Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
www.nat-hazards-earth-syst-sci.net/11/383/2011/
doi:10.5194/nhess-11-383-2011
© Author(s) 2011. CC Attribution 3.0 License.
Natural Hazards
and Earth
System Sciences
Rogue waves and downshifting in the presence of damping
A. Islas and C. M. Schober
Department of Mathematics, University of Central Florida, Orlando, FL, USA
Received: 2 October 2010 – Revised: 6 December 2010 – Accepted: 13 December 2010 – Published: 9 February 2011
Abstract.
Recently Gramstad and Trulsen derived a
new higher order nonlinear Schrödinger (HONLS) equation
which is Hamiltonian (Gramstad and Trulsen, 2011). We
investigate the effects of dissipation on the development
of rogue waves and downshifting by adding an additonal
nonlinear damping term and a uniform linear damping
term to this new HONLS equation. We find irreversible
downshifting occurs when the nonlinear damping is the
dominant damping effect. In particular, when only nonlinear
damping is present, permanent downshifting occurs for all
values of the nonlinear damping parameter β. Significantly,
rogue waves do not develop after the downshifting becomes
permanent. Thus in our experiments permanent downshifting
serves as an indicator that damping is sufficient to prevent
the further development of rogue waves. We examine the
generation of rogue waves in the presence of damping for
sea states characterized by JONSWAP spectrum. Using the
inverse spectral theory of the NLS equation, simulations of
the NLS and damped HONLS equations using JONSWAP
initial data consistently show that rogue wave events are well
predicted by proximity to homoclinic data, as measured by
the spectral splitting distance δ. We define δcutoff by requiring
that 95% of the rogue waves occur for δ < δcutoff . We find
that δcutoff decreases as the strength of the damping increases,
indicating that for stronger damping the JONSWAP initial
data must be closer to homoclinic data for rogue waves to
occur. As a result when damping is present the proximity
to homoclinic data and instabilities is more crucial for the
development of rogue waves.
Correspondence to: C. M. Schober
([email protected])
1
Introduction
Some theoretical models attribute the development of
rogue waves in deep water to a nonlinear focusing of
uncorrelated waves in a very localized region of the
sea. This focusing may be related to the BenjaminFeir (BF) instability, described to leading order by the
nonlinear Schrödinger (NLS) equation (Henderson et al.,
1999; Kharif and Pelinovsky, 2001, 2004). Homoclinic
orbits of modulationally unstable solutions of the NLS
equation which undergo large amplitude excursions away
from their target solution exhibit many of the observed
properties of rogue waves and can be used to model rogue
waves (Osborne et al., 2000; Calini and Schober, 2002).
A more accurate description of the water wave dynamics
is obtained by retaining higher order terms in the asymptotic
expansion for the surface wave displacement. A commonly
used higher order NLS equation is the Dysthe equation
(Dysthe, 1979). Homoclinic orbits of the unstable Stokes
wave have been shown to be robust to the higher order
corrections in the Dysthe equation. Laboratory experiments
conducted in conjunction with numerical simulations of
the Dysthe equation established that the generic long-time
evolution of initial data near an unstable Stokes wave
with two or more unstable modes is chaotic (Ablowitz
et al., 2000, 2001). Subsequent studies of the Dysthe
equation showed that for a rather general class of such
initial data, the modulational instability leads to high
amplitude waves, structurally similar to the optimal phase
modulated homoclinic solutions of the NLS equation, rising
intermittently above a chaotic background (Calini and
Schober, 2002; Schober, 2006). These earlier studies relating
homoclinic solutions and rogue waves ignored the fact
that the Dysthe equation is not Hamiltonian and neglected
damping which, even when weak, can have a significant
effect on the wave dynamics.
Published by Copernicus Publications on behalf of the European Geosciences Union.
384
A. Islas and C. M. Schober: Rogue waves and downshifting
Recently Gramstad and Trulsen (2011) brought the Dysthe
equation into Hamiltonian form obtaining a new higher order
nonlinear Schrödinger (HONLS) equation (Gramstad and
Trulsen, 2011). In this paper we investigate the effects
of dissipation on the development of rogue waves and
downshifting in one space dimension by adding a nonlinear
damping term (the β-term which damps the mean flow Kato
and Oikawa, 1995) and a uniform linear damping term to this
new HONLS equation. The governing equations and their
properties are set up in Sect. 2.
Downshifting in the evolution of nearly uniform plane
waves was first observed by Lake et al. (1977). Their
experiment showed that after the wavetrain becomes strongly
modulated, it recurs as a nearly uniform wavetrain with the
dominant frequency permanently downshifted. Permanent
downshifting has been confirmed in other laboratory
experiments on the evolution of the Stokes wave (Su,
1982; Huang et al., 1996). Conservative models have been
inadequate in describing permanent downshifting and several
damped models have been suggested, e.g. where the damping
is modeling wave breaking or eddy viscosity to capture
downshifting (Trulsen and Dysthe, 1990; Hara and Mei,
1991, 1994).
Our previous studies of rogue waves show that an neighborhood of the unstable plane wave (the same regime
in which Lake examined downshifting) is effectively a rogue
wave regime for the Dysthe equation since the likelihood
of obtaining a rogue wave is extremely high. We reexamine this issue in Sect. 3 using the new HONLS equation.
In Sect. 4, using initial data for an unstable modulated
wavetrain we examine whether irreversible downshifting
(although here downshifting of the wave number is
considered) occurs in our damped HONLS equation and
what characterizes the damped wave train evolution on a
short and long time scale.
We find that rogue waves may emerge in both the linear
and nonlinear damped regimes that were not present without
damping. Although damping decreases the growth rates
of the individual modes, the modes may coalesce due to
changes in their focusing times, thus resulting in larger waves
in the damped regime. Even so, on average, the strength
is smaller and fewer rogue waves occur when damping is
present. While downshifting is not expected in the linearly
damped case, we examine the effect of linear damping on
the development of rogue waves due to its widespread use in
modeling dissipative processes.
The nonlinear β-term models damping of the mean
flow and since it is large only near the crest ofthe
envelope, the damping is localized when the wavetrain is
strongly modulated. Due to the BF instability, irreversible
downshifting occurs when the nonlinear damping is the
dominant damping effect. In particular, when only nonlinear
damping is present, permanent downshifting occurs for all
values of the nonlinear damping parameter β, appearing
abruptly for larger values of β. Significantly, we find that
after permanent downshifting occurs, rogue waves do not
appear in the nonlinearly damped evolution. Thus in our
experiments permanent downshifting serves as an indicator
that there has been sufficient cumulative damping to inhibit
the further development of rogue waves.
Developing sea states, where nonlinear wave-wave
interactions continue to occur, are described by the Joint
North Sea Wave Project (JONSWAP) power spectrum. A
spectral quantity, the splitting distance δ between simple
periodic points of the Floquet spectrum of the associated
AKNS spectral problem of the initial condition may be
used to measure the proximity in spectral space to unstable
waves and homoclinic data of the NLS equation. In (Islas
and Schober, 2005; Schober, 2006), simulations of both
the NLS and Dysthe equations using JONSWAP initial data
consistently show that rogue wave events are well predicted
by proximity to homoclinic data, as measured by δ.
In Sect. 5 we examine the generation of rogue waves
in the presence of damping for sea states characterized by
JONSWAP spectrum. Using the damped HONLS equation
we find that both the strength and likelihood of rogue waves
occuring in a given simulation are typically smaller when
damping is present. We define δcutoff by requiring that 95%
of the rogue waves occur for δ < δcutoff . Significantly, we
find that δcutoff is generally decreasing as the strength of
the damping increases. Thus when damping is present the
JONSWAP initial data must be closer to instabilities and
homoclinic data for rogue waves to occur. The proximity to
homoclinic data and instabilities becomes more essential for
the development of rogue waves when damping is present.
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
2
2.1
Analytical background
A new damped higher order nonlinear Schrödinger
equation
The equations for inviscid deep water waves, as well as
the nonlinear Schrödinger (NLS) equation, can be derived
from a Lagrangian. Ideally, a higher order NLS equation
would retain this feature. The commonly used higher
order NLS equation (Dysthe, 1979), often referred to
as Dysthe’s equation, is not Hamiltonian and does not
conserve momentum. Recently, Gramstad and TRulsen
used the Zakharov equation enhanced with the Krasitskii
kernel (Krasitskii, 1994) to bring the Dysthe equation into
Hamiltonian form, obtaining a new higher order nonlinear
Schrödinger (HONLS) equation (Gramstad and Trulsen,
2011).
In order to examine rogue waves and downshifting in deep
water we consider the HONLS equation of Gramstad and
Trulsen with periodic boundary conditions, u(x,t) = u(x +
L,t), and add damping as follows:
www.nat-hazards-earth-syst-sci.net/11/383/2011/
A. Islas and C. M. Schober: Rogue waves and downshifting
iut + uxx + 2|u|2 u + i0u
+ i
h 1
uxxx − 8|u|2 ux − 2ui (1 + iβ) H |u|2
2
i x
= 0,
(1)
where H (f ) represents the Hilbert transform of f .
Throughout this paper “HONLS equation” refers to (1) with
0 = 0 and β = 0. The Hamiltonian for the HONLS equation
is given by
Z Ln
H=
−i|ux |2 + i|u|4 − ux u∗xx − u∗x uxx
4
0
h i o
dx.
+ 2|u|2 u∗ ux − uu∗x + i|u|2 H |u|2
(2)
x
Uniform linear damping occurs for 0 > 0 and β = 0 and has
been used extensively to study damping of wave trains with
comparisons to physcial wave-tank experiments (Segur et al.,
2005). Localized nonlinear damping of the mean flow occurs
for ,β > 0 and 0 = 0 and it was introduced to investigate
downshifting in deep water waves (Kato and Oikawa, 1995).
2.1.1
Energy, energy flux, and the spectral center
The mass or wave energy, E, and the momentum or total
energy flux, P , are defined by
Z L
Z L
|u|2 dx, and P = i
u∗ ux − uu∗x dx .
E=
0
0
Thus the total energy is conserved for the HONLS equation
(0 = β = 0) and is dissipated if either 0 or β is positive.
The total energy flux is related to the symmetry of the
Fourier modes since P can be written as
P = −2L
k |û−k |2 − |ûk |2 .
Since the evolution of the total energy flux is
Z
L
u
0
∗
ux − uu∗x
h i 0 + 2β H |u|2
dx ,
x
(4)
the momentum is conserved by the HONLS equation. In
contrast the momentum for the Dysthe equation exhibited
small oscillations in time (Islas and Schober, 2010).
www.nat-hazards-earth-syst-sci.net/11/383/2011/
km = −
1P
.
2E
(5)
The wave train is understood to experience a permanent
frequency downshift when the spectral center km decreases
monotonically in “time” or there is a permanent downshift of
the spectral peak kpeak (Trulsen and Dysthe, 1997a).
Given that the energy and momentum are conserved by
the HONLS equation, the spectral center, which is given by
their ratio, is also conserved. In the linearly damped model,
0 > 0,β = 0, from Eqs. (3) and (4) it follows that the energy
and momentum have a simple time dependence:
E(t) = E(0)exp(−20t)
P (t) = P (0)exp(−20t) .
(6)
(7)
Thus the spectral center km is constant in time for the linearly
damped HONLS equation and downshifting will not occur.
Linear stability of the Stokes wave
The HONLS equation admits a uniform wave train solution,
the Stokes wave,
2
ua (x,t) = ae2ia t .
(8)
Fourier analysis shows that this plane wave solution is
unstable to sideband perturbations. When its amplitude a
is sufficiently large, for 0 < π n/L < a, the solution of the
linearized NLS equation about ua has M linearly unstable
modes (UMs) ei(σn t+2π nx/L) with growth rates σn given by
σn2 = µ2n µ2n − 4a 2 , µn = 2π n/L ,
(9)
where M is the largest integer satisfying 0 < M < aL/π. We
refer to this as the M-unstable mode regime.
2.2
k=1
dP
= −2i
dt
Using the Fourier spectrum of u(x,t), ûk , two different
choices of diagnostic frequencies can be used to measure
downshifting. On the one hand, the dominant mode or
spectral peak intuitively corresponds to the k for which |ûk |
achieves its maximum and is denoted as kpeak . On the other
hand, Uchiyama and Kawahara defined the wave number for
the spectral center or mean frequency of the spectrum as
(Uchiyama and Kawahara, 1994):
2.1.2
To show damping occurs for positive β or 0, one finds that
Z L
h i dE
= −2
|u|2 0 + 2β H |u|2
dx .
x
dt
0
P
ikx ,
Expanding u and |u|2 in Fourier series, u = ∞
k=−∞ ûk e
P
ikx + B ∗ e−ikx ,
|u|2 = ∞
k=0 Bk e
k
#
"
∞
∞
X
X
dE
2
2
(3)
= −2L 0
|ûk | + 2β
k|Bk | .
dt
k=−∞
k=1
∞
X
385
Integrable theory of the nonlinear Schrödinger
equation
In previous studies of sea states described by the JONSWAP
power spectrum we showed a nonlinear spectral decomposition of the data provides relevant information on the
likelihood of rogue waves. Here we briefly review the
spectral theory of the nonlinear Schrödinger (NLS) equation,
iut + uxx + 2|u|2 u = 0 .
(10)
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
directly
fromfrom
the linearization.
EachEach
imaginary
The nonlinear spectral decomposition of an NLS initialcomputed
computed
directly
the linearization.
imaginary
condition
(or in (or
general
of an ensemble
of JONSWAP
initialinitialdouble
pointpoint
“labels”
the associated
unstable
mode
(Calini
condition
in general
of an ensemble
of JONSWAP
double
“labels”
the associated
unstable
mode
(Calini
data) isdata)
based
on theon
inverse
spectral
theorytheory
of theofNLS
is based
the inverse
spectral
the equaNLS equa-& Schober,
2002).
& Schober,
2002).
tion. For
boundary
conditions
u(x +u(x
L,t)
u(x,t),
tion.periodic
For periodic
boundary
conditions
+=
L,t)
= u(x,t),
386
A. Islas and C. M. Schober: Rogue waves and downshifting
the Floquet
spectrum
associated
with
an potential
NLS potential
the Floquet
spectrum
associated
with an
NLS
u (i.e.u (i.e.
the spectrum
the linear
operator
at u)becan
be described
the spectrum
of the of
linear
operator
L(x) atL(x)
u) can
described
λ− planeλ− plane
The
NLS
equation
is
equivalent
in of
terms
the Floquet
discriminant
ofsolvability
u, defined
astrace
the trace
in terms
the of
Floquet
discriminant
oftou,thedefined
as thecondition
λ − plane
of
the
AKNS
system,
the
pair
of
first-order
linear
systems
λ − plane
the transfer
of a fundamental
solution
of the of
transfer
matrixmatrix
of a fundamental
matrixmatrix
solution
Φ of Φ of
(Zakharov and Shabat, 1972):
(11)the
over
the interval
[0,L] (Ablowitz
& Segur,
1981):
(11) over
interval
[0,L] (Ablowitz
& Segur,
1981):
(x)
(t)
(11)
L φ = 0, L φ = 0 ,
−1
−1
∆(u;λ)
= Trace
Φ(x,t;λ)
+ L,t;λ)
.
∆(u;λ)
= Trace
Φ(x,t;λ)
Φ(x +Φ(x
L,t;λ)
.
for a vector-valued function φ. The operator L(x) is given by
the Floquet
spectrum
is defined
the region
Then, Then,
the Floquet
spectrum
isdefined
as theasregion
∂x + iλ −u
(x)
L =
∗
u{λ
∂I{λ
iλ
C|∆(u;λ)
I ∈ IR,−2
∈ IR,−2
≤2}.
∆ ≤ 2}.
x −∈
σ(u) =σ(u)
∈=C|∆(u;λ)
≤∆≤
and depends
x and t through
the of
potential
u andfor
on which
the
theoncontinuous
spectrum
uthose
are those
PointsPoints
of the of
continuous
spectrum
of u are
for which
(a)
(b)
spectral
parameter
λ.
the eigenvalues
of
the transfer
matrix
havemodulus,
unit modulus,
and
the eigenvalues
of
the
transfer
matrix
have
unit
and
The nonlinear
spectral
decomposition
of −2;
an NLS
initial
Fig.
1.Spectrum
Spectrum
of a)
(a)ofaa a)
generic
unstable
N-phase
solution
and
Fig.
1:
Spectrum
a generic
unstable
N-phase
solution
therefore
∆(u;λ)
isand
realbetween
and
between
and
in particular,
Fig.
1:
of
generic
unstable
N-phase
solution
therefore
∆(u;λ)
is real
2 ensemble
and 2−2;
inofparticular,
condition
(or
in
general
of
an
JONSWAP
(b)and
the plane
wave
solution.
The simpleThe
periodic
eigenvalues
are
b)
the
plane
wave
solution.
simple
periodic
eigenthe
real
line
is
part
of
the
continuous
spectrum.
Points
of
the
b) the plane wave solution. The simple periodic eigenthe real
line data)
is partisofbased
the continuous
spectrum.
Points
of of
thethe andlabeled
initial
on the inverse
spectral
theory
by are
circles
and the
double
points
by crosses.
values
labeled
by
circles
and
the
double
points
by
crosses.
L-periodic/antiperiodic
discrete
spectrum
of
u
are
those
for
L-periodic/antiperiodic
discrete
spectrum
of uconditions
are those u(x
for + values are labeled by circles and the double points by crosses.
NLS equation. For
periodic
boundary
which
the eigenvalues
of
the transfer
matrix
areequiva±1, equivawhichL,t)
the =
eigenvalues
of
the
transfer
matrix
are
±1,
u(x,t), the Floquet spectrum associated with an NLS
lently
∆(u;λ)
±2. Points
of the discrete
spectrum
(x) atwhich
lentlypotential
∆(u;λ)
=(i.e.
±2.=
Points
of the
which
u
the
spectrum
of discrete
the linearspectrum
operator L
u)
As a concrete example consider the plane wave solution,
are
embedded
in
a
continuous
band
of
spectrum
are
critical
2t
are embedded
in a continuous
spectrum
are critical
can be described
in terms band
of theof
Floquet
discriminant
of u,
ua (x,t) = ae2ia √
, whose Floquet discriminant is given by
points
for
the
Floquet
discriminant
(i.e.,
d∆/dλ
must
vanish
pointsdefined
for theas
Floquet
discriminant
(i.e.,matrix
d∆/dλofmust
vanish
the trace
of the transfer
a fundamental
a 2 + λ2 L). The resulting Floquet
spectrum
1(a,λ)
=
2cos(
at such points).
