DYNAMICALLY ORTHOGONAL REDUCED-ORDER MODELING OF STOCHASTIC NONLINEAR WATER WAVES

Transcription

1 DYNAMICALLY ORTHOGONAL REDUCED-ORDER MODELING OF STOCHASTIC NONLINEAR WATER WAVES Saviz Mowlavi Supervisors Prof. Themistoklis Sapsis (MIT) & Prof. François Gallaire (EPFL) Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the Ecole Polytechnique Fédérale de Lausanne February 215

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3 Acknowledgements Working under the guidance of Prof. Themistoklis Sapsis during these past six months at MIT has been an immense pleasure. His scientific advice, his open-mindedness, his availability and his endless patience, all contributed to making my research here a very enjoyable experience. I warmly thank him for all these reasons, and look forward to starting my Ph.D. under his supervision. It is through the passionate lectures of Prof. François Gallaire that I started developing a profound interest for the fascinating field of fluid mechanics, and it is under his supervision that I first entered the research world. He will long remain a source of inspiration for me. I owe him a big debt of gratitude and thank him for always being so helpful and supportive in my decisions. I would also like to thank Will Cousins who has always been willing to spend all his time answering my questions, my lab mates at SandLab for great company and my friends for providing me with refreshing moments of escape from work. Finally, words alone cannot express the gratitude that I feel towards my family for their unconditional love and support in all circumstances. I feel very lucky to have them as a family and they are the most precious thing I have in this world. i

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5 Abstract In this thesis, we implement a reduced-order framework for the stochastic evolution of nonlinear water waves governed by the nonlinear Schrördinger (NLS) equation and subject to random initial conditions. Our reduced-order model is based on the dynamically orthogonal (DO) equations introduced by Sapsis & Lermusiaux (29), and consists in the expansion of the stochastic solution on a few time-dependent deterministic modes that capture the subspace where the dominant stochastic fluctuations reside, while an associated set of stochastic coefficients describes the stochasticity within this subspace. Using a dynamical orthogonality condition for the modes, a closed set of coupled evolution equations for the mean state, the modes and the stochastic coefficients can be directly derived from the governing NLS equation. This reduced-order set of DO equations enables the efficient computation of the stochastic solution, and permits the visualization in phase space of its time-dependent structure. We benchmark this reduced-order model against two well-known cases, that of a uniform wavetrain undergoing Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence, and that of an ensemble of waves with Gaussian spectrum and random phases. In both cases, we obtain a very good agreement with results reported in the literature, validating our DO equations. Finally, we exploit the benefits of the DO framework to study the nonlinear evolution of an extreme wave subjected to small initial stochastic perturbations and we visualize its attractor in phase space. Key words: Water waves, nonlinear Schrödinger equation, reduced-order modeling, stochastic dynamical systems. iii

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7 Contents Acknowledgements Abstract i iii Introduction 1 1 Review of deep-water wave theory Governing equations Linear dispersion relation Energy considerations Weakly nonlinear envelope equation Weakly nonlinear narrow-band wavetrain Nonlinear Schrödinger equation Modulational instability and long-time evolution Uniform Stokes wave solution Benjamin-Feir instability Fermi-Pasta-Ulam recurrence NLS equation under the DO framework Review of probability theory and KL expansion Probability spaces and random variables Stochastic processes and random fields Karhunen-Loève orthogonal expansion Dynamically orthogonal NLS equation Dynamically orthogonal expansion On the choice of the inner product Dynamically orthogonal equations Stochastic energy transfers Numerical implementation Numerical schemes Initial condition formulation Diagonalization of the covariance matrix Overview of the code structure v

8 Contents 3 Preliminary results and validation Idealized Benjamin-Feir instability Random Gaussian wavenumber spectrum Statistical properties of the DO solution Structure of the DO solution Dynamics of an extreme wave Nonlinear focusing of localized wave packets Adaptivity of the DO modes Attractor of an idealized extreme wave Results with the MNLS equation Modified nonlinear Schrödinger and DO equations Idealized Benjamin-Feir instability Attractor of an idealized extreme wave Conclusions and perspectives 55 A Complex coefficients in the DO framework 57 A.1 Dynamically orthogonal equations with complex coefficients A.2 Diagonalization of the complex covariance matrix B Dynamically orthogonal MNLS equation 61 Bibliography 63 vi

9 Introduction Starting with the work of Stokes (1847), water waves have attracted the attention of scientists and researchers alike. Because nonlinearity enters through the boundary conditions of their governing equations, they are notoriously difficult to solve analytically and have continuously posed great challenges. This probably explains why it was not until the work of Lighthill (1965) followed by the experimental confirmation of Benjamin & Feir (1967) that weakly nonlinear deep-water uniform wavetrains were found to be unstable to small modulational perturbations, a surprising phenomenon that became known as the Benjamin-Feir instability. Subsequent research focused on understanding the long-time evolution and dynamics of these unstable wavetrains. A great progress in that direction was made by Zakharov (1968), who derived a simple governing equation for the envelope of weakly nonlinear narrow-band wavetrains to cubic order in nonlinearity, the nonlinear Schrödinger (NLS) equation. Through experiments and numerical simulations of the NLS equation, Lake et al. (1977) showed that the long-time behavior of the Benjamin-Feir instability included the generation of large coherent structures through spatial focusing of the energy of the wave field. Later, Dysthe (1979) and Trulsen & Dysthe (1996) introduced higher-order more accurate versions of the NLS equation. Nevertheless, in this thesis we will mainly use the NLS equation because of its reasonable accuracy in the one-dimensional setting that we consider (Yuen & Lake, 198). Water waves are characterized by the interplay between two mechanisms. Dispersive effects induce wave packet dispersion and mixing between different wavenumbers, while nonlinear effects create energy transfers between modes and can result in the spatial focusing of wave groups. Rogue waves are a dramatic example of the effects of nonlinearities. They are deepwater gravity waves of much larger height than one would expect given the sea state (Dysthe et al., 28). Because of their potential catastrophic impact (Liu, 27), they have recently started to be an area of active research (Onorato et al., 21). The Benjamin-Feir instability was believed to be the basic physical mechanism at their origin, for it gives rise to large structures. However, it applies to the idealized case of a uniform wavetrain while the ocean surface is an irregular surface with energy spread over a range of wavenumbers. Nevertheless, Alber (1978) showed that a similar type of instability persisted in random wavetrains characterized by a narrow Gaussian spectrum with random phases. Numerical simulations of the NLS equation and its higher-order derivatives confirmed that a sufficiently narrow spectrum leads to an increased occurrence of these extreme waves (Janssen, 23; Dysthe et al., 23). More 1

10 Contents recently, Ruban (213) and Cousins & Sapsis (215b,a) have shown that these extreme waves are in fact induced by the nonlinear focusing of spatially localized wave groups that exceed a certain critical length scale and amplitude. In this thesis, we adopt a reduced-order stochastic approach towards the modeling of nonlinear deep-water waves and we investigate the aforementioned phenomena in this reducedorder stochastic setting. Specifically, we use the dynamically orthogonal (DO) framework introduced by Sapsis & Lermusiaux (29) to derive reduced-order equations for the stochastic evolution of water waves governed by the NLS equation and subject to random initial conditions. This DO framework is based on the expansion of the stochastic solution on a few time-dependent deterministic modes with associated stochastic coefficients that efficiently describe stochastic fluctuations around the mean state as the system evolves in time. From the governing NLS equation and a dynamical orthogonality condition for the modes, can then be derived a set of explicit equations that allow for the coupled simultaneous evolution of (i) the mean state, (ii) the reduced-order subspace that contains the stochastic fluctuations (i.e. the spatio-temporal form of the modes) and (iii) the stochasticity within this subspace (i.e. the stochastic coefficients). Our choice of the DO framework follows from the time adaptivity of its modes (critical in the highly transient context of water waves) and from the fact that no prior knowledge on the modes is required. This thesis is structured as follows. In Chapter 1 we provide a review of the theory related to weakly nonlinear unidirectional waves on deep water. Then, in Chapter 2 we develop our DO reduced-order framework for the stochastic modeling of water waves and we detail its numerical implementation. In Chapter 3, we illustrate the use of the obtained DO equations through the computation of the stochastic solutions for two well-known situations. By comparing our results with those from the literature, we validate the accuracy of our DO equations. Finally, in Chapter 4 we study the nonlinear evolution of an extreme wave subjected to small initial stochastic perturbations and we visualize its structure in phase space. 2

11 Chapter 1 Review of deep-water wave theory In this first chapter, we provide a review of the theory related to weakly nonlinear unidirectional waves on deep water. After presenting in Section 1.1 the governing equations for the surface elevation, we show in Section 1.2 how perturbation methods can lead to a simplified nonlinear equation for the envelope of a weakly nonlinear narrow-band wavetrain. We then introduce in Section 1.3 the Benjamin-Feir modulational instability, which plays an important role in the dynamics of nonlinear wavetrains and we discuss its long-time evolution. 1.1 Governing equations Consider a two-dimensional system consisting of two layers of incompressible and inviscid fluid, water at the bottom and air at the top. Since we are concerned with deep-water gravity waves, we assume the water layer to be of infinite depth and we neglect surface tension effects. Although water is not inviscid, here viscosity is neglected because it is effective only for small-scale motion (Yuen & Lake, 198). The air layer is assumed to remain at rest (which is justified by the difference in densities), thus neglecting wind-wave interactions and effectively restricting the problem to the water region. An (x, z) coordinate system is chosen in such a way that the undisturbed water surface coincides with z = and gravity points in the negative z-direction. The fluid being irrotational, potential flow theory can be used and the system is characterized by the surface elevation η(x, t) of the water and its velocity potential φ(x, z, t). In the water domain, the velocity potential satisfies Laplace s equation 2 φ = for < z < η(x, t), (1.1) with the following boundary condition that enforces a vanishing velocity at the bed φ = when z. (1.2) z 3

12 Chapter 1. Review of deep-water wave theory There are two boundary conditions at the free surface. The kinematic boundary condition ensures that fluid particles cannot traverse the surface, by equating the normal component of the fluid velocity at the surface with that of the surface s motion η t + φ η x x φ = at z = η(x, t). (1.3) z Because we have neglected surface tension effects, the dynamic boundary condition states that the fluid pressure at the surface equals the atmospheric pressure. The Bernoulli equation applied at the surface therefore takes the following form φ t ( φ)2 + g z = at z = η(x, t), (1.4) where we have taken the atmospheric pressure to be zero without loss of generality, and g denotes the acceleration of gravity. The set of equations (1.1) to (1.4) constitutes the governing equations of gravity waves on deep water. Although Laplace s equation for the velocity potential φ(x, z, t) is linear, it applies to a domain for which one of the boundaries, the water surface η(x, t), is unknown a priori and itself part of the problem. Nonlinearity thus appears implicitly through the boundary conditions at the unknown surface Linear dispersion relation Restricting ourselves to small disturbances about the base state η = and φ =, the free surface boundary conditions can be linearized and applied at the undisturbed surface z =. Assuming periodic conditions in the horizontal direction, the variables η and φ may then be expanded in Fourier modes in x and t, leading to the linear solution η(x, t) = a cos(kx ωt), (1.5) φ(x, z, t) = ag ω ekz sin(kx ωt), (1.6) where the wave frequency ω is related to the wavenumber k through the well-known linear dispersion relation ω = g k. The phase velocity of the waves is given by c = ω/k = g /k while their group velocity is v g = ω/ k = c/2. The dependency of c on k is indicative of the dispersive nature of the system, a property that will remain important for finite-amplitude waves as linear dispersion will counteract nonlinearity Energy considerations Finally, we note that because of the absence of dissipation and external forcing, the governing equations (1.1) to (1.4) conserve the total energy of the fluid (see Janssen, 24), expressed as H = ρg L η η z dz dx ρ L ( φ) 2 dz dx, (1.7) 4

13 1.2. Weakly nonlinear envelope equation where the first term represents the potential energy of the fluid, the second term its kinetic energy and we have considered a periodic domain x D = [,L]. Similarly to an harmonic oscillator, we can think of the wave motion as caused by the resonance between the kinetic and potential energies. In fact, we can use the linear solution (1.5) (1.6) to show that these energies are equal to leading order, giving the following expression for the total energy H = 1 L 2 ρg η 2 dx + 1 L 4 ρg a2 (1 + 2kη + O (η 2 ))dx = 1 2 ρg a2 L + O (a 4 ). (1.8) 1.2 Weakly nonlinear envelope equation The linear solution (1.5) (1.6) describes a uniform wavetrain of perfect periodicity and constant infinitesimal amplitude. It provides a sufficient description for linear systems, as the properties of an entire wave system can then be determined by superposition. However, this no longer holds true for nonlinear systems like the one given by governing equations (1.1) (1.4), hence we rather seek a generalization of the idealized uniform wavetrain where we would allow for slow (with respect to the wavelength and wave period) variations in space and time of the amplitude, wavenumber and frequency Weakly nonlinear narrow-band wavetrain Specifically, we introduce the weakly nonlinear narrow-band wavetrain through the concept of a carrier wave with carrier wavenumber k and frequency ω, that is modulated by a small but finite and slowly varying complex envelope function u(x, t) = a(x, t)e iθ(x,t). The respective roles of the modulus a(x, t) and the phase θ(x, t) can be made explicit through the following (single-harmonic) expression for the surface elevation of the resulting wavetrain η(x, t) = Re{u(x, t)e i (k x ω t) } = Re{a(x, t)e iθ(x,t) e i (k x ω t) }, (1.9) where it is seen that a(x, t) represents a slowly varying modulation amplitude, and θ(x, t) is a slowly varying phase modulation function that describes small variations in wavenumber and frequency of the wavetrain about the carrier wavenumber and frequency. Specifically, we have the modulation wavenumber k and modulation frequency ω k = θ x k, ω = θ t ω, (1.1) both of which are small with respect to k and ω, hence the narrow-band designation as departures in wavenumber k = k + k and frequency ω = ω + ω of the wavetrain are restricted to a narrow-band window around the corresponding carrier properties. 5

