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 ofMaster of Science in Mechanical Engineering
at the
Ecole Polytechnique Fédérale de Lausanne
February 2015
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
Abstract
In this thesis, we implement a reduced-order framework for the stochastic evolution of non-
linear 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 (2009), and consists in the expansion of
the stochastic solution on a few time-dependent deterministic modes that capture the sub-
space 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 wave-
train 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
Contents
Acknowledgements i
Abstract iii
Introduction 1
1 Review of deep-water wave theory 3
1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Linear dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Energy considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Weakly nonlinear envelope equation . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Weakly nonlinear narrow-band wavetrain . . . . . . . . . . . . . . . . . . 5
1.2.2 Nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Modulational instability and long-time evolution . . . . . . . . . . . . . . . . . . 7
1.3.1 Uniform Stokes wave solution . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Benjamin-Feir instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Fermi-Pasta-Ulam recurrence . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 NLS equation under the DO framework 11
2.1 Review of probability theory and KL expansion . . . . . . . . . . . . . . . . . . . 11
2.1.1 Probability spaces and random variables . . . . . . . . . . . . . . . . . . . 11
2.1.2 Stochastic processes and random fields . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Karhunen-Loève orthogonal expansion . . . . . . . . . . . . . . . . . . . . 13
2.2 Dynamically orthogonal NLS equation . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Dynamically orthogonal expansion . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 On the choice of the inner product . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Dynamically orthogonal equations . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Stochastic energy transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.1 Numerical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.2 Initial condition formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Diagonalization of the covariance matrix . . . . . . . . . . . . . . . . . . . 23
2.4.4 Overview of the code structure . . . . . . . . . . . . . . . . . . . . . . . . . 24
v
Contents
3 Preliminary results and validation 27
3.1 Idealized Benjamin-Feir instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Random Gaussian wavenumber spectrum . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Statistical properties of the DO solution . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Structure of the DO solution . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Dynamics of an extreme wave 41
4.1 Nonlinear focusing of localized wave packets . . . . . . . . . . . . . . . . . . . . 41
4.2 Adaptivity of the DO modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3 Attractor of an idealized extreme wave . . . . . . . . . . . . . . . . . . . . . . . . 43
5 Results with the MNLS equation 47
5.1 Modified nonlinear Schrödinger and DO equations . . . . . . . . . . . . . . . . . 47
5.2 Idealized Benjamin-Feir instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Attractor of an idealized extreme wave . . . . . . . . . . . . . . . . . . . . . . . . 50
Conclusions and perspectives 55
A Complex coefficients in the DO framework 57
A.1 Dynamically orthogonal equations with complex coefficients . . . . . . . . . . . 57
A.2 Diagonalization of the complex covariance matrix . . . . . . . . . . . . . . . . . 58
B Dynamically orthogonal MNLS equation 61
Bibliography 63
vi
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, 1980).
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 deep-
water gravity waves of much larger height than one would expect given the sea state (Dysthe
et al., 2008). Because of their potential catastrophic impact (Liu, 2007), they have recently
started to be an area of active research (Onorato et al., 2001). 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, 2003; Dysthe et al., 2003). More
1
Contents
recently, Ruban (2013) and Cousins & Sapsis (2015b,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 nonlin-
ear deep-water waves and we investigate the aforementioned phenomena in this reduced-
order stochastic setting. Specifically, we use the dynamically orthogonal (DO) framework
introduced by Sapsis & Lermusiaux (2009) to derive reduced-order equations for the stochastic
evolution of water waves governed by the NLS equation and subject to random initial con-
ditions. 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 numeri-
cal 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
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, 1980). 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 = 0 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
∇2φ= 0 for −∞< z < η(x, t ), (1.1)
with the following boundary condition that enforces a vanishing velocity at the bed
∂φ
∂z= 0 when z →−∞. (1.2)
3
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
∂x
∂η
∂x− ∂φ
∂z= 0 at z = η(x, t ). (1.3)
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+ 1
2(∇φ)2 + g z = 0 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 govern-
ing 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.
1.1.1 Linear dispersion relation
Restricting ourselves to small disturbances about the base state η = 0 and φ = 0, the free
surface boundary conditions can be linearized and applied at the undisturbed surface z = 0.
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
ωekz 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 vg = ∂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.
1.1.2 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, 2004), expressed as
H = ρg∫ L
0
∫ η
−∞z dz dx + 1
2ρ
∫ L
0
∫ η
−∞(∇φ)2dz dx, (1.7)
4
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 = [0,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
2ρg
∫ L
0η2dx + 1
4ρg a2
∫ L
0(1+2kη+O (η2))dx = 1
2ρg a2L+O (a4). (1.8)
1.2 Weakly nonlinear envelope equation
The linear solution (1.5)–(1.6) describes a uniform wavetrain of perfect periodicity and con-
stant 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.
1.2.1 Weakly nonlinear narrow-band wavetrain
Specifically, we introduce the weakly nonlinear narrow-band wavetrain through the concept
of a carrier wave with carrier wavenumber k0 and frequency ω0, 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 ) = Reu(x, t )e i (k0x−ω0t ) = Rea(x, t )e iθ(x,t )e i (k0x−ω0t ), (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¿ k0, ∆ω=−∂θ
∂t¿ω0, (1.10)
both of which are small with respect to k0 and ω0, hence the ‘narrow-band’ designation
as departures in wavenumber k = k0 +∆k and frequency ω = ω0 +∆ω of the wavetrain are
restricted to a narrow-band window around the corresponding carrier properties.
5
Chapter 1. Review of deep-water wave theory
1.2.2 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= k0a ¿ 1 and
we require the modulation bandwidth to be of the same order of magnitude i.e. ∆k/k0 =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 ) = Reu(εx,εt )e i (k0x−ω0t ) +u2(εx,εt )e2i (k0x−ω0t ) + . . . , (1.11)
φ(x, z, t ) = Rev(εx, z,εt )ek0z e i (k0x−ω0t ) + v2(εx, z,εt )e2k0z e2i (k0x−ω0t ) + . . . , (1.12)
where the slowly varying complex functions u, u2, ..., v , v2, ... are O (ε/k0). 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 = 0, the equations can
be solved in orders of ε. At the first order, the usual linear dispersion relation ω0 =√
g k0 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
i
(∂u
∂t+ ω0
2k0
∂u
∂x
)− ω0
8k20
∂2u
∂x2 − 1
2ω0k2
0 |u|2u = 0, (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 u2 = k0u2/2, 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 u2 =O (ε2/k20), thus
the first harmonic provides a sufficient description of the surface elevation
η(x, t ) = Reu(εx,εt )e i (k0x−ω0t )+O (ε2/k20). (1.14)
In studying solutions to the NLS equation, it is helpful to use a reference frame moving at the
6
1.3. Modulational instability and long-time evolution
group velocity of the carrier wave, which is achieved through the change of coordinates
x∗ = x − ω0
2k0t . (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 =ω0t , x = k0x∗, u = k0u, ∆k = ∆k
k0, ∆ω= ∆ω
ω0, (1.16)
leading to the following nondimensionalized form of the NLS equation (1.13)
∂u
∂t=− i
8
∂2u
∂x2 − i
2|u|2u. (1.17)
and the surface elevation to lowest order is expressed as η(x, t ;ω) = Reu(x, t ;ω)e i (x−t/2). Note
that under this choice of length scale, the wave steepness ε= k0a becomes equivalent to the
nondimensional wave amplitude |u|. 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
1.3.1 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 ) = a0 e−(i /2)a20 t , a0 ∈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 ω0 = √g k0. This can be shown by writing the
corresponding expression for the surface elevation in dimensional form
η(x, t ) = Rea0 e−(i /2)k20 a2
0ω0t e i (k0x−ω0t )+O (a20)
= a0 cos[k0x −ω0(1+k20 a2
0/2)t ]+O (a20). (1.19)
We obtain a uniform wave with dispersion relation ω=ω0(1+k20 a2
0/2) that depends not only
on the wavenumber k0, but also on the finite amplitude a0. 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
∂2ω
∂k20
=− ω0
8k20
and − ∂ω
∂a20
=−1
2ω0k2
0 , (1.20)
7
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 k0 and zero amplitude (Yuen & Lake, 1982; Janssen, 2004).
1.3.2 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 ) = a0 (1+b1e i (∆kx−∆ωt ) + i b2e i (∆kx−∆ωt ))e−(i /2)a20 t , (1.21)
where |b1| and |b2| 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
∆ω2 = ∆k2
8
(∆k2
8−a2
0
). (1.22)
This indicates that perturbations with nondimensional wavenumber in the range
0 <∆k <∆kc = 2p
2a0 (1.23)
have positive growth rate σ= Im∆ω, hence are linearly unstable, while ∆kc = 2p
2a0 repre-
sents a cut-off wavenumber above which there is restabilization. The instability is maximum at
∆km = 2a0 with an associated nondimensional maximum growth rate of σm = a20/2, implying
that it occurs on a nondimensional timescale of O (a−20 ). 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 a0 tends to 0, thus recovering the linear result of
a marginally stable water surface around z = 0.