S equations
3 1b)
Rogue
waves
in the
NLS and
HONLS
at such
points).
matrix
solution 8 of (11) over the interval [0, L] (Ablowitz 3 (Fig.
consists
of
the
continuous
bands
IR
[−ia,ia],
and
Rogue
waves
in
the
NLS
and
HONLS
equations
the
transfer matrix only changes by conjugation
andSince
Segur,
1981):
Since
the
transfer
matrix only changes by conjugation
a discrete part containing the simple periodic/antiperiodic
when we shift in x or t, ∆ is independent of those variwhen we shift in x or t, ∆ is independent of those varieigenvalues
±ia, and solutions
the infiniteofsequence
double points
Homoclinic
the NLSofequation
as models
−1
ables.
An
important
consequence
this observation
is that3.1 3.1
Homoclinic solutions of the NLS equation as models
= Trace
8(x,t;λ)
8(x
+of
L,t;λ)
.
ables.1(u;λ)
An important
consequence
of this
observation
is that
for
rogue
waves
the Floquet discriminant is invariant under the NLS flow, 2 for rogue2 waves
λ = (j π/L) − a 2 ,
j ∈Z.
(12)
the Floquet discriminant is invariant under the NLS flow,
and thus
encodesspectrum
an infinite
family as
ofthe
constants
of motion j
Then,
the
Floquet
is
defined
region
and thus encodes an infinite family of constants of motion
The
NLS of
equation
admits modulationally
unstable
periodic
(parametrized by λ).
The
number
imaginary
points, obtained
for periodic
0≤j <
The
NLS
equation
admits double
modulationally
unstable
(parametrized
by
λ).
σ (u)
= {λ
∈ C|1(u;λ)
I
IR,−2
≤ 1spectrum
≤ 2}.
solutions
with
homoclinic
orbits
that
can
undergo
large
The
continuous
part∈of
Floquet
of a generic NLS aL/π,
with the orbits
number
of can
unstable
modes
(Eq.am9) amsolutions coincides
with homoclinic
that
undergo
large
The continuous part of Floquet spectrum of a generic NLS
plitude excursions
away
from
their
target
solution.
Such
hopotential consists of the real axis and of complex bands ter- computed
directlyaway
fromfrom
the their
linearization.
Each imaginary
Points
of the of
continuous
spectrum
ofcomplex
u are those
for
which plitude excursions
targetrogue
solution.
Such
hopotential
consists
the real
axis
bands
s and of
′ termoclinic
orbits
can
be
used
to
model
waves.
An
examminating
in simple
points
λ
(at
which
∆
=
±2,∆
=
6
0).
The
j matrix have unit′ modulus, and
s
double orbits
point can
“labels”
the to
associated
unstable
mode
the eigenvalues
of theλtransfer
moclinic
be
model
rogue
waves.
An(Calini
examminating
in simple
points
(at which
∆ = ±2,∆
6=
0). The
j those
ple
is provided
by used
the unstable
plane
wave
solution
ua (x,t) =
N
-phase
potentials
are
characterized
by
a
finite
number
and
Schober,
2002).
therefore
1(u;λ)
isthose
real and
between 2 by
anda −2;
innumber
particular, ple is provided
2
by
the
unstable
plane
wave
solution
u
(x,t)
=
N -phase
potentials
are
characterized
finite
a
2ia t
of
bands
of
continuous
spectrum
(or
a
finite
number
of
simwhere the number of unstable modes (UMs) is pro2
the of
realcontinuous
line is partspectrum
of the continuous
spectrum.
Points
of ae2iaae
t
of bands
(or
a
finite
number
of
simwhere
of unstable
modes
is propleL-periodic/antiperiodic
points). Fig. 1a shows discrete
the spectrum
of a typical
-phase
vided
by the
eq. number
(9). For
each UM
there(UMs)
is a correspondthe
spectrum
ofN
u -phase
areNthose
ple points).
Fig.
1a
shows
the
spectrum
of
a
typical
vided
by
eq.
(9).
For
each
UM
there
is
a
correspondpotential:
complex
criticalofpoints
(usuallymatrix
doubleare
points
homoclinic
A global
representation
of the homoRogue
waves inorbit.
the NLS
and HONLS
equations
for
which
the
eigenvalues
the transfer
±1, of 3 ing
potential:
complex
critical
points
(usually
′ double points of
ing homoclinic
orbit.
Aobtained
global representation
of the
the
discrete
spectrum
for
which
∆
=
0
and
∆”
=
6
0),
such
clinic
orbits
can
be
by
exponentiating
thehomolinear inequivalently
1(u;λ)
=
±2. Points
ofand
the ∆”
discrete
spectrum
′
the discrete
spectrum
for
which
∆
=
0
=
6
0),
such
clinic
orbits
can
be
obtained
by
exponentiating
the
linear
as
the
one
appearing
in
the
figure,
are
in
general
associated
stabilities
via
Bäcklund
transformations
(Matveev
&inSalle,
3.1
Homoclinic
solutions
of
the
NLS
equation
as
models
which are embedded in a continuous band of spectrum are
as the with
one appearing
in
the
figure,
are
in
general
associated
stabilities
via
Bäcklund
transformations
(Matveev
&
Salle,
linear
instabilities
of
u
and
label
its
homoclinic
orbits
1991;
McLaughlin
&
Schober,
1992).
Moreover,
for
NLS
for
rogue
waves
critical points for the Floquet discriminant (i.e., d1/dλ must
with linear
instabilities
of u and label its homoclinic orbits
1991;
McLaughlin
&
Schober,
1992).
Moreover,
for
NLS
(Ercolani
et
al.,
1990).
potentials
with
several
UMs,
iterated
Bäcklund
transformavanish at such points).
(ErcolaniAs
et al., 1990). example consider the plane wave solution,potentials
with
several
UMs,modulationally
iterated
Bäcklund
transformaThe
NLSwill
equation
admits
periodic
tions
generate
their
entire
stable
andunstable
unstable
manifolds,
Sincea concrete
the transfer
matrix only changes by conjugation
2ia2 t
As when
auaconcrete
example
consider
the
plane
wave
solution,
tions
will
generate
their
entire
stable
and
unstable
manifolds,
solutions
with
homoclinic
orbits
that
can
undergo
large up
comprised
of
homoclinic
orbits
of
increasing
dimension
(x,t)
=
ae
,
whose
Floquet
discriminant
is
given
by
we shift
in √
x or t, 1 is independent of those variables.
2
t
of
homoclinic
orbits
oftheir
increasing
dimension
up
ua (x,t)
=important
ae2ia
, whose
Floquet
discriminant
is given
bythe comprised
amplitude
excursions
away
from
target solution.
Such
λ2 L).
Floquet
spectrum
to
the
dimension
of
the
invariant
manifolds.
Such
highera2 +
∆(a,λ)
=√
2cos(
An
consequence
ofThe
thisresulting
observation
is
that
S
2 L). The resulting Floquet
spectrum
to
the
dimension
of
the
invariant
manifolds.
Such
highera2 + λof
∆(a,λ)
= 2cos(
homoclinic
orbits
can
be
used
to
model
rogue
waves.
An
(Fig.
1b)discriminant
consists
the
continuous
bands
I
R
[−ia,ia],
and
dimensional
homoclinic
orbits
are
also
known
as
combinaFloquet
is invariant
under
the
NLS
flow,
and
S
(Fig. 1b)
consists
of an
the
continuous
[−ia,ia],
and
dimensional
are also plane
knownwave
as combinaexample
ishomoclinic
provided
byorbits
the unstable
solution
a discrete
part
containing
thebands
simple
periodic/antiperiodic
tion homoclinic
orbits.
thus
encodes
infinite
family
of IR
constants
of motion
2t
a discrete
part containing
tion
orbits.
uahomoclinic
(x,t) = ae2ia
where the number of unstable modes
(parametrized
by λ). the simple periodic/antiperiodic
(UMs) is provided by Eq. (9). For each UM there is a
The continuous part of Floquet spectrum of a generic NLS
corresponding homoclinic orbit. A global representation of
potential consists of the real axis and of complex bands
the homoclinic orbits can be obtained by exponentiating the
terminating in simple points λsj (at which 1 = ±2,10 6=
linear instabilities via Bäcklund transformations (Matveev
0). The N-phase potentials are those characterized by a
and Salle, 1991; McLaughlin and Schober, 1992). Moreover,
finite number of bands of continuous spectrum (or a finite
for NLS potentials with several UMs, iterated Bäcklund
number of simple points). Figure 1a shows the spectrum of
transformations will generate their entire stable and unstable
a typical N-phase potential: complex critical points (usually
manifolds, comprised of homoclinic orbits of increasing
double points of the discrete spectrum for which 10 = 0 and
dimension up to the dimension of the invariant manifolds.
100 6 = 0), such as the one appearing in the figure, are in
general associated with linear instabilities of u and label its
Such higher-dimensional homoclinic orbits are also known
as combination homoclinic orbits.
homoclinic orbits (Ercolani et al., 1990).
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
www.nat-hazards-earth-syst-sci.net/11/383/2011/
A. Islas and C. M. Schober: Rogue waves and downshifting
387
3.1.2
Phase modulated rogue waves
As the number of UMs increases, the space-time structure of
the homoclinic solutions becomes more complex. When two
or more UMs are present the initial wave train can be phase
modulated to produce additional focusing.
The family of homoclinic orbits of the plane wave
potential with two UMs is given by an expression of the form
u(x,t) = ae2ia
2t
g(x,t)
,
f (x,t)
(15)
where the expression for f (x,t) and g(x,t) depend on the
two spatial modes cos(2nπ x/L), cos(2mπ x/L), and on
temporal exponential factors
q exp(σn t + ρn ), exp(σm t + ρm ),
Fig. 2. Amplitude √
plot of a homoclinic orbit of the Stokes wave
with a = 0.5, L = 2 2π.
3.1.1
One unstable mode rogue wave solutions
A single (i.e. lowest dimensional) homoclinic orbit of the
plane wave potential is given by
u(x,t) = ae−2ia
2t
1 + 2cos(px)eσn t+2iφ+ρ + Ae2σn t+4iφ+2ρ
1 + 2cos(px)eσ1 t+ρ + Ae2σn t+2ρ
(13)
p
where A = 1/cos2 φ, σn = ±p 4a 2 − p2 , φ = sin−1 (p/2a),
and p = µn = 2π n/L < a for some integer n. Each UM has
an associated homoclinic orbit characterized by mode p =
µn .
Figure 2 shows the space-time plot of the amplitude
|u(x,t)|
√ of a homoclinic orbit with one UM, for a = 0.5,
L = 2 2π and p = 2π/L. As t → ±∞, solution (13) limits
to the plane wave potential; in fact, the plane wave behavior
dominates the dynamics of the homoclinic solution for most
of its lifetime. As t approaches t0 = 0, nonlinear focusing
occurs due to the BF instability and the solution rises to
a maximum height of 2.4a. Thus, the homoclinic solution
with one UM can be regarded as the simplest model of rogue
wave.
An almost equally dramatic wave trough occurs close to
the crest of the rogue wave as a result of wave compression
due to wave dislocation. The amplitude amplification factor
is given by
Af =
maxx∈[0,L],t∈IR |u(x,t)|
≈ 2.4 .
limt→±∞ |u(x,t)|
www.nat-hazards-earth-syst-sci.net/11/383/2011/
(14)
with growth rates σl = µl µ2l − 4a 2 , µl = 2π l/L. (The
complete formulas can be found in Calini et al., 1996; Calini
and Schober, 2002.)
As in the one-UM case, this combination homoclinic
orbit decays to the plane wave potential as t → ±∞,
and the associated rogue wave remains hidden beneath
the background plane wave for most of its lifetime. The
temporal separation of the two spatial modes depends upon a
parameter ρ related to the difference ρn −ρm in the temporal
phases (Calini et al., 1996; Calini and Schober, 2002).
In turn, ρ affects the amplitude amplification factor.
Figure 3a–b shows the combination homoclinic orbit (15)
obtained with all parameters set equal except for ρ. In
Fig. 3a, ρ = .1, the modes are well separated, and the
amplitude amplification factor is roughly three. In Fig. 3b,
the value of ρ is approximately −0.65, corresponding to the
two UMs being simultaneously excited or coalesced. For this
value of ρ the amplitude amplification factor is maximal and
the rogue wave rises to a height of 4.1 times the height of the
carrier wave. The surface plot of |u(x,t)| has been rotated in
Fig. 3b to facilitate comparison with Fig. 4a.
Figure 3a shows focusing due to only weak amplitude
modulation of the initial wave train; the growth in amplitude
beginning at t ≈ 10 and at t ≈ 25 is due to the BF instability.
However, in Fig. 3b focusing due to both amplitude and
phase modulation occurs. The amplitude growth at t ≈ 10
is due to the BF instability, while the additional very rapid
focusing at t ≈ 18.4 is due to the phase modulation. In
general it is possible to select the phases in a combination
homoclinic orbit with N spatial modes so that any number n
(2 ≤ n ≤ N ) of modes coalesce at some fixed time.
3.2
Rogue waves in the higher order NLS equation
In this section we examine the robustness of homoclinic
solutions of the NLS equation, as well as frequency
downshifting, when higher order terms are included in
the wave dynamics. The HONLS equation (1) is solved
numerically using a smoothed exponential time differencing
integrator. The details of the method are provided in Sect. 4
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
388
A. Islas and C. M. Schober: Rogue waves and downshifting
4
3
2
1
0
246
245
244
5
0
243
Fig. 3. Amplitude plots of the combination homoclinic orbit: (a)
the phases are selected to produce well separated spatial modes; (b)
shows a coalesced homoclinic orbit.
and in references (Cox and Matthews, 2002; Khaliq et al.,
2009). As expected, we find permanent downshifting does
not occur in the HONLS equation. We find that the chaotic
background increases the likelihood and amplitude of rogue
waves as compared to predictions obtained with the NLS
equation.
We choose initial data that are small perturbations of the
unstable plane wave potential,
u(x,0) = a(1 + 0.1cosµx),
(16)
where the amplitude a is varied to be in the two or √
three
unstable mode regime with µ = 2π/L and L = 4 2π.
Figure 4a illustrates a prominent rogue wave solution of
Eq. (1), = 0.05,β = 0 = 0 for initial data (16) in the twoUM regime with a = 0.5, 456 < t < 461. The solution
rapidly becomes chaotic (around t = 27) with rogue waves
emerging intermittently afterwards. For example, at t ≈
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
−5
Fig. 4. Rogue waves solutions for the HONLS equation ( = .05,
0 = β = 0) when (a) two and (b) three unstable modes are present.
459 a rogue wave rises with a maximum wave amplitude
Umax ≈ 2.15. Comparing Fig. 4a with Fig. 3b one finds
that the structure of this rogue wave is similar to that of
the combination homoclinic solution (15) with coalesced
spatial modes obtained when ρ = −0.65, although the wave
amplification factor is slightly smaller. The symmetry
breaking effects of the higher order terms in the HONLS
equation prevent a complete spatial coalescence of the
nonlinear modes.
Numerical simulations of the HONLS equation in the
three-UM regime, e.g. initial condition (16) with a = 0.7,
show a similar phenomenon: after the onset of chaotic
dynamics, rogue waves rise intermittently above the chaotic
background (see Fig. 4b). At t ≈ 245 a rogue wave develops,
Umax ≈ 3.59, which is close to the coalesced homoclinic
solution of the NLS equation in the three-UM regime.
Extensive numerical experiments were performed for the
HONLS equation in the two- and three-UM regime, varying
www.nat-hazards-earth-syst-sci.net/11/383/2011/
A. Islas and C. M. Schober: Rogue waves and downshifting
(a)
STRENGTH
2
1.5
1
0
20
40
60
80
100
TIME
120
140
160
180
200
1
(b)
Fourier Modes: k=−3,−2,−1,0,1,2,3
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
TIME
0.1
(c)
0
SPECTRAL CENTER
favors
the occurrence
of +large
amplitude−rogue
u(x,0)
= 0.7(1
+ 0.1(cosµx
0.2(cos2µ(x
L/4) waves, as
compared to predictions obtained from the NLS equation.
− L/3)))),we varied the perturbation
(17)
To +0.38cos3µ(x
investigate downshifting
√
strength in the HONLS equation and the parameters in the
with L = 4 2π and ǫ = 0.05.
initial data in the two- and three-UM regime. Typical results
Figure
5a with
shows
time equation
evolution(1)ofare
theshown
strength
of 5
obtained
thethe
HONLS
in Fig.
u(x,t),
for initial data
Umax (t)
=
,
u(x,0) = 0.7(1 +S(t)
0.1(cosµx
+ 0.2(cos2µ(x
− L/4) (18)
Hs (t)
+ 0.38cos3µ(x − L/3)))),
(17)
where Umax (t)√= maxx∈[0,L] |u(x,t)| and Hs (t) is the signif= 4 2π and = 0.05.
icantwith
waveLheight,
defined as 4 times the standard deviation of
Figure 5a shows the time evolution of the strength of
the surface elevation. Rogue waves occur throughout the time
u(x,t),
series whenever the strength exceeds the threshold criteria of
Umax (t) the time evolution of the main Fourier
2.2. S(t)
Figure
= 5b shows
,
(18)
Hsfor
(t) k = 0,±1,...,±4. During each of the modmodes |Ak (t)|
ulation
stages
the
loses energy as the upper and
where
Umax
(t)zeroth
= maxmode
x∈[0,L] |u(x,t)| and Hs (t) is the signifilowercant
higher
become
Forstandard
0 < t < 200
the
waveharmonics
height, defined
as excited.
4 times the
deviation
total of
energy
and theelevation.
total energy
flux waves
are constant.
This althe surface
Rogue
occur throughout
lowsthe
thetime
energy
flow back
the zeroth
mode
seriestowhenever
thetostrength
exceeds
the keeping
threshold
the spectral
center
constant
(Fig.
5c).
The
plot
of
kpeak
criteria of
2.2. kFigure
5b
shows
the
time
evolution
of the
m
main
Fourier
modes
|A
(t)|
for
k
=
0,
±1,...,±4.
During
(Fig. 5d) confirms that permanent
frequency downshifting
k
of the modulation
stages
thewhen
zerothsteeper
mode loses
doeseach
not occur.
As expected,
even
wavesenergy
are
as
the
upper
and
lower
higher
harmonics
become
obtained, permanent frequency downshifting does not excited.
occur
0 < t <numerical
200 the total
energy and the
total energy
flux are
in theFor
HONLS
experiments,
indicating
the necesconstant.
This
allows
the
energy
to
flow
back
to
the
zeroth
sity of a dissipative or nonconservative process.
mode keeping the spectral center km constant (Fig. 5c). The
plot of kpeak (Fig. 5d) confirms that permanent frequency
downshifting
occur. As expected,
even
when steeper
4 Rogue
wavesdoes
andnotdownshifting
in the
presence
of
waves
are
obtained,
permanent
frequency
downshifting
does
damping
not occur in the HONLS numerical experiments, indicating
necessity
a dissipative
nonconservative of
process.
