Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model
QQuantification and prediction of extreme events in aone-dimensional nonlinear dispersive wave model
Will Cousins and Themistoklis P. Sapsis ∗ Department of Mechanical Engineering, Massachusetts Institute of Technology,77 Massachusetts Av., Cambridge, MA 02139September 12, 2018
Abstract
The aim of this work is the quantification and prediction of rare events characterized byextreme intensity in nonlinear waves with broad spectra. We consider a one-dimensional non-linear model with deep-water waves dispersion relation, the Majda-McLaughlin-Tabak (MMT)model, in a dynamical regime that is characterized by broadband spectrum and strong non-linear energy transfers during the development of intermittent events with finite-lifetime. Tounderstand the energy transfers that occur during the development of an extreme event weperform a spatially localized analysis of the energy distribution along different wavenumbersby means of the Gabor transform. A stochastic analysis of the Gabor coefficients reveals i) thelow-dimensionality of the intermittent structures, ii) the interplay between non-Gaussian statis-tical properties and nonlinear energy transfers between modes, as well as iii) the critical scales(or critical Gabor coefficients) where a critical amount of energy can trigger the formation ofan extreme event. We analyze the unstable character of these special localized modes directlythrough the system equation and show that these intermittent events are due to the interplayof the system nonlinearity, the wave dispersion, and the wave dissipation which mimics wavebreaking. These localized instabilities are triggered by random localizations of energy in space,created by the dispersive propagation of low-amplitude waves with random phase. Based onthese properties, we design low-dimensional functionals of these Gabor coefficients that allowfor the prediction of the extreme event well before the nonlinear interactions begin to occur.
Extreme or rare events have attracted substantial attention in various scientific fields both becauseof their catastrophic impact but also because of the serious lack of specialized mathematical toolsfor the analysis of the underlying physics. Important examples can be found in i) the environmentalfield: rogue waves in the ocean [1, 2, 3, 4], extreme weather and climate events [5, 6], and ii) theengineering field: overloads and failures in power grids [7, 8], stability loss and capsizing of ships inmild waves [9]. For all of the above applications it has now been well established that extreme eventsoccur much more frequently than it was initially believed and that their traditional characterizationas ‘rare events’ (especially in a Gaussian context where a rare event has practically zero probability) ∗ Corresponding author: [email protected], Tel: (617) 324-7508, Fax: (617) 253-8689 a r X i v : . [ n li n . C D ] J a n everely underestimates the frequency of their occurrence. Therefore, it is important to study themmore thoroughly and develop effective algorithms for their prediction.Extreme events refer to system responses with magnitude that is much larger than the typicaldeviation that characterizes the system response. Thus, from the very nature of these events itcan be concluded that traditional analysis tools restricted to second order statistics would not besufficient for their understanding. Apart from their intermittent properties, another manifestationof the non-Gaussian character of extreme events is the strong localization of energy in (physical ormodal) space – a situation that is inherently connected with non-linear dynamics and transient orpersistent instabilities, which has been shown (see e.g. [10, 11]) to be an important factor that canlead to non-Gaussian statistics.These characteristics also define the modeling challenges for the study of these systems withthe most important being the interplay of a few intermittent modes with a large number of modesthat act as ‘reservoir’ of energy for the former. This large set of modes is usually characterizedby a broadband spectrum consisting of dispersive waves with weakly non-Gaussian statistics thatpropagate and sporadically give rise to extreme, localized events. In contrast to this large setof waves, extreme events are characterized by strong nonlinear energy transfers and non–Gaussianstatistics. Therefore, we have on the one hand a nearly Gaussian ‘heat bath’ of waves that propagatein the presence of dispersion which leads to energy localization in random scales and places, andon the other hand a nonlinear mechanism that uses the former as excitation to generate extremeevents [12].It is clear from the above discussion that a mathematical framework able to handle problemscharacterized by extreme events should include higher order statistics and also should be able todeal with the inherent nonlinear character of the underlying dynamics. However, the computationalcost associated with these requirements would be enormous since i) the number of physical degreesof freedom is usually very large and ii) because the description of non-Gaussian properties andin particular the description of rare events that ‘live’ in the tails of the distribution requires asubstantial amount of realizations which is very hard to obtain and process in a direct Monte-Carloframework. In addition, a purely statistical understanding cannot provide a rigorous analysis ofthe underlying physical mechanisms.On