1.3.3 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
1.3. Modulational instability and long-time evolution
"k/a0
0 0.5 1 1.5 2 2.5 3<
/<m
0
0.2
0.4
0.6
0.8
1
Figure 1.1 – Normalized growth rate σ/σm of the Benjamin-Feir instability versus normalizedperturbation wavenumber ∆k/a0. Note that the instability is maximum at ∆km = 2a0 and thecut-off occurs at ∆kc = 2
p2a0.
Figure 1.2 – Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence in a slightly modu-lated 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 threedifferent 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,0) = a0(1+0.1cos∆km x), with a0 = 0.1, representing a uniform
wavetrain that is slightly modulated by the linearly most unstable modulation wavenumber
∆km = 2a0 = 0.2. 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 = 660 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 = 500 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
Chapter 1. Review of deep-water wave theory
k0 0.5 1 1.5 2
|F[2
]|
0
5
10
15
20
25
30
35
40
45Fourier transform of 2 at t = 500
t0 500 1000 1500 2000 2500 3000 3500
|F[u
]("
k)|
0
20
40
60
80
100
120Fourier modes of |u|
"k = 0"k = 1"km"k = 2"km"k = 3"km"k = 4"km
"km
2"km
Figure 1.3 – Benjamin-Feir instability and Fermi-Pasta-Ulam recurrence in a slightly mod-ulated uniform wavetrain. Left, Fourier spectrum F [η] of the surface elevation at t = 500,with k = 1 corresponding to the carrier wave and k = 1±∆km to the unstable modulation.Right, time evolution of the amplitudes of the carrier wave, the unstable modulation and itsharmonics obtained through a Fourier transform F [u] of the envelope. This time, ∆k = 0corresponds to the carrier wave and ∆k =∆km is the unstable modulation.
wavenumbers comprising the unstable initial modulation k = 1±∆km and its harmonics
k = 1±2∆km ,1±3∆km , ... 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 =∆km is growing at the expense of the carrier wave ∆k = 0. 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.
10
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
(2009), a novel order-reduction method for the solution of stochastic partial differential equa-
tions. 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 2.4.
2.1 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.
2.1.1 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 withΩ, that is and
loosely speaking, B is the collection of subsets ofΩ. The elements of B are random events and
consist of combinations of elementary eventsω. Finally, P is a probability measure defined on
B, i.e. P is a countably additive function that associates a non-negative number between 0
and 1 to each random event in B, and that assigns the value 1 to the sample spaceΩ. In loose
11
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 spaceΩ, divided by the total number of elementary
events inΩ.
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
dx, x ∈R. (2.2)
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)] =∫Ω
X (ω)dP (ω). (2.3)
2.1.2 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
2.1. Review of probability theory and KL expansion
field u(x, t ;ω) as the ensemble average over the elementary events ω
u(x, t ) = E [u(x, t ;ω)] =∫Ω
u(x, t ;ω)dP (ω), (2.4)
2.1.3 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 conju-
gate. For a fixed time t , u(x, t ;ω) is a random function of x and lives in the infinite-dimensional
Hilbert space L2 of all continuous, square integrable and complex-valued functions defined
on D , and having the following spatial inner product
⟨u1,u2⟩ =∫
Du1(x)u∗
2 (x)dx for all u1,u2 ∈ L2. (2.5)
Since L2 is an infinite-dimensional space, it is spanned by an infinite number of orthogonal
basis functions vi (x)∞i=1. Hence, if we want to write the random field u(x, t ;ω) (at a given time)
as a linear decomposition of deterministic fields multiplied by scalar stochastic coefficients by
projecting each realization u(x, t ;ω) onto the basis vi (x)∞i=1, then we need an infinite series
of the form
u(x, t ;ω) = u(x, t )+∞∑
i=1Yi (t ;ω)vi (x), (2.6)
where u(x, t ) is the mean field and Yi (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 u(x, t). The question now is whether one can, for a given time, find a
deterministic basis ui (x, t )∞i=1 that is optimal for u(x, t ;ω) (at that time), in the sense that a
finite-dimensional representation of the form
u(x, t ;ω) ' u(x, t )+s∑
i=1Yi (t ;ω)ui (x, t ), s <∞, (2.7)
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
maxui∈L2
E [|⟨u(x, t ;ω),ui (x, t )⟩|2]
⟨ui (x, t ),ui (x, t )⟩ , (2.8)
where | · | denotes the absolute value, and it is looked for an ui (x, t) such that the ensemble
average of the projection of u(x, t ;ω) onto ui (x, t) is maximized. Using variational calculus
techniques, this condition reduces to the following eigenvalue problem∫D
Rt (x, y)ui (x, t )dx =λi (t )ui (y, t ), (2.9)
13
Chapter 2. NLS equation under the DO framework
where Rt (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 ui (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
Yi (t ;ω) = ⟨u(x, t ;ω)− u(x, t ),ui (x, t )⟩, (2.10)
and they verify the following properties, relating to the eigenvalues λi (t )
E [|Yi (t ;ω)|2] =λi (t ), E [Yi (t ;ω)Y ∗j (t ;ω)] = 0 for i 6= j . (2.11)
Hence, the eigenvalues λi (t ) represent the variance (or spread) of the stochastic fluctuations
u(x, t ;ω)− u(x, t ) along each direction ui (x, t ) in phase space. As a consequence of the opti-
mality of the representation, the orthogonal directions ui (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. λi (t) ∼ e−ci for some positive c. It is therefore justified to define a threshold s <∞ and
neglect the directions ui (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 Rt (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 Yi (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
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 (2009) 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, 2015b,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, t0;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 & Lermu-
siaux (2009) 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.
2.2.1 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 t0
u(x, t0;ω) = u(x, t0)+s∑
i=1Yi (t0;ω)ui (x, t0), s <∞, (2.12)
where u(x, t0) is the initial mean, Yi (t0;ω) are the zero-mean initial stochastic coefficients
and ui (x, t0) 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
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) = u(x, t )+s∑
i=1Yi (t ;ω)ui (x, t ), (2.13)
where u(x, t) is the time-dependent mean, ui (x, t) are the time-dependent deterministic
modes describing the main directions of stochastic fluctuations at time t and Yi (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, t0;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, 2013),
they are negligible in a number of other cases (see Mantic-Lugo, Arratia & Gallaire, 2014, 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 Vs = spanui (x, t)si=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 Yi (t ;ω) and
the modes ui (x, t ) needs to be overcome. Sapsis & Lermusiaux (2009) showed that this can be
achieved by imposing the following dynamical orthogonality (DO) condition
dVs
dt⊥Vs ⇔
⟨∂ui (·, t )
∂t,u j (·, t )
⟩= 0, i , j = 1, ..., s, (2.14)
that restricts the time variation of the subspace Vs where stochasticity lives to be orthonormal
to itself. In other words, since variations of the stochastic fluctuations within Vs can be entirely
described by variations of the stochastic coefficients Yi (t ;ω), the DO condition imposes the
natural constraint that modes only move when the fluctuations evolve to new directions
not already included in Vs . The DO condition also implies the preservation of the initial
orthonormality of the modes ui (x, t ) since
∂
∂t⟨ui (·, t ),u j (·, t )⟩ =
⟨∂ui (·, t )
∂t,u j (·, t )
⟩+
⟨∂u j (·, t )
∂t,ui (·, t )
⟩= 0, i , j = 1, ..., s. (2.15)
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 u(x, t ), the modes ui (x, t ) and the stochastic coefficients
Yi (t ;ω). We emphasize that the time-dependent basis functions ui (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
2.2. Dynamically orthogonal NLS equation
2.2.2 On the choice of the inner product
Before proceeding to the derivation of equations for the mean, modes and stochastic coef-
ficients, 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 ui (x, t ) are orthogonal to each others, as well as the value
of the scalar coefficients Yi (t ;ω), for they result from the projection of the solution u(x, t ;ω)
onto the basis ui (x, t )si=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
⟨u1,u2⟩ =∫
Du1(x, t ;ω)u∗
2 (x, t ;ω)dx. (2.16)
This complex-valued inner product implies that the stochastic coefficients in the DO ex-
pansion (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., 2013; Sapsis et al.,
2013) 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,
2006; Adali et al., 2011; Cheong Took et al., 2012).
Therefore we opted to go the safer route by using the following real-valued inner product
⟨u1,u2⟩ = Re
∫D
u1(x, t ;ω)u∗2 (x, t ;ω)dx
, (2.17)
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 u1(x, t) and u2(x, t) = i u1(x, t) that are orthogonal
under inner product (2.17), since ⟨u1,u2⟩ = Re−i = 0. Together they span the subspace
spanu1,u2 = Y1 u1(x, t )+Y2u2(x, t ) | Y1,Y2 ∈R
= (Y1 + i Y2)u1(x, t ) | Y1,Y2 ∈R
= Z1u1(x, t ) | Z1 ∈C . (2.18)
It is therefore observed that u1(x, t) and u2(x, t) span the same subspace as would u1(x, t)
alone with a complex coefficient. Note that under the complex-valued inner product (2.16) we
would indeed have ⟨u1,u2⟩ = −i 6= 0, 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
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 (Reu(x, t ;ω), Imu(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.