14 Chapter 1. Review of deep-water wave theory Nonlinear Schrödinger equation We now seek to obtain the equation governing the evolution of the slowly varying complex envelope u(x, t). This can be done by taking advantage of both the small amplitude a and the slow time and space variation of u, through a perturbation expansion combined with a multiple-scales method. As a small parameter, we define the wave steepness ε = k a 1 and we require the modulation bandwidth to be of the same order of magnitude i.e. k/k = O (ε), meaning that u varies on the slow space and time scales εx and εt. We then employ the following harmonic expansions for η(x, t) and φ(x, z, t) η(x, t) = Re{u(εx,εt)e i (k x ω t) + u 2 (εx,εt)e 2i (k x ω t) +... }, (1.11) φ(x, z, t) = Re{v(εx, z,εt)e k z e i (k x ω t) + v 2 (εx, z,εt)e 2k z e 2i (k x ω t) +... }, (1.12) where the slowly varying complex functions u, u 2,..., v, v 2,... are O(ε/k ). Inserting these expansions into the governing equations (1.1) (1.4) and expanding the free-surface boundary conditions (1.3) (1.4) in Taylor series about the undisturbed surface z =, the equations can be solved in orders of ε. At the first order, the usual linear dispersion relation ω = g k for the carrier wave is recovered. At the second order, it is found that the envelope is advected at the group velocity of the carrier wave. Taking the perturbation expansion for the first harmonic to the third order gives the following evolution equation for the complex envelope ( u i t + ω ) u ω 2 u 2k x 8k 2 x ω k 2 u 2 u =, (1.13) where the first term indicates advection of the envelope at the group velocity of the carrier wave, the second term represents effects of dispersion, and the third term describes nonlinear effects to the lowest-order. This equation is the nonlinear Schrödinger (NLS) equation. It has widespread use in physics across a number of fields that involve nonlinear dispersive waves and was first obtained by Benney & Newell (1967) in a general context. For deep-water waves, it has been derived using a variety of methods, first by Zakharov (1968) using a spectral method, then by Hasimoto & Ono (1972) and Davey (1972) using multiple-scale methods and finally by Yuen & Lake (1975) through a Lagrangian variational approach from Whitham (1965). Finally, from the perturbation expansion for the second harmonic we have u 2 = k u 2 /2, implying that the dynamics of the higher harmonics are tied to the first one. The waves pertaining to the higher harmonics are therefore referred to as the bound waves, while those lying within the bandwidth of the first harmonic are called the free waves. Note also that u 2 = O(ε 2 /k 2 ), thus the first harmonic provides a sufficient description of the surface elevation η(x, t) = Re{u(εx,εt)e i (k x ω t) } + O (ε 2 /k 2 ). (1.14) In studying solutions to the NLS equation, it is helpful to use a reference frame moving at the 6

15 1.3. Modulational instability and long-time evolution group velocity of the carrier wave, which is achieved through the change of coordinates x = x ω 2k t. (1.15) Additionally, we use the length and time scales imposed by the wavelength and period of the carrier wave to introduce the following nondimensional variables t = ω t, x = k x, ũ = k u, k = k k, ω = ω ω, (1.16) leading to the following nondimensionalized form of the NLS equation (1.13) ũ t = i 8 2 ũ x 2 i 2 ũ 2 ũ. (1.17) and the surface elevation to lowest order is expressed as η( x, t;ω) = Re{ũ( x, t;ω)e i ( x t/2) }. Note that under this choice of length scale, the wave steepness ɛ = k a becomes equivalent to the nondimensional wave amplitude ũ. Unless specified otherwise, nondimensional variables will be used in the remainder of this thesis so we hereafter drop the prime symbols. 1.3 Modulational instability and long-time evolution Uniform Stokes wave solution As a first step in investigating the properties of weakly nonlinear wavetrains, we look at the simplest solution of the NLS equation (1.17), given by the following spatially constant envelope u(x, t) = a e (i /2)a2 t, a R, (1.18) which describes a uniform wavetrain similar to the linear solution (1.5), but with a nonlinear correction to the linear dispersion relation ω = g k. This can be shown by writing the corresponding expression for the surface elevation in dimensional form η(x, t) = Re{a e (i /2)k2 a2 ω t e i (k x ω t) } + O (a 2 ) = a cos[k x ω (1 + k 2 a2 /2)t] + O (a2 ). (1.19) We obtain a uniform wave with dispersion relation ω = ω (1 + k 2a2 /2) that depends not only on the wavenumber k, but also on the finite amplitude a. This nonlinear correction was already found by Stokes (1847) through a weakly nonlinear harmonic expansion of the uniform wavetrain. He also found that the higher harmonics resulted in an altered profile with sharp crests and flat troughs, resulting in the so-called Stokes wave. Note that the dispersion relation for the finite-amplitude uniform wave gives 1 2 ω 2 k 2 = ω 8k 2 and ω a 2 = 1 2 ω k 2, (1.2) 7

16 Chapter 1. Review of deep-water wave theory which shows that the second and third terms of the NLS equation (1.13) indeed relate to dispersion and nonlinear effects, respectively. This can be shown in a more rigorous way from a heuristic derivation of the NLS equation based on a Taylor expansion of the dispersion relation around wavenumber k and zero amplitude (Yuen & Lake, 1982; Janssen, 24) Benjamin-Feir instability The linear stability of this uniform wavetrain can be investigated by imposing small Fourier mode perturbations representing infinitesimal modulations in amplitude and phase u(x, t) = a (1 + b 1 e i ( kx ωt) + i b 2 e i ( kx ωt) )e (i /2)a2 t, (1.21) where b 1 and b 2 are infinitesimal real numbers and k, ω represent respectively the nondimensional modulation wavenumber and frequency around the carrier wavenumber and frequency. Linearizing the NLS equation about the uniform solution and solving the resulting eigenvalue problem results in the following dispersion relation ω 2 = k2 8 ( k 2 8 a2 ). (1.22) This indicates that perturbations with nondimensional wavenumber in the range < k < k c = 2 2a (1.23) have positive growth rate σ = Im ω, hence are linearly unstable, while k c = 2 2a represents a cut-off wavenumber above which there is restabilization. The instability is maximum at k m = 2a with an associated nondimensional maximum growth rate of σ m = a 2 /2, implying that it occurs on a nondimensional timescale of O (a 2 ). This instability is called the Benjamin- Feir (BF) or modulational instability and was initially discovered for very long perturbation wavelengths by Lighthill (1965), but it was Benjamin & Feir (1967) who first obtained equation (1.22) and found the restabilization at higher perturbation wavenumbers. A plot of the growth rate versus perturbation wavenumber is shown in Figure 1.1. Note that the instability region depends on the amplitude and disappears as a tends to, thus recovering the linear result of a marginally stable water surface around z = Fermi-Pasta-Ulam recurrence Through experiments in a wave tank and simulations of the NLS equation, Lake et al. (1977) studied the long-time behavior of the Benjamin-Feir instability. They showed that after an initial period of exponential growth, unstable modulations grow to a maximum and saturate before decaying, and the wavetrain returns to a nearly-uniform state, after which a new cycle starts. This surprising behavior had previously been discovered in another context by Fermi et al. (1955) and thus became known as the Fermi-Pasta-Ulam (FPU) recurrence. 8

17 1.3. Modulational instability and long-time evolution 1.8 </< m "k/a Figure 1.1 Normalized growth rate σ/σ m of the Benjamin-Feir instability versus normalized perturbation wavenumber k/a. Note that the instability is maximum at k m = 2a and the cut-off occurs at k c = 2 2a. Figure 1.2 Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence in a slightly modulated uniform wavetrain. Left, spatio-temporal evolution of the complex envelope modulus u(x, t). Right, surface elevation η(x, t) (blue) and envelope modulus u(x, t) (orange) at three different times. We illustrate the Benjamin-Feir instability and subsequent Fermi-Pasta-Ulam recurrence in Figures 1.2 and 1.3, in which we have computed the numerical solution to a periodic initial condition of the form u(x,) = a (1 +.1cos k m x), with a =.1, representing a uniform wavetrain that is slightly modulated by the linearly most unstable modulation wavenumber k m = 2a =.2. In the left-hand side plot of Figure 1.2, we plot the spatio-temporal evolution of the complex envelope modulus u(x, t). After an initial growth, the wavetrain is observed to reach a strongly modulated state before returning to its initial state and undergoing a new cycle. The right-hand side plot displays the surface elevation η(x, t) (blue) together with the envelope modulus u(x, t) (orange) at three different times, and shows that the strongly modulated state at t = 66 can be interpreted as a situation where energy from the carrier wave has focused in space to produce localized extreme waves. The Fourier spectrum F [η] of the surface elevation at t = 5 is shown in the left-hand side plot of Figure 1.3 and shows that the Benjamin-Feir instability manifests itself as pairs of growing sideband 9

18 F[2] F[u]("k) Chapter 1. Review of deep-water wave theory Fourier transform of 2 at t = "km "km k Fourier modes of u "k = "k = 1"km "k = 2"km "k = 3"km "k = 4"km t Figure 1.3 Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence in a slightly modulated uniform wavetrain. Left, Fourier spectrum F [η] of the surface elevation at t = 5, with k = 1 corresponding to the carrier wave and k = 1 ± k m to the unstable modulation. Right, time evolution of the amplitudes of the carrier wave, the unstable modulation and its harmonics obtained through a Fourier transform F [u] of the envelope. This time, k = corresponds to the carrier wave and k = k m is the unstable modulation. wavenumbers comprising the unstable initial modulation k = 1 ± k m and its harmonics k = 1 ± 2 k m,1 ± 3 k m,... Note that the carrier wave corresponds to k = 1 because of the nondimensionalization. In the right half of Figure 1.3, the amplitudes of the carrier wave and of the prescribed unstable modulation together with three of its harmonics are obtained through a Fourier transform F [u] of the envelope, and plotted versus time. It is clearly observed that the unstable modulation k = k m is growing at the expense of the carrier wave k =. The harmonics being linearly stable, they only appear as forced oscillations and are phase-locked to the initially prescribed modulation. In the case where some of the harmonics lie within the linearly unstable range (1.23), however, Yuen & Ferguson (1978) revealed that the long-time behavior of the solution is governed by both the initially prescribed unstable mode and its unstable harmonics, with the unstable mode and harmonics each taking turn dominating a recurrence cycle. As a conclusion, we would like to mention that in this chapter we have only provided a brief review of the classical theory of weakly nonlinear deep-water waves. Our goal was mainly to introduce the NLS equation to be used in the rest of this thesis, and to familiarize the reader with the notion of a slowly modulated narrow-band wavetrain and its envelope description. As a result, we have focused on the description of the idealized Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence of a uniform wavetrain, and we have deliberately omitted other important notions such as the evolution of more realistic narrow-band Gaussian spectra of waves or the nonlinear focusing of wave packets. These notions will be introduced in the later chapters, along with the presentation of our results. 1

19 Chapter 2 NLS equation under the DO framework In this chapter, we develop a reduced-order framework for the stochastic modeling of water waves governed by the nonlinear Schrödinger (NLS) equation (1.17). This framework is based on the dynamically orthogonal (DO) field equations first introduced by Sapsis & Lermusiaux (29), a novel order-reduction method for the solution of stochastic partial differential equations. In Section 2.1, we start with a brief review of probability spaces, stochastic processes and the Karhunen-Loève expansion. The DO framework is then presented in Section 2.2 and applied to the NLS equation, resulting in a set of equations for the coupled evolution of the mean state and the reduced-order stochastic fluctuations. In Section 2.3, we derive expressions for the rates of energy transfer between the different dynamical components of the solution. Finally, the numerical implementation of the equations is detailed in Section Review of probability theory and KL expansion We start by introducing a few definitions and concepts from probability theory and stochastic processes. Our goal is not to provide a comprehensive review of the subject (for this we refer to Sobczyk, 1991), but merely to introduce the usual notation and provide select notions that will prove useful in the following sections Probability spaces and random variables A central concept in the description of random phenomena is the so-called probability space (Ω,B,P ). The sample space Ω is the set of all elementary events ω Ω associated with the random phenomenon under consideration. B is the σ-algebra associated with Ω, that is and loosely speaking, B is the collection of subsets of Ω. The elements of B are random events and consist of combinations of elementary events ω. Finally, P is a probability measure defined on B, i.e. P is a countably additive function that associates a non-negative number between and 1 to each random event in B, and that assigns the value 1 to the sample space Ω. In loose 11