In thethelast
sectionofwe
saw that anorǫ-neighborhood
the unstable plane wave with two or more UMS is effectively a
www.nat-hazards-earth-syst-sci.net/11/383/2011/
2.5
−0.1
−0.2
−0.3
−0.4
−0.5
0
20
40
60
80
100
120
140
160
180
200
TIME
133
(d)
132
131
SPECTRAL PEAK
the NLS equation they occur only once or twice, depnding
the perturbation strength , the amplitude a and adding in
on whether
the modes are coalesced or not. In the damped
some random phases φi ’s in the initial data. In all cases, the
HONLS,
they
occur irregularly
over a emerges
short time
period but
coalesced homoclinic
NLS solution
generically
as a
eventually
are
damped
out.
structurally stable feature of the perturbed dynamics.
The The
chaotic
regime produces
focusing
effecoccurence
of rogue additional
wave in the
HONLSbyequation
tivelyis selecting
optimal
phase
modulations
and
the
chaotic
distinct from their occurence in the NLS or the damped
dynamics
singles
out theIn maximally
HONLS
equations.
the HONLScoalesced
equation, homoclinic
rogue waves
solutions
of theintermittently
unperturbedthroughout
NLS equation
physically
ob-In
will occur
very as
long
time series.
servable
rogueequation
waves. Moreover,
of larger
amthe NLS
they occur the
onlylikelihood
once or twice,
depnding
plitude
waves (see
e. g. Fig.
4) increases
for the
HONLS
on whether
the modes
are coalesced
or not.
In the
damped
HONLS,
they occur irregularly
over a short developed
time period in
but
equation,
as substantiated
by the diagnostics
eventually
are damped
section
5, correlating
waveout.
strengths in the NLS and HONLS
chaotic to
regime
produces
modelsThe
to proximity
homoclinic
data.additional
Thus, for focusing
initial databy
effectively
selecting
optimal
phase
modulations
and the
in the
neighborhood
of an
unstable
plane
wave the underout the equation
maximallyfavors
coalesced
lyingchaotic
chaoticdynamics
dynamicssingles
of the HONLS
the
homoclinic
solutions
of
the
unperturbed
NLS
equation
occurrence of large amplitude rogue waves, as compared toas
physically
observable
waves.
Moreover, the likelihood
predictions
obtained
fromrogue
the NLS
equation.
increases for
of
larger
amplitude
waves
(see
e.g.
Fig.
To investigate downshifting we varied
the4)perturbation
the HONLS equation, as substantiated by the diagnostics
strength ǫ in the HONLS equation and the parameters in the
developed in Sect. 5, correlating wave strengths in the NLS
initial data in the two- and three-UM regime. Typical results
and HONLS models to proximity to homoclinic data. Thus,
obtained with the HONLS equation (1) are shown in Fig. 5
for initial data in the neighborhood of an unstable plane wave
for initial
data
the underlying chaotic dynamics of the HONLS equation
389
130
129
128
127
126
125
0
20
40
60
80
100
120
140
160
180
200
TIME
Fig.
Γ = 0,0ǫ==0,0.05,
= 0:β time
Fig.5:5.The
TheHONLS
HONLSeq.,
equation,
=β
0.05,
= 0: evolutime
evolution
(a)strength
the strength
S(t),
mainFourier
Fourier mode,
tion
of (a)ofthe
S(t),
(b)(b)thethemain
mode,(c)(c)
thespectral
spectral center
(d) (d)
the the
spectral
peak kpeak
peak . kpeak .
the
centerkmk,mand
, and
spectral
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
4
(a)
3.5
3
2.5
STRENGTH
rogue
regime
thedownshifting
HONLS equation
the probabil4 wave
Rogue
wavesfor
and
in theas
presence
ity of obtaining
a rogue wave is high. In the numerical experiof damping
ments we choose initial data that are small perturbations of an
In theplane
last section
sawthethat
an -neighborhood
of the
unstable
wave towe
study
effects
of linear and nonlinwith two of
or rogue
more UMS
effectively
ear unstable
dampingplane
on thewave
development
wavesisand
downa rogue
regime
for the HONLS
HONLSequation
equationin as
the
shifting.
We wave
consider
the damped
nonprobabilityform
of with
obtaining
a rogueterms
waveonisthehigh.
In the
dimensional
the damping
same order
numerical
we choose
initial Consequently
data that are small
or smaller
thanexperiments
the higher order
NLS terms.
we
an unstabletoplane
thewhere
effects
varyperturbations
the dampingofcoefficients
be 0 wave
< ǫβ,toΓ study
< O(ǫ),
and nonlinear
damping
on the
development
of rogue
ǫ isof
thelinear
coefficient
of the higher
order
NLS
terms.
waves and downshifting. We consider the damped HONLS
In the numerical experiments equation (1) is solved using
equation in non-dimensional form with the damping terms
a smoothing fourth-order exponential time differencing inon the same order or smaller than the higher order NLS
tegrator (Cox & Matthews, 2002; Khaliq et al., 2009) that
terms. Consequently we vary the damping coefficients to be
avoids inaccuracies when inverting matrix polynomials by
0 < β, 0 < O(), where is the coefficient of the higher
using Pade approximations. We use N = 256 Fourier modes
order NLS terms.
in space and a fourth-order Runge-Kutta discretization in
In the numerical experiments Eq. (1) is solved using
time (∆t = 10−3 ). The high frequency modes were elimia smoothing fourth-order exponential time differencing
nated
at every(Cox
timeand
step
to avoid2002;
aliasing
by setting
ûk =that
0
Matthews,
Khaliq
et al., 2009)
integrator
for avoids
|k| > 120.
Eliminating
the
high
frequencies
does
not
have
inaccuracies when inverting matrix polynomials by
a significant
on the exchange
betweenmodes
the
using Padeeffect
approximations.
We useofNenergy
= 256 Fourier
dominant
low
wave
number
modes
and
thus
does
not
imin space and a fourth-order Runge-Kutta discretization
pactinour
results
and rogue
time
(1t related
= 10−3to
). frequency
The highdownshifting
frequency modes
were
waves.
eliminated at every time step to avoid aliasing by setting ûk =
0 fortemporal
|k| > 120.andEliminating
the highallows
frequencies
not
This
spatial resolution
for thedoes
three
have invariants
a significant
on the exchange
of energy
between
integral
of effect
the conservative
HONLS
equation,
E,
−8not
wave number
and of
thus
does
P , the
anddominant
H, to below
conserved
with anmodes
accuracy
O(10
),
−2
−2
impact
our results
related to frequency
downshifting
and
O(10
), O(10
), respectively.
The nonlinear
mode contentrogue
of thewaves.
data is numerically computed using the direct specThis temporal
and above,
spatial i.e.
resolution
allows
the (11)
three
tral transform
described
the system
of for
ODEs
integral invariants
conservative
HONLS ∆.
equation,
is numerically
solvedoftothe
obtain
the discriminant
The ze-E,
−8
to be
conserved
withwith
an accuracy
of O(10
ros Pof, and
∆ ±H,
2 are
then
determined
a root solver
based),
−2
−2
O(10 (Ercolani
), respectively.
TheThe
nonlinear
mode
on O(10
Muller’s),method
et al., 1990).
spectrum
is
−6
content with
of theandata
is numerically
usingthe
thespecdirect
computed
accuracy
of O(10computed
), whereas
−3ODEs
transform
i.e. the
system
tralspectral
quantities
we aredescribed
interestedabove,
in range
from
O(10of
) to
−1is numerically solved to obtain the discriminant 1. The
(11)
O(10 ).
zeros of 1 ± 2 are then determined with a root solver based
on Muller’s method (Ercolani et al., 1990). The spectrum
4.1 is The
HONLS
Equation
with Linear
−6 ), whereas the
computed
with
an accuracy
of O(10Damping
spectral quantities we are interested in range from O(10−3 )
−1 ).
O(10by
Wetobegin
examining the evolution of damped uniform
wavetrains with initially one to three pairs of sidebands ex4.1 It The
HONLS to
equation
linear damping
cited.
is important
note in with
comparisons
of the damped
and undamped dynamics that atypical cases may arise. Figbegin
bythe
examining
thethe
evolution
of (solid
damped
uniform
uresWe
6a-b
show
strength of
wavetrain
line)
and
wavetrains
with
initially
one
to
three
pairs
of
sidebands
Umax (dashed line) for the undamped and linearly damped
excited.
It is important
in comparisons
the
(Γ =
0.005) HONLS
equationto(ǫnote
= 0.05),
respectively,offor
damped
and
undamped
dynamics
that
atypical
cases
may
initial data√(16) in the two-UM regime: a = 0.5, µ = 2π/L
6a–b shows
theone
strength
the wavetrain
2π. Contrary
to what
might of
expect,
Fig. 6b
andarise.
L = 4 Figure
(solid
line)
and
U
(dashed
line)
for
the
undamped
and
shows a rogue wave max
occuring at t ≈ 23 in the damped case
linearly damped (0 = 0.005) HONLS equation ( = 0.05),
with an enhanced strength ≈ 2.8 that was not present in the
respectively, for initial data (16)
in the two-UM regime:
√ typically
undamped evolution (Fig. 6a). This
occurs for small
a = 0.5, µ = 2π/L and L = 4 2π. Contrary to what one
values of Γ or β where the smaller growth rates of the indimight expect, Fig. 6b shows a rogue wave occuring at t ≈ 23
vidual modes due to damping can be offset by a coalescence
in the damped case with an enhanced strength ≈ 2.8 that
of the modes due to changes in their focusing times.
A. Islas and C. M. Schober: Rogue waves and downshifting
2
1.5
1
0.5
Strength
Umax
0
0
5
10
15
20
25
30
35
40
45
50
4
(b)
3.5
3
2.5
STRENGTH
390
2
1.5
1
0.5
Strength
Umax
0
0
5
10
15
20
25
30
35
40
45
50
Fig. 6. Strength S(t) (solid line) and Umax (dashed line) for
Fig.
6: Strength S(t) (solid line) and Umax (dashed line) for
(a) HONLS equation = 0.05; (b) HONLS
equation with linear
(a)
HONLS
Equation ǫ = 0.05; (b) HONLS Equation with
damping, = 0.05, 0 = 0.006, for IC u0 = 0.5(1 + 0.1cosµx).
Linear Damping, ǫ = 0.05, Γ = 0.006, for IC u0 = 0.5(1 +
0.1cosµx).
was not present in the undamped evolution (Fig. 6a). This
typically occurs for small values of 0 or β where the smaller
growth rates of the individual modes due to damping can be
offset by a coalescence of the modes due to changes in their
focusing times.
Weexamined
examined the
the HONLS
HONLS equation with linear
We
linear damping
damping
for0.005
0.005≤≤Γ0≤≤0.05.
0.05. Although
Although individual simulations
for
simulations may
may
yieldlarger
largeramplification
amplification factors
factors in
in the
the damped
yield
damped regime,
regime, we
we
findthat
thatononaverage,
average,
presence
of linear
damping
find
in in
thethe
presence
of linear
damping
(see
(see 5.2
Sect.
for confirmation
withnonlinear
the nonlinear
spectral
sec.
for5.2
confirmation
with the
spectral
diagdiagnostics),
the
strength
is
smaller
and
fewer
rogue
waves
nostics), the strength is smaller and fewer rogue waves
ococcur.
Figure
7 shows
the results
obtained
Eq. (1)
cur.
Figure
7 shows
the results
obtained
withwith
equation
(1)
forǫ==0.05,
0.05, Γ0==0.01,
0.01, ββ =
=00 for
for initial
initial condition
for
condition (17)
(17) for
for
200. InInthe
thelinearly
linearlydamped
damped evolution,
evolution, Figs.
Fig. 7a,
00<<t t<<200.
7a, the
the
lastrogue
roguewave
wave
occurs
≈ 10;
in contrast,
in 5a
Fig.
5a
last
occurs
at tat≈t10;
in contrast,
in Fig.
rogue
rogue
waves
occur
throughout
the
unperturbed
HONLS
time
waves occur throughout the unperturbed HONLS time seseries. The total energy exponentially decays (Eq. 6) and the
ries.
The total energy exponentially decays (eq. 6) and the
Fourier modes are uniformly damped. The total energy flux
Fourier modes are uniformly damped. The total energy flux
is constant, keeping the spectral center km zero. Permanent
is constant, keeping the spectral center km
zero. Permanent
downshifting does not occur for the linearly damped HONLS
downshifting does not occur for the linearly damped HONLS
equation.
equation.
www.nat-hazards-earth-syst-sci.net/11/383/2011/
the main Fourier modes, 0 < t < 200.
ǫ = 0.05, β
For small
ergy and fl
4.2 The HONLS Equation with Nonlinear Damping
larger β. A
391
Fig. 10a), m
tion of the
Time = 22.7
downshifti
(dashed line ) of the nonlinearly damped HONLS equation
evolution
o
for ǫ = 0.05, β = 0.1, Γ = 0, for initial condition (16). At
tion (ǫ = 0
t = 22.7 the wavetrain is strongly modulated and we find that
alternate v
the β-term is significant only near the maximum crest of the
occurence
envelope. In general, steeper waves result in stronger damp- When b
ing and the effects of the nonlinear β-term are concentrated
the nonline
in space-time when the wavetrain is strongly modulated. manent do
To illustrate the effects of nonlinear damping on downlution of e
initial data
shifting we consider the solution of equation (1) with ǫ =
waves occ
0.05, β = 0.5 and Γ = 0, for initial data (17), 0 < t < 50. Figs.
smaller am
9(a-b) show the surface amplitude |u(x,t)| and the time evogrowth in t
lution
of the distribution
strength, respectively.
Twoline)
rogue waves
∗ )|∗(solid
)| (solid
ofβ-term
the occur
Fig.
Spatial
|u(x,t
Fig.
8.8:Spatial
distribution of of
|u(x,t
line) and and
of the
the spectra
∗ t < 10, before damping has a chance
early in
theattime
series,
β-term
(dashed
at tfor
=initial
22.7 for
initial (16),
condition
for (16),
= 0.05,
(dashed
line)
t ∗line)
= 22.7
condition
Charact
to
alter
the
ǫsignificantly
= 0.05,
Γ=
0. growth rate of the modes and preβfor
= 0.1,
0 = 0.β = 0.1,
To charact
vent further rogue waves from forming. Comparing Fig. 9b
waves we fi
and
9c, the onset
downshifting
occurs
when the first
TheFig.
perturbation
of theof
Hilbert
tranform term
in equation
nonlinear d
2
rogue
wave
at approximately
t ≈ 6 when
the β-term
]x . is referred
to as the “β-term”
and models
(1),
βu[H
|u|appears
amplitude
is
large.
This
produces
an
abrupt
rapid
decay
in
the
energy
nonlinear
damping
of
the
mean
flow.
Figure
(8)
shows
the
term is large. This produces an abrupt rapid decay in the
spatial
of |u(x,t
(solid
and
of9e),
theresulting
β-term
(Fig. distribution
9d) and
in∗ )|the
fluxline)
(Fig.
9e),
energy
(Fig.
9d) growth
and
growth
in
the
flux
(Fig.
resultingin a
A. Islas and C. M. Schober: Rogue waves and downshifting
3.5
(a)
1
0.9
3
0.8
STRENGTH
0.7
2.5
0.6
0.5
0.4
2
0.3
0.2
1.5
0.1
0
−10
1
0
20
40
60
80
100
120
140
160
180
200
TIME
1
(b)
0.9
0.8
FOURIER MODES
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
140
160
180
200
TIME
Fig.
HONLSeq.eq.with
with
linear
damping:
Γ = 0.01,
ǫ=
Fig.7:
7. The
The HONLS
linear
damping:
0 = 0.01,
= 0.05,
β = 0.β =
The
of (a) the
(b) the
main
0.05,
0. time
Theevolution
time evolution
of strength
(a) the S(t)
strength
S(t)
(b)
Fourier
0 <modes,
t < 200.0 < t < 200.
the
mainmodes,
Fourier
4.2 The HONLS Equation with nonlinear damping
4.2 The HONLS Equation with Nonlinear Damping
The perturbation
of the Hilbert tranform term in Eq. (1),
βu H |u|2 x . is referredTime
to as
the “β-term” and models
= 22.7
nonlinear damping of the mean flow. Figure 8 shows the
spatial distribution of |u(x,t ∗ )| (solid line) and of the β-term
(dashed line) of the nonlinearly damped HONLS equation
for = 0.05, β = 0.1, 0 = 0, for initial condition (16).
At t = 22.7 the wavetrain is strongly modulated and we
find that the β-term is significant only near the maximum
crest of the envelope. In general, steeper waves result in
stronger damping and the effects of the nonlinear β-term are
concentrated in space-time when the wavetrain is strongly
modulated.
To illustrate the effects of nonlinear damping on downshifting we consider the solution of Eq. (1) with =
0.05, β = 0.5 and 0 = 0, for initial data (17), 0 < t < 50.
∗
)| (solid
line) and
the
Fig.
8: Spatial
distribution
of |u(x,t
Figure
9a–b shows
the surface
amplitude
|u(x,t)|
andofthe
∗
β-term
(dashedofline)
at t = 22.7
for initial
condition
(16),
time evolution
the strength,
respectively.
Two
rogue waves
for
ǫ = 0.05,
0.1,
Γ =series,
0.
occur
early β
in=the
time
t < 10, before damping has
a chance to significantly alter the growth rate of the modes
and
further of
rogue
waves from
forming.
Theprevent
perturbation
the Hilbert
tranform
termComparing
in equation
2 the onset of downshifting occurs when the
9b
and
9c,
Fig.
(1), βu[H |u| ]x . is referred to as the “β-term” and models
first roguedamping
wave appears
approximately
t ≈ 6(8)
when
the βnonlinear
of theatmean
flow. Figure
shows
the
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−10
−8
−6
−4
−2
0
2
4
6
8
spatial distribution of |u(x,t∗ )| (solid line) and of the β-term
www.nat-hazards-earth-syst-sci.net/11/383/2011/
10
−8
−6
−4
−2
0
2
4
6
8
10
shiftshift
in theinspectral
centercenter
km (Fig.
inrapid
a rapid
the spectral
km 9f).
(Fig.The
9f).downshiftThe
ing
becomes
irreversible
after
t
≈
17
when
the
wavetrain
downshifting becomes irreversible after t ≈ 17 when
the is
demodulated
and
the
lower
modes
k
=
−1,−4
become
domwavetrain is demodulated and the lower modes k = −1,−4
inant.
In
the
demodulation
stage
the
β-term
is
small
so
become dominant. In the demodulation stage the β-term isthat
the energy
change
at a change
much slower
rate. slower
Thus km
small
so thatand
the flux
energy
and flux
at a much
remains
andnegative
does notand
upshift
rate.
Thusnegative
km remains
does back.
not upshift back.
Permanent
downshifting
is obtained
forvalues
all values
of β.
Permanent
downshifting
is obtained
for all
of β. As
an example
consider
the solution
of equation
(1) with
anAsexample
consider
the solution
of Eq. (1)
with = 0.05,
0.05,0β==0 0.04,
Γ = data
0 for(17),
initial
0 <small
t < 200.