2.2.3 Dynamically orthogonal equations
In this section, we follow the steps in Sapsis & Lermusiaux (2009) to derive the DO equations
that govern the evolution of all unknown quantities in the DO expansion (2.13), i.e. the
mean u(x, t ), the deterministic modes ui (x, t ) and the stochastic coefficients Yi (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 Yi (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)
∂t=− i
8
∂2u(x, t ;ω)
∂x2 − i
2|u(x, t ;ω)|2u(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, t0;ω) = u0(x;ω), x ∈ D, ω ∈Ω. (2.20)
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
∂u
∂t+ dYi
dtui +Yi
∂ui
∂t= F0 +Yi Fi +Yi Y j Fi j +Yi Y j Yk Fi j k , (2.21)
where repeated indices indicate summation from 1 to s, and F0, Fi , Fi j and Fi j k are complex
deterministic fields defined by
F0 =− i
8
∂2u
∂x2 − i
2|u|2u, Fi =− i
8
∂2ui
∂x2 − i u Reuu∗i − i
2|u|2ui ,
Fi j =− i
2Reui u∗
j u − i Reuu∗i u j , Fi j k =− i
2Reui u∗
j uk ,
(2.22)
where i , j ,k = 1, ..., s. The deterministic PDE governing the evolution of the mean field u(x, t )
is obtained by taking the ensemble average of the governing equation (2.21)
∂u
∂t= F0 +CYi Y j Fi j +MYi Y j Yk Fi j k , (2.23)
18
2.3. Stochastic energy transfers
where CYi Y j (t ) = E [Yi Y j ] is the time-dependent covariance matrix and MYi Y j Yk (t ) = E [Yi Y j Yk ]
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 ui (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 Yi (t ;ω)
dYi
dt= Ym ⟨Fm ,ui ⟩+ (YmYn −CYm Yn )⟨Fmn ,ui ⟩+ (YmYnYl −MYm Yn Yl )⟨Fmnl ,ui ⟩. (2.24)
Finally, we multiply the governing equation (2.21) with each of the stochastic coefficients
Yi (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 ui (x, t )∂ui
∂t= Hi −⟨Hi ,u j ⟩u j , (2.25)
where Hi is a complex deterministic field defined by Hi = E [L [u]Yk ]C−1Yi Yk
with L [u] the
right-hand side of the governing equation (2.21), and is expressed as
Hi = Fi +MYm Yn Yk C−1Yi Yk
Fmn +MYm Yn Yl Yk C−1Yi Yk
Fmnl , (2.26)
with MYi Y j Yk Yl (t) = E [Yi Y j Yk Yl ] the time-dependent matrix of fourth-order moments. The
mean u(x, t), the modes ui (x, t) and the stochastic coefficients Yi (t ;ω) are initialized at t0
through a KL expansion (2.12) of the initial condition u0(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 ui (x, t ) of the stochastic sub-
space Vs 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 = [0,L] can be expressed as
H (t ;ω) =∫ L
0η(x, t ;ω)2dx =
∫ L
0Reu(x, t ;ω)e i (x−t )2dx ' 1
2
∫ L
0|u(x, t ;ω)|2dx (2.27)
where the energy has been made nondimensional with ρg /k30 , 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
2E [⟨u,u⟩] = 1
2E [⟨u +Yi ui , u +Yi ui ⟩] = 1
2
(‖u‖2 +E [Yi Yi ])
(2.28)
19
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 2i ] of the stochastic coefficients,
while the term ‖u‖2 = ⟨u, u⟩ 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
εi = 1
2
d
dtE [Y 2
i ] = E
[Yi
dYi
dt
]= Ai i CYi Yi +Bi mn MYi Ym Yn +Ci mnl MYi Ym Yn Yl , (2.29)
(no sum on i ), where we have assumed that the covariance matrix CYi Y j has been diagonalized
at the present time instant (see Section 2.4.3 for the details) so that the stochastic coefficients
are uncorrelated. Ai i , Bi mn and Ci mnl are the deterministic fields appearing in equation (2.24)
Ai i =⟨− i
8
∂2ui
∂x2 − i u Reuu∗i − i
2|u|2ui ,ui
⟩, (2.30)
Bi mn =⟨− i
2Reumu∗
nu − i Reuu∗mun ,ui
⟩, (2.31)
Ci mnl =⟨− i
2Reumu∗
nul ,ui
⟩. (2.32)
The term Ai i can be considerably simplified since⟨− i
8
∂2ui
∂x2 ,ui
⟩= Re
−
∫ L
0
i
8
∂2ui
∂x2 u∗i dx
= Re
− i
8
∂ui
∂xu∗
i
∣∣∣∣L
0+
∫ L
0
i
8
∂ui
∂x
∂u∗i
∂xdx
= 0, (2.33)
and ⟨− i
2|u|2ui ,ui
⟩= Re
−
∫ L
0
i
2|u|2ui u∗
i dx
= 0, (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
Ai i =−Re
∫ L
0
i
2u2u∗
i2dx
, (2.35)
Bi mn =−Re
∫ L
0
i
2[ uumu∗
n + uu∗mun + u∗umun ]u∗
i dx
, (2.36)
Ci mnl =−Re
∫ L
0
i
2umu∗
nul u∗i dx
. (2.37)
By inspection, we observe that the linear term Ai i represents the rate of energy transfer
between the mean and mode i . The term Bi mn indicates modal energy production due to the
simultaneous interaction of mode i with the mean and two other modes, while Ci 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 MYi Ym Yn and MYi Ym Yn Yl (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
20
2.4. Numerical implementation
variance) of mode i thus boils down to
εi = εmean→i +εmean,mn→i +εmnl→i , (2.38)
with the following linear and nonlinear contributions
εmean→i = Ai i CYi Yi , εmean,mn→i = Bi mn MYi Ym Yn , εmnl→i =Ci mnl MYi Ym Yn Yl , (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
dt= E
[⟨∂u
∂t,u
⟩]= E
[⟨− i
8
∂2u
∂x2 ,u
⟩]+E
[⟨− i
2|u|2u,u
⟩]= E
[Re
−
∫ L
0
i
8
∂2u
∂x2 u∗dx
]+E
[Re
−
∫ L
0
i
2|u|2uu∗dx
]= E
[Re
− i
8
∂u
∂xu∗
∣∣∣∣L
0+
∫ L
0
i
8
∂u
∂x
∂u∗
∂xdx
]+E
[Re
−
∫ L
0
i
2|u|4dx
]= 0,
(2.40)
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 2i ]/2 =∑s
i=1 εi
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 Ci mnl =−Cmi l n ,
leading to∑s
i=1 εmnl→i = 0.
2.4 Numerical implementation
The DO equations for the mean (2.23), the modes (2.25) and the coefficients (2.24) are imple-
mented 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.
2.4.1 Numerical schemes
The deterministic PDEs for the mean and the modes are discretized on a grid of size 1024 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 second-
order 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
Chapter 2. NLS equation under the DO framework
for the stochastic coefficients is solved using a Monte-Carlo method with 103 to 104 particles
and is advanced in time with a 4th-order Runge-Kutta scheme, using a nondimensional time
step of 0.01. 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 compu-
tation 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 com-
putational 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.
2.4.2 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 t0
u(x, t0;ω) = u(x, t0)+s∑
i=1Yi (t0;ω)ui (x, t0), s <∞, (2.41)
where u(x, t0) is the initial mean, Yi (t0;ω) are the zero-mean initial stochastic coefficients and
ui (x, t0) 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 Rt0 (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
u(x, t0;ω) = u(x, t0)+N∑
n=1An(ω)e iθn (ω)e i∆kn x , N <∞, (2.42)
where the modulus An(ω) and phase θn(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
2.4. Numerical implementation
coefficients Yi (t0;ω). This issue can be overcome by expanding (2.42) as
u(x, t0;ω) = u(x, t0)+N∑
n=1An(ω)cosθn(ω)e i∆kn x +
N∑n=1
An(ω)sinθn(ω)e i (∆kn x+π/2), (2.43)
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
Y2n−1(t0;ω) = An(ω)cosθn(ω), u2n−1(x, t0) = e i∆kn x ,
Y2n(t0;ω) = An(ω)sinθn(ω), u2n(x, t0) = e i (∆kn 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∆kn x ,e i (∆kn x+π/2)⟩ = 0, 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).
2.4.3 Diagonalization of the covariance matrix
In the KL expansion (2.7), the stochastic coefficients are uncorrelated i.e. E [Yi (t ;ω)Y j (t ;ω)] = 0
for i 6= j , which is a consequence of the fact that the directions ui (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 CYi Y j =E [Yi (t ;ω)Y j (t ;ω)]. As a result, the modes ui (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 CYi Y j , which corresponds to applying a
rotation to the modes ui (x, t) such that they become aligned with the principal variance
directions and the coefficients Yi (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 ui and that
the covariance matrix C =CYi 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 = umVmi . Note that we have
⟨ui ,u′j ⟩ = ⟨ui ,umVm j ⟩ =Vm j ⟨ui ,um⟩ =Vi j , (2.46)
23
Chapter 2. NLS equation under the DO framework
We first show that the new basis u′i is orthonormal
⟨u′i ,u′
j ⟩ = ⟨umVmi ,unVn j ⟩ =Vmi Vn j ⟨um ,un⟩ =Vmi Vm j =V Tj mVmi = δi j . (2.47)
Then, the coefficients Y ′i 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)
and they are uncorrelated since
CY ′i Y ′
j= E [Y ′
i Y ′j ] = E [YmVmi YnVn j ] =V T
i mCmnVn j = Di j = δi jλi , (2.49)
where λi are the eigenvalues of C and the diagonal elements of D . We have therefore shown
that by diagonal decomposition of the covariance matrix CYi 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′i (x, t ) correspond to the directions of principal variance of the solution
u(x, t ;ω). We apply this procedure every time the solution is plotted or saved.