the other hand, order-reduction approaches based, for example, on Polynomial Chaos expan-sions or Proper Orthogonal decompositions have proven to be of limited applicability in nonlinearsystems with intermittency [13]. Due to their localized spatial and temporal character, extremeevents carry only small amounts of energy compared with other global modes that characterizethe full response field. Therefore, standard order-reduction techniques will most likely miss theessential parts of the extreme event dynamics.To simulate the dynamical mechanisms that lead to the generation of extreme events, we usethe MMT model, a one-dimensional nonlinear dispersive equation originally proposed by Majda,McLaughlin, and Tabak to assess the validity of weak turbulence theory [14]. MMT admits four-wave resonant interactions and, when coupled with large scale forcing and small scale damping,admits a rich family of spectra exhibiting direct and inverse cascades [15, 16]. Zakharov et. al.have also analyzed the MMT model in detail and have used large amplitude coherent structurespresent in MMT as models of extreme ocean waves [17, 18, 19]. In this work, we analyze in detail the‘solitonic’ coherent structures in the focusing MMT, which have also been investigated by Cai et.al. [16]. In their early stages, these localized structures resemble self-similar spatial collapses andrapidly transfer energy to small scales where it is dissipated [16]. We are particularly interestedin these localized structures as they generate states which are extreme compared to the benign2ackground out of which they arise.In the present work, we first aim to develop analytical and numerical tools in order to understandhow these localized extreme events are triggered by spatially localized perturbations in the MMTmodel. We illustrate that there is a critical spatial lengthscale and a critical amount of energyassociated with it that leads to the occurrence of extreme solutions. This critical scale is the resultof the interplay between wave dispersion, wave nonlinearity and selective dissipation that occursin high wavenumbers. For perturbations of a zero background state we are able to analyze thisphenomena directly by deriving a family of scale invariant solutions. However, the critical amountof energy depends also on the background energy level of the system, the effects of which we analyzenumerically. In contrast to the standard linearized analysis, which considers small Fourier modeperturbations about about a given state, the framework presented here considers spatially localizedperturbations that are not necessarily small.We illustrate that these extreme events are characterized by low-dimensionality and we use aspatially localized basis, a Gabor basis, with localization characteristics tuned according to theresults of the previous conclusions. Using the projected information of the extreme events to thislocalized basis we perform a statistical analysis of the Gabor coefficients to reveal the stronglynon-Gaussian character associated with the strongly nonlinear interactions of these modes duringan extreme event. Note that this statistical structure, which is directly connected to the nonlinearenergy transfers that take place, is otherwise ‘buried’ in the broad-band spectrum of the full wavefield and its only signature in the stochastic field response is the heavy tail statistics.Finally, we formulate predictive functionals that efficiently characterize the domain of attractionto the extreme event solutions. These predictive functionals are formulated in a probabilistic fashionin terms of the Gabor coefficients that correspond to the critical lengthscales. Given the currentinformation of the wavefield, they provide the probability of occurrence of an extreme event in alater time instant. Note that the propagation of waves (having random phases) in the presence ofdispersion creates conditions for localization of energy in arbitrary scales and positions in space.The formulated probabilistic functionals assess these random localizations of energy and quantifythe probability that they will lead to an occurrence of an extreme event in the future. We consider the following one-dimensional partial differential equation originally proposed by Ma-jda, McLaughlin, and Tabak [14] for the study of 1D wave turbulence: iu t = | ∂ x | α u + λ | ∂ x | − β/ (cid:18)(cid:12)(cid:12)(cid:12) | ∂ x | − β/ u (cid:12)(cid:12)(cid:12) | ∂ x | − β/ u (cid:19) + iDu (1)where u is a complex scalar. On the real line, the pseudodifferential operator | ∂ x | α is definedthrough the Fourier transform as follows: (cid:92) | ∂ x | α u ( k ) = | k | α (cid:98) u ( k ) . This operator may also be defined analogously on a periodic domain. The MMT equation was intro-duced on the basis of a simple enough model to test thoroughly the predictions of weak turbulencetheory. In the context of dispersive nonlinear waves it provides a prototype system with non-trivial3nergy transfers between modes or scales, non-Gaussian statistics with heavy tails, and intermittentevents with high intensity, while remaining accessible to high resolution simulations [14, 15, 16, 20].Therefore, it is an ideal basis to assess the performance of probabilistic quantification algorithmsfor the occurrence and prediction of extreme events. Even though the MMT model was originallyderived through a heuristic approach, it was later shown that it can be rigorously obtained as anapproximation of the fully nonlinear