20 Chapter 2. NLS equation under the DO framework terms, P can be interpreted as the function that counts the number of elementary events ω contained in a given subset of the sample space Ω, divided by the total number of elementary events in Ω. Various physical outcomes can occur in a random phenomena. In the simplest case, the outcome of a given random experiment can be represented by a real number, thus one can assign a real number X (ω) for each elementary event ω Ω. This function X (ω) that maps elements in Ω to values in R is called a random variable, and its probabilistic behavior is completely specified by its probability distribution F (x), defined as F (x) = P {ω X (ω) < x }, x R, ω Ω, (2.1) which characterizes the probability that X (ω) be less than x. For a continuous random variable, the distribution function is continuous everywhere and we call probability density f (x) the function given by the following derivative f (x) = df, x R. (2.2) dx Let us stress that the probability space (Ω,B,P ) is defined independently of any particular random phenomena, in the sense that the probabilistic behavior associated to the outcome of a given random experiment is embedded in the random variable X (ω) itself, and not in the elementary events ω which by definition happen with equal probability dp (ω). Finally, we have the following definition for the mean value or expectation of the random variable X (ω) E[X (ω)] = X (ω)dp (ω). (2.3) Ω Stochastic processes and random fields In physical applications, many random phenomena are also time dependent, in which case the outcome of a random experiment can be represented by a real function of time. This leads us to the notion of a stochastic process. Denoting t T the time, a stochastic process X (t;ω) is a function that maps elementary events ω Ω to elements in the space of all finite and real-valued functions of time. In this case, the function x(t) = X (t;ω) associated to a fixed elementary event ω Ω is called a realization of the process. We are ultimately interested in phenomena that are both time and space dependent. Denoting x D R the one-dimensional space variable, we generalize the notion of a stochastic process to that of a complex random field u(x, t;ω), that is, a function that maps elementary events ω Ω to elements in the space of all finite and complex-valued functions of space and time. Similarly to the mean value of a random variable, we can define the mean value of the random 12

21 2.1. Review of probability theory and KL expansion field u(x, t;ω) as the ensemble average over the elementary events ω ū(x, t) = E[u(x, t;ω)] = u(x, t;ω)dp (ω), (2.4) Ω Karhunen-Loève orthogonal expansion Consider a complex random field u(x, t;ω) that is continuous and square integrable, i.e. D E[u(x, t;ω)u (x, t;ω)]dx < for all t T, where the asterisk denotes the complex conjugate. For a fixed time t, u(x, t;ω) is a random function of x and lives in the infinite-dimensional Hilbert space L 2 of all continuous, square integrable and complex-valued functions defined on D, and having the following spatial inner product u 1,u 2 = u 1 (x)u2 (x)dx for all u 1,u 2 L 2. (2.5) D Since L 2 is an infinite-dimensional space, it is spanned by an infinite number of orthogonal basis functions {v i (x)}. Hence, if we want to write the random field u(x, t;ω) (at a given time) i=1 as a linear decomposition of deterministic fields multiplied by scalar stochastic coefficients by projecting each realization u(x, t;ω) onto the basis {v i (x)}, then we need an infinite series i=1 of the form u(x, t;ω) = ū(x, t) + Y i (t;ω)v i (x), (2.6) where ū(x, t) is the mean field and Y i (t;ω) are zero-mean and complex-valued scalar stochastic coefficients that carry information on the stochastic fluctuations of the random field u(x, t;ω) around the mean ū(x, t). The question now is whether one can, for a given time, find a deterministic basis {u i (x, t)} that is optimal for u(x, t;ω) (at that time), in the sense that a i=1 finite-dimensional representation of the form u(x, t;ω) ū(x, t) + i=1 s Y i (t;ω)u i (x, t), s <, (2.7) i=1 would approximate members of u(x, t;ω) better than representations of the same dimension in any other basis. This statement can be formalized (see Holmes et al., 1996) as follows E[ u(x, t;ω),u i (x, t) 2 ] max, (2.8) u i L 2 u i (x, t),u i (x, t) where denotes the absolute value, and it is looked for an u i (x, t) such that the ensemble average of the projection of u(x, t;ω) onto u i (x, t) is maximized. Using variational calculus techniques, this condition reduces to the following eigenvalue problem D R t (x, y)u i (x, t)dx = λ i (t)u i (y, t), (2.9) 13

22 Chapter 2. NLS equation under the DO framework where R t (x, y) = E[u(x, t;ω)u (y, t;ω)] is the time-dependent autocorrelation function of the random field u(x, t;ω) and the desired optimal basis is given by the orthogonal set of eigenfunctions u i (x, t). The zero-mean stochastic coefficients in representation (2.7) are then obtained by the projection of the stochastic fluctuations to the optimal basis Y i (t;ω) = u(x, t;ω) ū(x, t),u i (x, t), (2.1) and they verify the following properties, relating to the eigenvalues λ i (t) E[ Y i (t;ω) 2 ] = λ i (t), E[Y i (t;ω)y j (t;ω)] = for i j. (2.11) Hence, the eigenvalues λ i (t) represent the variance (or spread) of the stochastic fluctuations u(x, t;ω) ū(x, t) along each direction u i (x, t) in phase space. As a consequence of the optimality of the representation, the orthogonal directions u i (x, t) are aligned with the principal directions of the variance of u(x, t;ω) (i.e. they capture the dominant fluctuations of u(x, t;ω)), and it is often the case that the variance along subsequent directions decreases exponentially i.e. λ i (t) e ci for some positive c. It is therefore justified to define a threshold s < and neglect the directions u i (x, t) with i > s, along which u(x, t;ω) has negligible spread. The study of the random field u(x, t;ω) is therefore restricted to the s-dimensional subspace spanned by the eigenmodes associated to the s largest eigenvalues. This idea is at the foundation of reduced-order modeling. Depending on the phenomenon under study, the value of s can be very small, in which case we say that the system has a low-dimensional attractor. Note that since R t (x, t) is self-adjoint and positive definite, we are assured that equation (2.9) will always admit a countable infinity of positive eigenvalues and orthogonal eigenfunctions. The linear orthogonal decomposition (2.7) with the optimal basis functions given by the eigenvalue problem (2.9) is known as the Karhunen-Loève (KL) expansion (Loève, 1945). It has found important applications, notably in fluid mechanics for the reduced-order modeling and analysis of statistically stationary turbulent flows, where it bears the name Proper Orthogonal Decomposition (POD) (Berkooz et al., 1993; Holmes et al., 1996). In this context, the ensemble averages can be replaced by time averages over a single experimental run, and the time dependence of the mean and the modes in the decomposition (2.7) is removed. The procedure then involves two steps. First, a relevant set of optimal basis functions is found by capturing experimental or numerical snapshots of a flow at different times, constructing the resulting empirical autocorrelation function and solving the resulting eigenvalue problem (2.9). Then, the low-dimensional deterministic dynamics are obtained by projecting the governing (Navier- Stokes) equations on the previously obtained finite-dimensional basis, resulting in a set of coupled equations for the time evolution of the coefficients Y i (t;ω). Although the POD has proved successful in identifying coherent structures and their dynamics in turbulent flows (Aubry et al., 1988), it has two main limitations: (i) it uses time-independent basis functions and is therefore not suitable for the description of transient phenomena, and (ii) it relies on experiments or numerical simulations to derive the set of optimal basis functions. 14

23 2.2. Dynamically orthogonal NLS equation To overcome the aforementioned limitations of the POD and other methods in the context of highly transient stochastic systems, Sapsis & Lermusiaux (29) introduced the dynamically orthogonal (DO) equations, a novel and time-adaptive reduced-order framework for the solution of systems governed by generic stochastic partial differential equations (SPDEs). Since transient phenomena and intermittent instabilities are often observed in deep-water gravity waves (Cousins & Sapsis, 215b,a), we opt to use the DO equations for the derivation in the next section of a reduced-order framework for the modeling of stochastic water waves. 2.2 Dynamically orthogonal NLS equation Let s consider a complex random field u(x, t;ω) representing the (nondimensional) complex envelope of a deep-water weakly nonlinear narrow-band wavetrain, as defined in Chapter 1. We assume that u(x, t;ω) is continuous and square integrable (which makes sense from a physical viewpoint). While this complex envelope is governed by the deterministic NLS equation (1.17), we are interested in studying its evolution under random initial conditions that follow some given probability distribution. Stochasticity therefore enters the problem through the initial condition, which is specified in the form of a large ensemble of initial realizations u(x, t ;ω) that follow a given probability distribution. We then want to simultaneously evolve in time all realizations, so that at each time instant we can recover the full stochastic solution u(x, t;ω) and its associated statistics. This can be readily done with a Monte-Carlo approach but will result in a high computational cost, so we instead opt to use the DO reduced-order framework introduced in Sapsis & Lermusiaux (29) to do this in an efficient way. Our choice of the DO framework instead of other order-reduction methods follows from both its adaptivity and the fact that no prior knowledge on the form of the basis functions is required Dynamically orthogonal expansion The basis idea behind the DO method goes as follows. We first suppose that the random initial condition can be accurately represented by a truncated KL expansion (2.7) at initial time t s u(x, t ;ω) = ū(x, t ) + Y i (t ;ω)u i (x, t ), s <, (2.12) i=1 where ū(x, t ) is the initial mean, Y i (t ;ω) are the zero-mean initial stochastic coefficients and u i (x, t ) are the initial set of orthonormal basis functions or modes. Note that s defines the dimensionality of the subspace containing the initial stochastic fluctuations. Next, we assume that the system (in this case governed by the NLS equation) retains a low-dimensional attractor as time evolves, i.e. its stochastic solution u(x, t;ω) at time t can still be accurately represented by a truncated KL expansion of similar dimension. The DO method then provides a set of coupled equations, directly derived from the system governing equation, for the time 15

24 Chapter 2. NLS equation under the DO framework evolution of all quantities involved in (2.12), in such a way that the approximate full stochastic solution at time t can be written in the form of the following DO expansion u(x, t;ω) = ū(x, t) + s Y i (t;ω)u i (x, t), (2.13) where ū(x, t) is the time-dependent mean, u i (x, t) are the time-dependent deterministic modes describing the main directions of stochastic fluctuations at time t and Y i (t;ω) are the time-dependent zero-mean stochastic coefficients. The DO solution given by (2.13) aims at being close enough to an s-truncated KL expansion of the exact stochastic solution at time t, as would be obtained from a direct Monte-Carlo simulation on the system governing equation with initial realizations u(x, t ;ω). The discrepancy between the two is caused by the effects that dynamics along the neglected directions i > s may have on the resolved directions i s and the mean. Although these effects can be large in turbulent systems (Sapsis & Majda, 213), they are negligible in a number of other cases (see Mantič-Lugo, Arratia & Gallaire, 214, for a dramatic example in the case of the flow behind a cylinder, where as a further approximation the single mode that is used is computed as a quasilinear approximation around the mean). The restriction of the stochastic dynamics to the subspace V s = span{u i (x, t)} s i=1 containing the dominant fluctuations can thus be a good approximation, and results in a much better computational efficiency than a full Monte-Carlo simulation. In deriving explicit equations for all unknown quantities in the DO expansion (2.13), the redundancy stemming from the allowed time variation of both the coefficients Y i (t;ω) and the modes u i (x, t) needs to be overcome. Sapsis & Lermusiaux (29) showed that this can be achieved by imposing the following dynamical orthogonality (DO) condition dv s dt V s ui (, t) t i=1,u j (, t) =, i, j = 1,..., s, (2.14) that restricts the time variation of the subspace V s where stochasticity lives to be orthonormal to itself. In other words, since variations of the stochastic fluctuations within V s can be entirely described by variations of the stochastic coefficients Y i (t;ω), the DO condition imposes the natural constraint that modes only move when the fluctuations evolve to new directions not already included in V s. The DO condition also implies the preservation of the initial orthonormality of the modes u i (x, t) since t u ui (, t) u j (, t) i (, t),u j (, t) =,u j (, t) +,u i (, t) =, i, j = 1,..., s. (2.15) t t In the next subsections, it will be shown how the insertion of the DO expansion (2.13) together with the DO condition (2.14) in the NLS governing equation can lead to a closed and exact set of coupled equations for the mean ū(x, t), the modes u i (x, t) and the stochastic coefficients Y i (t;ω). We emphasize that the time-dependent basis functions u i (x, t) are not chosen a priori and are able to dynamically evolve to adapt to temporal changes in the dominant stochastic fluctuations, thereby remedying two shortcomings of the POD. 16