2.4.4 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 u(x, t ), the modes ui (x, t ) and the stochastic coefficients
Yi (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 CYi Y j and higher-order moments matrices MYi Y j Yk and MYi Y j Yk Yl
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
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 2.4.3 and leading to uncorrelated
stochastic coefficients.
25
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 enve-
lope u(x, t0) = a0, is linearly unstable to small Fourier mode perturbations with modulation
wavenumber in the range 0 < ∆k < ∆kc = 2p
2a0. If there is only one prescribed unstable
modulation wavenumber ∆k and if its harmonics fall outside the unstable regime (that is
∆k >∆kc /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 frame-
work for the illustration and validation of our DO stochastic equations. Indeed, we can then
consider random initial conditions consisting of a constant envelope a0 perturbed by Fourier
modes with deterministic small amplitude but random phase
u(x, t0;ω) = a0 +N∑
n=1Ane iθn (ω)e i∆kn x , N <∞, (3.1)
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
Chapter 3. Preliminary results and validationM
ean
-0.1
0
0.1
t = 0
Mod
e 1
-0.1
-0.05
0
0.05
0.1
Mod
e 2
-0.1
-0.05
0
0.05
0.1
Mod
e 3
-0.1
-0.05
0
0.05
0.1
x0 10 20 30
Mod
e 4
-0.1
-0.05
0
0.05
0.1
-0.1
0
0.1
t = 800
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
x0 10 20 30
-0.1
-0.05
0
0.05
0.1
-0.1
0
0.1
t = 1400
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
x0 10 20 30
-0.1
-0.05
0
0.05
0.1
Figure 3.1 – Mean and DO modes at various times for the stochastic BF instability and FPUrecurrence 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 = 50 ·2π, weonly plot the solution over a portion of the domain corresponding to the wavelength of thelinearly most unstable wavenumber ∆km = 2a0 = 0.2.
grow, in such a way that the modes should reveal the dominant components of the dynamics.
We therefore consider initial condition (3.1) with a0 = 0.1. We choose to use N = 2 Fourier
modes of wavenumbers ∆k1 = 0.2 and ∆k2 = 0.4, in such a way that the first Fourier mode cor-
responds to the linearly most unstable wavenumber (given by ∆km = 2a0) while its harmonics
and the second Fourier mode are stable. The random phases θn(ω) are drawn from indepen-
dent and uniform distributions on [0,2π], and the deterministic amplitudes An are assigned
the infinitesimal value 0.0036. For the initial DO expansion (2.12), the mean is assigned the
uniform wave component u(x, t0) = a0, while the modes ui (x, t0) and stochastic coefficients
Yi (t0;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 = 50 ·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
∆k1 = 0.2 present in the initial condition implies that the solution will remain periodic with
period 2π/∆k1, therefore we only show a portion of the total domain corresponding to this
28
3.1. Idealized Benjamin-Feir instability
t0 200 400 600 800 1000 1200 1400 1600 1800 2000
E[Y
i2]
10 -3
10 -2
10 -1
100
101
MeanMode 1Mode 2Mode 3Mode 4Deterministic
Figure 3.2 – Energy of the mean ⟨u, u⟩ and the modes E [Y 2i ] for the stochastic BF instability
and FPU recurrence with random phase in the initial modulation. The dashed lines showthe corresponding energies for a deterministic simulation of an equivalent initial conditionu(x,0) = 0.1+ 0.0036cos∆km x with ∆km = 0.2, with the energies obtained as the normal-ized squared modulus of the Fourier coefficients of wavenumber ∆k = 0 (carrier wave), 0.2(unstable modulation) and 0.4 (stable harmonic) (similarly to Figure 1.3).
wavelength. In Figure 3.2, we show the energy present in the mean ⟨u, u⟩ together with the
stochastic energy present in the modes E [Y 2i ]. Focusing on the modes at t = 800 in Figure
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 ∆k1 = 0.2 as was assigned in the initial condition, while modes 3 and 4 represent
the same stable modulation ∆k2 = 0.4. 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 ∼ 400, 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 sub-
sequent 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 quantita-
tive comparison, we computed the solution to a deterministic initial condition of the form
u(x,0) = 0.1+0.0036cos∆km x where ∆km = 0.2, 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 trans-
form of the envelope. The normalized resulting squared modulus of the Fourier coefficients
for ∆k = 0 (carrier wave), 0.2 (unstable modulation) and 0.4 (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 = 0.2 (contained in the DO modes 1 and 2), and the time at which it occurs.
29
Chapter 3. Preliminary results and validation
Figure 3.3 – Stochastic attractor at various times of the stochastic BF instability and FPUrecurrence with random phase in the initial modulation. The attractor is represented in termsof 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 correspondingsurface elevation η(x, t ;ω) = Reu(x, t ;ω)e i (x−t/2).
However, from time t ∼ 1100, 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 = 1400 that lose any kind of regular structure. Moreover, from t ∼ 1900, 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, 1990), 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 Y1(t ;ω) = A(t)cosθ(ω) and Y2(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) = Reu(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
30
3.1. Idealized Benjamin-Feir instability
mem
ber
with
max
imum
2
-0.2-0.1
00.10.2
t = 0
x0 10 20 30
mem
ber
with
min
imum
2
-0.2-0.1
00.10.2
-0.2-0.1
00.10.2
t = 800
x0 10 20 30
-0.2-0.1
00.10.2
-0.2-0.1
00.10.2
t = 1100
x0 10 20 30
-0.2-0.1
00.10.2
Figure 3.4 – Two realizations at various times of the stochastic BF instability and FPU recur-rence with random phase in the initial modulation. Both the surface elevation (blue) andcomplex envelope modulus (orange) are represented. The top row shows the realization withmaximum surface elevation η(x, tω) at t = 800, while the bottom row shows the one withminimum surface elevation at that time. Note that we only plot a portion of the domaincorresponding to the wavelength of the linearly most unstable wavenumber ∆km = 2a0 = 0.2.
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 = 800. 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 (2013) 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
Chapter 3. Preliminary results and validation
Mod
e 1
#10 -3
-2
-1
0
1
2"mean!i
Mod
e 2
#10 -3
-2
-1
0
1
2
Mod
e 3
#10 -4
-2
-1
0
1
2
t200 400 600 800 1000 1200
Mod
e 4
#10 -4
-2
-1
0
1
2
#10 -4
-5
0
5"mean;mn!i
#10 -4
-5
0
5
#10 -4
-4
-2
0
2
4
t200 400 600 800 1000 1200
#10 -4
-4
-2
0
2
4
#10 -5
-1
-0.5
0
0.5
1"mnl!i
#10 -5
-2
-1
0
1
#10 -5
-1
0
1
2
t200 400 600 800 1000 1200
#10 -5
-1
-0.5
0
0.5
1
Figure 3.5 – Modal energy production in every mode due to (i) linear interaction with themean flow (first column), (ii) nonlinear simultaneous interaction with the mean and twoother modes (second column) and (iii) nonlinear interaction with three other modes (thirdcolumn) for the stochastic BF instability and FPU recurrence with random phase in the initialmodulation. Recall that the solution is only valid up to t ∼ 1100.
3.2 Random Gaussian wavenumber spectrum
The Benjamin-Feir instability provides an idealized framework for the investigation of instabil-
ities 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. There-
fore, 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, t0;ω) =N /2∑
n=−N /2+1
√2∆k1F (∆kn)e iθn (ω)e i∆kn x , (3.2)
where the Fourier coefficients have random uncorrelated phases θn(ω) drawn from a uniform
distribution on [0,2π], and deterministic amplitude proportional to the square root of the
following Gaussian wavenumber spectrum
F (∆kn) = ε2
σp
2πexp
(−∆k2
n
2σ2
), (3.3)
where ∆kn = n∆k1 is the modulation wavenumber with ∆k1 the wavenumber discretization,
ε is the average wave steepness and σ is the relative bandwidth of the spectrum. The av-
erage steepness is defined as the standard deviation of the surface elevation, and it can be
32
3.2. Random Gaussian wavenumber spectrum
verified that we indeed have ε=√
E [η(x, t0;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 = 2p
2ε
σ, (3.4)
is less than 1. In the opposite case, numerical simulations of the NLS equation (Janssen, 2003;
Dysthe et al., 2003) and experiments (Onorato et al., 2005) 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 = 10 Fourier modes with a
wavenumber discretization ∆k1 = 0.06. The spectrum is assigned a fixed initial width σ= 0.1,
and we consider seven different values for the wave steepness ε ranging from 0.025 to 0.1,
giving a range of values for the initial BFI from 0.72 to 2.87 (in this regard note that most
sea states have a BFI < 1, Dysthe et al., 2008). 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 = 1.43. 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 θn(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 ui (x, t0) as per the relations (2.44), since two real-valued
stochastic coefficients Yi (t0;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 = 50 ·2π. Note
33
Chapter 3. Preliminary results and validation
"k-1 -0.5 0 0.5 1
F("
k)
0
0.002
0.004
0.006
0.008
0.01
0.012Gaussian spectrum with BFI = 1.43
discretecontinuous
x0 20 40 60 80 100
rand
om p
hase
s
-0.2
-0.1
0
0.1
0.2
zero
pha
ses
-0.2
-0.1
0
0.1
0.2|u| (orange) and 2 (blue)
Figure 3.6 – Initial condition for the DO simulation of a Gaussian random wave field forthe case BFI = 1.43. Left, discretized Gaussian wavenumber spectrum and its continuousequivalent. Right, two corresponding realizations of the surface elevation η(x, t0;ω) (blue) andenvelope modulus |u(x, t0;ω)| (orange), either when all phases are equal to zero (top) or withrandom uncorrelated phases (bottom).