βǫ==0.04,
for initial
0 <data
t < (17),
200. For
small
β there
is a decay
slow gradual
decay
in the
βFor
there
is a slow
gradual
in the total
energy
andtotal
flux enrather
the abrupt
obtained
largerobtained
β. As a for
ergy than
and flux
ratherrapid
than decay
the abrupt
rapidfordecay
result
wave
at t ≈
51 (Fig.
10a),atmuch
largertheβ.last
Asrogue
a result
theappears
last rogue
wave
appears
t ≈ 51 (
later
formuch
β = 0.5.
the0.5.
evolution
of of
thethe
main
Fig.than
10a),
laterThe
thanplot
for of
β=
The plot
evoluFourier
(Fig.
10b) modes
shows (Fig.
permanent
downshifting
tion of modes
the main
Fourier
10b) shows
permanent
occurs
at t ≈ 65.
Figure
which
shows11,
thewhich
evolution
of the
downshifting
occurs
at 11,
t ≈ 65.
Figure
shows
kpeak
for theof
nonlinearly
HONLSdamped
equationHONLS
( = 0.05,
evolution
kpeak for damped
the nonlinearly
equa0tion
= 0)(ǫwith
β
=
0.04
or
β
=
0.5,
provides
an
alternate
view
= 0.05, Γ = 0) with β = 0.04 or β = 0.5, provides an
ofalternate
permanent
downshifting
for downshifting
β 6= 0 and itsfor
occurence
at its
view
of permanent
β 6= 0 and
later
times
for
smaller
values
of
β.
occurence at later times for smaller values of β.
When
nonlinear damping
damping are
are present
presentand
Whenboth
both linear
linear and
and nonlinear
and
the
nonlinear
damping
is
dominant,
the
mechanism
forperthe nonlinear damping is dominant, the mechanism for
permanent
downshifting
is
the
same.
Figure
12
shows
the
manent downshifting is the same. Fig. (12) shows the sosolution
of equation
Eq. 1 with(1)
=with
0.05,ǫ β
0.5, β
0=
= 0.005
initial for
lution of
==
0.05,
0.5, Γfor
= 0.005
data
(17),
0
<
t
<
50.
Comparing
to
Fig.
9,
the
rogue
waves
initial data (17), 0 < t < 50. Comparing to Fig. (9), the rogue
occur
at occur
approximately
the same time
with slightly
smaller
waves
at approximately
the same
time with
slightly
amplitudes.
An
abrupt
rapid
decay
in
the
energy
and
growth
smaller amplitudes. An abrupt rapid decay in the energy and
in the flux occurs as before leading to a rapid shift in the
growth in the flux occurs as before leading to a rapid shift in
spectral center km .
the spectral center km .
Characterization of the nonlinearly damped evolution:
Characterization
of the
nonlinearly
damped
evolution
To characterize the
effects
of nonlinear
damping
on rogue
waves we fix ǫ = 0.05 and Γ = 0 in equation (1) and vary the
Tononlinear
characterize
the effects
of nonlinear
rogue
damping
coefficient,
0 < β <damping
0.75, asonwell
as the
and
vary
waves
we
fix
=
0.05
and
0
=
0
in
Eq.
(1)
amplitude of the initial data in the three-UM regime: the
nonlinear damping coefficient, 0 < β < 0.75, as well as the
u(x,0) = a(1 + 0.01cosµx),
(19)
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
392
A. Islas and C. M. Schober: Rogue waves and downshifting
(a)
(a)(a)
(a)
(a)
(a)
(e)
(e)
(e)
(e)
(e)
(e)
11
1 1
1
1
0.8
0.8
0.8 0.8
0.8
0.8
FLUX
FLUX
FLUX
FLUX
FLUX
FLUX
0.6
0.6
0.6 0.6
0.6
0.6
0.4
0.4
0.4 0.4
0.4
0.4
0.2
0.2
0.2 0.2
0.2
0.2
00
0 0
0
0
0 0
0 0
0
0.1
0.1
0
0.1 0.1
0.1
2.4
2.4
2.4 2.4
2.4
(b)
(b)
(b)
(b)
(b)
(b)
1.4
1.2
1.2 1.2
1.2
1.2
55
5
5
5
5
10
10
10 10
10
15
15
15 15
15
20
20 20
20
20
10
15
20
10
10
10
10
15
15
15
15
20
20 20
20
10
15
20
10
10 10
10
10
10
15
15 15
15
15
15
20
20 20
20
20
20
1
0.9
1
0.9
0.9 0.9
0.9
0.8
0.9 0.8
0.8
0.8
25
30
25
30 30
25
25
30
TIME
TIME
25
TIME 30
TIME
TIME
25
TIME
30
35
35 35
35
35
40
40 40
40
40
45
4545
45
45
50
5050
50
35
40
45
50
50
(c)
(c)
(c)
(c)
(c)
(c)
FOURIER
MODES
FOURIER
MODES
FOURIER
MODES
MODES
FOURIERMODES
MODES
FOURIER
0.8
0.7
0.8 0.7
0.7
0.7
0.7
0.6
0.7 0.6
0.6
0.6
0.6
0.5
0.6 0.5
0.5
0.5
0.5
0.4
0.5 0.4
0.4
0.4
0.3
0.4 0.3
0.3
0.3
0.3
0.2
0.3 0.2
0.2
0.2
11
1
0.1
55
5
5
5
5
1
1
25
25
25
TIME
25
TIME
25
TIME
TIME
TIME
25
TIME
30
35
40
45
50
30
30
30
30
35
35
35
35
40
40
40
40
45
45
45
45
50
50
50
50
30
35
40
45
50
30
30 30
30
30
30
35
35 35
35
35
35
40
40 40
40
40
40
45
45 45
45
45
45
50
50 50
50
50
50
1
(d)
(d)
(d)
(d)
(d)
(d)
0.8
0.8 0.8
0.8
0.8
ENERGY
ENERGY
ENERGY
ENERGY
ENERGY
ENERGY
0.8
0.6
0.6 0.6
0.6
0.6
0.6
0.4
0.4 0.4
0.4
0.4
0.4
0.2
0.2 0.2
0.2
0.2
0.2
0
0
0 0
00 0
00
0
0
0
15
20
35
35
35 35
35
40
40
40 40
40
45
45
45 45
45
50
50
50 50
50
30
35
40
45
50
(f)
(f)
(f)
(f)
(f)
0.1
0
0
0 0
0
−0.4
−0.5
−0.5 0
−0.5−0.5
00
−0.50
1.2
0.2
10
30
30
30 30
30
−0.3
−0.4
−0.4
−0.4−0.4
−0.4
1.6
1.4
1.4 1.4
1.4
1.4
0.2 0.1
0.1
0.1
0.1
0
0.1 00
00
0 0
00
0
0
0
5
25
25
25
25
TIME
TIME
25
TIME
TIME
TIME
25
TIME
−0.2
−0.3
−0.3
−0.3−0.3
−0.3
1.8
1.6
1.6 1.6
1.6
1.6
0.4
20
20
20 20
20
−0.1
−0.2
−0.2
−0.2−0.2
−0.2
2
1.8
1.8 1.8
1.8
1.8
1
11 10
0 0
10
0
1 1
110 1
15
15
15 15
15
0
−0.1
−0.1
−0.1−0.1
−0.1
2
2
STRENGTH
STRENGTH
STRENGTH
STRENGTH
STRENGTH
STRENGTH
2.2
2
2
2
1010
10 10
10
SPECTRAL
CENTER
SPECTRAL
CENTER
SPECTRAL
CENTER
SPECTRAL
CENTER
SPECTRALCENTER
CENTER
SPECTRAL
2.4
2.2
2.2
2.2 2.2
2.2
55
5 5
5
5
5
5
5
5
5
25
25
25
TIME
25
TIME
25
TIME
TIME
25
TIME
TIME
Fig.
9: a-d
Nonlinearly
damped
HONLS
eq.
ǫǫ =
0.05,
=
Fig.9:
a-dNonlinearly
Nonlinearlydamped
dampedHONLS
HONLSeq.
eq.ǫǫ=
=0.05,
0.05,βββ
β=
Fig.
9:9:a-d
Nonlinearly
Fig.
damped
HONLS
eq.
=
0.05,
==
Fig.
9:Γa-d
a-d
Nonlinearly
damped
HONLS
eq.
ǫ=
0.05,
βevo=
0.5,
=
0:
a)
Surface
amplitude
|u(x,t)|,
and
the
time
Fig.
9.Γ
Nonlinearly
damped
HONLS
equationeq.
and
=ǫthe
0.05,
β =evo0.5,
0.5,
=
Surface
amplitude
|u(x,t)|,
time
evoFig.
a-d
Nonlinearly
damped
HONLS
=the
0.05,
βevo=
0.5,
=
0:0:a)
a)a)Surface
amplitude
|u(x,t)|,
and
time
0.5,
Γ9:Γ
=
0:
amplitude
|u(x,t)|,
and
the
time
=of
0:
a) Surface
Surface
amplitude
|u(x,t)|,
and
the
time
evolution
(b)
the
strength
S(t),
(c)
the
main
Fourier
modes,
00.5,
=
0:Γ
(a)
surface
amplitude
|u(x,t)|,
and
the
time
evolution
of
(b)
lution
of
(b)
the
strength
S(t),
(c)
the
main
Fourier
modes,
0.5,
Γ
=
0:
a)
Surface
amplitude
|u(x,t)|,
and
the
time
evolution
of
(b)
the
strength
S(t),
(c)
the
main
Fourier
modes,
lution
of
(b)
the
strength
S(t), data
(c) the
the
main Fourier
Fourier modes,
modes,
lution
of total
(b)
the
strength
S(t),
(c)
main
(d)
the
energy,
for
initial
(17).
the
strength
S(t),
(c)
the
main
Fourier
modes,
(d)Fourier
the totalmodes,
energy,
(d)
the
total
energy,
initial
data
(17).
lution
of
(b)
the
strength
S(t),
(c)(17).
the
main
(d)
the
total
energy,
forfor
initial
data
(d)
the
total
energy,
for
(d)
initial
data
(17).
for
dataenergy,
(17), thefortime
evolution
of (e) the total energy flux,
(d)initial
the total
initial
data (17).
and (f) the spectral center km for initial data (17).
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
−0.5
0
555
5
5
10
101010
10
15
15
1515
15
20
20
2020
20
0
5
10
15
20
25
25
25
25
TIME
TIME
25
TIME
TIME
TIME
25
TIME
30
35
40
45
50
30
30 30
30
35
35 35
35
40
40 40
40
45
45 45
45
50
50 50
50
30
35
40
45
50
Fig.
e-f
Nonlinearly
damped
HONLS
eq.
ǫǫ =
=
0.05,
=
Fig.
9.
Fig.
9:9:
e-f
damped
eq.
Fig.9:
9:Continued.
e-fNonlinearly
Nonlinearlydamped
damped HONLS
HONLSeq.
eq.ǫ ǫ=
= 0.05,
0.05,β ββ==
Fig.
Fig.
9:Γ e-f
e-f0:Nonlinearly
Nonlinearly
dampedofHONLS
HONLS
eq. energy
ǫ = 0.05,
0.05,
β and
=
0.5,
=
the
time
evolution
(e)
the
total
flux,
0.5,Γ9:
=
thetime
timeevolution
evolution
of(e)
(e)the
thetotal
total
energy
flux,
0.5,
ΓΓ=e-f
0:0:Nonlinearly
the
evolution
of
(e)
total
energy
flux,
Fig.
damped
HONLS
eq.energy
ǫ = 0.05,
β and
=
0.5,
0:
of
0.5,
Γ=
=spectral
0: the
the time
time
evolution
of
(e) the
the
total
energy flux,
flux, and
and
(f)
center
k
for
initial
data
(17).
(f)the
the
spectral
center
kmm
forinitial
initial
data
(17).
0.5,
Γthe
=
0: the time
evolution
of
(e) data
the
total
energy flux, and
(f)
the
spectral
center
km
for
initial
data
(17).
(f)
spectral
center
k
for
(17).
m
(f) the spectral center km for initial data (17).
(f) the spectral center km
for initial data (17).
amplitude of the initial data in the three-UM
regime:
√
√2π. Figure 13
√
√
with
0.57
<
a
<
0.67,
µ
=
2π/L,
L
=
4
√
with0.57
0.57
0.67,µµµ==
=2π/L,
2π/L,LLL==
= 4√
4 2π.
2π. Figure
Figure13
with
0.57
<<
<<0.67,
0.67,
2π.
13
with
aaaa
<
u(x,0)
=as<
a<
+
(19)
(1
with
0.57
<0.01cosµx),
0.67, of
µ β:
= 2π/L,
2π/L,
L = 44value
2π.of Figure
Figure
13
shows,
function
a)
the
final
the
spectral
shows,
asa<aaafunction
function
of
β:a)2π/L,
a)the
thefinal
final
of
the
spectral
with
0.57
afunction
< 0.67,ofof
µβ:β:
=
L =value
4value
2π.
Figure
13
shows,
as
a)
the
final
value
of
the
spectral
shows,
as
of
the
spectral
√of the
shows,
as
a, function
of β: a)strength
the finalmax
value
spectral
center
k
b)
the
maximum
S(t),
c) the
t∈[0,200]
center0.57
kmm
,<
b)the
the
maximum
strength
max
S(t),
shows,
function
of β:
a)
the final
value
the
spectral
with
athe
<maximum
0.67,
µ=
2π/L,
L
=
4 of
2π.
Figure
13
center
b)
maximum
strength
max
S(t),
c)the
the
center
kkkas
,,a,b)
strength
max
S(t),
c)
m
t∈[0,200]
m
t∈[0,200]
t∈[0,200]
center
b)
the
maximum
strength
max
S(t),
c)d)
the
m of rogue waves obtained for 0 <
number
t
<
200,
and
the
t∈[0,200]
number
rogue
waves
obtained
for
0<
<
t<
<
200,
andd)
center
km
,ofrogue
b)
the waves
maximum
strength
max
S(t),
c)
number
of
obtained
for
0
<
t
<
200,
and
the
number
of
rogue
waves
obtained
for
0
t
200,
and
d)
the
shows,
as
a
function
of
β:
(a)
the
final
value
of
the
spectral
t∈[0,200]
number
ofthe
rogue
waves
obtained
forsolid
0 < tcurve
< 200,
and d) the
time
ofof
last
rogue
wave.
The
represents
the
timeof
last
rogue
wave.strength
The
solid
curve
represents
number
waves
obtained
for
0max
<curve
tcurve
<
200,
and
d)(c)
time
last
rogue
wave.
The
solid
represents
the
time
ofof
the
last
rogue
wave.
The
solid
represents
the
center
kthe
,rogue
(b)
the
maximum
S(t),
the
mthe
t∈[0,200]
time
of
the
last
rogue
wave.
The
solid
curve
represents
the
averages
over
the
initial
data.
In
the
numerical
experiments
averages
over
the
initial
data.
In
the
numerical
experiments
time
of the
last
rogue
wave.
The
solid
curve
represents
thethe
averages
over
the
initial
data.
In
the
numerical
experiments
averages
over
the
initial
data.
In
the
numerical
experiments
number
of
rogue
waves
obtained
for
0
<
t
<
200,
and
(d)
averages
overthe
thefollowing:
initial data. In the numerical experiments
we
observe
weobserve
observe
the
following:
averages
over
the
initialwave.
data. In
thesolid
numerical
we
the
following:
we
observe
the
following:
time
of the
last
rogue
The
curve experiments
represents the
we
observe
the
following:
we
observe
the
following:
averages over the initial data. In the numerical experiments
1. The
spectral
center
k
(t)
decreasing
function
of
Thespectral
spectral
centerkkm
km
(t)isisis
isa aadecreasing
decreasingfunction
functionofof
oft ttt
we
the following:
1.1.1.observe
The
center
(t)
The
spectral
center
(t)
decreasing
function
1. The
spectral
center
kmmm
(t)
isisaanot
decreasing
function
of
t
for
all
β
>
0.
Figure
13a
showing
the
time
evoforall
allβββ>>>0.0.0.
Figure
13aisisisisnot
not
showingthe
thetime
timeevoevo1. for
The
spectral
center
km13a
(t)
anot
decreasing
function
ofevot
Figure
showing
for
all
Figure
13a
showing
the
time
for
all
β
>
0.
Figure
13a
is
not
showing
the
time
evolution
of
k
,,Figure
(see
e.g.
Fig.
9f
for
a typical
example),
1. lution
The
center
kmFig.
(t)
is
decreasing
function
lution
of
k0.m
(seee.g.
e.g.
Fig.
9fafor
for
typical
example),
for
allspectral
βof
>
13a
is not
showing
the example),
time
evo- of
of
kkm
, ,(see
9f9f
a aatypical
m
lution
(see
e.g.
Fig.
for
typical
example),
lution
of
kβmm
,>(see
e.g.
Fig.
9f
for
a 200,
typical
example),
but
rather
the
final
value
of
k
at
t
=
where
smaller
t
for
all
0.
Figure
13a
is
not
showing
the
time
m
but
rather
the
final
value
of
k
at
t
=
200,
where
smaller
lution
of kthe
, final
(see
e.g.
for
a200,
typical
example),
but
rather
value
ofofkkm9f
atat
t=
where
smaller
m
but
rather
the
final
valueFig.
200,
where
smaller
but
rather
the
final
value
of Fig.
ktommm
at9f
t tfor
==200,
where
smaller
values
mean
aam ,downshift
aaat
lower
mode.
Thus,
perevolution
of
k
(see
e.g.
a
typical
example),
values
mean
downshift
to
lower
mode.
Thus,
perbut
rather
the
final
value
of
k
t
=
200,
where
smaller
values
mean
aadownshift
totom
a lower
mode.
Thus,
pervalues
mean
downshift
lower
mode.
Thus,
pervalues
mean
downshift
to kaamlower
mode.
Thus,
permanent
downshifting
is observed
all
>
0 with
the
but
rather
theaa final
value
of
atfor
t for
=mode.
200,
smaller
manent
downshifting
observed
for
allβββ
β>where
>
with
the
values
mean
downshift
to
a lower
Thus,
permanent
downshifting
isisisobserved
all
0 00with
the
manent
downshifting
observed
for
all
>
with
the
manent
downshifting
is
observed
for
all
β
>
0
with
the
onset
occuring
rapidly
for
larger
values
of
β.
values
mean rapidly
arapidly
downshift
to values
afor
lower
mode.