25 2.2. Dynamically orthogonal NLS equation On the choice of the inner product Before proceeding to the derivation of equations for the mean, modes and stochastic coefficients, we remark that the choice of the inner product has important implications on the quantities involved in the DO expansion (2.13) of the solution. Indeed, the inner product defines the way the basis functions u i (x, t) are orthogonal to each others, as well as the value of the scalar coefficients Y i (t;ω), for they result from the projection of the solution u(x, t;ω) onto the basis {u i (x, t)} s i=1. Since u(x, t;ω) is complex-valued, it appears logical to consider the standard complex-valued spatial inner product defined in equation (2.5) as u 1,u 2 = u 1 (x, t;ω)u2 (x, t;ω)dx. (2.16) D This complex-valued inner product implies that the stochastic coefficients in the DO expansion (2.13) will also be complex-valued. Note that previous DO schemes have always concerned real-valued systems where all quantities are real (Choi et al., 213; Sapsis et al., 213) and this is, to our knowledge, the first time that an attempt at using complex coefficients and complex fields within the DO framework is carried out. While we managed to derive the DO equations with complex coefficients (see Appendix A), complex statistics have to be dealt with when analyzing the resulting solution, and this is very much still an area of active research (for an overview of some of the complications involved, see Eriksson & Koivunen, 26; Adali et al., 211; Cheong Took et al., 212). Therefore we opted to go the safer route by using the following real-valued inner product { u 1,u 2 = Re u 1 (x, t;ω)u2 }, (x, t;ω)dx (2.17) D so that the stochastic coefficients in the DO expansion (2.13) will be real-valued. To better understand the implications of using a real-valued inner product in a space of complex-valued functions, consider two basis functions u 1 (x, t) and u 2 (x, t) = i u 1 (x, t) that are orthogonal under inner product (2.17), since u 1,u 2 = Re{ i } =. Together they span the subspace span{u 1,u 2 } = {Y 1 u 1 (x, t) + Y 2 u 2 (x, t) Y 1,Y 2 R} = {(Y 1 + i Y 2 )u 1 (x, t) Y 1,Y 2 R} = { Z 1 u 1 (x, t) Z 1 C}. (2.18) It is therefore observed that u 1 (x, t) and u 2 (x, t) span the same subspace as would u 1 (x, t) alone with a complex coefficient. Note that under the complex-valued inner product (2.16) we would indeed have u 1,u 2 = i, confirming that the two directions are not orthogonal when considering complex coefficients. Therefore the use of the real-valued inner product (2.17) with real stochastic coefficients, while implying no loss of generality, results in a higher number of basis functions required to span a given subspace for the stochastic fluctuations. 17

26 Chapter 2. NLS equation under the DO framework As a side note, when using the real-valued inner product (2.17) with real coefficients, we can make an analogy between the space of complex functions u(x, t;ω) and that of real 2D vector fields defined as (Re{u(x, t;ω)},im{u(x, t;ω)}). Indeed, in this case it is easily seen that the real inner product (2.17) is equal to the standard inner product on the space of real-valued 2D vector fields. The two formulations are therefore completely equivalent. On the other hand, the use of the complex-valued inner product (2.16) results in complex coefficients that have the ability to swap the real and imaginary parts of the complex fields they are multiplying, something that real coefficients cannot do hence there is no analogy with real 2D vector fields in this case Dynamically orthogonal equations In this section, we follow the steps in Sapsis & Lermusiaux (29) to derive the DO equations that govern the evolution of all unknown quantities in the DO expansion (2.13), i.e. the mean ū(x, t), the deterministic modes u i (x, t) and the stochastic coefficients Y i (t;ω) for i = 1,..., s. As discussed in the previous section, we consider the real-valued inner product (2.17), which implies that the stochastic coefficients Y i (t;ω) are real. Recall that the nondimensional complex envelope u(x, t;ω) is governed by the deterministic NLS equation (1.17) u(x, t;ω) = i t 8 2 u(x, t;ω) x 2 i 2 u(x, t;ω) 2 u(x, t;ω), x D, t T, ω Ω, (2.19) subject to periodic boundary conditions, and to the following random initial conditions with known probability distribution u(x, t ;ω) = u (x;ω), x D, ω Ω. (2.2) We begin by inserting the DO expansion (2.13) in the NLS equation (2.19), leading to the following governing equation for all unknown quantities ū t + dy i dt u u i i + Y i = F + Y i F i + Y i Y j F i j + Y i Y j Y k F i j k, (2.21) t where repeated indices indicate summation from 1 to s, and F, F i, F i j and F i j k are complex deterministic fields defined by F = i 2 ū 8 x 2 i 2 ū 2 ū, F i = i 8 F i j = i 2 Re{u i u j }ū i Re{ūu i }u j, F i j k = i 2 Re{u i u j }u k, 2 u i x 2 i ū Re{ūu i } i 2 ū 2 u i, (2.22) where i, j,k = 1,..., s. The deterministic PDE governing the evolution of the mean field ū(x, t) is obtained by taking the ensemble average of the governing equation (2.21) ū t = F +C Yi Y j F i j + M Yi Y j Y k F i j k, (2.23) 18

27 2.3. Stochastic energy transfers where C Yi Y j (t) = E[Y i Y j ] is the time-dependent covariance matrix and M Yi Y j Y k (t) = E[Y i Y j Y k ] is the time-dependent matrix of third-order moments of the stochastic coefficients. Next, we project the governing equation (2.21) onto each of the modes u i (x, t). Using the DO condition, the orthonormality of the modes and the zero-mean property of the coefficients, we get a set of s coupled stochastic differential equations (SDEs) for the stochastic coefficients Y i (t;ω) dy i dt = Y m F m,u i + (Y m Y n C Ym Y n ) F mn,u i + (Y m Y n Y l M Ym Y n Y l ) F mnl,u i. (2.24) Finally, we multiply the governing equation (2.21) with each of the stochastic coefficients Y i (t;ω), we apply the ensemble average operator and we use the SDEs for the stochastic coefficients to obtain a set of s coupled deterministic PDEs governing the evolution of the DO modes u i (x, t) u i t = H i H i,u j u j, (2.25) where H i is a complex deterministic field defined by H i = E[L [u]y k ]C 1 Y i Y k with L [u] the right-hand side of the governing equation (2.21), and is expressed as H i = F i + M Ym Y n Y k C 1 Y i Y k F mn + M Ym Y n Y l Y k C 1 Y i Y k F mnl, (2.26) with M Yi Y j Y k Y l (t) = E[Y i Y j Y k Y l ] the time-dependent matrix of fourth-order moments. The mean ū(x, t), the modes u i (x, t) and the stochastic coefficients Y i (t;ω) are initialized at t through a KL expansion (2.12) of the initial condition u (x;ω). We have thus derived an exact set of fully coupled evolution equations for these quantities, in the sense that no approximation other than the truncation at finite size of the DO expansion has been used. 2.3 Stochastic energy transfers The spread of the stochastic fluctuations along each directions u i (x, t) of the stochastic subspace V s is varying with time, owing to flows of energy (variance) between modes and the mean. In this section, we derive expressions for these rates of stochastic energy transfers between a given DO mode, the mean and the other modes. Recall from equation (1.8) that the potential and kinetic energies of the wavetrain are equal to leading order, so that the stochastic total energy over the domain D = [,L] can be expressed as H (t;ω) = L η(x, t;ω) 2 dx = L Re{u(x, t;ω)e i (x t) } 2 dx 1 2 L u(x, t;ω) 2 dx (2.27) where the energy has been made nondimensional with ρg /k 3, and the last equality follows from the slow space variation of u(x, t;ω). The last term can be expressed in terms of the inner product, so that we can make use of the DO expansion (2.13) to decompose the average energy in the solution as contributions from the mean and the modes E (t) = E[H ] = 1 2 E[ u,u ] = 1 2 E[ ū + Y i u i,ū + Y i u i ] = 1 2 ( ū 2 + E[Y i Y i ] ) (2.28) 19

28 Chapter 2. NLS equation under the DO framework where the last equality follows from the orthonormality of the DO modes, and shows that the energy contained in the modes is equal to the variance E[Y 2 ] of the stochastic coefficients, i while the term ū 2 = ū,ū represents the energy contained in the mean. In order to study the flow of energy between the different modes and the mean, we use equation (2.24) for the stochastic coefficients to write the rate of change of the stochastic energy contained in mode i ε i = 1 [ d 2 dt E[Y 2 i ] = E dy i Y i dt ] = A i i C Yi Y i + B i mn M Yi Y m Y n +C i mnl M Yi Y m Y n Y l, (2.29) (no sum on i ), where we have assumed that the covariance matrix C Yi Y j has been diagonalized at the present time instant (see Section for the details) so that the stochastic coefficients are uncorrelated. A i i, B i mn and C i mnl are the deterministic fields appearing in equation (2.24) A i i = i 2 u i 8 x 2 i ū Re{ūu i } i 2 ū 2 u i,u i, (2.3) B i mn = i2 Re{u mu n }ū i Re{ūu m }u n,u i, (2.31) C i mnl = i2 Re{u mu n }u l,u i. (2.32) The term A i i can be considerably simplified since i 2 { u L i 8 x 2,u i 2 } u { i i = Re 8 x 2 u i dx = Re i u i 8 x u i L L i u i u } i + 8 x x dx =, (2.33) and i2 { L } i ū 2 u i,u i = Re 2 ū 2 u i u i dx =, (2.34) where we have taken advantage of the periodicity in the boundary conditions and the realness of the inner product. After some work, we obtain the following expressions { L i A i i = Re 2 ū2 u i { L i B i mn = Re 2 [ūu mun + ūu m u n + ū u m u n ]u i dx { L i C i mnl = Re 2 u mun u l u i dx 2 dx }, (2.35) }, (2.36) }. (2.37) By inspection, we observe that the linear term A i i represents the rate of energy transfer between the mean and mode i. The term B i mn indicates modal energy production due to the simultaneous interaction of mode i with the mean and two other modes, while C i mnl involves the interaction of mode i with three other modes. These nonlinear four-mode interactions arise due to the cubic term in the NLS equation and depend on the non-gaussian statistics of the system, for they are associated with the high-order moments M Yi Y m Y n and M Yi Y m Y n Y l (note that, however, for Gaussian statistics there can still be nonlinear energy transfers localized in phase space that don t manifest in the variance). The modal energy production (in terms of 2

29 2.4. Numerical implementation variance) of mode i thus boils down to with the following linear and nonlinear contributions ε i = ε mean i + ε mean,mn i + ε mnl i, (2.38) ε mean i = A i i C Yi Y i, ε mean,mn i = B i mn M Yi Y m Y n, ε mnl i = C i mnl M Yi Y m Y n Y l, (2.39) (no summation on i ). Finally, note that there is no energy dissipation and the sum of the average energy contained in the mean and the modes is conserved [ de u dt = E = E ] t,u [ i 2 u 8 [ { = E Re x 2,u L [ { = E Re i u 8 =, ] [ + E i ] 2 u 2 u,u }] [ { L + E Re i 2 u 8 x 2 u dx L L i u u + 8 x x dx x u }] + E }] i 2 u 2 uu dx [ { L Re }] i 2 u 4 dx (2.4) which is the well-known energy conservation property of the NLS equation (Zakharov & Shabat, 1972). Therefore, the average variation in total stochastic energy d/dt s i=1 E[Y 2 i ]/2 = s i=1 ε i is only caused by interactions between the mean and the modes, and we anticipate the sum of all nonlinear interactions between the modes to be zero. Indeed we have C i mnl = C mi l n, leading to s i=1 ε mnl i =. 2.4 Numerical implementation The DO equations for the mean (2.23), the modes (2.25) and the coefficients (2.24) are implemented numerically in the MATLAB software and solved in a coupled fashion. To increase the speed efficiency, we implemented specific parts of the code in the C language through the MEX interface. The various details and hurdles associated with the numerical implementation are given in the subsections that follow, and an overview of the steps followed by the code is provided in the last subsection Numerical schemes The deterministic PDEs for the mean and the modes are discretized on a grid of size 124 points and a semi-implicit Euler scheme is used for time advancement of the solution. Indeed, the diffusion operator that appears in the forcing fields in equation (2.22) is built with a secondorder finite-difference scheme and is treated implicitly when it appears in the linear terms in equations (2.23) and (2.25), while the nonlinear terms are treated explicitly. The set of SDEs 21

30 Chapter 2. NLS equation under the DO framework for the stochastic coefficients is solved using a Monte-Carlo method with 1 3 to 1 4 particles and is advanced in time with a 4th-order Runge-Kutta scheme, using a nondimensional time step of.1. Finally, it should be mentioned that while the DO condition (2.14) implies the preservation of the orthonormality of the modes, numerical rounding errors lead to a deviation from that state. Therefore, orthonormality is enforced by applying a stabilized Gram-Schmidt process to the modes at each time step, and adjusting the stochastic coefficients for the new basis so that the solution itself remains the same. Since the stochastic coefficients are expressed as a large ensemble of realizations, the computation of various statistical quantities such as the variance or the joint probability distribution of the coefficients is straightforward. In addition, this allows for the direct recovery from the DO expansion (2.13) of the complex envelope solution u(x, t;ω) corresponding to any ensemble member, enabling the study of individual envelope realizations or statistics such as the probability density function of the surface elevation. Note that provided s is low enough, the DO method can lead to significantly increased computational efficiency compared to a Monte-Carlo simulation of the governing NLS equation for all realizations, as the mean and the modes only require the solution of s + 1 expensive PDEs, while the stochastic coefficients that need to be evolved for all realizations are given by a simpler s-dimensional ODE Initial condition formulation It was seen in equation (2.12) that in general the initial condition is formulated in terms of a truncated KL expansion at time t s u(x, t ;ω) = ū(x, t ) + Y i (t ;ω)u i (x, t ), s <, (2.41) i=1 where ū(x, t ) is the initial mean, Y i (t ;ω) are the zero-mean initial stochastic coefficients and u i (x, t ) are the initial set of orthonormal modes. In practice, instead of computing the initial modes from the eigenvalue problem (2.9) for a given autocorrelation function R t (x, y), the computation is initialized by directly assigning a shape to the modes and realizations of a given probability distribution to the random coefficients. In general, we formulate the initial condition in terms of a Fourier series with coefficients having random modulus and phase N u(x, t ;ω) = ū(x, t ) + A n (ω)e iθn(ω) e i k n x, N <, (2.42) n=1 where the modulus A n (ω) and phase θ n (ω) follow a desired probability distribution and N is the finite number of Fourier modes that are present in the initial condition. However, since we use the real inner product (2.17), we can only assign real values to the initial stochastic 22