k0 1 2
E[|F
[2]|2
]
0
20
40
60
80
100t0, BFI = 0.72
k0 1 2
tf
k0 1 2
E[|F
[2]|2
]
0
200
400
600
800
1000
1200
1400t0, BFI = 2.87
k0 1 2
tf
Figure 3.7 – Initial and final time wavenumber spectra E [|F [η]|2] for initial BFI = 0.72 (left)and BFI = 2.87 (right). The final time is taken as t f = 2/ε2.
that the minimum wavenumber allowed by the domain is ∆kmi n = 2π/L = 0.02, thus equal
to one third the wavenumber discretization ∆k1 = 0.06 that we assign to our initial Gaussian
spectrum. We could have used a finer wavenumber discretization ∆k1, but spanning the same
spectral width σ= 0.1 would have resulted in an unreasonably high number of DO modes and
prohibitive computational cost.
3.2.1 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 wavenum-
ber spectra given by E [|F [η]|2] for two different initial values of BFI = 0.72 (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., 2003), results are shown for a final time defined as t f = 2/ε2.
In accordance with results in the literature, it is observed that the case BFI = 0.72 < 1 is stable
while the case BFI = 2.87 > 1 is unstable and relaxes to a new spectrum that appears to be
34
3.2. Random Gaussian wavenumber spectrum
Initial time BFI0 0.5 1 1.5 2 2.5 3
Fin
al ti
me
BF
I
0
0.5
1
1.5
DO simulationJanssen (2003)Dysthe et al. (2003)
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 (2003) andDysthe et al. (2003) are also shown.
stable. Note that the spikes reflect the fact that the wavenumber resolution ∆kmi n of the
domain is one third the wavenumber discretization ∆k1 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
ε(t f ) =√
E [η(x, t f ;ω)2] and σ(t f ) =∫∆k2E [|F [η(x, t f ;ω)]|2]d∆k∫
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, 2003; Dysthe et al., 2003), 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 = 0.72 (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 0 corresponds to a Gaussian surface, while a value greater than 0
35
Chapter 3. Preliminary results and validation
2/0-5 0 5
10 -6
10 -4
10 -2
100t0, BFI = 0.72
2/0-5 0 5
tf
2/0-5 0 5
10 -6
10 -4
10 -2
100t0, BFI = 1.43
2/0-5 0 5
tf
Figure 3.9 – Probability density function at initial and final times of the surface elevationnormalized by ε=
√E [η(x, t ;ω)2], for initial BFI = 0.72 (left) and BFI = 1.43 (right). The final
time is taken as t f = 2/ε2. For reference, the corresponding Gaussian distribution is alsoshown in orange dotted line in each case.
Initial time BFI0 0.5 1 1.5 2 2.5 3
Fin
al ti
me
kurt
osis
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 3.10 – Final time kurtosis versus initial BFI, where the kurtosis is calculated usingequation (3.6) and is representative of the deviation from normality of the distribution of thesurface elevation.
is representative of a higher probability of extreme waves due to predominant nonlinear
effects. The final time kurtosis is plotted in Figure 3.10 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
3.2. Random Gaussian wavenumber spectrum
Mea
n
-0.1
-0.05
0
0.05
0.1BFI = 0.72
Mod
e 1
-0.1
-0.05
0
0.05
0.1
Mod
e 2
-0.1
-0.05
0
0.05
0.1
Mod
e 3
-0.1
-0.05
0
0.05
0.1
x0 50 100
Mod
e 4
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1BFI = 1.00
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
x0 50 100
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1BFI = 1.43
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
x0 50 100
-0.1
-0.05
0
0.05
0.1
Figure 3.11 – Mean and first four DO modes at final time t f = 2/ε2, for different initial valuesof the BFI. Both the complex envelope modulus (blue) and real part (orange) are represented.Note that while the domain size is L = 50 ·2π, we only plot the solution over one third of thewhole domain because of the effective periodicity induced by the wavenumber discretization∆k1 = 0.06 of the initial condition.
3.2.2 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/ε2. Note that
only one third of the whole domain size is represented, since the wavenumber discretization
∆k1 = 3∆kmi 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 = 0.72
correspond to sinusoidal modulations of wavenumber 0.06, equal to the Benjamin-Feir linearly
most unstable wavenumber ∆km = 2a0 for a uniform wave of amplitude a0 = 0.03, a value
close to the actual r.m.s amplitude E [a] =√
2E [η2] = 0.035 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 = 0.72,
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
Chapter 3. Preliminary results and validation
t0 1000 2000
E[Y
i2]
0
0.02
0.04
0.06
0.08
0.1BFI = 0.72
MeanMode 1Mode 2etc
t0 500 1000 1500
0
0.05
0.1
0.15
0.2BFI = 1.00
MeanMode 1Mode 2etc
t0 200 400 600 800
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4BFI = 1.43
MeanMode 1Mode 2etc
Figure 3.12 – Evolution of the energy in the mean and the modes for different initial values ofthe BFI. Recall that the mean is initially assigned the modulation wavenumber 0.
Figure 3.13 – Stochastic attractor at final time of the solution for different initial values of theBFI. The attractor is represented in terms of the 3D scatter plot of the first three stochasticcoefficients for all realizations. In addition, each realization is assigned a color indicative ofthe maximum value of its corresponding surface elevation η(x, t ;ω) = Reu(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, al-
lowing for the visualization in phase space of the time-dependent structure of the ensem-
ble 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) = Reu(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
3.2. Random Gaussian wavenumber spectrum
mem
ber
with
max
imum
2
-0.2
-0.1
0
0.1
0.2BFI = 0.72
x0 50 100
mem
ber
with
min
imum
2
-0.1
-0.05
0
0.05
0.1-0.2
-0.1
0
0.1
0.2BFI = 1.00
x0 50 100
-0.1
-0.05
0
0.05
0.1-0.4
-0.2
0
0.2
0.4BFI = 1.43
x0 50 100
-0.1
-0.05
0
0.05
0.1
Figure 3.14 – Two realizations of the stochastic solution at final time for different initial valuesof the BFI. The top row shows the realization with maximum surface elevation η(x, t ;ω) whilethe bottom row shows that with minimum surface elevation. Note that we only plot thesolution over one third of the whole domain because of the effective periodicity induced bythe wavenumber discretization ∆k1 = 0.06 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 = 1.00 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, 2013) 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 3.11.
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 3.12.
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 inter-
connected 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 com-
putational 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
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 u(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, 2009; Cousins & Sapsis, 2015b,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.
40
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, 2013). 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 = 1.43. A spatially localized wave
packet around x = 140 at time 0 (top row) is observed to focus in an extreme wave and reaches
a maximum amplitude of 0.37 at t = 270 (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, 2015b,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
Chapter 4. Dynamics of an extreme wave
t = 0
-0.4
-0.2
0
0.2
0.4
t = 2
70
-0.4
-0.2
0
0.2
0.4
x0 50 100 150 200 250 300
t = 4
00
-0.4
-0.2
0
0.2
0.4
Figure 4.1 – Focusing wave packet in a deterministic initial condition generated as a Gaussianspectrum of waves (3.3) with BFI = 1.43.
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
u(x, t0;ω) = u(x, t0)+3∑
n=1Ane iθn (ω)e i∆kn x , (4.1)
where the mean u(x, t0) 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 ∆kn = 0.2n, deterministic amplitude An = 0.001 and random uncorrelated
phases θn(ω) drawn from a uniform distribution on [0,2π]. As before, the stochastic coeffi-
cients 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 vari-
ous 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 = 270 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
4.3. Attractor of an idealized extreme wave
Mea
n
-0.2
-0.1
0
0.1
0.2t = 0
Mod
e 1
-0.1
-0.05
0
0.05
0.1
Mod
e 2
-0.1
-0.05
0
0.05
0.1
Mod
e 3
-0.1
-0.05
0
0.05
0.1
x100 150 200
Mod
e 4
-0.1
-0.05
0
0.05
0.1
-0.2
-0.1
0
0.1
0.2t = 100
-0.1
0
0.1
0.2
-0.1
0
0.1
0.2
-0.1
-0.05
0
0.05
0.1
x100 150 200
-0.1
0
0.1
0.2
-0.2
0
0.2
0.4t = 270
-0.2
0
0.2
0.4
-0.4
-0.2
0
0.2
0.4
-0.1
-0.05
0
0.05
0.1
x100 150 200
-0.1
0
0.1
0.2
Figure 4.2 – Mean and first four DO modes at various times for the initial wave field of Figure4.1 (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, 2015b), 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, t0) =A0sech(x/L0), 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, 2015b) 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 A0 is larger than the critical value
A0,c = 1/(p
2L0), leading to an extreme wave, or broaden and decay otherwise. Here, we
consider a length scale L0 = 7 and amplitude A0 = 0.15 giving roughly similar properties to
43
Chapter 4. Dynamics of an extreme waveM
ean
0
0.05
0.1
0.15t = 0
Mod
e 1
-0.4
-0.2
0
0.2
0.4
Mod
e 2
-0.1
0
0.1
0.2
0.3
Mod
e 3
-0.2
-0.1
0
0.1
0.2
x100 150 200
Mod
e 4
-0.1
0
0.1
0.2
-0.4
-0.2
0
0.2
0.4t = 300
-0.2
0
0.2
0.4
0.6
-0.4
-0.2
0
0.2
0.4
-0.2
0
0.2
0.4
0.6
x100 150 200
-0.2
0
0.2
0.4
-0.05
0
0.05
0.1
0.15t = 500
-0.4
-0.2
0
0.2
0.4
-0.2
0
0.2
0.4
-0.2
0
0.2
0.4
x100 150 200
-0.4
-0.2
0
0.2
0.4
Figure 4.3 – Mean and first four DO modes at various times for a localized wave group ofthe form A0sech(x/L0) initially subject to small localized stochastic perturbations. Both thecomplex envelope modulus (blue) and real part (orange) are represented. Note that t = 300would 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 0.24 around t = 300.
