estimation of the carrier crest length in the FloWave facility. Experiment 1d (θ = 90 • ) shows that, for high angle experiments, a reasonable region in which to expect full sideband-carrier interactions occupies approximately 10 wavelengths centred about the basin origin. However, the effective length is extended significantly to more than 20 wavelengths for crossing angles up to 30 • , the region of greatest interest in Part II. As expected, for waves in the x-direction (θ = 0 • ), the region covers all wave gauge locations. The results from the Part I tests were interpolated in order to estimate the finite-crest effect at all crossing angles.
All experiments in Part II were performed with constant values of carrier frequency, f 0 = 1.5 Hz, carrier amplitudes a 0 = b 0 = 0.018 m, and initial sideband amplitude a δ,0 = 0.003 m, giving a depth parameter k 0 d = 18, and steepness of a single carrier, k 0 a 0 = 0.16. Figure 1a shows the expected growth rates, crossing angles, and sideband wavenumbers for the Part II tests. A simple system of four plane waves, consisting of two carrier waves propagating at ±θ to the x-axis, and two sidebands propagating along the x-axis was used as input to the wave generation software. Explicitly, we thus have, where x 0 is the x-position of the wavemaker along y = 0 (the axis of propagation of the sidebands).

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The relatively high carrier frequency was chosen to slow group velocity, increasing the effective 168 evolution distance. The carrier amplitude was subsequently calculated to give a moderate steepness of 169 k 0 a 0 = 0.16, required for prominent instability but to avoid breaking. Each experiment was repeated 3 170 times.   The numerical solution in Figure 6a also shows significant growth and follows the average of the 209 upper and lower sideband amplitudes well, displaying many of the same characteristics (such as FPU 210 recurrence). However, the lower sideband grows much more quickly than the upper sideband, which 211 is subject to initial growth followed by considerable attenuation, a feature not predicted by the NLSE    The split-step method (also known as the Fourier method) takes advantage of the fact that the 260 linear and nonlinear components can be separated and then solved exactly [56].  The split-step method is second-order accurate in ∆t and to all orders in ∆x, it is unconditionally stable 266 [57].

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First, the CNLSE is rearranged and split into its linear and nonlinear components (here only (6) is considered for brevity), The nonlinear component is integrated forwards in the time domain as follows, whereas the linear component is Fourier-transformed, and then integrated in time to give,Â i+1 =Â i e −i∆tαk 2 . (A5) Combining the linear and nonlinear components, at each time step we have the explicit expression, The same process is applied to 7. The results of advancing A and B individually are combined in the 268 current time step to give the full system state to be passed to the next step.   Figure A3. Comparison of the evolution of sideband amplitude along the centreline of the basin for experiments 2i-l (Part II) from measurements, numerical solutions (crosses) of the CNLSE (thin blue and red lines) and linear stability analysis (thin black lines). Lower and upper sidebands are indicated in red and blue, respectively. Error bars and dashed lines represent one standard deviation from the mean across repeats for the measured data and the CNLSE solution, respectively. Thick lines represent carrier wave amplitudes from the seeded (Part II, dark grey) and unseeded (Part I, light grey) experiments.