31 2.4. Numerical implementation coefficients Y i (t ;ω). This issue can be overcome by expanding (2.42) as N N u(x, t ;ω) = ū(x, t ) + A n (ω)cosθ n (ω)e i k n x + A n (ω)sinθ n (ω)e i ( k n x+π/2), (2.43) n=1 n=1 resulting in an expansion similar to the DO initial condition (2.41), where the DO modes and the real-valued stochastic coefficients are given by Y 2n 1 (t ;ω) = A n (ω)cosθ n (ω), u 2n 1 (x, t ) = e i k n x, Y 2n (t ;ω) = A n (ω)sinθ n (ω), u 2n (x, t ) = e i ( k n x+π/2), (2.44) and a number of modes s = 2N is required because of the realness of the stochastic coefficients (as was thoroughly discussed in Section 2.2.2, enabling the use of complex coefficients would eliminate this drawback). Note that we indeed have e i k n x,e i ( k n x+π/2) =, confirming the fact that a given complex Fourier mode is spanned by two orthogonal directions under the real-valued inner product. The values given in (2.44) will be used to initiate the quantities in the DO solution for initial conditions of the type (2.42) Diagonalization of the covariance matrix In the KL expansion (2.7), the stochastic coefficients are uncorrelated i.e. E[Y i (t;ω)y j (t;ω)] = for i j, which is a consequence of the fact that the directions u i (x, t) are aligned with the principal directions of variance of u(x, t;ω). On the other hand, in the DO expansion (2.13) the stochastic coefficients are not constrained to remain uncorrelated, so that even when the simulation is initiated with uncorrelated coefficients, over a finite time they will develop some correlation and off-diagonal terms will appear in the covariance matrix C Yi Y j = E[Y i (t;ω)y j (t;ω)]. As a result, the modes u i (x, t) are no longer aligned with the principal directions of variance of the DO solution u(x, t;ω). This problem can nonetheless be easily overcome by diagonalizing the covariance matrix C Yi Y j, which corresponds to applying a rotation to the modes u i (x, t) such that they become aligned with the principal variance directions and the coefficients Y i (t;ω) become uncorrelated, while the full solution u(x, t;ω) remains intact. Let us describe the reasoning behind the procedure. Suppose that we have modes u i and that the covariance matrix C = C Yi Y j has off-diagonal elements at a given time instant. Since it is real symmetric, we are assured of the existence of the following diagonal decomposition C = V DV T (2.45) where T denotes the transpose, V is formed by the eigenvectors of C hence is orthogonal, and D is a diagonal matrix containing the eigenvalues of C. Since V is orthogonal, it can be used as a rotation matrix and we define a new basis with u i = u mv mi. Note that we have u i,u j = u i,u m V m j = V m j u i,u m = V i j, (2.46) 23

32 Chapter 2. NLS equation under the DO framework We first show that the new basis u is orthonormal i u i,u j = u mv mi,u n V n j = V mi V n j u m,u n = V mi V m j = V T j m V mi = δ i j. (2.47) Then, the coefficients Y i and they are uncorrelated since in the new basis are given by the following projection Y i = Y j u j,u i = Y j u j,u i = Y j V j i, (2.48) C Y i Y = E[Y j i Y j ] = E[Y mv mi Y n V n j ] = V T i m C mnv n j = D i j = δ i j λ i, (2.49) where λ i are the eigenvalues of C and the diagonal elements of D. We have therefore shown that by diagonal decomposition of the covariance matrix C Yi Y j, we can always define a new rotated basis such that the stochastic coefficients in the new basis become uncorrelated. The new directions u (x, t) correspond to the directions of principal variance of the solution i u(x, t;ω). We apply this procedure every time the solution is plotted or saved Overview of the code structure Here we provide an overview of the structure of the code. The DO solution is first initialized by assigning a value to the mean field ū(x, t), the modes u i (x, t) and the stochastic coefficients Y i (t;ω), for example by following (2.44). We then apply a Gram-Schmidt process to the initial modes to make sure they are orthonormal. The code then enters a loop where the solution is advanced in time, and that consists of the following steps, in order: 1. The zero-mean property of the stochastic coefficients is enforced to avoid deviations due to rounding errors. 2. The covariance matrix C Yi Y j and higher-order moments matrices M Yi Y j Y k and M Yi Y j Y k Y l of the stochastic coefficients are calculated. The calculation is implemented in C (through the MEX interface) and takes advantage of the symmetries in the moments. 3. The deterministic forcing fields in equation (2.22) are calculated. 4. The stochastic coefficients are advanced in time using a 4th-order Runge-Kutta scheme, where the calculation of the right-hand side of equation (2.24) is implemented in C (through the MEX interface). 5. The mean field is advanced in time through equation (2.23) and a semi-implicit Euler scheme, where the diffusion term is treated implicitly in the linear term only. 6. The deterministic forcing fields in equation (2.26) are calculated. 7. The modes are advanced in time through equation (2.25) and the same semi-implicit Euler scheme as for the mean. 24

33 2.4. Numerical implementation 8. The modes are orthonormalized using a stabilized Gram-Schmidt process and the adjusted stochastic coefficients are calculated. 9. At some of the time steps, the solution is plotted and/or saved after the modes have been rotated following the procedure from Section and leading to uncorrelated stochastic coefficients. 25

34

35 Chapter 3 Preliminary results and validation In this chapter, we illustrate the use of the DO reduced-order model introduced in the previous chapter, by presenting simulation results for situations that are well documented in the literature. These situations will also provide us with a way to benchmark our results and validate the accuracy our DO reduced-order equations. Specifically, we simulate in Section 3.1 the evolution of a uniform wavetrain undergoing semi-stochastic Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence. In Section 3.2, we compute the stochastic evolution of a random Gaussian spectrum of waves and investigate the properties of the resulting solution. 3.1 Idealized Benjamin-Feir instability Recall from Section 1.3 that a uniform wavetrain, represented by a spatially constant envelope u(x, t ) = a, is linearly unstable to small Fourier mode perturbations with modulation wavenumber in the range < k < k c = 2 2a. If there is only one prescribed unstable modulation wavenumber k and if its harmonics fall outside the unstable regime (that is k > k c /2), then the long time evolution of this perturbation is very well understood. After initially undergoing Benjamin-Feir (BF) instability, the unstable modulation will grow and decay repeatedly in a Fermi-Pasta-Ulam (FPU) recurrence cycle that is illustrated in Figures 1.2 and 1.3. Since this behavior is the same regardless of the phase difference between the modulation and the carrier wave, this situation therefore provides us with a simple framework for the illustration and validation of our DO stochastic equations. Indeed, we can then consider random initial conditions consisting of a constant envelope a perturbed by Fourier modes with deterministic small amplitude but random phase N u(x, t ;ω) = a + A n e iθn(ω) e i k n x, N <, (3.1) n=1 and we expect that all realizations will evolve similarly and according to the deterministic results. The stochastic fluctuations introduced by the phase randomness should persist and 27

36 Mode 4 Mode 3 Mode 2 Mode 1 Mean Chapter 3. Preliminary results and validation t = t = t = x x x Figure 3.1 Mean and DO modes at various times for the stochastic BF instability and FPU recurrence with random phase in the initial modulation. Both the complex envelope modulus (blue) and real part (orange) are represented. Note that while the domain size is L = 5 2π, we only plot the solution over a portion of the domain corresponding to the wavelength of the linearly most unstable wavenumber k m = 2a =.2. grow, in such a way that the modes should reveal the dominant components of the dynamics. We therefore consider initial condition (3.1) with a =.1. We choose to use N = 2 Fourier modes of wavenumbers k 1 =.2 and k 2 =.4, in such a way that the first Fourier mode corresponds to the linearly most unstable wavenumber (given by k m = 2a ) while its harmonics and the second Fourier mode are stable. The random phases θ n (ω) are drawn from independent and uniform distributions on [,2π], and the deterministic amplitudes A n are assigned the infinitesimal value.36. For the initial DO expansion (2.12), the mean is assigned the uniform wave component ū(x, t ) = a, while the modes u i (x, t ) and stochastic coefficients Y i (t ;ω) carry the stochastic perturbations and are initialized with the relations (2.44). Note that each Fourier mode has to be represented with two DO modes because of the real-valued coefficients, resulting in a total of 4 DO modes. A periodic domain of size L = 5 2π is used but its size doesn t affect the solution since the latter is periodic and non-localized. The solution for the mean and the modes at various times is plotted in Figure 3.1, where both the modulus (blue) and real part (orange) are represented. The smallest wavenumber k 1 =.2 present in the initial condition implies that the solution will remain periodic with period 2π/ k 1, therefore we only show a portion of the total domain corresponding to this 28

37 E[Y i 2 ] 3.1. Idealized Benjamin-Feir instability Mean Mode 1 Mode 2 Mode 3 Mode 4 Deterministic t Figure 3.2 Energy of the mean ū,ū and the modes E[Y 2 ] for the stochastic BF instability i and FPU recurrence with random phase in the initial modulation. The dashed lines show the corresponding energies for a deterministic simulation of an equivalent initial condition u(x,) = cos k m x with k m =.2, with the energies obtained as the normalized squared modulus of the Fourier coefficients of wavenumber k = (carrier wave),.2 (unstable modulation) and.4 (stable harmonic) (similarly to Figure 1.3). wavelength. In Figure 3.2, we show the energy present in the mean ū,ū together with the stochastic energy present in the modes E[Y 2 ]. Focusing on the modes at t = 8 in Figure i 3.1, we observe that the modes keep their initial pairing indicative of a randomness in the phase of the associated structure. Modes 1 and 2 represent the same linearly most unstable modulation k 1 =.2 as was assigned in the initial condition, while modes 3 and 4 represent the same stable modulation k 2 =.4. The spatially constant mean still represents a uniform carrier wave. From Figure 3.2, it is observed that the stochastic energy in the linearly unstable modes 1 and 2 grows, saturates then decays. The stable modes 3 and 4 are slaved to the unstable modulation and experience the same process from t 4, albeit to a lesser degree. Meanwhile, the energy of the mean follows the inverse tendency since the modes are growing at its expense. Note that the modulations grow as stochastic fluctuations (since they are represented in the modes) because of the phase randomness that they have been assigned in the initial condition. These observations are in perfect accordance with the deterministic BF instability and subsequent FPU recurrence shown in Figures 1.2 and 1.3, where the unstable modulations are growing at the expense of the uniform carrier wave, before decaying. For a quantitative comparison, we computed the solution to a deterministic initial condition of the form u(x,) = cos k m x where k m =.2, i.e. similar in structure to one realization of the random DO initial condition (3.1) (only without the phase randomness of the modulation), and we retrieve the energy of the carrier wave and the modulations by means of a Fourier transform of the envelope. The normalized resulting squared modulus of the Fourier coefficients for k = (carrier wave),.2 (unstable modulation) and.4 (stable harmonic) are shown as dashed lines in Figure 3.2. The agreement between the stochastic DO computation and the deterministic calculation is remarkable, particularly the maximum energy of the unstable modulation k =.2 (contained in the DO modes 1 and 2), and the time at which it occurs. 29

38 Chapter 3. Preliminary results and validation Figure 3.3 Stochastic attractor at various times of the stochastic BF instability and FPU recurrence with random phase in the initial modulation. The attractor is represented in terms of the 3D scatter plot of the first three stochastic coefficients for all realizations. In addition, each realization is assigned a color indicative of the maximum value of its corresponding surface elevation η(x, t;ω) = Re{u(x, t;ω)e i (x t/2) }. However, from time t 11, the DO solution is observed in Figure 3.2 to deviate from its deterministic counterpart, which can also be seen in Figure 3.1 in the shapes of the modes at t = 14 that lose any kind of regular structure. Moreover, from t 19, the energy in the modes begin oscillating, a sign that the numerical solution is collapsing. We tried refining the mesh, using a smaller time step or a larger number of Monte-Carlo realizations, but to no avail. The source of the problem thus remains unknown and could be related either to the numerics or to the DO equations themselves. It should be noted that the NLS equation has been found to cause numerical instabilities in similar situations (Herbst & Ablowitz, 1989; Ablowitz & Herbst, 199), while numerical instabilities have also been encountered in other systems using reduced-order models based on the KL expansion (Kirby & Armbruster, 1992). Since the stochastic coefficients are expressed as a large ensemble of realizations, it is possible to visualize the time-dependent structure in phase space of the ensemble solution. In Figure 3.3, we therefore display the stochastic attractor of the solution, represented as the 3D scatter plot of the first three stochastic coefficients for all realizations. The coefficients associated to modes 1 and 2 (containing the linearly unstable modulation) are observed to be of the form Y 1 (t;ω) = A(t)cosθ(ω) and Y 2 (t;ω) = A(t)sinθ(ω), with amplitude A(t) approximately equal for all realizations and undergoing growth then decay. This shows that the unstable modulation contained in the first two modes is growing and decaying at a similar rate for all realizations, but with a random phase shift θ(ω), as can be expected since all realizations have been assigned the same initial perturbation amplitude in the initial condition (3.1). In addition, each realization in Figure 3.3 is assigned a color indicative of the maximum value of the corresponding surface elevation, obtained as η(x, t;ω) = Re{u(x, t;ω)e i (x t/2) } where the envelope u(x, t;ω) can be reconstructed from the DO expansion (2.13). Since the amplitude of the modulation is the same between all members of the ensemble, differences in surface elevation only arise from the stochastic phase difference between the modulation and the 3

