Article under revision for the Journal of Atmospheric Sciences: “Using local dynamics to explain analog forecasting of chaotic systems”

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Linear waves and wave groups

The motion of gravity waves can be described assuming incompressible and inviscid water and wind, irrotational water, and small steepness of the waves (i.e., H/λ 1). From these physical assumptions, Navier-Stokes equations allow to derive the relation between the wave period and wavelength of a sinusoidal gravity wave, called the dispersion relation of linear gravity waves (we will come back to the term “linear” later). Expressed as a function of wavenumber k = 2π/λ and pulse ω = 2π/T , where λ is the wavelength and T the wave period, this dispersion relation reads: ω2 = gk , where g is the earth’s gravitational acceleration at the ocean surface. The velocity of this
sinusoidal wave is given by the ratio ω/k, and is thus a growing function of λ as stated earlier (∝ √λ). This description is valid at first order in wave steepness kH, where H is the wave height. This approximation allows to describe the so-called linear waves, because the equations of wave motion have been linearized in the limit kH → 0. Non-linear effects arise from solutions to the same physical equations, but for higher orders of expansion in wave steepness. There is no interaction between linear waves of different wavelengths: they simply “cross” each other, and the total surface elevation is the sum of the contributions of each wave. Therefore, an ocean surface of sufficiently small steepness can be described as an infinite (or large) sum of sinusoidal waves that propagate according to the dispersion relation of linear gravity waves. From a statistical point of view, the probability distri-bution of the elevation of such a sea-surface is Gaussian (see the classical monograph of Longuet-Higgins, 1957). Therefore, this type of sea-surface is called a Gaussian sea.
This type of surface is therefore usually analyzed using Fourier decompositions. The “wave spectrum” is obtained from the square amplitude of the elevation’s Fourier trans-form. The integral of the wave spectrum gives the total wave energy. The latter remains constant in the absence of energy input (from wind, or other sources listed above) and wave energy dissipation. The range of frequencies (or wavenumbers) over which the spec-trum is significantly high defines the spectral bandwidth. From the previous chapter, it follows that swell seas are narrow-banded and wind seas are wide-banded.
The hypothesis of narrow spectrum allows for several approximations, including the description of waves using wave groups or wave packets. It is well known that that the sum of two sinusoïds with similar frequencies can be viewed as the product of a sinusoïd with the same frequency as the original waves (called the carrier wave) and a low-frequency amplitude or envelope. The same can be said of a large (or infinite) sum of sinusoïds with similar frequencies. Thus, swell seas are organized in groups, a phenomenon that surfers know well: a large swell wave is usually followed by several other large swell waves.

Non-linear wave

Solving the same physical equations as for linear waves, but relaxing a little the hypothesis of small steepness (i.e. kH → 0), one can find new solutions that correspond to expansions with respect to the small parameter kH. The solution including the next expansion term after linear waves is called second-order non-linear waves. In this solution, two waves of wavenumbers k1 and k2 can interact to create waves with wavenumbers equal to the sum and difference of original wavenumbers k1 + k2 and |k1 − k2|. The first created wave has a high frequency and thus changes the shape of waves, while the second has a low frequency and thus changes local mean levels of sea-surface elevation (see, for instance, the figures of Forristall, 2000). In the case of an initially individual, sinusoïdal, linear wave of wavenumber k, second-order non-linearities add a high-frequency wave of wavenumber 2k, which travels at the same velocity as the initial linear wave. The resulting wave is called the Stokes wave, and has sharper crests and wider troughs than the linear wave. These added waves that are the consequence of second-order, two-wave interactions, travel at the same velocity then the linear waves they originate from, and are therefore termed bound waves. Consequently, the latter do not respect the dispersion relation of linear waves. Another important aspect of second-order wave physics is that, as in the linear case, there is no exchange of energy between waves of different wavelengths. The wave-spectrum thus remains time-invariant, which is not the case for higher-order non-linear ocean waves.
In the following, the term higher-order non-linear waves designates solutions of the same wave-physics equations as above, but including terms in the wave-steepness expan-sion that are higher than second-order terms. These include three-wave and four-wave interactions. Such higher-order non-linear waves exhibit exchanges of energy between different wavelengths, and therefore the spectrum of such waves varies in time, even when there is no energy input or dissipation. A consequence of these interactions is the Benjamin-Feir (1967) instability, which causes any individual linear wave solution to be unstable to side-band perturbations. This mechanism may be responsible for the forma-tion of waves of large height. The non-linear Schrödinger equation (NLSE) provides a narrow-band approximation of the time evolution of the wave envelope. The NLSE is valid at third order in wave steepness, and is therefore said to correspond to “weakly non-linear” ocean waves. This equation is used in Chapter IV to evaluate the effect of higher-order non-linearities on our prediction method.