Thus,
onset
occuring
for
larger
values
of
β.
manent
downshifting
is
observed
all
β
>
0
with
the
onset
occuring
for
larger
of
β.
onsetoccuring
occuringrapidly
rapidlyfor
forlarger
largervalues
valuesofofβ.β.
onset
permanent
downshifting
is observed
onset
occuring
rapidly for larger
values for
of β.all β > 0 with
2. the
Figures
13b-c
show
that
for
small
values
of
β
the
averonset
occuring
rapidly
for
larger
values
β.averFigures
13b-c
show
that
for
small
values
ofβof
βthe
the
aver2.2.2.Figures
13b-c
show
that
for
small
values
ofof
Figures
13b-cshow
show
thatfor
fortime
small
values
the
aver2. Figures
13b-c
that
small
values
of ββnumber
the
average
maximum
strength
in
and
average
of
age
maximum
strength
in
time
and
average
number
of
2. age
Figures
13b-c
show
that
for
small
values
of
β
the
avermaximum
strength
in
time
and
average
number
age
maximum
strength
intime
time
and
average
number
of
2. age
Figure
13b–cexhibit
showsirregular
that
forbehavior.
small
values
of obtain
βof
maximum
strength
in
and
average
number
ofthe
rogue
waves
We
may
rogue
waves
exhibit
irregular
behavior.
We
may
obtain
age
maximum
strength
in
time
and
average
number
of
rogue
waves
exhibit
irregular
behavior.
We
may
obtain
rogue
waves
exhibit
irregular
behavior.
We
mayobtain
obtain
average
maximum
strength
in
time
andWe
average
number
rogue
waves
exhibit
irregular
behavior.
may
more
or
larger
rogue
waves
in
the
damped
regime
than
moreor
orlarger
larger
rogue
wavesinin
inthe
thedamped
damped
regime
than
rogue
waves
exhibit
irregular
behavior.
Weregime
may
obtain
more
rogue
waves
than
more
or
larger
rogue
waves
the
damped
regime
than
of
rogue
waves
exhibit
irregular
behavior.
We
may
more
or
larger
rogue
waves
in
the
damped
regime
than
without
damping
for
this
time
frame
due
to
changes
in
without
damping
for
this
time
frame
due
to
changes
in
more
or
larger
rogue
waves
in
the
damped
regime
than
without
damping
for
this
time
frame
due
to
changes
inin
without
damping
for
this
time
frame
due
to
changes
without
damping
for
this
time
frame
due
to
changes
in
obtain
more
or
larger
rogue
waves
in
the
damped
regime
the
focusing
time
and
coalescence
of
the
modes.
Howthe
focusing
time
and
coalescence
of
the
modes.
Howwithout
damping
for
this
time
frame
due
to
changes
in
the
focusing
time
and
coalescence
of
the
modes.
Howthe
focusing
time
and
coalescence
the
modes.
Howthe
focusing
time
and
coalescence
ofofthe
modes.
Howthan
without
damping
for this time
due
to
changes
ever
for
large
β,
the
maximum
strength
in
time
and
numever
for
large
β,the
the
maximum
strength
in
time
and
numthe
focusing
time
and
coalescence
offrame
the
modes.
However
for
large
β,β,
maximum
strength
inin
time
and
numever
for
large
the
maximum
strength
time
and
numever
for
large
β,
the
maximum
strength
in time
and
numin
the
focusing
time
and
coalescence
of
the
modes.
ber
of
rogue
waves
are,
in
general,
decreasing.
beroffor
ofrogue
roguewaves
waves
are,iningeneral,
general,
decreasing.
ever
large
β,
the
maximum
strength
in
time
and
number
are,
decreasing.
ber
ofrogue
rogue
waves
are,the
general,decreasing.
decreasing.
ber
of
waves
are,
iningeneral,
However
forwaves
largeare,
β,
maximum
strength in time and
ber
of rogue
in general,
decreasing.
number of rogue waves are, in general, decreasing.
www.nat-hazards-earth-syst-sci.net/11/383/2011/
A. Islas and C. M. Schober: Rogue waves and downshifting
393
33
55
(a)
(a)
2.8
2.8
22
SPECTRAL
PEAK
SPECTRAL PEAK
33
2.4
2.4
STRENGTH
STRENGTH
2.6
2.6
2.2
2.2
22
1.8
1.8
11
00
−1
−1
1.6
1.6
−2
−2
1.4
1.4
−3
−3
1.2
1.2
−4
−4
11
00
(a)
(a)
44
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
−5
−5
00
10
10
20
20
30
30
40
40
TIME
TIME
11
60
60
70
70
80
80
90
90
100
100
55
(b)
(b)
0.9
0.9
(b)
(b)
44
0.8
0.8
FOURIER
FOURIER MODES,
MODES, kk == 0,
0, −4
−4
33
0.7
0.7
SPECTRAL
SPECTRAL PEAK
PEAK
22
0.6
0.6
0.5
0.5
0.4
0.4
11
00
−1
−1
0.3
0.3
−2
−2
0.2
0.2
−3
−3
0.1
0.1
−4
−4
00
00
50
50
TIME
TIME
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
TIME
TIME
−5
−5
00
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
100
100
TIME
TIME
Fig. 10. The HONLS equation with nonlinear damping = 0.05,
Fig.
10: The
TheHONLS
HONLSeq.
eq. with
with nonlinear
nonlineardamping
damping ǫǫ=
=0.05,
0.05,
Fig.
10:
β = 0.04, 0 = 0: time evolution of (a) the strength S(t) and (b) the
0.04,
Γ
=
0:
time
evolution
of
(a)
the
strength
S(t)
and
ββmain
==0.04,
Γ
=
0:
time
evolution
of
(a)
the
strength
S(t)
and
Fourier modes for initial data (17).
(b)the
themain
mainFourier
Fouriermodes
modesfor
forinitial
initialdata
data(17).
(17).
(b)
Fig.11.
11:The
The
evolution
of kkpeak
for
the nonlinearly
nonlinearly
damped
Fig.
11:
The
evolution
the
peak
Fig.
evolution
of kof
for
thefor
nonlinearly
damped damped
HONLS
peak
HONLSequation
equation
with0ǫǫ=
=0.05,
Γ=
=β00=
and
(a)orββ(b)
=0.04
0.04
orfor
(b)
equation
with
= 0.05,
=
00.05,
and Γ
(a)
0.04
β = .5
HONLS
with
and
(a)
=
or
(b)
(17).
β=
=.5
.5data
forinitial
initial data
data (17).
(17).
βinitial
for
3. For all values of the initial amplitude, the first rogue
For all
all values of
of the
the initial
initial amplitude, the
the first rogue
rogue
3.3. For
wave isvalues
only slightly
delayedamplitude,
(t ≈ 6) for all β first
considered
wave
is
only
slightly
delayed
(t
≈
6)
for
all
β
considered
wave
is
only
slightly
delayed
(t
≈
6)
for
all
β
considered
(not shown). The time of the last rogue wave, Tl ,
(not
shown).
The
timefor
of the
the
lastβ.
rogue
wave,
may
(not
time
of
last
rogue
wave,
TTll,, may
mayshown).
slightlyThe
increase
small
As β
increases,
Tl
slightly
increase
for
small
β.
As
β
increases,
T
signifslightly
increase
for
small
β.
As
β
increases,
T
signifl
l waves
significantly decreases, indicating fewer rogue
icantly
decreases,
indicating
fewer
roguepair
waves
occur.
icantly
rogue
waves
occur. decreases,
For β >indicating
β∗ only fewer
the
initial
of occur.
rogue
For
β
>
β
only
the
initial
pair
of
rogue
waves
develop
For
β
>
β
only
the
initial
pair
of
rogue
waves
develop
∗
∗
waves develop
(Fig. 13d) as large β values trigger
(Fig.
13d)as
aslarge
largeββdamping
valuestrigger
trigger
sufficient
nonlinear
(Fig.
13d)
values
sufficient
nonlinear
sufficient
nonlinear
that inhibit
further
rogue
damping
that
inhibit
further
rogue
wave
formation.
damping
that
inhibit
further
rogue
wave
formation.
wave formation.
0 < β < 0.75. Figure 14a shows the comparison for initial
data (19) with a = 0.63: we observe that for all values
β rogue
rogue waves
waves do
do not
not occur
occur after the
the downshifting
downshifting becomes
becomes
βof
β rogue waves
do not after
occur after
the downshifting
irreversible.
Further,
rogue
waves
do
not
develop
after
perirreversible.
Further, rogue
waves
do waves
not develop
perbecomes irreversible.
Further,
rogue
do notafter
develop
manent downshifting
downshifting occurs
occurs for any
any other pair
pair of
of parameparamemanent
after permanent
downshiftingfor
occurs other
for any other
pair of
ter values
values (a,β)
(a,β) considered
considered in
in the
the experiments.
experiments. This
This is
is sumter
parameter values (a, β) considered in the experiments. sumThis
marized in
in Fig.
Fig. 14b
14b which
which compares,
compares, for
for 00 <
< ββ <
< 0.75, the
the
marized
is summarized
in Fig.
14b which compares,
for 0.75,
0<β<
average time
time of
of the
the last
last rogue
rogue wave
wave (box)
(box) with
with the
the average
average
average
0.75, the average time of the last rogue wave (box) with the
time at
at which
which downshifting
downshifting isis irreversible
irreversible (x),
(x), where
where the
the avavtime
average
time at which downshifting
is irreversible
(x), where
eragesare
areover
overthe
thesix
six simulations
simulations with
with initial
initial data
data amplitude
amplitude
erages
the averages are over the six simulations with initial data
0.57<
<aa<
<0.67.
0.67. These results
results imply
imply permanent
permanent downshiftdownshift0.57
amplitude
0.57 <These
a < 0.67. These
results imply permanent
ingserves
servesas
asan
anindicator
indicatorthat
thatthe
thecumulative
cumulativeeffects
effects of
of dampdamping
downshifting serves as an indicator that the cumulative
ingare
are sufficient
sufficient to
to prevent
preventthe
the further
further development
development of
of rogue
rogue
ing
effects of damping are sufficient to prevent the further
waves.
waves.
development of rogue waves.
we
consider
thethe
casecase
whenwhen
both linear
linear
and nonlinnonlinFinally we
weconsider
consider
both and
linear
and
Finally
the
case
when
both
ear
damping
are
present
and
the
nonlinear
damping
is
domnonlinear
damping
are
present
and
the
nonlinear
damping
ear damping are present and the nonlinear damping is dominant
to allow
allowtopermanent
permanent
downshifting
to occur.
occur. Figure
Figure
15
is
dominant
allow permanent
downshifting
to
occur.
inant
to
downshifting
to
15
compares
the
times
for
permanent
downshifting
and
the
last
15
compares
the
times
for
permanent
downshifting
Figure
compares the times for permanent downshifting and the last
rogue
wave
for
the damped
damped
HONLS
eqn. with
with ǫǫ=
= 0.05,Γ
0.05,Γ=
=
and
the
lastfor
rogue
wave
forHONLS
the damped
HONLS
equation
rogue
wave
the
eqn.
0.005
and
0
<
β
<
0.75.
The
same
relation
between
downwith
=
0.05,
0
=
0.005
and
0
<
β
<
0.75.
The
same
0.005 and 0 < β < 0.75. The same relation between downshifting and
and the
the last
last
rogue wave
wave
are
observed
to wave
hold, are
i.e.
relation
between
downshifting
andare
theobserved
last rogue
shifting
rogue
to
hold,
i.e.
rogue
waves
do
not
occur
after
the
downshifting
is
permaobserved
to
hold,
i.e.
rogue
waves
do
not
occur
after
the
rogue waves do not occur after the downshifting is permanent. However,
However,
due
to the
the additional
additional
linear
damping
there is
is
downshifting
is due
permanent.
However,linear
due to
the additional
nent.
to
damping
there
a
longer
time
lag
between
the
last
rogue
wave
and
permanent
linear
damping
there
is
a
longer
time
lag
between
the
last
a longer time lag between the last rogue wave and permanent
downshifting.
rogue
wave and permanent downshifting.
downshifting.
Relationof
ofDownshifting
Downshifting and
and Rogue
Rogue Waves
Waves :: ItIt isis sigsigRelation
nificant
in
Figs.
9b-c
and
10a-b
that
permanent
downshifting
nificant
in
Figs.
9b-c
and
10a-b
that
permanent
downshifting
Relation of downshifting and rogue waves
occursafter
afterthe
thelast
last rogue
roguewave
wave in
in the
the time
time series.
series. In
In other
other
occurs
words,
roguewaves
waves
do
not9b–c
occurand
after10a–b
the downshifting
downshifting
beIt is significant
in do
Figs.
that permanent
words,
rogue
not
occur
after
the
becomes
irreversible
these
examples.
To generalize
generalize
thistime
redownshifting
occurs
after examples.
the
last rogue
wave in the
comes
irreversible
inin these
To
this
result
weanalyze
analyze
theprevious
previousrogue
serieswaves
ofnumerical
numerical
experiments
series.
In other
words,
do notexperiments
occur after
sult
we
the
series
of
using
initialdata
data (19)
(19)
with 0.57
0.57
<aa<
<0.67
0.67
to determine
determine
1)
the downshifting
becomes
irreversible
in these
examples.
using
initial
with
<
to
1)
the
time
of
the
last
rogue
wave
and
2)
the
last
time
k
To
generalize
this
result
we
analyze
the
previous
series
of
peak
the time of the last rogue wave and 2) the last time kpeak isis
equal
indicating permanent
permanent
downshifting,
as the
the
pernumerical
using initial
data (19) with
0.57
<
equal
toto kk00, ,experiments
indicating
downshifting,
as
perturbation
strength
β
is
varied.
a
<
0.67
to
determine
(1)
the
time
of
the
last
rogue
wave
and
turbation strength β is varied.
Figure
14 time
compares
theequal
time to
which
downshifting
be(2)
the last
kpeak the
is
permanent
Figure
14
compares
time
atat kwhich
downshifting
be0 , indicating
comes
irreversible
(labeled
by
an
x)
with
the
time
at
which
downshifting,
as
the
perturbation
strength
β
is
varied.
comes irreversible (labeled by an x) with the time at which
thelast
lastrogue
rogue
wave
occurs(labeled
(labeled
byataabox)
box)for
fordownshifting
theHONLS
HONLS
14wave
compares
the
timeby
which
Figure
the
occurs
the
equation
with
only
nonlinear
damping,
ǫ
=
0.05,Γ
=0which
0 and
and
becomes
irreversible
(labeled
by
an
x)
with
the
time
at
equation with only nonlinear damping, ǫ = 0.05,Γ =
0
<
β
<
0.75.
Fig.
14a
shows
the
comparison
for
initial
the
last
rogue
wave
occurs
(labeled
by
a
box)
for
the
HONLS
0 < β < 0.75. Fig. 14a shows the comparison for initial
data
(19) with
a==0.63
0.63
we observe
observe that
that
for
all values
values
of
equation
with aonly
nonlinear
damping,
=for
0.05,0
= 0 and
data
(19)
:: we
all
of
www.nat-hazards-earth-syst-sci.net/11/383/2011/
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
0
A. Islas and C. M. Schober: Rogue waves and downshifting
1.2
5
5
0
10
10
5
15
15
10
20
20
15
20
1
1
1
25
25
TIME
TIME
25
TIME
30
30
35
35
30
40
40
35
40
0
−0.250
50
0.1
0.1
0
0.3
0.4
β 0.4
β 0.4
β
0.5
0.5
0.6
0.6
0.5
0.6
a = 0.67
2.9
2.9
2.6
2.6
2.6
1.6
−0.05
2.5
2.5
2.5
2.4
2.4
2.4
1.2
5
5
0
10
10
10
5
0
5
15
15
15
10
20
20
20
15
25
25
TIME
25
TIME
TIME
20
30
30
30
25
35
35
35
40
40
40
30
35
45
45
45
40
45
1
0.8
0.7
0.6
0.4
3
6
6
6
2.9
3
3
3
0.3
0.2
0.2
0.2
0
7
7
7
4
4
4
0.5
In this
this section we
we examine the
the generation of
of rogue waves
waves in
In
In thissection
section weexamine
examine thegeneration
generation ofrogue
rogue waves in
in
the
presence
of
damping
for
sea
states
characterized
by the
the
the
thepresence
presenceofofdamping
dampingfor
forsea
seastates
statescharacterized
characterized by
by the
Joint North
North Sea Wave
Wave Project (JONSWAP)
(JONSWAP) spectrum:
Joint
Joint NorthSea
Sea WaveProject
Project (JONSWAP)spectrum:
spectrum:
2
f −f0 2
1
2
f
−f
−1(
0.07 ff ≤
0)
αg22 − 554 ( fff00f )444 e−
21( σf
≤f
f −f
0 0) 2
αg
2
σf
−
0
(
)
0.07 f ≤ff000
S(f )) =
= αg 5 ee−−4 54((f f0)) γ
γ ee 2 σf0 ,, σ
σ=
= 0.07
S(f
0.09 ff >
>f
S(f ) =(2πf
γ
, σ = 0.09
(2πf ))55 e
0.09 f >ff000
TIME
(2πf )
0.1
0.1
0.1
−0.25
3.1
5
5
5
Damped random
random oceanic sea
sea states
555 Damped
Damped randomoceanic
oceanic seastates
states
0.57
0.59
0.61
0.63
0.65
0.67
(a)
−0.15
8
8
8
Number
Waves
Number
ofRogue
Rogue
Waves
Number
ofofRogue
Waves
0.9
=
=
=
=
=
=
−0.1
9
9
9
50
TIME
Fig. 12:
12: Nonlinearly
Nonlinearly damped
damped HONLS
HONLS
eq. ǫǫ =
= 0.05, β
β=
= 0.5,
0.5,
Fig.
eq.
Fig.
12: Nonlinearly
dampedof
HONLS
eq.
ǫ =0.05,
0.05,
β(b)
=(b)
0.5,
Γ
=
0.005:
the
time
evolution
(a)
the
strength
S(t),
the
ΓΓ==0.005:
the
time
evolution
ofof(a)
the
strength
S(t),
(b)
the
0.005:
the
time
evolution
(a)
the
strength
S(t),
(b)
the
main
Fourier
modes,
for
initial
data
(17).
main
mainFourier
Fouriermodes,
modes,for
forinitial
initialdata
data(17).