39 member with minimum 2 member with maximum Idealized Benjamin-Feir instability t = t = t = x x x Figure 3.4 Two realizations at various times of the stochastic BF instability and FPU recurrence with random phase in the initial modulation. Both the surface elevation (blue) and complex envelope modulus (orange) are represented. The top row shows the realization with maximum surface elevation η(x, tω) at t = 8, while the bottom row shows the one with minimum surface elevation at that time. Note that we only plot a portion of the domain corresponding to the wavelength of the linearly most unstable wavenumber k m = 2a =.2. carrier wave. To illustrate this, we plot in Figure 3.4 the surface elevation (blue) and envelope modulus (orange) at various times of the two realizations that have the maximum (top row) and minimum (bottom row) surface elevation at t = 8. Their envelope is indeed observed to be modulated to a similar degree, with differences in surface elevation resulting from the phase shift between complex envelope and carrier wave. Finally, in Figure 3.5 we illustrate the flows of stochastic energy between modes and the mean. Using the expressions derived in Section 2.3, we calculate the energy production in every mode due to (i) its linear interaction with the mean (first column), (ii) its nonlinear simultaneous interaction with the mean and two other modes (second column) and (iii) its nonlinear interaction with three other modes (third column). In the first column it is seen that DO modes 1 and 2 (corresponding to the linearly unstable modulation) are from the onset receiving energy from the mean in a linear fashion, before giving it back after saturation of the modulation. The same happens for DO modes 3 and 4 (corresponding to the linearly stable modulation), although in this case the transfer of energy is delayed since they are initially linearly stable. The third column shows that there are nonlinear exchanges of energy occurring between the unstable and stable modulations, although these are one to two orders of magnitude smaller than the exchanges of energy with the mean. Sapsis (213) proved that linear transfers of energy from the mean to the modes result in the uniform increase or decrease of the attractor in phase space, while only nonlinear energy exchanges are able to cause local changes in its shape. This is indeed the case here, since the attractor is mainly expanding and contracting uniformly in phase space and energy transfers mostly occur in a linear fashion. 31

40 Mode 4 Mode 3 Mode 2 Mode 1 Chapter 3. Preliminary results and validation # # # # " mean!i t # # # # " mean;mn!i t # # # # " mnl!i t Figure 3.5 Modal energy production in every mode due to (i) linear interaction with the mean flow (first column), (ii) nonlinear simultaneous interaction with the mean and two other modes (second column) and (iii) nonlinear interaction with three other modes (third column) for the stochastic BF instability and FPU recurrence with random phase in the initial modulation. Recall that the solution is only valid up to t Random Gaussian wavenumber spectrum The Benjamin-Feir instability provides an idealized framework for the investigation of instabilities of water waves. In practice, however, the ocean surface is not merely a uniform wavetrain and energy is rather distributed over a range of wavenumbers around the carrier wave. Therefore, a more realistic setup would be to consider the evolution of a narrow-band wave field consisting of a Gaussian random distribution of waves around the carrier wavenumber. The resulting non-uniform wavetrain can be written in terms of the complex envelope as u(x, t ;ω) = N /2 n= N /2+1 2 k1 F ( k n )e iθ n(ω) e i k n x, (3.2) where the Fourier coefficients have random uncorrelated phases θ n (ω) drawn from a uniform distribution on [, 2π], and deterministic amplitude proportional to the square root of the following Gaussian wavenumber spectrum F ( k n ) = ( ) ε2 σ 2π exp k2 n 2σ 2, (3.3) where k n = n k 1 is the modulation wavenumber with k 1 the wavenumber discretization, ε is the average wave steepness and σ is the relative bandwidth of the spectrum. The average steepness is defined as the standard deviation of the surface elevation, and it can be 32

41 3.2. Random Gaussian wavenumber spectrum verified that we indeed have ε = E[η(x, t ;ω) 2 ] at any x location. Note that the wave field defined in (3.2) leads to a Gaussian distribution of the surface elevation since the Fourier wave components are uncorrelated. Based on the NLS equation, Alber (1978) investigated the stability of the random wave field (3.2) under a narrow-bandwidth and near-gaussian statistics approximation, and found that the ensemble-averaged Gaussian spectrum (3.3) is stable when the ratio of the wave steepness to the relative bandwidth, defined as the Benjamin-Feir index BFI = 2 2ε σ, (3.4) is less than 1. In the opposite case, numerical simulations of the NLS equation (Janssen, 23; Dysthe et al., 23) and experiments (Onorato et al., 25) have shown that when the initial BFI > 1, the ensemble-averaged spectrum relaxes on a time scale of O(ε 2 ) to a wider stable spectrum (characterized by a final time BFI 1), while in the meantime the so-called random version of the Benjamin-Feir instability occurs. The latter manifests as the focusing of localized wave packets in the irregular wavetrain due to the increased effect of nonlinearities, resulting in large coherent structures (similar to those observed in the deterministic Benjamin-Feir instability in Figure 1.2) and creating heavy-tailed statistics for the surface elevation. The random Gaussian spectrum of waves (3.3) thus provides us with a full stochastic setting in which we can benchmark our DO equations. We assign the stochastic initial condition (3.2) to our initial DO expansion (2.12), and with a single DO simulation we study the properties of the resulting ensemble solution and compare them with results obtained from Monte-Carlo simulations in the literature. Specifically, we consider initial condition (3.2) in the case of N = 1 Fourier modes with a wavenumber discretization k 1 =.6. The spectrum is assigned a fixed initial width σ =.1, and we consider seven different values for the wave steepness ε ranging from.25 to.1, giving a range of values for the initial BFI from.72 to 2.87 (in this regard note that most sea states have a BFI < 1, Dysthe et al., 28). The resulting discrete Gaussian spectrum (3.3) and its continuous equivalent are shown in the left-hand side plot of Figure 3.6 for the case BFI = The importance of having random and uncorrelated phases is shown in the right-hand side plot, where we compare two realizations of the surface elevation (blue) and envelope modulus (orange) corresponding to this Gaussian spectrum, either when all phases are equal to zero (top) or with random uncorrelated phases (bottom). The randomness of the phases θ n (ω) creates mixing between the different wavenumbers and leads to the desired irregularity for the surface elevation. For the initial DO expansion (2.12), each Fourier mode in (3.2) is assigned to two DO modes u i (x, t ) as per the relations (2.44), since two real-valued stochastic coefficients Y i (t ;ω) are necessary to reproduce the phase randomness of the Fourier coefficients. Without loss of generality, we decide to assign a deterministic phase equal to zero to the modulation wavenumber zero so that we can assign it to the mean, resulting in a total number of 18 DO modes. As before, we use a periodic domain of size L = 5 2π. Note 33

42 E[ F[2] 2 ] E[ F[2] 2 ] random phases F("k) zero phases Chapter 3. Preliminary results and validation Gaussian spectrum with BFI = 1.43 discrete continuous "k u (orange) and 2 (blue) x Figure 3.6 Initial condition for the DO simulation of a Gaussian random wave field for the case BFI = Left, discretized Gaussian wavenumber spectrum and its continuous equivalent. Right, two corresponding realizations of the surface elevation η(x, t ;ω) (blue) and envelope modulus u(x, t ;ω) (orange), either when all phases are equal to zero (top) or with random uncorrelated phases (bottom) t, BFI = k k t f t, BFI = k k Figure 3.7 Initial and final time wavenumber spectra E[ F [η] 2 ] for initial BFI =.72 (left) and BFI = 2.87 (right). The final time is taken as t f = 2/ε 2. t f that the minimum wavenumber allowed by the domain is k mi n = 2π/L =.2, thus equal to one third the wavenumber discretization k 1 =.6 that we assign to our initial Gaussian spectrum. We could have used a finer wavenumber discretization k 1, but spanning the same spectral width σ =.1 would have resulted in an unreasonably high number of DO modes and prohibitive computational cost Statistical properties of the DO solution We begin by investigating the stability properties of the ensemble-averaged spectrum with respect to the BFI. In Figure 3.7, we plot the initial and final time ensemble-averaged wavenumber spectra given by E[ F [η] 2 ] for two different initial values of BFI =.72 (left) and BFI = 2.87 (right). Since spectral changes for unstable values of the BFI are supposed to occur on a timescale of O (ε 2 ) (Dysthe et al., 23), results are shown for a final time defined as t f = 2/ε 2. In accordance with results in the literature, it is observed that the case BFI =.72 < 1 is stable while the case BFI = 2.87 > 1 is unstable and relaxes to a new spectrum that appears to be 34

43 Final time BFI 3.2. Random Gaussian wavenumber spectrum DO simulation Janssen (23) Dysthe et al. (23) Initial time BFI Figure 3.8 Final versus initial BFI, where the final value is calculated using equations (3.4) and (3.5). Results from Monte-Carlo simulations of the NLS equation from Janssen (23) and Dysthe et al. (23) are also shown. stable. Note that the spikes reflect the fact that the wavenumber resolution k mi n of the domain is one third the wavenumber discretization k 1 used in the initial condition. These spectrum relaxation results are shown in a more quantitative way and for all considered initial values of the BFI in Figure 3.8, where we plot the final time BFI as a function of the initial one. The final time BFI, indicative of the broadening of the spectrum, is calculated with equation (3.4) where the final time steepness and spectral width are given by k 2 E[ F [η(x, t ε(t f ) = E[η(x, t f ;ω) 2 f ;ω)] 2 ]d k ] and σ(t f ) = E[ F [η(x, t f ;ω)] 2 ]d k. (3.5) While no clear bifurcation is found, it is indeed observed that values of the BFI initially larger than 1 lead to a broadening of the spectrum in such a way that the final time BFI approaches 1. The results from the DO simulations are also compared with Monte-Carlo simulations of the NLS equation from the literature (Janssen, 23; Dysthe et al., 23), and show a reasonable agreement with these authors. Apart from the relaxation of the spectrum, a value of the BFI larger than 1 also leads to increased focusing of wave packets due to the increased effect of nonlinearities, resulting in a frequent occurrence of extreme waves. In Figure 3.9, we plot the ensemble-averaged probability density function of the surface elevation normalized by ε = E[η(x, t;ω) 2 ], at initial and final times and for initial BFI =.72 (left) and initial BFI = 1.43 (right). For reference, the corresponding Gaussian distribution in each case is also shown in orange dotted line. The deviation from normality of the final time pdf for to BFI = 1.43 is indicative of strong nonlinear effects resulting in a large occurrence of focusing wave groups, in accordance with results from the literature. We can get a more quantitative picture of the probability of extreme waves with the kurtosis, defined as C = E[η(x, t;ω)4 ] 3E[η(x, t;ω) 2 1, (3.6) ] which is a measure of the deviation of the pdf of the surface elevation from the Gaussian distribution. A value of corresponds to a Gaussian surface, while a value greater than 35

44 Final time kurtosis pdf pdf Chapter 3. Preliminary results and validation 1 t, BFI =.72 t f 1 t, BFI = 1.43 t f / 2/ / / Figure 3.9 Probability density function at initial and final times of the surface elevation normalized by ε = E[η(x, t;ω) 2 ], for initial BFI =.72 (left) and BFI = 1.43 (right). The final time is taken as t f = 2/ε 2. For reference, the corresponding Gaussian distribution is also shown in orange dotted line in each case Initial time BFI Figure 3.1 Final time kurtosis versus initial BFI, where the kurtosis is calculated using equation (3.6) and is representative of the deviation from normality of the distribution of the surface elevation. is representative of a higher probability of extreme waves due to predominant nonlinear effects. The final time kurtosis is plotted in Figure 3.1 versus the initial BFI, and is observed to increase with the value of the BFI, corroborating results from the literature. We have therefore shown that the DO solution for the Gaussian wavenumber spectrum gives accurate statistical (i.e. ensemble-averaged) results compared with Monte-Carlo simulations from the literature. It should however be noted that solving the DO equations for this situation of a random Gaussian spectrum is computationally very costly, because of the large number of DO modes required to represent the large number of random wavenumbers needed in the initial condition (18 in our case). For computations directed towards the obtention of purely statistical results of the kind that were shown in this section, it appears preferable to use a traditional Monte-Carlo method. Nevertheless, we observed Section 3.1 that a strength of the DO method lies in its ability to reveal the time-dependent dominant directions of stochastic fluctuations, plus the structure in phase space of the stochastic solution. Therefore, we investigate in the following section the structure of our DO solutions of random Gaussian wave fields, similarly to what was done in Section 3.1 for the idealized Benjamin-Feir instability. 36