We now study the evolution of this idealized localized wave group subject to small stochastic
perturbations. We consider the following random initial condition
u(x, t0;ω) = A0sech(x/L0)+ sech(x/L0)3∑
n=1Ane iθn (ω)e i∆kn x , (4.2)
where A0 = 0.15, L0 = 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 ∆kn = 0.02n, deterministic amplitude An = 0.002 and random uncorrelated
phases θn(ω) drawn from a uniform distribution on [0,2π]. 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
4.3. Attractor of an idealized extreme wave
t0 50 100 150 200 250 300 350 400 450 500
E[Y
i2]
10 -4
10 -3
10 -2
10 -1
100
MeanMode 1Mode 2Mode 3Mode 4Mode 5Mode 6
Figure 4.4 – Energy of the mean ⟨u, u⟩ and the modes E [Y 2i ] for a localized wave group of the
form A0sech(x/L0) subject to small localized stochastic perturbations.
Figure 4.5 – Stochastic attractor at various times for the solution to a localized wave groupof the form A0sech(x/L0) initially subject to small localized stochastic perturbations. Theattractor is represented in terms of the 3D scatter plot of the first three stochastic coefficientsfor all realizations. In addition, each realization is assigned a color indicative of the maximumvalue 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 = 300 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
Chapter 4. Dynamics of an extreme wave
Mod
e 1
#10 -4
-1
0
1
2
3"mean!i
Mod
e 2
#10 -5
-4
-2
0
2
Mod
e 3
#10 -5
-3
-2
-1
0
1
t100 200 300 400 500
Mod
e 4
#10 -6
-4
-2
0
2
4
#10 -5
-5
0
5
10
15"mean;mn!i
#10 -5
-5
0
5
#10 -5
-1
0
1
2
3
t100 200 300 400 500
#10 -6
-5
0
5
10
#10 -5
-10
-5
0
5"mnl!i
#10 -5
-5
0
5
10
#10 -5
-2
0
2
4
t100 200 300 400 500
#10 -6
-4
-2
0
2
4
Figure 4.6 – Modal energy production in the first four modes due to (i) linear interaction withthe mean flow (first column), (ii) nonlinear simultaneous interaction with the mean and twoother modes (second column) and (iii) nonlinear interaction with three other modes (thirdcolumn) for the solution to a localized wave group of the form A0sech(x/L0) initially subjectto small localized stochastic perturbations.
mem
ber
with
max
imum
|u|
-0.2
0
0.2
t = 0
x100 150 200
mem
ber
with
min
imum
|u|
-0.2
0
0.2
-0.2
0
0.2
t = 300
x100 150 200
-0.2
0
0.2
-0.2
0
0.2
t = 500
x100 150 200
-0.2
0
0.2
Figure 4.7 – Two realizations at various times of the solution to a localized wave group ofthe form A0sech(x/L0) initially subject to small localized stochastic perturbations. Both thesurface elevation (blue) and complex envelope modulus (orange) are represented. The toprow shows the realization with maximum envelope modulus |u(x, t ;ω)| at t = 300, while thebottom 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 ∼ 200 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 = 300. No
big differences in the degree of focusing at t = 300 are observed between the two.
46
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 4.3.
5.1 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= k0a leading to the NLS equation one step
further to O (ε4), the MNLS equation is obtained. In addition to being more accurate for values
of the wave steepness larger than 0.15, 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, 1980; Janssen, 2004), 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, 2011), up to a time scale t ∼ 10ε−2, one order of magnitude
larger than that for the NLS equation (Trulsen et al., 2001).
The MNLS equation is presented in Appendix B, together with its corresponding DO equations.
Except for the deterministic forcing fields F0, Fi , Fi j and Fi j k appearing in the DO evolution
47
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
0.01, 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., 2003).
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 per-
turbations 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 en-
velope of amplitude a0 = 0.1 that is perturbed with one unstable Fourier mode of modulation
wavenumber ∆k1 = 0.2 and one stable Fourier mode of wavenumber ∆k2 = 0.4. The Fourier
modes are assigned independent random phases and deterministic small amplitude equal to
0.0036, 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 ⟨u, u⟩ together with the stochastic energy present in the modes E [Y 2i ]. The
observations from Section 3.1 are still valid here. In a nutshell, the DO modes 1 and 2 at t = 800
in Figure 5.1 still represent the linearly unstable modulation ∆k1 = 0.2 that was assigned in
the initial condition, while modes 3 and 4 represent the stable modulation ∆k2 = 0.4, 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,0) =0.1+0.0036cos∆km x where∆km = 0.2, 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 = 0 (carrier wave), 0.2 (unstable modulation) and 0.4 (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
5.2. Idealized Benjamin-Feir instability
Mea
n
-0.1
0
0.1
t = 0M
ode
1
-0.1
-0.05
0
0.05
0.1
Mod
e 2
-0.1
-0.05
0
0.05
0.1
Mod
e 3
-0.1
-0.05
0
0.05
0.1
x0 10 20 30
Mod
e 4
-0.1
-0.05
0
0.05
0.1
-0.1
0
0.1
t = 800
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
x0 10 20 30
-0.1
-0.05
0
0.05
0.1
-0.1
0
0.1
t = 1400
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
0
0.05
0.1
x0 10 20 30
-0.1
-0.05
0
0.05
0.1
Figure 5.1 – Mean and DO modes at various times for the stochastic BF instability and FPUrecurrence with random phase in the initial modulation, under the MNLS equation. Both thecomplex envelope modulus (blue) and real part (orange) are represented. Note that while thedomain size is L = 50·2π, we only plot the solution over a portion of the domain correspondingto the wavelength of the linearly most unstable wavenumber ∆km = 2a0 = 0.2.
t0 200 400 600 800 1000 1200 1400 1600 1800 2000
E[Y
i2]
10 -3
10 -2
10 -1
100
101
MeanMode 1Mode 2Mode 3Mode 4Deterministic
Figure 5.2 – Energy of the mean ⟨u, u⟩ and the modes E [Y 2i ] for the stochastic BF instability and
FPU recurrence with random phase in the initial modulation, under the MNLS equation. Thedashed lines show the corresponding energies for a deterministic simulation of an equivalentinitial condition u(x,0) = 0.1+0.0036cos∆km x with ∆km = 0.2, with the energies obtained asthe normalized squared modulus of the Fourier coefficients of wavenumber ∆k = 0 (carrierwave), 0.2 (unstable modulation) and 0.4 (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
Chapter 5. Results with the MNLS equation
equation is also happening here, as can be seen from the shape of the modes at t = 1400 which
deviate from deterministic theory, or the oscillations in the energy levels from t ∼ 1900 in
Figure 5.2.
As a conclusion, we have shown that our DO equations for MNLS produce results in accor-
dance with deterministic Benjamin-Feir and Fermi-Pasta-Ulam recurrence theory. While the
quantitative agreement between our DO simulation and deterministic results is not as excel-
lent 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, t0) = A0sech(x/L0) with L0 = 7 and A0 = 0.15. A
deterministic simulation of the MNLS equation reveals that these values lead to a maximum
focusing amplitude of 0.21 at t = 410. 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, 2015b).