Rogue waves

In order to go beyond this brief introduction, the reader is referred to the numerous literature reviews on the topic of oceanic rogue waves (Kharif and Pelinovsky, 2003; Müller et al., 2005; Dysthe et al., 2008; Adcock and Taylor, 2014; Slunyaev, 2017).
The concept of rogue wave emerges from sailors’ stories. From the “mythical” point of view, a wave is rogue when it emerges from nowhere, disappears in the same fashion, and is abnormally large compared to its neighbourgs7. Rogue waves were consider a sailors’ tale until scientific research revealed that waves with such properties can be measured in the ocean, reproduced in laboratory and numerical experiments, and correspond to some analytical solutions of physical equations. Numerous offshore accidents have been attributed to rogue waves. The hazardous potential of rogue waves is a motivation for research on this topic. A quantitative definition has been retained by the scientific community, saying that a wave is “rogue” if its crest-to-trough wave height is at least twice as large as the significant wave height: H > 2 Hs , (I.1).
where the significant wave height Hs can be defined as four times the standard deviation of the sea-surface elevation (statistics computed from samples which are assumed to originate from a constant wave spectrum, which may involve practical issues). Note that the exact threshold is arbitrary. It correspond to relatively rarely observed waves. Similar criteria are based on crest height rather than crest-to-trough wave height. Assuming linear waves with Gaussian probability distribution and a relatively narrow spectrum, one finds that one wave out of 104 is rogue according to the definition of H > 2Hs. The average number of encountered rogue waves varies when including other types of wave physics.
Several physical phenomenon are candidates to explain the occurrence of oceanic rogue waves. They include the linear superposition of a large number of ocean waves or “lin-ear focusing”, the crossing of sea-states from different directions, the aforementioned Benjamin-Feir instability, wave-current interactions, wind blowing over high amplitude waves, and the influence of bathymetry on waves. In Chapter IV, we use the linear approximation of ocean waves. Due to the narrow-banded character of swell waves, we assume that the surface can be studied from the evolution of isolated wave packets. The effects of higher-order non-linearities are studied through the non-linear-Schrödinger equation.

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Analogs in dynamical systems and local dimensions

In ergodic systems, trajectories come back infinitely close to their initial condition after a sufficiently long time (Poincaré, 1890). Furthermore, if the dynamical system has an attractor set A, then all trajectories converge to this subset of the phase-space (Milnor, 1985). Analog methods are based on the idea that if one is provided with a long enough trajectory of the system of interest, one will find analog states close to any point z of the attractor A.
The trajectory from which the analogs are taken is called the “catalog” C, and can either come from numerical model output or reprocessed observational data. It can be seen either as a trajectory from a discrete dynamical system, or as evenly-spaced time samples from a continuous dynamical system. In any case, the catalog has a finite number of elements noted L := card(C). This catalog size may be divided by a typical correlation time-scale so that elements of the catalog can be considered independent (Van Den Dool, 1994). In fact, for the analogs of a given target z to be considered independent, it is enough that the typical distance between two analogs of z be smaller than the typical distance between an analog and its time-successor. The structure of the attractor, expressed by the systems’ invariant measure µ, condi-tions the structure of the catalog and the ability to find analogs. In particular, Van Den Dool (1994) and Nicolis (1998) studied the role of the attractor’s dimension. Let Bz,r the ball centered on z ∈ A and of radius r, then dz,r := log µ(Bz,r) , (II.1) log r. defines the finite-resolution (r-resolution) local dimension at point z. Note that for ergodic measures, µ(Bz,r) can be approximated by counting the number of times a given trajectory enters Bz,r (this is the consequence of the ergodic theorem of Birkhoff 1931). If µ is ergodic and limr→0 dz,r exists, then µ is said to be exact dimensional and the limit is independent of z (Young, 1982). This typical value of the local dimension is noted D1.