45 Mode 4 Mode 3 Mode 2 Mode 1 Mean 3.2. Random Gaussian wavenumber spectrum BFI = x BFI = x BFI = x Figure 3.11 Mean and first four DO modes at final time t f = 2/ε 2, for different initial values of the BFI. Both the complex envelope modulus (blue) and real part (orange) are represented. Note that while the domain size is L = 5 2π, we only plot the solution over one third of the whole domain because of the effective periodicity induced by the wavenumber discretization k 1 =.6 of the initial condition Structure of the DO solution First, we show in Figure 3.11 the final time shape of the mean and first four DO modes, for three different values of the initial BFI, where again the final time is defined as t f = 2/ε 2. Note that only one third of the whole domain size is represented, since the wavenumber discretization k 1 = 3 k mi n of the initial condition leads to a solution that has a triple periodicity within the whole domain. The modes appear more regular when the initial BFI < 1 (corresponding to a stable ensemble-averaged spectrum), and we note that the DO modes 1 and 2 for BFI =.72 correspond to sinusoidal modulations of wavenumber.6, equal to the Benjamin-Feir linearly most unstable wavenumber k m = 2a for a uniform wave of amplitude a =.3, a value close to the actual r.m.s amplitude E[a] = 2E[η 2 ] =.35 corresponding to this irregular wave field. This being said, for higher BFI the situation is less clear and the modes are difficult to interpret. In Figure 3.12, we show the evolution from initial to final time of the energy present in the mean and the modes, for the same three values of the initial BFI. It is observed that for BFI =.72, the energy of the DO modes 1 and 2 (that have a regular sinusoidal shape) is set apart from the others, confirming their special status. For higher BFI, the modes converge to an energy level 37

46 Chapter 3. Preliminary results and validation BFI =.72.1 Mean Mode 1 Mode 2 etc E[Y2i ].8 BFI = 1..2 Mean Mode 1 Mode 2 etc.15 BFI = Mean Mode 1 Mode 2 etc t 5 1 t t Figure 3.12 Evolution of the energy in the mean and the modes for different initial values of the BFI. Recall that the mean is initially assigned the modulation wavenumber. Figure 3.13 Stochastic attractor at final time of the solution for different initial values of the BFI. The attractor is represented in terms of the 3D scatter plot of the first three stochastic coefficients for all realizations. In addition, each realization is assigned a color indicative of the maximum value of its corresponding surface elevation η(x, t ; ω) = Re {u(x, t ; ω)e i (x t /2) }. that is closer with the others, in such a way that there is not really a dominant direction in the stochastic fluctuations. This corroborates observations from Figure 3.11, where no definite trend was observed in the shape of the modes. Recall that the stochastic coefficients are expressed as a large ensemble of realizations, allowing for the visualization in phase space of the time-dependent structure of the ensemble solution. In Figure 3.13, we display the stochastic attractor of the final time solution for the same three values of the initial BFI, represented as the 3D scatter plot of the first three stochastic coefficients for all realizations. In addition, each realization is assigned a color indicative of the maximum value of the corresponding surface elevation, obtained as η(x, t ; ω) = Re {u(x, t ; ω)e i (x t /2) } where the stochastic envelope u(x, t ; ω) can be reconstructed from the DO expansion (2.13). As can be expected from the similar energy levels present in all the modes, no particular structure can be observed and there doesn t appear to be a clear correlation between the maximum of the surface elevation for a given realization and the value of the corresponding stochastic coefficients. 38

47 member with minimum 2 member with maximum Random Gaussian wavenumber spectrum BFI = x BFI = x BFI = x Figure 3.14 Two realizations of the stochastic solution at final time for different initial values of the BFI. The top row shows the realization with maximum surface elevation η(x, t;ω) while the bottom row shows that with minimum surface elevation. Note that we only plot the solution over one third of the whole domain because of the effective periodicity induced by the wavenumber discretization k 1 =.6 of the initial condition. Finally, for each of the same three values of the initial BFI, we plot in Figure 3.14 the surface elevation (blue) and envelope modulus (orange) of the two realizations that have the maximum (top row) and minimum (bottom row) surface elevation at the final time. The realizations with maximum surface elevation for BFI = 1. and 1.43 show localized wave groups that have focused, sucking energy from the nearby wave field. These focusing wave groups are known to appear for BFI > 1 (Ruban, 213) and are responsible for the heavy-tailed statistics of the surface elevation in these higher-energy wave fields. Being able to observe them in individual realizations of the reduced-order DO stochastic solution therefore constitutes a check of the validity of our DO framework. However, it appears difficult to establish a precise relationship between the shape of these wave groups and that of the modes from Figure This is because the focusing wave groups observed in Figure 3.14 are events of a local nature that appear at random locations in the domain, therefore they cannot be captured as such by the DO modes. The absence of a clear relationship between the shape of the realizations and that of the modes also confirms that the dynamics of these irregular wave fields don t possess any dominant component on a global scale, as could be inferred from the similar levels of energy present in all the modes in Figure As a conclusion, the advantages brought by the DO framework in the context of the stochastic solution to a Gaussian random spectrum of waves are limited. There are mainly two interconnected reasons for that. First, by assigning all the stochastic components of the initial condition into the DO modes, a high number of them is required, resulting in a high computational cost as the latter mostly scales to the cube of the number of modes. Second, the Gaussian stochastic wave field is such that energy is spread over a large number of modes and there is no clear dominant component in the stochastic dynamics on the global scale. This means that (i) the high number of modes required to initialize the solution is still needed as time evolves, and (ii) the modes don t reveal any global dominant tendency in the stochastic fluctuations and the solution does not possess a clear structure in phase space. 39

48 Chapter 3. Preliminary results and validation Based on these observations, it looks like the DO framework is better suited for computing and analyzing the nonlinear evolution of a given deterministic wave field that would be assigned to the mean ū(x, t), and perturbed by stochastic perturbations that can be contained in a reasonable number of DO modes. This is in essence what was done in Section 3.1, where in order to study the stochastic evolution of the idealized Benjamin-Feir instability, we assigned a uniform wavetrain to the mean and perturbed it with small stochastic perturbations contained in the modes. At the same time, we observed in Figure 3.14 that while there are no dominant components in the global dynamics of these irregular wave fields, local events constituting of the focusing of localized wave groups are nevertheless present. These wave groups have recently attracted attention in the literature (Adcock & Taylor, 29; Cousins & Sapsis, 215b,a) due to the extreme waves that can result from them. Therefore, in the following chapter we focus our attention to these spatially localized wave groups, and study their nonlinear evolution when subjected to small stochastic perturbations. 4

49 Chapter 4 Dynamics of an extreme wave In this chapter, we exploit the benefits of the DO framework to study the nonlinear evolution of deterministic wave fields under small initial stochastic perturbations. Specifically, we concentrate on the focusing behavior of spatially localized wave groups induced by nonlinear effects of the NLS equation, resulting in extreme waves. This known phenomenon is presented in Section 4.1. Then, in Section 4.2 we investigate the ability of the DO modes to track the emergence of these extreme waves out of a Gaussian spectrum of background waves. Finally, in Section 4.3 we study the structure in phase space of an idealized extreme wave (i.e. without any background wave field) subject to small initial perturbations. The results presented in this chapter are still a work in progress. 4.1 Nonlinear focusing of localized wave packets In Section 3.2, we mentioned that in a random Gaussian spectrum of waves (3.2) characterized by a BFI > 1, increased effects of nonlinearities lead to the focusing of spatially localized wave packets of moderate amplitude, resulting in extreme waves and heavy-tailed statistics for the distribution of the surface elevation (Ruban, 213). An example of such an extreme event formation in shown in Figure 4.1, where the initial wave field is generated through a Gaussian spectrum (3.3) with random phases and BFI = A spatially localized wave packet around x = 14 at time (top row) is observed to focus in an extreme wave and reaches a maximum amplitude of.37 at t = 27 (middle row), before eventually fading away at a larger time (bottom row). This focusing of the localized wave group is due to the nonlinear term of the NLS equation, which counterbalances the inverse effect of the dispersive term. More specifically, it has recently been shown (Cousins & Sapsis, 215b,a) that there exists a critical length scale and amplitude for the wave packet above which nonlinear effects become predominant in such a way that it will be likely to focus, giving rise to an extreme wave. We are now interested in taking advantage of the DO framework to investigate the nonlinear evolution of these focusing wave groups when they are subject to small stochastic initial 41

50 t = 4 t = 27 t = Chapter 4. Dynamics of an extreme wave x Figure 4.1 Focusing wave packet in a deterministic initial condition generated as a Gaussian spectrum of waves (3.3) with BFI = perturbations, and to analyze the resulting structure of the stochastic solution. As mentioned in the concluding remarks of Chapter 3, we do this by assigning the wave field of interest to the mean component of the initial DO expansion (2.12), while the modes and stochastic coefficients carry small stochastic perturbations of a desired shape. 4.2 Adaptivity of the DO modes We first investigate the ability of the DO modes to adaptively track the emergence of a localized extreme wave in the mean. We consider the following random initial condition 3 u(x, t ;ω) = ū(x, t ) + A n e iθn(ω) e i k n x, (4.1) where the mean ū(x, t ) is assigned the initial wave field of Figure 4.1 (which contains the focusing wave group), and the Fourier modes represent small stochastic perturbations with wavenumbers k n =.2n, deterministic amplitude A n =.1 and random uncorrelated phases θ n (ω) drawn from a uniform distribution on [,2π]. As before, the stochastic coefficients and modes of the initial DO expansion (2.12) are initiated through the relations (2.44), resulting in 6 DO modes. The solution for the mean and the first four DO modes at various times is plotted in Figure 4.2, where both the modulus (blue) and real part (orange) are represented. While we have assigned global Fourier mode perturbations to the initial DO modes, at time t = 27 when the extreme wave occurs in the mean the first two DO modes are observed to converge towards the location of the extreme wave. This reveals that the dominant 42 n=1

51 Mode 4 Mode 3 Mode 2 Mode 1 Mean 4.3. Attractor of an idealized extreme wave t = t = x x x t = 27 Figure 4.2 Mean and first four DO modes at various times for the initial wave field of Figure 4.1 (contained in the mean) subject to small stochastic perturbations (contained in the modes). Both the complex envelope modulus (blue) and real part (orange) are represented. directions of stochastic fluctuations at that time are concentrated within the extreme wave itself. This is not surprising considering that the focusing properties of localized wave groups depend in a sensitive manner on their initial amplitude (Cousins & Sapsis, 215b), meaning that the initial stochastic perturbations will be more amplified at the location of the focusing wave group than elsewhere in the domain. Still, these results show the nice time-adaptive property of the DO modes, able to track the main directions of stochastic fluctuations even these are highly time-dependent. Such a property is completely absent from order-reduction schemes that rely on fixed basis functions such as the POD. 4.3 Attractor of an idealized extreme wave We now study the evolution of an isolated wave packet of the idealized form u(x, t ) = A sech(x/l ), without any background spectrum of waves. This idealized shape has been extensively studied in the literature (Yuen & Lake, 1975; Peregrine, 1983; Dysthe & Trulsen, 1999; Cousins & Sapsis, 215b) as a prototype model for the focusing wave groups observed in irregular wave fields such as the one in Figure 4.1. Indeed, such a wave packet will either focus and grow in amplitude when its initial amplitude A is larger than the critical value A,c = 1/( 2L ), leading to an extreme wave, or broaden and decay otherwise. Here, we consider a length scale L = 7 and amplitude A =.15 giving roughly similar properties to 43

52 Mode 4 Mode 3 Mode 2 Mode 1 Mean Chapter 4. Dynamics of an extreme wave t = x t = x t = x Figure 4.3 Mean and first four DO modes at various times for a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations. Both the complex envelope modulus (blue) and real part (orange) are represented. Note that t = 3 would be the time of maximum focusing for the unperturbed wave group. those of the focusing localized wave group initially observed in Figure 4.1. A deterministic simulation reveals that these values lead to a maximum focusing of.24 around t = 3. We now study the evolution of this idealized localized wave group subject to small stochastic perturbations. We consider the following random initial condition 3 u(x, t ;ω) = A sech(x/l ) + sech(x/l ) A n e iθn(ω) e i k n x, (4.2) where A =.15, L = 7 and the Fourier modes are mollified with the same sech function as assigned to the mean. Hence they represent localized small stochastic perturbations, with wavenumbers k n =.2n, deterministic amplitude A n =.2 and random uncorrelated phases θ n (ω) drawn from a uniform distribution on [,2π]. The solution for the mean and the modes at various times is plotted in Figure 4.3, where both the modulus (blue) and real part (orange) are represented. The initial perturbation implies that every realization corresponds to a slightly different initial shape for the localized wave group, in such a way that the DO modes evolve in a time-dependent local optimal basis to describe the dominant directions in the fluctuations of the realizations around the mean. In Figure 4.4, we show the energy present in the mean together with the stochastic energy present in the modes. We observe that the 44 n=1