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 = 300 for NLS,
t = 410 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
50
5.3. Attractor of an idealized extreme wave
Mea
n
0
0.05
0.1
0.15t = 0
Mod
e 1
-0.4
-0.2
0
0.2
0.4
Mod
e 2
-0.1
0
0.1
0.2
0.3
Mod
e 3
-0.2
-0.1
0
0.1
0.2
x100 150 200
Mod
e 4
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1
0.2t = 410
-0.2
0
0.2
0.4
-0.2
0
0.2
0.4
-0.4
-0.2
0
0.2
0.4
x100 150 200
-0.4
-0.2
0
0.2
0.4
-0.05
0
0.05
0.1
0.15t = 680
-0.4
-0.2
0
0.2
0.4
-0.4
-0.2
0
0.2
0.4
-0.2
0
0.2
0.4
x100 150 200
-0.2
0
0.2
0.4
Figure 5.3 – Mean and first four DO modes at various times for a localized wave group ofthe form A0sech(x/L0) initially subject to small localized stochastic perturbations, underthe MNLS equation. Both the complex envelope modulus (blue) and real part (orange) arerepresented. Note that t = 410 would be the time of maximum focusing for the unperturbedwave group.
t0 100 200 300 400 500 600
E[Y
i2]
10 -4
10 -3
10 -2
10 -1
100
MeanMode 1Mode 2Mode 3Mode 4Mode 5Mode 6
Figure 5.4 – Energy of the mean ⟨u, u⟩ and the modes E [Y 2i ] for a localized wave group of
the form A0sech(x/L0) subject to small localized stochastic perturbations, under the MNLSequation.
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 = 410. 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
Chapter 5. Results with the MNLS equation
Figure 5.5 – Stochastic attractor at various times for the MNLS solution to a localized wavegroup of the form A0sech(x/L0) initially subject to small localized stochastic perturbations.The attractor is represented in terms of the 3D scatter plot of the first three stochastic coef-ficients for all realizations. In addition, each realization is assigned a color indicative of themaximum value of its corresponding envelope modulus |u(x, t ;ω)|.
Mod
e 1
#10 -4
0
0.5
1
1.5
2"mean!i
Mod
e 2
#10 -5
-2
-1
0
1
Mod
e 3
#10 -6
-15
-10
-5
0
5
t200 400 600
Mod
e 4
#10 -6
-2
-1
0
1
2
#10 -5
-5
0
5
10
15"mean;mn!i
#10 -5
-5
0
5
#10 -5
-1
0
1
2
t200 400 600
#10 -6
-4
-2
0
2
4
#10 -5
-10
-5
0
5"mnl!i
#10 -5
-5
0
5
10
#10 -6
-5
0
5
10
t200 400 600
#10 -6
-2
0
2
4
6
Figure 5.6 – Modal energy production in the first four modes due to (i) linear interaction withthe mean flow (first column), (ii) nonlinear simultaneous interaction with the mean and twoother modes (second column) and (iii) nonlinear interaction with three other modes (thirdcolumn) for the MNLS solution to a localized wave group of the form A0sech(x/L0) initiallysubject to small localized stochastic perturbations.
the degree of focusing at t = 410 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 = 410 than the bottom one, a
situation that reverses at t = 680.
52
5.3. Attractor of an idealized extreme wave
mem
ber
with
max
imum
|u|
-0.2
0
0.2
t = 0
x100 150 200
mem
ber
with
min
imum
|u|
-0.2
0
0.2
-0.2
0
0.2
t = 410
x100 150 200
-0.2
0
0.2
-0.2
0
0.2
t = 680
x100 150 200
-0.2
0
0.2
Figure 5.7 – Two realizations at various times of the MNLS solution to a localized wave groupof the form A0sech(x/L0) initially subject to small localized stochastic perturbations. Both thesurface elevation (blue) and complex envelope modulus (orange) are represented. The toprow shows the realization with maximum envelope modulus |u(x, t ;ω)| at t = 410, while thebottom row shows that with minimum envelope modulus.
53
Conclusions and perspectives
In this thesis, we have implemented a reduced-order stochastic framework based on the
dynamically orthogonal (DO) equations (introduced in Sapsis & Lermusiaux, 2009), for the
stochastic evolution of water waves governed by the nonlinear Schrödinger (NLS) equation
and subject to random initial conditions. Using a generalized time-dependent truncated
Karhunen-Loève expansion, we decomposed the stochastic solution in a mean state and
stochastic fluctuations, the latter being described by a finite number of deterministic modes
and associated stochastic coefficients. A set of explicit and coupled equations was then derived
for the time evolution of these quantities, allowing for the efficient computation of the full
stochastic solution, since only the stochastic coefficients are solved in a Monte-Carlo fashion.
Additionally, expressions were derived to quantify the transfers of energy (in a variance sense)
between modes and the mean state.
We benchmarked our DO equations against two well-known cases, at the same time illustrating
their use. First, we considered a uniform wavetrain perturbed by unstable modulations with
small deterministic amplitude but random phases. In perfect accordance with deterministic
results, the stochastic solution was observed to undergo Benjamin-Feir instability followed by
Fermi-Pasta-Ulam recurrence. The DO modes reproduced exactly the shape of the linearly
unstable modulation and its harmonics, while the variance of the stochastic coefficients was
observed to increase and decrease at the exact same rate as the unstable modulation of an
equivalent deterministic simulation. Energy transfers showed the flow of energy from the
mean (containing the uniform wave) to the modes then back to the mean, in accordance
with Fermi-Pasta-Ulam recurrence. The stochastic solution was observed to possess a low
dimensional attractor in phase space.
Second, we considered the case of a Gaussian spectrum of waves with random phases, leading
to an irregular wave field for each realization. Relevant statistical properties of the DO solution
were compared with ensemble-averaged results from full Monte-Carlo simulations and a
reasonable agreement between the two was obtained. However, the need to reproduce a large
number of wavenumbers with independent random phases resulted in a high number of DO
modes and a high computational cost. Moreover, the modes were observed to all converge to
similar levels of energy, meaning that no dominant direction in the stochastic dynamics of
these irregular wave fields could be perceived on a global scale. However, we observed in some
55
Chapter 5. Results with the MNLS equation
individual realizations of the higher-energy wave fields that concentration of energy in the
form of localized wave packets resulted in local extreme waves, phenomenon that is known in
literature (Ruban, 2013). Due to their appearance at random locations in the domain and their
presence in only some of the realizations, these extreme waves could not be captured as such
by the DO modes. This, combined with the issue of computational cost, indicated that the DO
framework was not suited for the solution of a stochastic wave field with Gaussian spectrum
and random phases.
Instead, we turned our attention to a single one of these extreme waves, this time using
the DO framework as a means to investigate its nonlinear evolution under small stochastic
perturbations of its initial shape. We first showed that in this case the DO modes are able to
adaptively track the emergence of the extreme wave out of a Gaussian spectrum background
of waves. We then focused on an isolated wave packet of idealized shape as a prototype model
for these extreme waves and investigated its nonlinear evolution when initially subjected to
small localized stochastic perturbations. The resulting stochastic solution was observed to fall
on a low-dimensional attractor in phase space, and the modes revealed the effect of initial
perturbations on the nonlinear evolution of the wave group.
As an immediate next step, we want to consider a higher-order version of the NLS equation,
the modified nonlinear Schrödinger (MNLS) equation introduced by Dysthe (1979). Since it is
accurate to a higher order in amplitude, its use would be more appropriate for the investigation
of the nonlinear evolution of extreme waves. The DO equations for the MNLS equation can be
found in Appendix B and initial results for the case of Section 3.1 have already been obtained,
validating the equations and their implementation. Another possible future direction concerns
the use of a complex-valued inner product hence complex-valued stochastic coefficients in
the DO expansion (2.13). As has been discussed in Section 2.2.2, this choice would lead
to a reduction in the number of required modes, but dealing with the resulting solution is
not straightforward as was seen in Appendix A. A third possible line of work would be the
integration of real-life measurements with our reduced-order model, following a Kalman
filtering algorithm. Finally, recall that a problem of the DO modes resides in their inability to
track localized events that appear at different random locations between the realizations. A
possible way around this problem could be to use the following expansion instead of (2.13)
u(x, t ;ω) = u(x + c(t ;ω), t ;ω) = ¯u(x + c(t ;ω), t )+s∑
i=1Yi (t ;ω)ui (x + c(t ;ω), t ), (5.1)
where the time-dependent stochastic shift c(t ;ω) would allow the modes to track dynamics
appearing at different locations between the realizations. Additional equations are then
needed for c(t ;ω) in order to close the problem. Similar issues have been investigated in the
context of the POD (Rowley & Marsden, 2000), but these concerned statistically stationary
systems so our problem presents a greater challenge.
56
Appendix A
Complex coefficients in the DOframework
Here, we consider the complex-valued inner product (2.5) and we derive the resulting DO
equations for the NLS equation. As was thoroughly discussed in Section 2.2.2, this complex
inner product implies that the stochastic coefficients in the DO expansion (2.13) are also
complex-valued, hence we hereafter refer to them as Zi (t ;ω). First, we derive the DO equations
in Section A.1. We then show issues associated with the removal of the correlations between
the complex coefficients in Section A.2.
A.1 Dynamically orthogonal equations with complex coefficients
As was done in Section 2.2.3, we begin by inserting the DO expansion (2.13) with complex
stochastic coefficients Zi (t ;ω) in the NLS equation (2.19), leading to the following governing
equation for all unknown quantities
∂u
∂t= F0 +Zi Fi +Z∗
i Gi +Zi Z j Fi j +Zi Z∗j Gi j +Zi Z j Z∗
k Fi j k , (A.1)
where the asterisk denotes the complex conjugate, and the deterministic fields on the right-
hand side are defined as
F0 =− i
8
∂2u
∂x2 − i
2|u|2u, Fi =− i
8
∂2ui
∂x2 − i |u|2ui , Gi =− i
2u2u∗
i ,
Fi j =− i
2u∗ui u j , Gi j =−i uui u∗
j , Fi j k =− i
2ui u j u∗
k .