wave system equations [21].In the present work the parameter α is set to 1/2 as this matches the dispersion relation fordeep water waves ω = | k | . Setting α = 2 and β = 0 in 1 yields the nonlinear Schr¨odinger equation(ignoring the dissipation term). As in [14] we include dissipation at small scales (modeling e.g.wave breaking in the context of water waves) through a selective Laplacian operator Du, definedin Fourier space: (cid:99) Du ( k ) = (cid:26) − ( | k | − k ∗ ) ˆ u ( k ) | k | > k ∗ | k | ≤ k ∗ Similar dissipation models have been used in more realistic settings involving ocean water waves[22]. The critical wavenumber is taken as k ∗ = 500 which is a value that is large enough so that itallows for the development of nonlinear instabilities that lead to extreme waves and small enoughto create energy cascades in higher wavenumbers and thus, allow for these waves to exist only forfinite-time.We choose λ = −
4, which corresponds to the focusing case and gives rise to four-wave resonantinteractions [14] which are relevant for ocean gravity waves. The latter cannot resonate in orderlower than four if we exclude the short wave gravity-capillary region of the spectrum where three-wave interactions can occur [22]. Note that even though a connection with the fully nonlinear wavesystem would require λ > u has a nearly Gaussian distribution. For a linearmodel, the distribution would remain nearly Gaussian as the modes evolve independently. Inter-estingly, even in simulations of the focusing nonlinear Schr¨odinger equation we find that u remainsnearly Gaussian. However, for the MMT model we find that the distribution u develops heavy tailswith a power law decay rate (see Figure 1).The heavy tails in solutions of MMT are induced by the intermittent formation and subsequentcollapse of localized extreme events arising out of a nearly Gaussian background. Figure 2 displaysthe origination and disappearance of such an extreme event. In their early stages these extremeevents resemble the collapses that are present in focusing MMT with no dissipation. In thesecollapses, which have been described by Cai et. al. [16], energy is dramatically transferred tosmaller scales and the solution experiences a singularity in finite time. In our simulations, the smallscale dissipation included in (1) (modeling wave breaking in the context of water waves) ensuresthat u remains regular for all times. Collapse dynamics have been found to induce heavy tailedstatistics in other situations as well, such as the damped-driven quintic 1D nonlinear Schr¨odingerequation [24]. 4 −4 −2 P D F V a l ue Re(u)
MMTNLS
Figure 1: Probability density for the real part of u for simulations of NLS and MMT ( α = 1 / , β = 0)Figure 2: Example of an extreme event arising out of a weakly non-Gaussian ‘heat bath’ of dispersivewaves with random phase. 5 .1 Numerical simulation and computation of statistics We solve (1) for x ∈ [0 , π ] with periodic boundary conditions using a Fourier method in spacecombined with a 4th order Runge-Kutta exponential time differencing scheme [25, 20]. This schemerequires evaluation of the function φ ( z ) = ( e z − /z . Naive computation of φ can suffer fromnumerical cancellation error for small z [26]. We use a Pad´e approximation code from the EXPINTsoftware package, which does not suffer from such errors [27]. We use 2 Fourier modes with atime step of 10 − ; results in this work were insensitive to further refinement in grid size.For the statistical studies performed in this work, we evolve a sum of 31 complex exponentialswith independent, uniformly distributed random phases. We compute statistics by averaging overtime and space over 300 ensembles, each spanning 100 time units ( t = 100 to t = 200). There isno external forcing in our simulations and all the energy of the system comes through the initialconditions while dissipation occurs whenever an extreme event takes place. To this end, we do notobserve an exact statistical steady state in our simulations, but after an initial transient where amoderate amount of energy is dissipated through selective damping, the solution settles to a nearly(or very slowly varying) statistical steady state where the L norm decays slowly (see Figure 3). Wefocus on this slowly varying regime where roughly 2-4 extreme events occur per simulation in thetime window t ∈ [100 , L norm of the solution: after 100 time units the decay rate becomessmall. Right: Locations of extreme events in an ensemble of simulations. In this section we examine the role of spatially localized energy in the formation of an extremeevent. More specifically, we define the energy E of a solution as E (cid:44) r = ˆ (cid:12)(cid:12) u ( x ) (cid:12)(cid:12) dx, where r is the L norm of the solution, which is conserved by undamped MMT [17]. In theundamped MMT equation, localized initial data with energy above some critical level leads to a6nite time blowup [16, 17]. Here we examine how this critical energy level varies with the degreeof initial energy localization, as well as the energy of the background state, in the presence ofselective dissipation. Both of these parameters are important to determine the critical scale thatis most sensitive for the formation of an extreme event. For the zero background case, we are ableto analytically determine this relationship by deriving a scale invariant family of solutions. Weinvestigate the non-zero background case numerically. Zero background energy: scale invariant solutions.