(17).
a
a
a
a
a
a
−0.2
2.3
2.3 0
0
2.3
0
50
50
50
(b)
(b)
2.9
2.7
2.7
2.7
0.7
(b)
3
2.8
2.8
2.8
0.7
0.7
a = 0.57
a
a=
= 0.59
0.57
a
a=
= 0.61
0.59
a
a=
= 0.63
0.61
a
a=
= 0.65
0.63
a
a=
= 0.67
0.65
3
3
1.4
FOURIER MODES
0.3
0.3
0.2
0
3.1
1.8
1
0.2
0.2
0.1
3.1
3.1
2
STRENGTH
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
00
00
45
Maximum
Strength
Maximum
Strength
Maximum
Strength
FOURIER
FOURIERMODES
MODES
FOURIER MODES
0.9
0.8
0.8
2.2
0.8
0.7
0.7
0.7
0.6
0.6
−0.25
−0.25
50
50
(b)
(b)
(b)(a)
2.4
0.9
0.9
45
45
Spectral Center
1
10
0
1
Maximum Strength
S
SpeS
−0.2
394
0.1
0.3
0.3
0.3
0.2
0.4
β 0.4
β 0.4
β 0.3
0.5
0.5
0.5
β
0.4
0.6
0.6
0.6
0.5
0.7
0.7
0.7
a
a
a
a
a
a
a
a
a
a
a
a
=
=
=
=
=
=
=
=
=
=
=
=
0.6
0.57
0.59
0.57
0.61
0.59
0.63
0.61
0.65
0.63
0.67
0.65
0.67
(c)
(c)
(c)
0.7
a
a
a
a
a
a
=
=
=
=
=
=
0.57
0.59
0.61
0.63
0.65
0.67
(b)
2.8
2.7
2.6
2
2
2
2.5
1
10
0
1
0
0.1
2.40.1
0.2
0.1
0
5
10
15
20
25
30
35
40
45
50
150
150
150
Time
Last
Rogue
Wave
Time
Last
Rogue
Wave
Time
ofofof
Last
Rogue
Wave
0
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0
0.1
0.2
0.5
0.5
0.5
0.4
β 0.4
β 0.4
β
0.6
0.6
0.6
2.3
0.3
β
0.4
0.5
0.7
0.7
0.7
a = 0.57
a
0.57
a=
= 0.6
0.59
a
a=
= 0.59
0.61
a
a=
= 0.61
0.63
a
a=
= 0.63
0.65
a
a=
= 0.65
0.67
a = 0.67
0.7
(d)
9
Here ff is
is spatial
spatial frequency,
frequency, ffnn =
=k
knn /2π,
/2π, ff00 is
is the
the dominant
dominant
Here
Here
f12:
is12.
spatial
frequency,
fnHONLS
=HONLS
kn /2π,
feq.
dominant
Nonlinearly
damped
equation
=
ββ
==
0.5,0.5,
0 is
Fig.Fig.
Nonlinearly
damped
ǫthe
=0.05,
0.05,
frequency
determined
by
the
wind
speed
at
a
specified
height
(d)
frequency
determined
the
speed
atataastrength
specified
height
0 0.005:
= 0.005:
the time
timeby
evolution
of (a)
the strength
S(t), (b)
the main
(d)
frequency
determined
by
thewind
wind
speed
specified
height
Γ
=
the
evolution
of
(a)
the
S(t),
(b)
above Fourier
the sea
seamodes,
surface
and
g data
is gravity.
gravity.
Developing storm
storm dydy- the
above
the
surface
and
g
is
Developing
for
initial
(17).
above
sea surface
and
is gravity.
Developing
storm dymainthe
Fourier
modes,
forg initial
datafor
(17).
namics
are
governed
by this
this
spectrum
a range of
of param(c)
namics
namicsare
aregoverned
governedby
by thisspectrum
spectrumfor
foraarange
range ofparamparameters
γ
and
α.
Here
γ
is
the
peakedness
parameter
and
α
eters
etersγγand
andα.α. Here
Hereγγisisthe
thepeakedness
peakednessparameter
parameter and
and αα
is related
related to
to the
the amplitude
amplitude and
and energy
energy content.
content. The
The putative
putative
is
is 5related
to
the
amplitude
and
energy
content.
The
putative
oceanic
sea states
region Damped
of validityrandom
for the
the NLS
equation
and its
its higher order
order
region
regionofofvalidity
validityfor
for theNLS
NLSequation
equationand
and itshigher
higher order
(a)
generalizations
is
2
<
γ
<
8
and
0.008
<
α
<
0.02
generalizations
isis2we
0.008
<<αα<<0.02
generalizations
2<<γexamine
γ<<88and
andthe
0.008
0.02
In
this
section
generation
of
rogue
waves in
The
initial
data
for
the
surface
elevation
isis of
of
the
form
The
initial
for
elevation
is
Thepresence
initialdata
data
forthe
the0surface
surface
elevation
of the
the form
form
the
of
damping
for
sea
states
characterized
by the
a = 0.57
(Onorato
etetal.,
al.,
2001)
(Onorato
et
2001)
a = 0.59
(Onorato
al.,
2001)
Joint North
Sea
Wave
Project
(JONSWAP)
spectrum:
0
a = 0.61
N
a
=
0.57
a = 0.63
NN
X
Fig.
13:
The
HONLS
eqn.
with
nonlinear
damping 00 <
<β
β<
<
X
2
f
−f
X
Fig.
13:
The
HONLS
eqn.
with
nonlinear
a
=
0.59
0
2
Fig. 13: The HONLS eqn. with a =β0.65 damping
4 cos(k
−1(
f0 C
)
η(x,0)
=
x
−
φ
),
(20)
5 −0.05
αg
2
σf
0.07
f
≤
f
n
n
n
0
a
=
0.61
a<
= 0.67
0
η(x,0)
=
C
cos(k
x
−
φ
),
(20)
−
ecos(k
(
)
n
n
n
0.75,
u
=
a(1
+
0.01cosµx),
0.57
a
<
0.67,
(a)
spectral
η(x,0)
=
C
x
−
φ
),
(20)
4
f
0
n
n
n
0.75,
0.57 < a < 0.67, (a) spectral
S(f ) =
e n=1 γ
, σ=
0.75, uu00 =
= a(1
a(1+
+0.01cosµx),
0.01cosµx),
a = 0.63
0.09 f > f0 center
n=1
(2πf )5
n=1
kkm
at
tt =
=
200,
(b)
max
S(t), (c)
(c) number
number of
of
a (b)
=
0.65
m
t∈[0,200] S(t),
center
k
at
t
200,
max
center
at
=
200,
(b)
max
t∈[0,200]
Fig.
13.
Continued.
m
t∈[0,200]
−0.05
a = 0.67
rogue
waves,
and
(d)
time
of
last
rogue
wave
as
a
function
where
k = 2πn/L,
the
random
phases
φφnn are
are uniformly
uniformly
rogue waves,
waves, and
and (d)
(d) time
time of
of last rogue wave as a function
where
the
φ
where
2πn/L,frequency,
therandom
random
phases
Herekknnfn=is=2πn/L,
spatial
fnphases
= kn /2π,
the dominantrogue
n fare
0 isuniformly
of
β.
The
solid
curves
represent
thestates
averages.
distributed
on
(0,2π)
and
the
spectral
amplitudes
C
=
of
β.
The
solid
curves
represent
the
averages.
n
of
β.
The
solid
curves
represent
distributed
on
(0,2π)
spectral
amplitudes
CCnn ==
−0.1
distributed
on
(0,2π) and
and
the wind
spectral
amplitudes
frequency
determined
by the
the
speed
at a specified
height 5 Damped random oceanic sea
(d)
above the sea surface and g is gravity. Developing storm dy- In this section we examine the generation of rogue waves in
−0.1
β
namics are governed by this spectrum
for a range of param- the presence of damping for sea states characterized by the
eters γ and α. Here
γ
is
the
peakedness
parameter and α Joint North Sea Wave Project (JONSWAP) spectrum:
−0.15
Fig.
13.
The
HONLS
equation
with
nonlinear
damping
β<
is related to the amplitude and energy content. The0 <
putative
0.75, u0 = a(1 + 0.01cosµx), 0.57 < a < 0.67, (a) spectral center
(
4 − 1 f −f0 2
region
validity
NLS equation and its higher order
0.07 f ≤ f0
σf0
2
αg 2 − 54 ff0
km −0.15
atoft =
200, (b)for
maxthe
t∈[0,200] S(t), (c) number of rogue waves,
e
S(f
)
=
e
γ
,
σ
=
(b)
generalizations
is
2
<
γ
<
8
and
0.008
<
α
<
0.02
and (d) time of last rogue wave as a function of β. The solid curves
(2πf )5
0.09 f > f0
−0.2
represent
the averages.
The
initial
data for the surface elevation is of the form
(Onorato et al., 2001)
Here f is spatial frequency, fn = kn /2π, f0 is the dominant
8
50
50
50
0
Number of Rogue Waves
100
100
100
a
a
a
a
a
a
=
=
=
=
=
=
0.57
0.59
0.61
0.63
0.65
0.67
7
6
5
4
3
−0.05
0.2
0.2
0.2
0.3
0.3
0.3
0.4
β 0.4
0.4
β
β
0.5
0.5
0.5
0.6
0.6
0.6
0.3
0.4
0.5
0.7
0.7
0.7
−0.1
1
0
0.1
0.2
0.6
0.7
150
a
a
a
a
a
a
−0.15
−0.2
50
0.1
0.1
0.1
2
−0.25
0
0.1
3.1
3
0.2
0.3
0.4
0.5
0.6
Time of Last Rogue Wave
Spectral Center
0
00
0
0
0
Spectral Center
0
−0.15
−0.2
−0.2
1.4
1.2
1.2
Spectral Center
Maximum Strength
a)
1.6
1.4
1.4
0.7
a
a
a
a
a
a
=
=
=
=
=
=
0.57
0.59
0.61
0.63
0.65
0.67
(a)
=
=
=
=
=
=
0.57
0.59
0.61
0.63
0.65
0.67
(a)
100
50
2.9
2.8
0
2.7
−0.2
0
0.1
0.2
0.3
β
0.4
0.5
0.6
0.7
2.6
N
−0.25X
η(x,0)
=
Cn 11,
cos(k
),
n x − φ2011
n0.2
0 Sci.,
0.1
Nat. Hazards Earth Syst.
383–399,
Fig. 13: The HONLS eqn. with nonlinear damping 0 < β <
0.5
0.6
0.7
0.75, uwww.nat-hazards-earth-syst-sci.net/11/383/2011/
a(1 + 0.01cosµx),
0.57 < a < 0.67, (a) spectral
0=
β
n=1
center
k
at
t
=
200,
(b)
max
−0.25
m
t∈[0,200] S(t), (c) number of
0
0.1
0.2
0.3
0.4
0.5 rogue waves,
0.6
0.7
and (d) time ofa =last
0.57rogue wave as a function
where kn = 2πn/L,3.1the random phases φn are uniformly
(20)
2.5
0.3
0.4
2.4
2.3
0
0.1
0.2
0.3
β
0.4
0.5
0.6
β
0.7
a = 0.59
A. Islas and C. M. Schober: Rogue waves and downshifting
395
200
200
200
200
180
180
160
160
Last Rogue
Last Rogue
DS onsetDS onset
180
160
160
140
140
140
140
120
120
120
120
Time
100
(a)(a)
100
100
Time
180
Time
Time
Last Rogue
Last Rogue
DS onset
DS onset
100
80
80
80
80
60
60
60
60
40
40
40
40
20
20
20
20
0
0
0
0
0.1
0.1 0.2
0.2 0.3
0.3 0.4
β
β
0.4 0.5
0.5 0.6
0.6 0.7
0
0.7
200
200
(a) (a)
0
0
200
200
180
180
160
160
0
0.1
0.1 0.2
0.2 0.3
0.3 0.4
β
β
0.4 0.5
0.5 0.6
160
160
140
140
140
140
120
120
120
120
Time
(b)(b)
100
100
Time
180
100
80
80
80
60
60
60
60
40
40
40
40
20
20
20
20
0
0
0
0.1
0.1 0.2
0.2 0.3
0.3 0.4
β
β
0.4 0.5
0.5 0.6
0.6 0.7
0.7
damped
HONLS
eqn.
with
Fig.Fig.
14:
The
nonlinear
damped
HONLS
eqn.
with
ǫ = ǫ==
Fig.14:
14. The
The nonlinear
nonlinear
damped
HONLS
equation
with
0.05,0
=00 and
and
<<
0.75.
TheThe
timetime
downshifting
is irreversible
0.05,Γ
0.75.
downshifting
is irre0.05,Γ
= 0=
and
0 <00β<<
<ββ0.75.
The
time
downshifting
is irre(x)
and
the
time
the
last
rogue
wave
occurs
(box)
as
a
function
of β,as
versible
(x)
and
the
time
the
last
rogue
wave
occurs
(box)
versible (x) and the time the last rogue wave occurs (box) as
u
=
a(1
+
0.01cosµx),
where
(a)
a
=
0.63
and
(b)
averaged
0
a function
u0a(1
=+
a(1
+ 0.01cosµx),
where
=over
0.63
a function
of β,ofuβ,
0.01cosµx),
where
(a) a(a)
= a0.63
0=
0.57 < a < 0.67.
(b) averaged
< 0.67.
and and
(b) averaged
overover
0.570.57
< a < a0.67.
frequency determined by the wind speed at a specified height
0
0
0
(b) (b)
100
80
0
0.7
Last Rogue
Last Rogue
DS onsetDS onset
180
Time
Time
Last Rogue
Last Rogue
DS onset
DS onset
0.6 0.7
0
0.1
0.1 0.2
0.2 0.3
0.3 0.4
β
β
0.4 0.5
0.5 0.6
0.6 0.7
0.7
Fig.
damped
HONLS
eqn.
ǫ0==0.05,Γ
= 0.005
Fig.
15:
The
damped
HONLS
eqn.
with
ǫ = 0.05,Γ
= 0.005
Fig.
15.15:
TheThe
damped
HONLS
eqn.
with
=with
0.05,
0.005
and
0 <and
The
time
downshifting
is irreversible
the time
<
< 0.75.
The
downshifting
isand
irreversible
and
0β<<0β0.75.
<β0.75.
The
timetime
downshifting
is (x)
irreversible
(x) (x)
the
last
rogue
wave
occurs
(box)
as
a
function
of
β
and
0
=
0.005,
and
the
time
the
last
rogue
wave
occurs
(box)
as
a
function
and the time the last rogue wave occurs (box) as a function of of
= a(1 + 0.01cosµx), where (a) a = 0.63 and (b) averaged over
0β
and
= 0.005,
a(1
+ 0.01cosµx),
where
= 0.63
β uand
Γ =Γ0.005,
u0 =ua(1
0.01cosµx),
where
(a) a(a)
= a0.63
0 =+
0.57 < a < 0.67.
(b) averaged
< 0.67.
and and
(b) averaged
overover
0.570.57
< a < a0.67.
5.1 Nonlinear spectral diagnostic for determining rogue
p pabove the sea surface and g is gravity. Developing storm
relation
to the
solution
of the is “close”
S(fnS(f
)/2L.
relation
of ηof toη the
solution
of the
is “close”
in spectral
the spectral
sense
an unstable
solution.
in the
sense
to antounstable
solution.
Con-Conn )/2L.TheThe
waves
dynamicsSchrödinger
are governedequation
by this spectrum
forgiven
a range
of=
nonlinear
u(x,t)
is
by
η
nonlinear
Schrödinger
equation
u(x,t)
is
given
by
η
=
sequently,
δ
measures
the
proximity
in
the
spectral
plane
sequently,
δ
measures
the
proximity
in
the
spectral
plane
to to
α. Here γ is the peakedness parameter and
ikx ikx
γ and
1parameters
√1 √
Re
iue
.
Re
iue
.
double
corresponding
instabilities
complex
double
points
and and
theiratheir
corresponding
instabilities
2kis related to the amplitude and energy content. The putative
In complex
previous
work
wepoints
proposed
nonlinear
spectral
decom2k α
and
can
be
used
as
a
criterium
for
predicting
the
strength
and
can
be
used
as
a
criterium
for
predicting
the
strength
and and
position
of
the
JONSWAP
data
and
introduced
the
splitting
region of validity for the NLS equation and its higher order
occurrence
of
rogue
waves
(Islas
&
Schober,
2005).
occurrence
of
rogue
waves
(Islas
&
Schober,
2005).
Nonlinear
spectral
diagnostic
for
determining
rogue
5.1 5.1
Nonlinear
spectral
diagnostic
for
determining
rogue
distance
between
two
simple
points,
δ(λ
,λ
)
=
|λ
−
λ
+ −
+
−|
generalizations is 2 < γ < 8 and 0.008 < α < 0.02
waves
to
measure
the
proximity
to
homoclinic
data
of
the
NLS
waves
We begin
by determining
the nonlinear
spectrum
of JONWe begin
by determining
the nonlinear
spectrum
of JONThe initial data for the surface elevation is of the form
equation
(Islas
and
Schober,
Schober,
2006). Briefly,
SWAP
initial
data
given
by (20)
for combinations
SWAP
initial
data
given
by 2005;
(20)
for
combinations
of αof=α =
(Onorato et al., 2001)
homoclinic
solutions arise
appropriate
degeneration
In previous
we proposed
a nonlinear
spectral
decompo- 0.008,0.012,0.016,0.02,
0.008,0.012,0.016,0.02,
= 1,2,4,6,8.
For
In previous
workwork
we proposed
a nonlinear
spectral
decompoandasand
γ an
= γ1,2,4,6,8.
For
eacheach
suchsuch
N
X
of
a
finite
gap
solution,
i.e.
as
δ(λ
,λ
)
→
0
the
resulting
+performed,
−performed,
sition
of
the
JONSWAP
data
and
introduced
the
splitting
dispair
(γ,α),
fifty
simulations
were
each
with
sition
of
the
JONSWAP
data
and
introduced
the
splitting
dispair
(γ,α),
fifty
simulations
were
each
with
a a
η(x,0) =
Cn cos(kn x − φn ),
(20)
double
point
is
complex.
Such
a
potential
possesses
no
tance
between
two
simple
points,
δ(λ
,λ
)
=
|λ
−
λ
|
to
different
set
of
randomly
generated
phases.
As
expected,
tance
between
two
simple
points,
δ(λ
,λ
)
=
|λ
−
λ
|
to
different
set
of
randomly
generated
phases.
As
expected,
+ +
− − + +− −
n=1
unstable
modes (simple
pointsonand
points
measure
the proximity
to homoclinic
of NLS
the NLS
equa- thelinear
the spectral
configuration
depends
onreal
thedouble
energy
α and
measure
the proximity
to homoclinic
datadata
of the
equaspectral
configuration
depends
the
energy
α and
the the
where kn = 2π n/L, the random phases φn are uniformly
are
in
general
associated
with
neutrally
stable
modes),
(Islas
& Schober,
2005;
Schober,
2006).
Briefly,
peakedness
γ. However,
the phase
information
is just
as imtiontion
(Islas
& Schober,
2005;
Schober,
2006).
Briefly,
ho- ho- peakedness
γ. However,
the phase
information
is just
as imdistributed
on (0,2π) and the spectral amplitudes Cn =
although the solution is “close” in the spectral sense to an
√ solutions
moclinic
solutions
arise
as
an
appropriate
degeneration
of
a
portant
to
the
final
determination
of
the
spectrum.
Typical
exmoclinic
arise
as
an
appropriate
degeneration
of
a
portant
to
the
final
determination
of
the
spectrum.
Typical
exS(fn )/2L. The relation of η to the solution of the nonlinear
unstable solution. Consequently, δ measures the proximity
double
amples
gap
solution,
i.e.
as
δ(λ
,λ
)
→
0
the
resulting
amples
of
the
numerically
computed
nonlinear
spectrum
for
finitefinite
gap
solution,
i.e.
as
δ(λ
,λ
)
→
0
the
resulting
double
of
the
numerically
computed
nonlinear
spectrum
for
1
ikx
+
−
+
−
Schrödinger equation u(x,t) is given by η = √ Re iue
.
in the spectral plane to complex double points and their
2k linear
point
is complex.
a potential
possesses
no
unsta- JONSWAP
JONSWAP
initial
= 0.016)
different
point
is complex.