(A.2)
Following the procedure of Section 2.2.3, the equation for the mean field writes
∂u
∂t= F0 +CZi Z j Fi j +CZi Z∗
jGi j +MZi Z j Z∗
kFi j k , (A.3)
57
Appendix A. Complex coefficients in the DO framework
where CZi Z j = E [Zi Z j ] and CZi Z∗j= E [Zi Z∗
j ] are respectively the complex pseudo-covariance
and covariance matrices, and MZi Z j Z∗k= E [Zi Z j Z∗
k ] is the matrix of third-order moments. The
stochastic coefficients obey the following evolution equation
dZi
dt= Zm ⟨Fm ,ui ⟩+Z∗
m ⟨Gm ,ui ⟩+ (Zm Zn −CZm Zn )⟨Fmn ,ui ⟩+ (Zm Z∗
n −CZm Z∗n
)⟨Gmn ,ui ⟩+ (Zm Zn Z∗l −MZm Zn Z∗
l)⟨Fmnl ,ui ⟩.
(A.4)
Finally, the evolution of the basis functions is given by
∂ui
∂t= Hi −⟨Hi ,u j ⟩u j , (A.5)
where Hi is defined by E [L [u]Zk ]C−1Zk Z∗
i= Hi and has the following expression
Hi = Fi +Gm CZ∗m Z∗
kC−1
Zk Z∗i+Fmn MZm Zn Z∗
kC−1
Zk Z∗i
+Gmn MZm Z∗n Z∗
kC−1
Zk Z∗i+Fmnl MZm Zn Z∗
l Z∗k
C−1Zk Z∗
i,
(A.6)
with MZm Zn Z∗l Z∗
k= E [Zm Zn Z∗
l Z∗k ] the matrix of fourth-order moments. While we were able to
derive these equations in a straightforward manner, we will see some of the complications
introduced by complex coefficients in the next section.
A.2 Diagonalization of the complex covariance matrix
Consider the DO solution at a given time. Similarly to what is done in Section 2.4.3 for real-
valued coefficients, we want to remove the second-order correlations between the complex-
valued stochastic coefficients. The second-order statistics are described by a complex covari-
ance matrix Ci j = E [Zi Z∗j ] and a complex pseudo-covariance matrix Ci j = E [Zi Z j ], and both
matrices may have non-zero off-diagonal elements. The covariance matrix is Hermitian and
can therefore be decomposed as
C =V DV † (A.7)
where the columns of V contain the eigenvectors of C and V satisfies V V † =V †V = I with †
denoting the complex conjugate, and D is a diagonal matrix containing the (real) eigenvalues
of C . We now define a new basis with u′i = umVmi . Note that
⟨ui ,u′j ⟩ = ⟨ui ,umVm j ⟩ =V ∗
m j ⟨ui ,um⟩ =V ∗i j ⇒ Vi j = ⟨u′
j ,ui ⟩. (A.8)
We first show that the new basis u′i is orthonormal
⟨u′i ,u′
j ⟩ = ⟨umVmi ,unVn j ⟩ =Vmi V ∗n j ⟨um ,un⟩ =Vmi V ∗
m j =V †j mVmi = δi j . (A.9)
58
A.2. Diagonalization of the complex covariance matrix
The coefficients Z ′i in the new basis are given by
Z ′i = ⟨Z j u j ,u′
i ⟩ = Z j ⟨u j ,u′i ⟩ = Z j V ∗
j i , (A.10)
and they are uncorrelated with respect to the new covariance matrix
C ′i j = E [Z ′
i Z ′∗j ] = E [ZmV ∗
mi Z∗n Vn j ] =V †
i mCmnVn j = Di j = δi jλi , (A.11)
where λi are the eigenvalues of C and the diagonal elements of D . Note that, however, the new
pseudo-covariance matrix remains correlated. Indeed, we have
C ′i j = E [Z ′
i Z ′j ] = E [ZmV ∗
mi ZnV ∗n j ] =V †
i mCmnV ∗n j (A.12)
which is not necessarily diagonal. To better understand the implications of this, we consider
what we would have if we were to represent the same solution with twice the number of
real-valued coefficients. Following the discussion in Section 2.2.2, we write Zi = Xi + i Yi and
Z ′i = X ′
i + i Y ′i , where Xi , Yi , X ′
i , Y ′i would be the equivalent real-valued stochastic coefficients.
If we were to directly work with these real-valued coefficients, there would be one single
real-valued covariance matrix four times the size of the equivalent complex covariance or
pseudo-covariance matrices (since there would be twice the number of coefficients), but we
would be able to diagonalize it, resulting in completely uncorrelated coefficients i.e. E [X ′i X ′
j ] =E [Y ′
i Y ′j ] = E [X ′
i Y ′j ] = E [Y ′
i X ′j ] = 0 for i 6= j . However, in the case of the complex formulation
for which we only managed to diagonalize the complex covariance matrix C ′i j , we have
C ′i j = E [X ′
i X ′j ]+E [Y ′
i Y ′j ]− i E [X ′
i Y ′j ]+ i E [Y ′
i X ′j ] = δi jλi , (A.13)
C ′i j = E [X ′
i X ′j ]−E [Y ′
i Y ′j ]+ i E [X ′
i Y ′j ]+ i E [Y ′
i X ′j ] 6= δi jλi , (A.14)
from which we see that in order to eliminate the correlation between all the X ′i and Y ′
i , we
would need both the covariance and pseudo-covariance matrices C ′ and C ′ to be diagonal.
Since this cannot be achieved with our diagonalization procedure, there will still be correla-
tion between the real and imaginary parts X ′i and Y ′
i of the complex coefficients, while that
correlation could have been completely removed in the equivalent formulation involving only
real-valued coefficients and one single real-valued covariance matrix.
As a side note, remark that in the original KL expansion (2.7) with complex coefficients, the
coefficients only verify E [Zi Z∗j ] = 0 for i 6= j , meaning that only their covariance matrix is
diagonalized, and not their pseudo-covariance matrix. This therefore corresponds to what we
are able to do, but doesn’t explain the difference with the equivalent real-valued formulation
for which we are moreover able to remove the correlations between the real and imaginary
parts of the equivalent complex coefficients. In any case, much more understanding of the
subject must be done before we can feel confident to proceed with using the complex-valued
inner product and coefficients. This issue of diagonalization of complex statistics is indeed an
area of active research (Eriksson & Koivunen, 2006; Adali et al., 2011; Cheong Took et al., 2012).
59
Appendix B
Dynamically orthogonal MNLSequation
The modified nonlinear Schrödinger equation (MNLS) was obtained by Dysthe (1979) through
a similar multiple-scales expansion (1.12) as for the NLS equation, but taking the perturbation
expansion one step further to fourth order in wave steepness ε= k0a. In a reference frame
moving at the group velocity of the carrier wave, the resulting equation for the complex
envelope of the first harmonic writes
∂u
∂t=− i
8
∂2u
∂x2 + 1
16
∂3u
∂x3 − i
2|u|2u − 3
2|u|2 ∂u
∂x− 1
4u2 ∂u∗
∂x− i u
∂φ
∂x
∣∣∣∣z=0
(B.1)
where time has been nondimensionalized withω0 and space with k0, the asterisk ∗ denotes the
complex conjugate and φ is the zeroth harmonic velocity potential, which can be expressed
in terms of u by solving Laplace’s equation with appropriate boundary conditions (given in
Dysthe, 1979; Trulsen & Dysthe, 1996) to give
∂φ
∂x
∣∣∣∣z=0
=−1
2F−1|k|F |u|2, (B.2)
where F denotes the Fourier transform. The corresponding reduced-order DO equations are
derived in a similar fashion as those for the NLS equation. Using the real-valued inner product
(2.17), the equations for the quantities involved in the DO expansion (2.13) are exactly the
same, except for the deterministic fields F0, Fi , Fi j and Fi j k which become in this case
F0 =− i
8
∂2u
∂x2 + 1
16
∂3u
∂x3 − i
2|u|2u − 3
2|u|2 ∂u
∂x− 1
4u2 ∂u∗
∂x+ i
2F−1|k|F |u|2u, (B.3)
Fi =− i
8
∂2ui
∂x2 + 1
16
∂3ui
∂x3 − i
2|u|2ui − i u Reuu∗
i − 3
2|u|2 ∂ui
∂x−3
∂u
∂xReuu∗
i
− 1
4u2 ∂u∗
i
∂x− 1
2u∂u∗
∂xui + iF−1|k|F Reuu∗
i u + i
2F−1|k|F |u|2ui , (B.4)
61
Appendix B. Dynamically orthogonal MNLS equation
Fi j =− i
2u Reui u∗
j − i Reuu∗i u j − 3
2
∂u
∂xReui u∗
j −3Reuu∗i ∂u j
∂x− 1
4
∂u∗
∂xui u j
− 1
2uui
∂u∗j
∂x+ i
2F−1|k|F Reui u∗
j u + iF−1|k|F Reuu∗i u j , (B.5)
Fi j k =− i
2Reui u∗
j uk −3
2Reui u∗
j ∂uk
∂x− 1
4ui u j
∂u∗k
∂x+ i
2F−1|k|F Reui u∗
j uk , (B.6)
where i , j ,k = 1, ..., s.
62
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