53 E[Y i 2 ] 4.3. Attractor of an idealized extreme wave 1 Mean Mode Mode 2 Mode 3 Mode 4 Mode Mode t Figure 4.4 Energy of the mean ū,ū and the modes E[Y 2 ] for a localized wave group of the i form A sech(x/l ) subject to small localized stochastic perturbations. Figure 4.5 Stochastic attractor at various times for the solution to a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations. The attractor is represented in terms of the 3D scatter plot of the first three stochastic coefficients for all realizations. In addition, each realization is assigned a color indicative of the maximum value of its corresponding envelope modulus u(x, t;ω). mean is transferring energy to the modes as time evolves, showing that the initial stochastic fluctuations of the realizations around the mean become amplified over time. In Figure 4.5, we display the stochastic attractor of the solution, represented as the 3D scatter plot of the first three stochastic coefficients for all realizations. It is observed that the solution falls on an attractor of low effective dimensionality. Indeed, the attractor for the first three stochastic coefficients appears as a locally two-dimensional structure in the three-dimensional space of all possible realizations. We have also colored each realization according to the maximum value of its corresponding envelope modulus, and we note that at t = 3 a clear correlation emerges between the maximum of the envelope modulus for a given realization (i.e. the degree of focusing of the wave group) and the associated stochastic coefficients. In Figure 4.6 we illustrate the flows of stochastic energy between modes and the mean. Specifically, we represent the energy production in every mode due to (i) its linear interaction with the mean (first column), (ii) its nonlinear simultaneous interaction with the mean and two other modes (second column) and (iii) its nonlinear interaction with three other modes (third column). 45

54 member with minimum u member with maximum u Mode 4 Mode 3 Mode 2 Mode 1 Chapter 4. Dynamics of an extreme wave # #1-5 2 " mean!i # #1-5 5 " mean;mn!i # #1-5 1 " mnl!i # # t -5 # # t -5 # # t Figure 4.6 Modal energy production in the first four modes due to (i) linear interaction with the mean flow (first column), (ii) nonlinear simultaneous interaction with the mean and two other modes (second column) and (iii) nonlinear interaction with three other modes (third column) for the solution to a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations. t = t = 3 t = x x x Figure 4.7 Two realizations at various times of the solution to a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations. Both the surface elevation (blue) and complex envelope modulus (orange) are represented. The top row shows the realization with maximum envelope modulus u(x, t;ω) at t = 3, while the bottom row shows that with minimum envelope modulus. Energy transfers appear to be dominated by the first mode, as could be inferred from Figure 4.4. The spikes in the energy transfers associated with DO mode 4 around t 2 are an artifact due to its crossing with another mode (see Figure 4.4). Finally, we plot in Figure 4.7 the surface elevation (blue) and envelope modulus (orange) at various times of the two realizations that have the maximum (top row) and minimum (bottom row) envelope modulus at t = 3. No big differences in the degree of focusing at t = 3 are observed between the two. 46

55 Chapter 5 Results with the MNLS equation In this chapter, we present results from a higher-order version of the NLS equation, the modified nonlinear Schrödinger (MNLS) equation that we briefly review in Section 5.1, along with the corresponding DO reduced-order equations. Similarly to what was done in Section 3.1 for the NLS equation, in Section 5.2 we validate our new DO equations for the MNLS equation through the simulation of a uniform wavetrain undergoing semi-stochastic Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence. Finally, in Section 5.3 we investigate the behavior under the MNLS equation of an idealized extreme wave subject to small stochastic initial perturbations and we compare the results to the corresponding ones for the NLS equation from Section Modified nonlinear Schrödinger and DO equations The modified nonlinear Schrödinger (MNLS) equation was introduced by Dysthe (1979) with the aim of relaxing the steepness limitation of the original NLS equation. Taking the perturbation expansion in the wave steepness ε = k a leading to the NLS equation one step further to O (ε 4 ), the MNLS equation is obtained. In addition to being more accurate for values of the wave steepness larger than.15, this new equation corrects an important shortcoming of the NLS equation in two dimensions. Indeed, the instability region of a uniform wave subject to two-dimensional perturbations is infinite in extent (in the perturbation wavenumber plane) for the NLS equation (Yuen & Lake, 198; Janssen, 24), which is a highly unphysical result. Meanwhile, the same two-dimensional instability does have a high-wavenumber cutoff under the MNLS equation (Trulsen & Dysthe, 1996). Finally, the MNLS equation has been shown to compare favorably to experiments and simulations of the governing fully nonlinear equations (Lo & Mei, 1985; Goullet & Choi, 211), up to a time scale t 1ε 2, one order of magnitude larger than that for the NLS equation (Trulsen et al., 21). The MNLS equation is presented in Appendix B, together with its corresponding DO equations. Except for the deterministic forcing fields F, F i, F i j and F i j k appearing in the DO evolution 47

56 Chapter 5. Results with the MNLS equation equation (2.21) and which can be found in the Appendix, the DO equations for MNLS are exactly the same as those presented in Section 2.2 for the NLS equation. Hence the expressions for energy transfers can be directly derived from the results of Section 2.3, and the numerical implementation procedure described in Section 2.4 is still valid. Note however that, this time, all equations (i.e. for the mean, the modes and the stochastic coefficients) are advanced in time using an explicit 4th-order Runge-Kutta scheme with a nondimensional time step of.1, and the spatial derivatives are calculated with a spectral method. As is usually done wen numerically solving the MNLS equation, modulation wavenumbers k greater than 1 in the mean and the modes are deleted at each time step (Dysthe et al., 23). 5.2 Idealized Benjamin-Feir instability In order to validate the DO equations for MNLS and their numerical implementation, we compute the stochastic evolution of a uniform plane wave subject to small Fourier mode perturbations with deterministic amplitude but random phase. This situation is expected to lead to Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence, and was already considered in Section 3.1 in the context of the NLS equation. Here, we use the same initial condition (3.1) under the same parameter values as in Section 3.1, that is, we consider a spatially constant envelope of amplitude a =.1 that is perturbed with one unstable Fourier mode of modulation wavenumber k 1 =.2 and one stable Fourier mode of wavenumber k 2 =.4. The Fourier modes are assigned independent random phases and deterministic small amplitude equal to.36, resulting in a total of 4 DO modes. The solution for the mean and the modes at various times is plotted in Figure 5.1, where both the modulus (blue) and real part (orange) are represented. In Figure 5.2, we show the energy present in the mean ū,ū together with the stochastic energy present in the modes E[Y 2 ]. The i observations from Section 3.1 are still valid here. In a nutshell, the DO modes 1 and 2 at t = 8 in Figure 5.1 still represent the linearly unstable modulation k 1 =.2 that was assigned in the initial condition, while modes 3 and 4 represent the stable modulation k 2 =.4, and the spatially constant mean still contains the uniform carrier wave. Figure 5.2 shows that the stochastic energy in the linearly unstable modes 1 and 2 grows then decays while the contrary happens to the energy of the mean (i.e. the uniform wave), in perfect accordance with Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence. For a quantitative comparison with deterministic theory and similarly to what we did in Section 3.1, we computed the MNLS solution to a deterministic initial condition of the form u(x,) =.1+.36cos k m x where k m =.2, i.e. similar in structure to one realization of the random DO initial condition, and we retrieve the energy of the carrier wave and the modulations by means of a Fourier transform of the envelope. The normalized resulting squared modulus of the Fourier coefficients for k = (carrier wave),.2 (unstable modulation) and.4 (stable harmonic) are shown as dashed lines in Figure 5.2. This time, the agreement between the stochastic DO computation and the deterministic calculation is not as good as what we 48

57 E[Y i 2 ] Mode 4 Mode 3 Mode 2 Mode 1 Mean 5.2. Idealized Benjamin-Feir instability t = t = t = x x x Figure 5.1 Mean and DO modes at various times for the stochastic BF instability and FPU recurrence with random phase in the initial modulation, under the MNLS equation. Both the complex envelope modulus (blue) and real part (orange) are represented. Note that while the domain size is L = 5 2π, we only plot the solution over a portion of the domain corresponding to the wavelength of the linearly most unstable wavenumber k m = 2a = Mean Mode 1 1 Mode 2 Mode 3 Mode 4 Deterministic t Figure 5.2 Energy of the mean ū,ū and the modes E[Y 2 ] for the stochastic BF instability and i FPU recurrence with random phase in the initial modulation, under the MNLS equation. The dashed lines show the corresponding energies for a deterministic simulation of an equivalent initial condition u(x,) = cos k m x with k m =.2, with the energies obtained as the normalized squared modulus of the Fourier coefficients of wavenumber k = (carrier wave),.2 (unstable modulation) and.4 (stable harmonic) (similarly to Figure 1.3). obtained for the NLS equation, and the reason for this discrepancy has yet to be understood. Note also that the collapse of the numerical solution observed in Section 3.1 for the NLS 49

58 Chapter 5. Results with the MNLS equation equation is also happening here, as can be seen from the shape of the modes at t = 14 which deviate from deterministic theory, or the oscillations in the energy levels from t 19 in Figure 5.2. As a conclusion, we have shown that our DO equations for MNLS produce results in accordance with deterministic Benjamin-Feir and Fermi-Pasta-Ulam recurrence theory. While the quantitative agreement between our DO simulation and deterministic results is not as excellent as what was obtained for the NLS equation, we nevertheless proceed with the investigation of extreme waves in the following section. 5.3 Attractor of an idealized extreme wave Here we investigate the behavior under the MNLS equation of an idealized extreme wave subject to small stochastic initial perturbations. We consider the exact same case as that studied in Section 4.3 in the context of the NLS equation. Specifically, we study the evolution of an idealized wave packet of the form u(x, t ) = A sech(x/l ) with L = 7 and A =.15. A deterministic simulation of the MNLS equation reveals that these values lead to a maximum focusing amplitude of.21 at t = 41. The lower growth rate and maximum focusing amplitude for MNLS as compared to NLS have both been previously reported in the literature (Dysthe, 1979; Cousins & Sapsis, 215b). We now study the behavior of this idealized localized wave group subject to small stochastic initial perturbations. We consider the same random initial condition (4.2) as in Section 4.3, we compute its evolution under the MNLS equation in the DO framework and we compare the obtained stochastic solution with the NLS results from Section 4.3. The solution for the mean and the modes at various times is plotted in Figure 5.3, where both the modulus (blue) and real part (orange) are represented. While the shape of the modes is overall similar to the NLS results of Figure 4.3, the asymmetric profile of the mean in the case of MNLS is a major difference with NLS and reflects the observed appearance of a tail during the evolution of certain envelope solitons (Yuen & Lake, 1975). In Figure 5.4, we show the energy present in the mean together with the stochastic energy present in the modes, and they are very similar to those observed for NLS in Figure 5.4. In Figure 5.5, we display the stochastic attractor of the solution, represented as the 3D scatter plot of the first three stochastic coefficients for all realizations. Additionally, each realization is colored according to the maximum value of its corresponding envelope modulus. Comparing this attractor to the NLS one of Figure 5.5 shows that both attractors are roughly similar at the time of maximum focusing (t = 3 for NLS, t = 41 for MNLS), while differences arise between NLS and MNLS at a later time. In Figure 5.6 we illustrate the flows of stochastic energy between modes and the mean. Specifically, we represent the energy production in every mode due to (i) its linear interaction with the mean (first column), (ii) its nonlinear simultaneous interaction with the mean and two other modes (second column) and (iii) its nonlinear interaction with three other modes (third column). These energy transfer plots are qualitatively similar to the NLS results of Figure 4.6. Finally, we 5

59 E[Y i 2 ] Mode 4 Mode 3 Mode 2 Mode 1 Mean 5.3. Attractor of an idealized extreme wave t = x t = x t = x Figure 5.3 Mean and first four DO modes at various times for a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations, under the MNLS equation. Both the complex envelope modulus (blue) and real part (orange) are represented. Note that t = 41 would be the time of maximum focusing for the unperturbed wave group. 1 Mean Mode Mode 2 Mode 3 Mode 4 Mode Mode t Figure 5.4 Energy of the mean ū,ū and the modes E[Y 2 ] for a localized wave group of i the form A sech(x/l ) subject to small localized stochastic perturbations, under the MNLS equation. plot in Figure 5.7 the surface elevation (blue) and envelope modulus (orange) at various times of the two realizations that have the maximum (top row) and minimum (bottom row) envelope modulus at t = 41. The tail observed in the mean in Figure 5.3 is reflected in these individual realizations. Note that such dail does not appear in the NLS realizations of Figure 4.7 that maintain a regular symmetric shape. As was observed in the case of NLS, no big differences in 51

60 Mode 4 Mode 3 Mode 2 Mode 1 Chapter 5. Results with the MNLS equation Figure 5.5 Stochastic attractor at various times for the MNLS solution to a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations. The attractor is represented in terms of the 3D scatter plot of the first three stochastic coefficients for all realizations. In addition, each realization is assigned a color indicative of the maximum value of its corresponding envelope modulus u(x, t;ω). # #1-5 1 " mean!i # #1-5 5 " mean;mn!i # #1-5 1 " mnl!i # # t -5 # # t -5 # # t Figure 5.6 Modal energy production in the first four modes due to (i) linear interaction with the mean flow (first column), (ii) nonlinear simultaneous interaction with the mean and two other modes (second column) and (iii) nonlinear interaction with three other modes (third column) for the MNLS solution to a localized wave group of the form A sech(x/l ) initially subject to small localized stochastic perturbations. the degree of focusing at t = 41 are observed between the two realizations since the initial perturbations are small. However, here a difference in shape is observed between the two realizations, the top one having a more pronounced tail at t = 41 than the bottom one, a situation that reverses at t =