We begin our analysis by focusing onlocalized perturbations when we have zero energy background in the system. More specifically, weconsider a family of initial data of the form u ( x,
0) = u ( x ; c, L ) = ce − x/L ) and determine howthe critical energy level required for blow-up depends on the length scale L . To do so, we derive an L -parametric family of solutions w L , L >
0, defined by the scaling of a given solution u ( x, t ) w L ( x, t ) = 1 L p u (cid:18) xL , tL q (cid:19) . To determine p and q , we plug this anzatz into MMT with no dissipation, which gives: iL p + q u t = 1 L p + α | ∂ x | α u + λL p − β | ∂ x | − β/ (cid:18)(cid:12)(cid:12)(cid:12) | ∂ x | − β/ u (cid:12)(cid:12)(cid:12) | ∂ x | − β/ u (cid:19) . So w is also a solution to MMT if q = α and p = ( α + β ) /
2, giving us the following family ofsolutions: w L ( x, t ) = 1 L ( α + β ) / u (cid:18) xL , tL α (cid:19) Therefore, if for a reference lengthscale L = 1 , we have the critical energy norm r crit (1) (associatedwith an initial condition u ( x ; c ∗ , L will be r crit ( L ) = 1 L α + β ˆ u (cid:16) xL ; c ∗ , (cid:17) dx = L − α − β r crit (1) . Hence, the critical energy norm r crit ( L ) required to initiate a blow-up is given by r crit ( L ) = L (1 − α − β ) / r crit (1) . (2)We consider the special case α = 1 / , β = 0, which gives r crit ( L ) = √ Lr crit (1) . (3)Since the above function decreases to 0 as L becomes small, in the deep water wave dispersion caseonly a small amount of localized energy is sufficient to initiate a blow-up. This fact holds as longas the exponent of L in (2) is positive, meaning β < − α , or simply β < / α = 1 /
2. Pushkarev and Zakharov [19] use β = − β = − r crit ( L ) = L / r crit (1), so this relationship ( r crit decreasing as L decreases) would presumably be even stronger than the β = 0 case we consider.Note that for the case that selective dissipation is present, very localized amounts of energy willbe rapidly dissipated. In particular, if energy is too localized, then the selective Laplacian damping7s dominant compared with the instability of the nonlinear terms and the amplitude of u decreasesrelative to its initial state. However, for values of L that are not excessively small, we have a rapidgrowth of the amplitude of u that leads to an energy cascade (see next section) to smaller scalesand subsequent dissipation by the selective Laplacian. In this way the high wavenumber dampingprevents the formation of a singularity due to continuous energy transfer and accumulation toinfinitesimally small scales and results in a finite lifetime for the extreme event (Figure 2).Therefore, in the damped MMT, for each localization scale L that is not excessively small weexpect there to be a critical amount of energy that will trigger a nonlinear instability resulting inan extreme event. We expect that, except for excessively small values of L , the above analysis willstill hold and the dissipation will only become relevant in the late stages of an extreme event whereit prevents the formation of a singularity. We quantify the critical energy for the damped systemusing two different measures. First, we compute the finite-time divergence of nearby (in terms ofenergy) initial perturbations through the quantity | ∂ r q ( r, L ) | (cid:44) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂r max x,t | u ( x, t ; r ) | max x u ( x, r ) (cid:12)(cid:12)(cid:12)(cid:12) This quantity is displayed by a color plot in Figure 4. We use the sharp ridge of | ∂ r q | to determinethe critical energy level at which the transition to extreme events occurs. Additionally, we determinethe critical energy level by determining the set of values ( r, L ) at which q ( r, L ) > .
5. The blackcurve in Figure 4 outlines the region where q exceeds this threshold value. This curve comparesfavorably with the results from the first method. Also in Figure 4 we present with a red dashedcurve the critical energy norm for the undamped system, given by (3). We emphasize that eventhough Figure 4 was generated by numerically solving (1) on a domain of size 16 π with periodicboundary conditions, these results do not change if the domain size is increased further due to thelocalization of these examples. This behavior contrasts sharply with similar experiments of thenonlinear Schrodinger equation, where the values of | q | never become large and the sharp gradientseen in Figure 4 does not occur.We note that for the case of the damped system the critical energy norm closely resembles theanalytical prediction (3), which is a result of the interplay between dispersion and nonlinearity.This is the case until we reach the critical scale L c , below which dissipation is important and noextreme solutions can occur. To this end this spatial scale L c is the most sensitive to localizedperturbations i.e. it can be triggered with the lowest amount of energy, and it is essentially thesmallest scale where dissipation is still negligible. The existence of this critical scale that triggersextreme events is the result of the synergistic action of dispersion, nonlinearity, and small scaledissipation. Case of finite background energy.