SuchSuch
a potential
possesses
no linear
unstainitial
datadata
(γ =(γ4,=α4,
=α0.016)
withwith
two two
different
ble modes
(simple
points
double
points
aregeneral
in general realizations
realizations
of random
the random
phases
are shown
in Figs.
ble modes
(simple
points
and and
real real
double
points
are in
of the
phases
are shown
in Figs.
16a 16a
and and
www.nat-hazards-earth-syst-sci.net/11/383/2011/
Nat.
EarthisSyst.
Sci., 11,
383–399,
2011
associated
neutrally
stable
modes),
although
the solution 17a.17a.
Since
theHazards
NLS
spectrum
is symmetric
with
respect
to the
Since
the
NLS
spectrum
symmetric
with
respect
to the
associated
withwith
neutrally
stable
modes),
although
the solution
396
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
A. Islas and C. M. Schober: Rogue waves and downshifting
.. ..
. .
. .. .
λ+
λ+
λ_
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
λ_
. .
0
0
−0.5 −0.4
0
0.1
−0.5 −0.3
−0.4 −0.2
−0.3 −0.1
−0.2 −0.1
0
2
(a) (a)
. .
0.2
0.1
0.3
0.2
0.4
0.3 0.5
0.4
0.5
λ+
. .
.. ..
.. ..
. .
λ+
λ_
2
λ_
. .
0
0
−0.5 −0.4
0
0.10
−0.5 −0.3
−0.4 −0.2
−0.3 −0.1
−0.2 −0.1
2
(a) (a)
0.2
0.1
0.3
0.2
(b) (b)
1.5
0.5
0.5
100
UMAX & 2.2*SH
1
1
0
1.5
UMAX & 2.2*SH
UMAX & 2.2*SH
UMAX & 2.2*SH
1.5
1
20
10
30
20
40
30
50
40
60
50
7060
8070
9080
100
90
0.5
2
(b) (b)
1.5
0.4
0.3 0.5
0.4
0.5
100
1
0
0.5
100
2010
3020
4030
5040
6050
7060
8070
9080
10090
100
Fig.
17.(a)
(a)
Spectrum
and(b)
(b)
evolution
of
from
Fig. 16:
(a)
Spectrum
andand
(b)
ofUU
to toFig.
17:
Spectrum
and
evolution
of UUmax
from
Fig.
16.
(a) Spectrum
(b)evolution
evolution
of
to “near”
Fig.
16:
(a)
Spectrum
and
(b) evolution
of“near”
U“near”
Fig.
17:
(a)
Spectrum
and
(b) evolution
of “far”
U“far”
“far” from
max
max
max
max
max
homoclinic
data.
homoclinic
data.
homoclinic
data.
homoclinic
data.
homoclinic data.
homoclinic data.
and
canupper
be used
a criterium
for curve.curve.
real axis
only
spectrum
in thein
upper
half
complex
λ-plane
that when
the correspond
nonlinear
spectrum
is
realcorresponding
axisthe
only
theinstabilities
spectrum
the
halfascomplex
λ-plane
Fig.shows
16b shows
thatmay
when
the nonlinear
spectrum
is
WeFig.
find 16b
that
JONSWAP
data
to “semipredicting
the
strength
and
occurrence
of
rogue
waves
(Islas
is displayed.
near
homoclinic
data,
U
exceeds
2.2H
(a
rogue
wave
is displayed.
max Umax exceeds
nearsolutions,
homoclinic
wave
stable”
i.e. data,
JONSWAP
data canS2.2H
be viewed
as
S (a rogue
and Schober, 2005).
develops
at t ≈of
40).
that
theunstable
nonlinear
perturbations
N-phase
solutions
onewhen
or more
We find
that
JONSWAP
data may
correspond
to “semidevelops
at
t ≈Fig.
40).17b
Fig.shows
17bwith
shows
that
when
the nonlinear
WeWe
find
that
JONSWAP
data
may
correspond
to
“semibegin by determining the nonlinear spectrum of
modes
(compare
Fig.
16ahomoclinic
with data,
Fig. 1a,
theUisspectrum
of
is far is
from
homoclinic
U
significantly
stable”
solutions,
i.e. JONSWAP
data can becan
viewed
as per-as per-spectrum
max
spectrum
far
from
data,
is significantly
stable”
solutions,
i.e.data
JONSWAP
be viewed
max
JONSWAP
initial
given by data
(20) for combinations
of
an
unstable
N-phase
solution).
In
Fig.
16a
the
splitting
below
2.2H
and
a
rogue
wave
does
not
develop.
As
a
result
turbations
of
N
-phase
solutions
with
one
or
more
unstable
S
below
2.2H
and
a
rogue
wave
does
not
develop.
As
a result
turbations
of N -phase solutions
one or more
S
α = 0.008,0.012,0.016,0.02,
andwith
γ = 1,2,4,6,8.
Forunstable
each
distance
δ(λ
,λ
)
≈
0.07,
while
in
Fig.
17a
δ(λ
,λ
)
≈
+
−
+
−
we
can
correlate
the
occurrence
of
rogue
waves
characterized
modesmodes
(compare
Fig.
16a
with
Fig.
1a,
the
spectrum
of
an
we can correlate the occurrence of rogue waves characterized
(compare
16a with Fig.
the spectrum
of an
such pair
(γ , α), Fig.
fifty simulations
were1a,
performed,
each with
for
fixed α with
and the
γ asproximity
randomlysoJONSWAP
spectrum
to are
homoclinic
unstable
-phase
solution).
In Fig.
dis- dis-by0.2.
byThus,
JONSWAP
spectrum
withthe
the phases
proximity
to homoclinic sounstable
N -phase
solution).
In 16a
Fig. the
16asplitting
theAssplitting
aN
different
set
of randomly
generated
phases.
expected,
varied,
the
spectral
distance
δ
of
typical
JONSWAP
data from
lutions
of
the
NLS
equation.
tance tance
δ(λ
)≈
0.07,
whilewhile
independs
Fig.in 17a
,λ
)≈
0.2.
lutions
of
the
NLS
equation.
the+ ,λ
spectral
configuration
on δ(λ
the
energy
α
the
−+
−+
δ(λ
,λ−
) ≈ 0.07,
Fig.
17a+
δ(λ
,λand
)
≈
0.2.
−
homoclinic data varies significantly and can be quite “near”
Thus,Thus,
for
fixed
α and
as the
are information
randomly
varied,
peakedness
γ .αγHowever,
the
is just
as
for fixed
and
γ asphases
the phase
phases
are randomly
varied,
as in Fig. 16a, or “far” from homoclinic data as in Fig. 17a.
important
to
the
final
determination
of
the
spectrum.
Typical
the spectral
distance
δ
of
typical
JONSWAP
data
from
hoexperiments
with the
the spectral distance δ of typical JONSWAP data from ho-5.2 JONSWAP
5.2 JONSWAP
experiments
withNLS
the and
NLSHONLS
and HONLS
The most striking feature is that irrespective of the values
examples
of varies
the
numerically
computed
nonlinear
spectrum
moclinic
data varies
significantly
and
can
quite
“near”
as as
equations
moclinic
data
significantly
andbecan
be quite
“near”
equations
of α and γ , in simulations of the NLS equation (10) extreme
for JONSWAP
initial
(γ data
= 4, as
α in
= Fig.
0.016)
with two
in Fig.in16a,
or
“far”
from
homoclinic
Fig.
16a,
or “far”
fromdata
homoclinic
data
as in17a.
Fig.
17a.
waves develop for JONSWAP initial data that is “near”
different realizations of the random phases are shown in
The most
striking
feature
is thatisirrespective
of theofvalues
The
results
of hundreds
of simulations
of the
and and
The
most
striking
feature
that
irrespective
the
values
The
results
of
of simulations
ofNLS
the isNLS
NLS
homoclinic
data,hundreds
whereas
the
JONSWAP
data
that
Figs. 16a and 17a. Since the NLS spectrum is symmetric
of α and
in simulations
of theofNLS
equation
(10) extreme
HONLS
equations
consistently
show
thatdoes
proximity
to ho-to hoof with
αγ,and
γ,
in
simulations
the
NLS
equation
(10)
extreme
HONLS
equations
consistently
show
that
proximity
“far”
from
NLS
homoclinic
data
typically
not
generate
respect to the real axis only the spectrum in the upper
waveswaves
develop
for JONSWAP
initialinitial
data that
“near”
NLS NLSmoclinic
data
isdata
a crucial
indicator
of rogue
wave wave
events.events.
extreme
waves.
Figures
and 17b
show
theofcorresponding
develop
for JONSWAP
dataisthat
is “near”
moclinic
is a16b
crucial
indicator
rogue
half complex
λ-plane
is displayed.
homoclinic
data,
whereas
the
JONSWAP
data
that
is
“far”
Figure
18
provides
a
synthesis
of
250
random
simulaevolution
of
U
(U
=
max
|u(x,t)|),
obtained
max a synthesis
x∈[0,L] of 250 random simulahomoclinic data, whereas the JONSWAP data that is “far”
Figure 18max
provides
withtions
equation.
Umax
isand
given
by(b-c)
the
curve
from from
NLS NLS
homoclinic
data typically
does not
ex- ex-tions
ofthe(a)NLS
NLS
equation
ofand
the HONLS
homoclinic
data typically
doesgenerate
not generate
ofthe
(a)
the NLS
equation
of solid
(b-c)
the HONLS
and
as
a
reference
the
threshold
for
a
rogue
wave,
2.2
tremetreme
waves.
Figs. 16b
the corresponding
evo- evo-equation
(β = Γ
and ǫ and
= 0.05,
respecwaves.
Figs.and
16b17b
andshow
17b show
the corresponding
equation
(β==0)Γ for
= 0)ǫ =
for0.025
ǫ = 0.025
ǫ = times
0.05,
respeclutionlution
of Umax
(U
=
max
|u(x,t)|),
obtained
with
tively,
using
JONSWAP
initial
data
for
different
(γ,α)
pairs
max
of Umax (Umax =x∈[0,L]
maxx∈[0,L]|u(x,t)|), obtained with
tively, using JONSWAP initial data for different (γ,α) pairs
the NLS
equation.
UmaxSyst.
given
by 383–399,
thebysolid
curve curve
and andwith γwith
= 1,2,4,6,8
and αand
= 0.008,0.012,0.016,0.02
and ranHazards
Earth
11,
www.nat-hazards-earth-syst-sci.net/11/383/2011/
theNat.
NLS
equation.
Uis
is given
the2011
solid
γ=
1,2,4,6,8
α = 0.008,0.012,0.016,0.02
and ranmaxSci.,
as a reference
the threshold
for a for
rogue
wave,wave,
2.2 times
the thedomlydomly
generated
phases.
Our considerations
are restricted
as a reference
the threshold
a rogue
2.2 times
generated
phases.
Our considerations
are restricted
the significant
wave wave
height,
2.2HS2.2H
, is given
by the dashed
to semi-stable
N -phase
solutions
near tonear
unstable
solutions
the significant
height,
to semi-stable
N -phase
solutions
to unstable
solutions
S , is given by the dashed
A. Islas and C. M. Schober: Rogue waves and downshifting
4
4
4
(a)
(a)
(a)
MAXIMUM
STRENGTH
MAXIMUM
STRENGTH
MAXIMUM
STRENGTH
3.5
3.5
3.5
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.50
1.50
40
4
4
0.2
0.2DISTANCE (ε = 0)
SPLITTING
0.2DISTANCE (ε = 0)
SPLITTING
SPLITTING DISTANCE (ε = 0)
0.4
0.4
0.4
(b)
(b)
(b)
3.5
MAXIMUM
STRENGTH
MAXIMUM
STRENGTH
MAXIMUM STRENGTH
3.5
3.5
3
3
3
2.5
2.5
2.5
2
2
2
1.5
0
1.5
1.50
40
4
0.2
SPLITTING DISTANCE (ε = 0.025)
0.2
SPLITTING0.2
DISTANCE (ε = 0.025)
SPLITTING DISTANCE (ε = 0.025)
4
0.4
0.4
0.4
(c)
(c)
(c)
3.5
3.5
MAXIMUM STRENGTH
MAXIMUM
STRENGTH
MAXIMUM
STRENGTH
3.5
3
3
3
2.5
2.5
2.5
2
2
2
1.5
1.50
0
1.5
0
0.2
SPLITTING0.2
DISTANCE (ε = 0.05)
SPLITTING DISTANCE (ε = 0.05)
0.2
SPLITTING DISTANCE (ε = 0.05)
0.4
0.4
0.4
Fig.
18: Maximum
strength versus
versus the
the splitting
splitting distance
distance
Fig.
Maximum
strength
Fig.Fig.18:
18:
Maximum
strength
the
splitting
distance
18.)Maximum
strength
versusversus
the splitting
distance
δ(λequation
,λ− )
+
δ(λ
,λ
for
a)
the
NLS
equation
and
the
HONLS
+ ,λ− ) for a) the NLS equation and the HONLS equation
δ(λ
+ ,λ
− )the
for
(a)
NLS
equation
and
the
HONLS
equation
with
(b)
δ(λ
for
a)
the
NLS
equation
and
the
HONLS
equation
+b) −
with
ǫǫ =
0.025,
and
c) ǫǫ =
= 0.05
0.05 ..
with
b)
=
0.025,
and
c)
=0.025,
and
(c)
=
0.05.
with b) ǫ = 0.025, and c) ǫ = 0.05 .
the the significant wave height, 2.2Hs , is given by the
dashed curve. Figure 16b shows that when the nonlinear
spectrum is near homoclinic data, Umax exceeds 2.2Hs (a
rogue wave develops at t ≈ 40). Figure 17b shows that
when the nonlinear spectrum is far from homoclinic data,
Umax is significantly below 2.2Hs and a rogue wave does
not develop. As a result we can correlate the occurrence of
rogue waves characterized by JONSWAP spectrum with the
proximity to homoclinic solutions of the NLS equation.
www.nat-hazards-earth-syst-sci.net/11/383/2011/
mum strength,
attained distance
in time during
t∈[0,20]
simulation,
as aamax
function
ofS(t),
the splitting
splitting
δ(λ+ ,λ
,λone
− ).
simulation,
as
function
of
the
distance
δ(λ
+ ,λ− ).
simulation,
as
a
function
of
the
splitting
distance
δ(λ
+
− ).
A
horizontal
line
at
U
/H
=
2.2
indicates
the
reference
max
S
A
horizontal
line
at
U
2.2
indicates
the
reference
max /H
S=
A
horizontal
line
at
U
/H
=
2.2
indicates
the
reference
maxformation.
S
strength for
for rogue
rogue wave
wave
We identify
identify two
two criticritistrength
formation.
We
strength
for
rogue
wave
formation.
We
identify
two
397
cal values
values δδ1 (ǫ)
(ǫ) and δδ2 (ǫ)
(ǫ) that clearly
clearly show
show that
that (i)
(i) if
if δδ criti< δδ1
cal
1 (ǫ) and
2 (ǫ) that
cal
values
δ
and
δ
that
clearly
show
that
(i)
if δis<
<exδ11
1
2
(near
homoclinic
data)
the
likelihood
of
a
rogue
wave
(near
homoclinic
data)
the
likelihood
of
a
rogue
wave
is
ex(near
homoclinic
thewith
ofand
a rogue
is ex5.2 tremely
JONSWAP
NLSlikelihood
HONLS
high;experiments
(ii) data)
if δδ1 <
<
δ likelihood
< δδthe
ofwave
obtaining
2 , the
tremely
high;
(ii)
if
<
likelihood
of
obtaining
1 <δ
2 ,, the
tremely
high;
(ii)
if
δ
δ
<
δ
the
likelihood
of
obtaining
2
equations
rogue
waves decreases
decreases1 as
as δδ increases
increases
and, (iii)
(iii) if
if δδ >
> δ2 the
rogue
waves
and,
rogue
waves
decreases
as
δ
increases
and,
(iii) if small.
δ > δδ22 the
the
likelihood
of
a
rogue
wave
occurring
is
extremely
likelihood
of
aa rogue
wave
occurring
is
extremely
small.
likelihood
of
rogue
wave
occurring
is
extremely
small.
The results
of simulations
of the
NLS and
As α
α of
andhundreds
γ are
are varied
varied
the behavior
behavior
is robust.
robust.
The maximaxiAs
and
γ
the
is
The
HONLS
equations
consistently
show
that
proximity
to
As
α
and
γ
are
varied
the
behavior
is
robust.
The
maximum wave
wave strength
strength and
and the
the occurrence
occurrence of
of rogue
rogue waves
waves
are
mum
homoclinic
data is
a crucialand
indicator
of rogue wave
events.waves are
mum
wave
strength
the
occurrence
of
rogue
are
well
predicted
by
the
proximity
to
homoclinic
solutions.
The
well
the
proximity
homoclinic
solutions.
The
18predicted
provides by
a synthesis
of 250to
random
simulations
Figure
well
predicted
by
the
proximity
to
homoclinic
solutions.
The
individual
plots
of and
the of
strength
vs.HONLS
the splitting
splitting
distance δδ
individual
of
the
strength
vs.
the
of (a)
the NLS plots
equation
(b–c) the
equationdistance
individual
plots
of
the
strength
vs.
the
splitting
distance
for
particular
pairs
(γ,α)
are
qualitatively
theusing
same
regard-δ
(β =for
0=
0) for =pairs
0.025(γ,α)
and are
= 0.05,
respectively,
particular
qualitatively
the
same
regardfor particular
pairs
(γ,α)
are qualitatively
the same
regardless
of the
the
pairdata
chosen.
Comparing
Figs.
18a-b
we
find
that
JONSWAP
initial
for different
(γ , α)Figs.
pairs 18a-b
with γwe
= find
less
of
pair
chosen.
Comparing
that
less
of
the
pair
chosen.
Comparing
Figs.
18a-b
we
find
that
enhanced
focusing
and
increased
wave
strength
occur
in
the
1,2,4,6,8
and
α
=
0.008,0.012,0.016,0.02
and
randomly
enhanced
focusing
and
increased
wave
strength
occur
in
the
enhanced
focusing
and
increased
wave
strength
occur
in
the
generated
phases.
Our
considerations
are
restricted
to
chaotic
HONLS
evolution
as
compared
with
the
NLS
evochaotic
HONLS
evolution
as
compared
with
the
NLS
evochaotic Figure
HONLS18solutions
evolution
as as
compared
with
the strength
NLS evo-ǫ
semi-stable
N-phase
near
to the
unstable
solutions
lution.
shows that
that
perturbation
lution.
Figure
18
shows
as
the
perturbation
strength
ǫ
lution.
Figure
18
shows
that
as
the
perturbation
strength
of the
NLS
with
one
UM.