Simple scaling of analog-to-target distance with dimension

Using extreme value theory and dynamical systems theory, Caby et al. (2019) showed that dz,r can be estimated using the empirical distribution of points inside a ball of expo-nentially decreasing radius. This empirical distribution is actually exactly the cumulative distribution function of the best available analogs. It then follows from Caby et al. (2019) that, for regular enough measures, we have the approximate scaling: rk(z) ∼ k1/d, (II.2).
where d = dz,rK is the local dimension at finite resolution rK (the largest analog-to-target distance). An application of this method to the three-variable system of Lorenz (1963) is given in Fig. II.1. Eq. II.2 gives an important point of our analysis, which is the scaling of rk with k, and is approximately given by a power-law with exponent 1/d. However, this formula comes from a work on local dimensions, not analog-to-target distances. It is therefore not surprising that some of the elements required for our study are missing. In particular, this scaling does not give the constant in front of k1/d, in which resides the relation to the catalog size, a crucial point for analog applications. Also, it only gives a mean or typical value of rk, while our objective is to evaluate the probability distribution of rk, or at least the probability of departures from this mean scaling. The next section gives theoretical elements to evaluate the full probability distribution of rk(z) from the local dimension, the catalog size, and the analog number k.

Table of contents :

Résumé en français
I Mathematical and physical frameworks 
1 Dynamical systems
1.1 Flow
1.2 Divergence of trajectories (chaos)
1.3 Recurrences and ergodicity
2 Analogs
2.1 Motivation and definitions
2.2 Applications of analogs
2.3 Strengths and weaknesses
3 Particle filtering
4 Extreme value theory
5 Ocean waves
5.1 Generalities
5.2 Linear waves and wave groups
5.3 Non-linear waves
5.4 Rogue waves
II How near are the best analogs? 
1 Introducting the article
2 Article in preparation for Entropy: “Probability distributions for analogto- target distances”
2.1 Abstract
2.2 Introduction
2.3 Theory
2.4 Consequences for applications of analogs
2.5 Numerical experiments
2.6 Conclusion
3 Complementary analysis and perspectives
3.1 The case of partial and noisy observations
3.2 From analog distances to analog forecasting: an illustration
4 Summary
IIIHow far from the truth are analog forecasts? 
1 Introducing the article
2 Article under revision for the Journal of Atmospheric Sciences: “Using local dynamics to explain analog forecasting of chaotic systems”
2.1 Abstract
2.2 Introduction
2.3 Analog forecasting
2.4 Successor-to-future state distance
2.5 Consequences for analog forecasting operators
2.6 Discussion
2.7 Conclusion
3 Complementary analysis
3.1 Supplementary material from conference paper
3.2 Splitted analog forecasts and finite-differences
3.3 The influence of additive noise
3.4 Analog short-term ocean wave forecasting from point-measurements
4 Summary
IV Forecast of extreme ocean waves from crest velocities 
1 Introducing the article
2 Article published in Natural Hazards: “Wave group focusing in the ocean: estimations using crest velocities and a Gaussian linear model”
2.1 Abstract
2.2 Introduction
2.3 Theory for linear Gaussian wave packets
2.4 Numerical experiments of linear waves
2.5 Non-linear effects
2.6 Conclusion
Data accessibility
3 Summary
V Can analogs forecast extreme events? 
1 Introduction
2 Analog forecasting of heavy-tailed random variables
2.1 Context
2.2 One-dimensional toy model
2.3 Crude analog forecasting with particle filters
2.4 Analog-to-target distances
2.5 Perspectives
3 Summary
Conclusion and perspectives
Lorenz systems
Product of Hessian with vectors


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