We now consider the formation of an extreme event out ofa background with non-zero energy, that is, we evolve initial data of the form u ( x,
0) = b + ce − x/L ) .We first consider the case of small ratio cb (cid:28) u byperforming a linearized stability analysis about the plane wave solution of MMT, u ( x, t ) = be − iλb t .For the nonlinear Schr¨odinger equation with periodic boundary conditions, this plane wavesolution is unstable to Fourier mode perturbations of wavenumber n when the following conditionis satisfied: n < L x π (cid:112) − λb , where L x is the domain width. The above is known as the Benjamin-Feir instability and has beenstudied extensively by many authors [28, 29, 30, 31].8igure 4: Critical energy norm of a localized perturbation that leads to the formation of an ex-treme event for the undamped (red dashed curve) and the damped MMT model in the absenceof background energy. The latter is described in terms of the finite-time divergence q of nearbytrajectories (color map) and the maximum value of the response field | u | . In the context of the undamped MMT equation, the Benjamin-Feir instability can be generalized.In particular one can show that for the case β = 0, the plane wave solution is unstable if λ < n < L x π (cid:0) − λb (cid:1) /α . (4)Clearly, for α = 2, the above result agrees exactly with the classical Benjamin-Feir instabilitycriterion of NLS.We emphasize that although the Benjamin-Feir modulation instability is present in both thefocusing MMT and the NLS, its manifestation is not the same in each case. We illustrate this factby numerical experiments involving no selective damping. For both equations, we take λ = − L x = 2 π , meaning that the critical value of b in (4) is roughly 0.35. Values of b larger than thisadmit at least one unstable mode, and positive b less than this value have no unstable modes. Weevolve initial data of the form u ( x,
0) = b + (cid:15) cos( x ) with b ≈ .
34 and b ≈ .
36. For each valueof b , we set (cid:15) = 0 .
01. When b ≈ .
34, the small perturbation does not grow for the MMT or theNLS, agreeing with the linearized analysis (see Figure 5). For the NLS, unstable perturbations ofthis kind initiate a nearly-periodic orbit where large, but bounded, coherent structures repeatedlyappear and subsequently dissolve in a Fermi-Pasta-Ulam-like recurrent cycle [32, 31]. However, inthe MMT, these unstable perturbations grow continuously and collapse into a singularity in finitetime (see Figure 5). This mechanism of collapse initiation via modulation instability, which has alsobeen studied by Cai et. al. [16], has significant implications for our critical energy analysis for thenonzero background case. Although including high frequency damping will prevent the formation ofa singularity, such damping will not prevent an extreme event from occurring since it only becomes9igure 5: Benjamin-Feir instability for the NLS equation (top) and the MMT model (bottom) with( α = 0 . , β = 0 , λ = − . relevant after energy has been transferred to the small scales; that is, after an extreme event hasalready occurred (as in Figure 2). Thus, if b and L x satisfy the Benjamin-Feir instability criterion(4) with n = 1, initial conditions of the form u ( x,
0) = b + ce − x/L ) will initiate an extreme eventfor any c > ce − x/L ) required to initiatean extreme event for various values of b and L . For b = 0, this critical amplitude is precisely thecurve described above and displayed in Figure 4, and for large enough b this critical amplitudeis infinitesimal due to the Benjamin-Feir instability. For intermediate values of b , the criticalamplitude presents a smooth transition between the two theoretically understood regimes. For b = 0, we noted previously that the more localized the energy is, the smaller amount of this energyis required to initiate a blowup. This fact remains true for nonzero background energy b untilthe point where b is large enough so that a Benjamin-Feir instability occurs, which in this case( L x = 8 π, λ = − , α = 1 /
2) occurs at b = 0 .
25. picture is another manifestation that we canhave extreme responses well below the Benjamin-Feir energy threshold as a result of the interplaybetween nonlinearity, dispersion and dissipation.
So far we have examined the conditions that lead to extreme wave solutions. In this section we willstudy the nonlinear interactions taking place during the occurrence of an extreme event using toolsfrom stochastic analysis. We choose to project the solution u onto an appropriate (localized) set10igure 6: Critical energy norm of localized perturbations that lead to extreme events in the presenceof background energy for the damped MMT model ( α = 0 . , β = 0).of modes. Given the localized character of extreme events, global basis elements such as Fouriermodes will not be able to describe effectively their dynamical properties since, despite their largeamplitude, extreme events carry very small portion of energy of the overall field spectrum.To this end, it is more informative to choose a set of modes which incorporate the localizedcharacter of the extreme events. We use the following family of Gabor basis elements consisting ofcomplex exponentials multiplied by Gaussian window functions: v n ( x ; x c ) (cid:44) exp (cid:20) − d ( x, x c ) L (cid:21) e i πnx/L , n = 0 , , , ... (5)where d ( x, x c ) = min( | x − x c | , π −| x − x c | ) expresses the distance from the center point x c measuredin the periodic domain. We then compute the Gabor projection coefficients of the solution. Y n ( x c , t ) (cid:44) (cid:104) u ( x, t ) , v n ( x ; x c ) (cid:105) / || v n ( x ; x c ) || , where (cid:104)· , ·(cid:105) denotes the standard L inner product. We noted in Section 3 that there is a criticalscale L c ≈ .
01 that is most sensitive to the formation of an extreme event, in that the requiredenergy to trigger an extreme event is smallest at this particular scale. In practice, we have observedthat these extreme events typically originate by energy localization in a slightly larger scale than L c . This motivates our choice of L = π/ ≈ L c , which is still extremely sensitive to smallperturbations (Figure 4). After a sufficient amount of energy is localized in this scale, an extremepeak is then produced as energy is transferred into the smaller scales until it reaches the scale at11igure 7: Energy cascade during an extreme event. Top row: extreme event shown in ( x, t ) spacetogether with the high wavenumber energy E h = (cid:80) n ≥ (cid:107) Y n (cid:107) . Second row: Gabor basis elements v n ( x ; 0) . Third row: modulus of the Gabor coefficients (cid:107) Y n ( x c , t ) (cid:107) . which selective dissipation is present. We have chosen L = 0 .
031 based on this argument with thegoal of using the coefficient Y as an indicator of an upcoming extreme event.In Figure 7 we present the Gabor coefficients in space and time during the occurrence of apair of extreme events. More specifically, in the first row (left) we show the extreme waves in the( x, t ) space. The Gabor basis elements are shown in the second row and the Gabor coefficients areshown in the third row. The Euclidian sum of the oscillatory Gabor coefficients, E h = (cid:80) n ≥ (cid:107) Y n (cid:107) expresses the energy that ‘lives’ in high wavenumbers and this is shown in the top-right panel.For the Gabor coefficients that correspond to the oscillatory basis elements ( Y , Y ), we notethat away from the region of the extreme events, wave components propagate almost independently,according to the dispersion relation, in a close to linear fashion. The energy of the non-oscillatorymode (expressed through the Gabor coefficient Y ) presents a more static (non-propagating) be-havior with high intensity that builds up before the extreme event. During the strongly non-linearphase (of the extreme response), the Gabor coefficients of the oscillatory basis elements are notgoverned by the dispersion relation anymore, but they also present a more static (non-propagating)behavior characterized by a large build up of their energy. This is the result of a strong, nonlinearenergy cascade initiated from the unstable lengthscale L , described by the non-oscillatory mode v , and ending to higher wavenumbers where it is dissipated (Figure 7). Statistics and energy transfers during the dissipation phase.
The strong nonlinearenergy transfer from the unstable scale L to smaller scales is manifested by the non-Gaussianstatistics of the Gabor coefficients during an extreme event. The connection between nonlinearenergy transfers and non-Gaussian statistics has been rigorously established in [10, 11, 33] in thecontext of viscous turbulent flows. Consider an orthonormal set of modes v i , i = 0 , , , ..., n c u ( x, t ) . Then, the MMT equation with α = 0 . , β = 0 u t = − i | ∂ x | / u − iλ | u | u + Du will take the projected form for each v i dY i dt = (cid:68)(cid:104) D − i | ∂ x | / (cid:105) u, v i (cid:69) + (cid:88) k Y k (cid:68)(cid:104) D + | ∂ x | / (cid:105) v k , v i (cid:69) (6) − iλ (cid:42)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + (cid:88) k Y k v k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) u + (cid:88) k Y k v k (cid:33) , v i (cid:43) . The growth rate of the energy of Y i will have the form d | Y i | dt = dY i dt Y ∗ i + dY ∗ i dt Y i = 2 Re (cid:20) dY i dt Y ∗ i (cid:21) . Note that in equation (6) the first line on the right hand side involves either energy conservingterms such as wave dispersion or negative definite terms such as dissipation (which occurs for highwavenumbers only). All the other contributions towards changes of | Y i | will only occur throughthe nonlinear interactions of the modes: d | Y i | dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) NL = − iλ (cid:42)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u + (cid:88) k Y k v k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) u + (cid:88) k Y k v k (cid:33) , v i (cid:43) Y ∗ i where the bar denotes ensemble average. We focus on the energy cascade regime from the mode v to higher wavenumber modes during an extreme event.13igure 9: Isosurfaces of the joint pdf of the imaginary part of Y , Y , and Y near to (left) and awayfrom (right) extreme events.We compute the Gabor coefficients for 400 values of x c equally distributed between 0 and 2 π ,with L ≈ L c . . We then classify each point ( x c , t ) into two regimes: points nearby an extreme eventand points away from extreme events. For each of these groups, we compute the joint statistics ofthe Gabor coefficients using data from an ensemble of MMT simulations with random initial data(details of these simulations are given in Section 2.1). Figure 9 displays the joint statistics of theimaginary parts of Y , Y , and Y near (left subplot) and far (right subplot) from extreme events.Away from extreme events the isosurfaces of the probability density function are elliptical,indicating that the Gabor coefficients are nearly Gaussian in this regime. Moreover, the timeoscillatory character of the wave components results in zero average value (in the ensemble sense)for all the corresponding Gabor coefficients Y i = 0 , i = 1 , , ... (for both regimes). Due to this fact,as well as the Gaussian distribution of the coefficients Y i in the non-extreme events regime, theaverage change of their energy due to nonlinear interactions, becomes zero.On the other hand, the statistics near extreme events are highly non-Gaussian, with Y exhibitinga bimodal distribution. The real parts of the Gabor coefficients are distributed similarly. This non-Gaussian distribution is directly related to the energy cascade from the non-oscillatory mode to thestrongly dissipative, high wavenumber modes. Dynamics during the built-up phase of the extreme event.
In Figure 10, we show howan extreme event trajectory emerges out of the Gaussian background describing the heat bath ofwaves propagating under the dominant effect of the dispersion relation. From the same figure it isclearly illustrated how extreme events are associated with large values of | Y | . The nature of thisassociation is particularly interesting: | Y | becomes large just before (and after) extreme events. Anexample of this behavior is displayed in Figure 11, where we observe the increase of | Y | while theoverall response field | u | has regular values. This growth continues until we have an extreme eventand it is followed by a sudden drop that is associated with an energy transfer to high wavenumbersillustrated by the energy E h (as described previously). This agrees with observations by Cai et.al. [16] that these extreme events form by focusing energy to high wavenumbers until saturationat a critical scale, at which point they radiate energy back to larger scales. We are particularly14igure 10: The red curves show trajectories of the imaginary parts of Y , Y , and Y during extremeevents. The blue surface is an isosurface of the joint density function for the imaginary parts of Y , Y , and Y containing 97% of the total probability–this shape is dominated by the Gaussian randomwaves that propagate with random phase under the effect of dispersion and weak nonlinearity.interested in the predictive utility of the localized energy | Y | buildup that occurs before extremeevents, often 1-2 time units in advance for the considered set of parameters.This phenomenon agrees with our analysis from Section 3, where we showed that a sufficientamount of localized energy is sufficient to trigger an extreme event. These localizations of energyoccur randomly through the dispersive propagation of waves that have random phases. Due to thelocalized nature of v , the associated Gabor coefficient | Y | measures such localized energy and isthus an indicator of an extreme event in the near future. When the extreme event occurs, energy istransferred into the more oscillatory modes ( v , v , . . . ), but the Gabor coefficients associated withthese modes lack predictive utility since they grow simultaneously with, rather than prior to, theextreme event (see Figure 11). The Gabor coefficient Y is a measure of energy localized at a particularly sensitive length scale, atwhich only a small amount of energy is necessary to trigger an extreme event. Thus, large valuesof | Y | often indicate that an extreme event will occur in the near future. We now use this fact todevelop short-term predictive capacity for extreme events. To do so, we first compute, for variousvalues of Y , the following family of probability distributions: F Y ( U ) (cid:44) P max | x ∗ − x c |
5. This value is greater than twice thesignificant wave height–here taken to be four times the typical deviation of the wave field (and isconsistent with the informal definition of rogue waves in the ocean [1]). This probability, displayedin the right half of Figure 12, is simply F Y (2 .
5) from (7).There is a clear bifurcation in the distributions displayed in Figure 12. When | Y | > .
1, thelikelihood of an upcoming extreme event increases dramatically. We may use the definition of || v || to compute the energy level at which this bifurcation occurs at the critical length scale L = 0 . . || v || = 1 . (cid:115) ˆ e − x/L ) dx = 1 . √ L (cid:112) π/ ≈ . L = 0 .
031 initiated extreme events if the energy levelexceeds approximately 0.2 (see Figure 4). These results from Section 3 were performed for localizedstates with a zero background, but the agreement of this analysis with the bifurcation energy level inFigure 12 is significant. Specifically, it suggests that the same localized energy instability discussedin Section 3 for toy examples triggers the formation of extreme events out of more complex states.We now analyze the performance of the computed extreme event probability data from Figure 12as a predictive scheme. At a given time, we compute | Y ( x c , t ) | for various values of x c and use ourcompiled statistics (Figure 12) to estimate the probability of an extreme event. If this probability16igure 12: Left: Family of conditional densities of maximum nearby | u | given current value of | Y | .Right: Probability of an extreme event in near future given | Y | .exceeds 0.8, we predict that an extreme event will occur. Choosing a larger probability thresholdvalue would decrease our rate of false positives; choosing a smaller value would increase this rate butwould have the benefit of increasing the amount of time by which extreme events are predicted inadvance. We found that a probability threshold of 0.8 provides a reasonable balance between thesetwo effects (false positive rate versus advanced warning time). Essentially the predictive schememeasures the probability that a given combination of phases between wave components (the currentform of the wave field) belongs to the domain of attraction of an extreme wave.We tested this scheme on 50 simulations of (1). These simulations were not used to computethe statistics in (7) and Figure 12. In these simulations, we predicted an extreme event 191 times,and 155 correctly predicted an extreme event, meaning that the false positive rate was only 18.9%.There was only 1 extreme event that was not predicted by our scheme, which means that the falsenegative rate was less than 1%. As mentioned in Section 4, in addition to preceding extreme events,large values of Y can occur after extreme events as energy is being transferred to larger scales.However, large values of | Y | in this particular situation do not actually imply that an extremeevent is forthcoming. To avoid false positive predictions that such behavior would generate, we“turn off” our predictive scheme in the spatial region nearby the extreme event for the following 1time unit.In Figure 13 we present the spatial distribution of the probabilistic predictor (left) and theactual wave field (right) for one random realization. As we observe the computed criterion capturesaccurately not only the temporal but also the spatial position of the extreme wave. The sameconclusions can be drawn from Figure 14 where our prediction scheme is often able to predictextreme events a full 1-2 time units in advance . We have examined the synergistic activity of nonlinearity, dispersion, and dissipation towards theformation of extreme events in a one-dimensional prototype system that possesses four-wave res-onant interactions, the focusing MMT equation. The latter provides a relatively simple model of17igure 13: Left: Spatial distribution of probability for a nearby extreme event. Right: | u | as afunction of space and time. The above figure shows the spatial skill of our predictive scheme
100 101 102 103 104 105 106 107 108 109 11001234 Time m a x | u | m a x | P EE | max|u|max|P EE | Figure 14: Spatial maximum values of | u | and extreme event probability, showing that our predictivescheme is able to give advance warning of extreme events.18xtreme events arising out of a nearly Gaussian background with broad-band spectrum, mimickingin this way many features of rogue waves in the ocean. Through analytical and numerical toolswe have shown that the MMT is highly sensitive to localized perturbations of a particular criticallength scale (Figure 4), which we analyze thoroughly. We show the existence of a family of solutionswith a scale-invariance property and based on this fact we quantify the required localized amountof energy that triggers an extreme event. Although the existence of a critical energy level forextreme events is certainly related to the modulation instability, our analysis illustrates that evenzero-backroung-energy states can lead to an extreme event if a localized perturbation of appropriatelengthscale and intensity is applied. These localized perturbations can occur randomly through thedispersive propagation of waves that have with random relative phase.We have illustrated that these extreme events are characterized by low-dimensionality and wehave use a spatially localized basis, a Gabor basis to describe their characteristics. By performing astatistical analysis of the Gabor coefficients we have been able to develop an inexpensive predictivescheme that is reliable with few false positives and false negatives. Furthermore, our scheme showsa high degree of spatial skill and issues warnings in advance (often 1-2 time units before the extremeevent). Future research efforts include the extension of the prediction window by combining thepresented approach with nonlinear filtering techniques [34]. We are also interested on applyingthe presented framework in more realistic two dimensional nonlinear wave models and a currentresearch effort is focused on the wave equation by Trulsen et. al. [21] which, like the MMT model,includes the exact dispersion relation for gravity waves over deep water. Acknowledgments.
This research effort is funded by the Naval Engineering Education Centerthrough grant 3002883706; We are grateful to Dr. Craig Merrill (technical point of contact for thisproject) for numerous motivating discussions and support. We would like to thank Prof. AndrewMajda who suggested the MMT model as a prototype system for extreme events as well as fornumerous other stimulating comments. We are grateful to Dr. Ian Grooms for providing a versionof his numerical solver for the MMT system, as well as to Lake Bookman for helpful discussions.
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