Each
circle
represents
the
increases, the
the maximum
maximum wave
wave strength
strength and
and the
the likelihood
likelihood of
ofǫ
increases,
increases,
theoccuring
maximum
wave
strength
the
likelihood of
maximum
strength,
maxt∈[0,20]
attained
inand
timeincreases.
during
rogue
waves
in aaS(t),
given
simulation
waves
occuring
in
given
simulation
increases.
one rogue
simulation,
as
a
function
of
the
splitting
distance
rogue waves occuring in a given simulation increases.
δ(λ+ ,λ− ). A horizontal line at Umax /Hs = 2.2 indicates the
5.3 JONSWAP
JONSWAP experiments
experiments with
with the
the damped HONLS
HONLS
5.3
reference
strength for rogue
wave formation.
identify HONLS
5.3 equations
JONSWAP
experiments
with theWedamped
damped
equations
two critical
values δ1 () and δ2 () that clearly show that (i)
equations
if δ < δ1 (near homoclinic data) the likelihood of a rogue
Inisthis
this
sectionhigh;
we show
show
that,
in
of damping,
damping,
the
wave
extremely
(ii) if that,
δ1 < in
δ <the
δ ,presence
the likelihood
of
In
we
of
the
In this section
section
we show
that,
in the
the2 presence
presence
of damping,
the
nonlinear
spectrum
and
the
proximity
to
homoclinic
data
is
obtaining
rogue
waves
decreases
as
δ
increases
and,
(iii)
if
nonlinear
spectrum
and
the
proximity
to
homoclinic
data
is
nonlinear
spectrum
and
the
proximity
to
homoclinic
data
is
likewise
an
important
indicator
of
the
maximum
strength
in
δ > likewise
δ2 the likelihood
of
a
rogue
wave
occurring
is
extremely
an
important
indicator
of
the
maximum
strength
in
likewise
an
important
indicator
of
the
maximum
strength
in
small.
time and
and likelihood
likelihood of
of rogue
rogue waves.
waves.
time
time
and
likelihood
ofthe
rogue
waves. iswave
As
α
and
γ
are
varied
the
behavior
robust.
The vs.
Figure
19
shows
maximum
strength
vs. the
the
Figure
19
shows
the
maximum
wave
strength
Figure
19
shows
the
maximum
wave
strength
vs. the
maximum
wave
strengthδ(λ
and
the
occurrence
of
rogue
waves
splitting
distance
,λ
)
for
250
random
simulations
of
+ − ) for 250 random simulations of
splitting
distance
δ(λ
+ ,λ
− ) for 250 random simulations of
splitting
distance
δ(λ
,λ
are well
predicted
by
the
proximity
to
homoclinic
solutions.
+
−
the damped
damped HONLS
HONLS equation
equation (ǫ
(ǫ =
= 0.05),
0.05), with
with either
either (a)
(a)
the
damped
HONLS
equation
(ǫ =
0.05),
with
either
(a)
Thethe
individual
plots
of the
strength
vs.
the
splitting
distance
linear
damping,
β
=
0,Γ
=
0.025,
or
with
(b)
nonlinear
linear
damping,
0,Γ
=
0.025,
or
with
(b)
nonlinear
δ for
particular
pairs β
(γ=
, α)
are
qualitatively
the same
linear
damping,
β
=
0,Γ
=
0.025,
or
with
(b)
nonlinear
damping β
β=
= 0.5,Γ
0.5,Γ =
= 0.
0. As
As before,
before, JONSWAP
JONSWAP initial
initial data
data
damping
18a–b we
regardless
of the
pair
chosen.
Comparing
Fig.
damping
β
=
0.5,Γ
=
0.
As
before,
JONSWAP
initial
was
used
for different
different
(γ,α)
pairs with
with
γ=
=
1,2,4,6,8 data
and
used
for
(γ,α)
pairs
γ
1,2,4,6,8
and
findwas
that
enhanced
focusing
and
increased
wave
strength
was
used for different (γ,α)and
pairs
with γgenerated
= 1,2,4,6,8
and
α
=
0.008,0.012,0.016,0.02
randomly
phases.
occur
in 0.008,0.012,0.016,0.02
the chaotic HONLS evolution
asrandomly
comparedgenerated
with the phases.
α
=
and
α
=
0.008,0.012,0.016,0.02
and
randomly
generated
phases.
Each
circle represents
represents
the
maximum
strength
attained during
during
shows
that as the
perturbation
NLSEach
evolution.
Figure 18 the
circle
maximum
Eachsimulation,
circle represents
the
maximum strength
strength attained
attained during
one
0
<
t
<
20.
strength
increases, 0the
wave strength and the
one
ttmaximum
<
20.
oneInsimulation,
simulation,
0is<
<evident
<occuring
20. thatinthe
likelihood
of
rogue
waves
a given
simulation
Fig.
19
it
strength
and likelihood
likelihood
In Fig.
19
the
and
Fig.waves
19 it
it is
is evident
evidentinthat
that
the strength
strength
andislikelihood
increases.
of In
rogue
occuring
a given
simulation
typically
of
waves
occuring
in
simulation
is
of rogue
roguewhen
waves
occuring
in aa given
given
simulation
is typically
typically
smaller
damping
is
present
(compare
to
results
of the
the
smaller
when
damping
is
present
(compare
to
results
of
smaller when
damping
is
present
(compare
to
results
ofsuch
the
HONLS
equation
(Fig.
18c).
A
cutoff
vaue
of
δ
exists
HONLS
equation
(Fig.
18c).
A
cutoff
vaue
of
δδ exists
such
HONLS
equation
(Fig.
18c).
A
cutoff
vaue
of
exists
such
5.3 that
JONSWAP
experiments
with
the
damped
HONLS
rogue waves
waves typically do
do not
not occur
occur if
if δδ >
> δδcutof f .. We
We
that
rogue
cutof f . We
equations
that
rogue δwaves typically
typically
do not
occur
if
δthe
> δrogue
cutof fwaves
determine
by
requiring
that
95%
of
cutof f by requiring that 95% of the rogue waves
determine
δδcutof
f by requiring that 95% of the rogue waves
determine
cutof
f
occur
for
δ
<
δcutof
example, the
the cutoff
cutoff values
values for
for the
the
f . For example,
occur
for
δ
<
δ
cutof
f .. For
In this
section
we
show
that,
in theexample,
presence
of0.02)
damping,
the HONLS
occur
for
δ
<
δ
For
the
cutoff
values
for the
cutof
f nonlinear
linear
(Γ
=
0.02)
and
(ǫβ
=
damped
linear
(Γ
0.02)
and
=
0.02)
nonlinear
and
thenonlinear
proximity (ǫβ
to homoclinic
data is HONLS
damped
linear spectrum
(Γ =
=
and
nonlinear
(ǫβ
=respectively,
0.02) damped
damped
HONLS
damped
equation
are0.02)
δcutof
≈ 0.16,0.17,
0.16,0.17,
which
are
f
damped
equation
are
δ
≈
respectively,
are
likewise
an important
indicator
of the maximum
strength inwhich
cutof
f ≈ 0.16,0.17,
equation
are
δ
respectively,
which
are
cutof
f
less
than
the cutoff
cutoff
value
for the
the undamped
undamped HONLS
HONLS equation
equation
than
the
value
for
timeless
and
likelihood
of
rogue
waves.
undamped
less than
the cutoff≈value
the undamped
HONLSthat
equation
undamped
where,
δcutof
0.2. for
Significantly,
this implies
implies
when
f
undamped
where,
0.2.
Significantly,
this
when
Figure
19δδcutof
shows
maximum
wave strength
vs. thethat
f the≈
where,
≈
0.2.
Significantly,
this
implies
that
when
cutof
f
damping
is
present
the
JONSWAP
initial
data
must
be
closer
damping
is
present
the
initial
data
must
be
closer
splitting
distance
δ(λ+ ,λ
for 250 random
simulations
− ) JONSWAP
damping
is
present
the
JONSWAP
initial
data
must
be
closer
to homoclinic
homoclinic
data and
and
instabilities
to obtain
obtain
rogue
waves.
of the
damped HONLS
equation
( = 0.05),
withrogue
eitherwaves.
to
data
instabilities
to
to Figure
homoclinic
data and
instabilities
to obtain
rogue
waves.
damped
damped
20
shows
δ
as
a
function
of
the
damping
(a) linear
damping,
β
=
0,
0
=
0.025,
or
with
(b)
nonlinear
cutof
f
damped
Figure
20
shows
δδcutof
aa function
of
the
damping
f as
Figure
20
shows
as
function
of
the
damping
cutof
fǫβ JONSWAP
parameters
Γ
(circles)
and
(boxes).
For
each
of
the ten
ten
damping
β
=
0.5,0
=
0.
As
before,
initial
data
parameters
Γ
(circles)
and
ǫβ
(boxes).
For
each
of
the
parameters
Γ
(circles)
and
ǫβ
(boxes).
For
each
of
the
ten
wasvalues
used
for
different
(γ
,
α)
pairs
with
γ
=
1,2,4,6,8
values of
of Γ
Γ and
and of
of ǫβ,
ǫβ, 00 <
< Γ,ǫβ
Γ,ǫβ <
< 0.04,
0.04, 250
250 simulations
simulations
of Γ and of ǫβ, 0 <and
Γ,ǫβ
< 0.04,generated
250 simulations
and values
α = 0.008,0.012,0.016,0.02
randomly
phases. Each circle represents the maximum strength
attained during one simulation, 0 < t < 20.
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
the
da
δcu
inc
of
damped
HONLS
equation
carried
398
A. Islas
and
C. M. Schober:
waveswere
and downshifting
of the
the
damped
HONLSRogue
equation
were
carried out.
out. We
We find
find
damped
ity
damped is generally decreasing as the strength of the
that
that δδcutof
cutof f
f is generally decreasing as the strength of the
dam
damping
damping increases.
increases. The
The cutoff
cutoff value
value in
in δδ for
for the
the nonlinnonlin(a)
early
(a)
early damped
damped case
case is
is not
not monotonically
monotonically decreasing
decreasing as
as atypatypical
cases
exist
where
a
small
increase
in
damping
may
ical cases exist where a small increase in damping may be
be
Re
offset
to
changes
in
the
offset by
by aa coalescence
coalescence of
of the
the modes
modes due
due
to
changes
in
the
damped
damped indicates that
focusing
Ab
focusing times.
times. The
The overall
overall decay
decay in
in δδcutof
cutof ff indicates that
for
rogue
waves
to
occur
the
JONSWAP
initial
data
must
lie
for rogue waves to occur the JONSWAP initial data must lie (
in
in aa shrinking
shrinking neighborhood
neighborhood of
of the
the homoclinic
homoclinic data.
data. Thus
Thus R
the
proximity
to
instabilities
and
homoclinic
data
is
more
the proximity to instabilities and homoclinic data is more eses-Ab
sential
sential for
for the
the development
development of
of rogue
rogue waves
waves when
when damping
damping is
is (
d
present.
present.
Ab
The
The observations
observations from
from the
the nonlinear
nonlinear spectral
spectral diagnostic
diagnostic i
complement
the
earlier
results
characterizing
the
nonlinear
complement the earlierΓ or
results
characterizing
the
nonlinear
εβ
Cal
damped
evolution.
In
section
4,
the
numerical
simulations
damped evolution. In section 4, the numerical simulations (
(b)
damped damped
start
very
close
to
data
(for
initial
(b)
20.The
The
damped
equation
(
=(ǫ
0.05):
δcutoff δdata
as a (19)
start
very
closeHONLS
to homolinic
homolinic
data
(for
initial
data
(19) the
the n
Fig.Fig.
20:
damped
HONLS
equation
=−6
0.05):
cutof f
splitting
distance
is
on
the
order
of
10
,
an
effective
rogue
−6
function
of 0 (circles)
andisβon
(boxes).
splitting
distance
the
order
of
10
,
an
effective
rogue
Cal
as a function of Γ (circles) and ǫβ (boxes)
wave
regime).
The
first
pair
of
rogue
waves
were
not
sigwave regime). The first pair of rogue waves were not sig- l
nificantly
Co
nificantly altered
altered by
by the
the damping.
damping. However
However after
after permanent
permanent
downshifting
occured
any
further
rogue
waves
were
inhiboffset
by
a
coalescence
of
the
modes
due
to
changes
in
the
downshifting occured any further
rogue waves were inhib- f
damped
ited.
Here
we
are
to
initial
focusing
decay in δcutoff
indicates that
ited. times.
Here The
we overall
are generalizing
generalizing
to JONSWAP
JONSWAP
initial data
data
which
are
not
necessarily
close
to
homoclinic
data.
We
for rogue
waves
to
occur
the
JONSWAP
initial
data
must
lie
which are not necessarily close to homoclinic data. We are
are
in amonitoring
shrinking neighborhood
of the
homoclinic
data.
the
strength
in
as
aa function
monitoring
the maximum
maximum
strength
in time
time
as Thus
function of
of
the δproximity
to instabilitiesissue
and homoclinic
data is more of rogue
δ which
which is
is aa separate
separate issue from
from the
the total
total number
number of rogue
essential
for the
development of rogue waves when damping
waves
waves in
in aa given
given simulation.
simulation.
is present.
Finally
we
remark
that
we
examined
rogue
Finally
we
remark
thatnonlinear
we recently
recently
examined
rogue waves
waves
The observations from the
spectral
diagnostic
and
downshifting
in
the
damped
Dysthe
equation
(Islas
&
and
downshifting
in
the
damped
Dysthe
equation
(Islas
&
complement the earlier results characterizing the nonlinear
Schober,
2010).
While
the
Dysthe
equation
does
not
conFig.
19:
Maximum
strength
vs.
splitting
distance
δ(λ
,λ
)
+
−
Schober,
2010).In While
equation
does not conevolution.
Sect. 4,thetheDysthe
numerical
simulations
Fig.Fig.
19:19.Maximum
strength
splitting
distance
,λ− ) damped
Maximum strength
vs. vs.
splitting
distance
δ(λ+ ,λ−δ(λ
) for+the
we
obtained
qualitatively
similar
forHONLS
the HONLS
HONLS
equation
(ǫ =
=either
0.05)
startserve
very momentum,
close to homolinic
(for initial
data (19)
serve
momentum,
we have
havedata
obtained
qualitatively
similar rereequation equation
( = 0.05) with
(a) with
linear either
damping,(a)
β =linear
0,
for
the
(ǫ
0.05)
with
either
(a)
linear
−6
sults.
For
the
nonlinearly
damped
Dysthe
equation,
irredamping,
β
=
0,
Γ
=
0.025,
or
(b)
nonlinear
damping
β
=
0
=
0.025,
or
(b)
nonlinear
damping
β
=
0.5,
0
=
0.
the
splitting
distance
is
on
the
order
of
10
,
an
effective
sults.
For
the
nonlinearly
damped
Dysthe
equation,
irredamping, β = 0, Γ = 0.025, or (b) nonlinear damping β =
versible
all
β,
rogue
wave downshifting
regime).
The also
first occurs
pair of for
rogue
waves of
were
0.5, Γ
Γ=
= 0.
0.
versible
downshifting
also
occurs
for
all values
values
of
β, although
although
0.5,
a
slightly
longer
timescale.
Rogue
waves
do
not
not on
significantly
altered
by
the
damping.
However
after
on a slightly longer timescale. Rogue waves do not occur
occur in
in
In Fig. 19 it is evident that the strength and likelihood
permanent
downshifting
occured
any further
rogue wavesSimilarly,
the
time
series
after
the
downshifting
is
permanent.
the
time
series
after
the
downshifting
is
permanent.
Similarly,
of rogue waves occuring in a given simulation is typically
wereδ damped
inhibited.
Here we
are generalizing
to JONSWAP
damped
decreases
as
strength
of
f generally
δcutof
generally
decreases
as the
the
strength
of the
the damping
damping
cutof
f which
smaller when damping is present (compare to results of the
initial
data
are
not
necessarily
close
to
homoclinic
increases
suggesting
the
heightened
importance
of
proximincreases
suggesting
heightened
importance
HONLS equation, Fig. 18c). A cutoff vaue of δ exists such
data.ity
We
are
monitoring
thethe
maximum
strength
in
time waves
asof
a proximto
homoclinic
data
when
obtaining
rogue
ity toofhomoclinic
data when
rogue
waves in
in the
the
that rogue waves typically do not occur if δ > δcutoff . We
function
δ which is a separate
issueobtaining
from the total
number
damped
Dysthe
regime.
determine δcutoff by requiring that 95% of the rogue waves
damped
Dysthe
regime.
of rogue waves in a given simulation.
4
4
0.25
Γ
β
0.2
cutoff
3
3
δ
MAXIMUM
STRENGTH
MAXIMUM
STRENGTH
3.5
3.5
2.5
2.5
0.15
2
2
1.5
1.5 0
0
4
0.1
0.2
0.2
SPLITTING
DISTANCE
SPLITTING DISTANCE
0.4
0.4
0.2
SPLITTING DISTANCE
0.2
SPLITTING DISTANCE
0.4
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
4
3.5
MAXIMUM STRENGTH
MAXIMUM STRENGTH
3.5
3
3
2.5
2.5
2
2
1.5
1.5
0
0
0.25
0.25
0.4
Γ
δδ
cutoff
cutoff
occur for δ < δcutoff . For example, the cutoff values forβΓβ the
linear (0 = 0.02) and nonlinear (β = 0.02) damped HONLS
damped
equation are δcutoff ≈ 0.16,0.17, respectively, which are
0.2
less0.2
than the cutoff value for the undamped HONLS equation
undamped
≈ 0.2. Significantly, this implies that when
where, δcutoff
damping is present the JONSWAP initial data must be
closer to homoclinic data and instabilities to obtain rogue
0.15
0.15
waves.
damped
Figure 20 shows δcutoff as a function of the damping
parameters 0 (circles) and β (boxes). For each of the ten
values
of 0 and of β, 0 < 0, β < 0.04, 250 simulations
0.1
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.1 0
of the
HONLS
equation
We find
0 damped
0.005
0.01
0.015
0.025carried
0.03 out.
0.035
0.04
Γ 0.02
or εβwere
Γ or εβ
damped
that δcutoff is generally decreasing as the strength of the
damped
increases.
cutoff value
in δ (ǫ
for the
nonlinearly
damped
Fig.damping
20: The
The
dampedThe
HONLS
equation
f
Fig.
20:
damped
HONLS
equation
(ǫ =
= 0.05):
0.05): δδcutof
cutof f
damped
case
is (circles)
not monotonically
decreasing as atypical
as
a
function
of
Γ
and
ǫβ
(boxes)
as acases
function
of
Γ
(circles)
and
ǫβ
(boxes)
exist where a small increase in damping may be
Nat. Hazards Earth Syst. Sci., 11, 383–399, 2011
Finally we remark that we recently examined rogue waves
and References
downshifting in the damped Dysthe equation (Islas
and References
Schober, 2010). While the Dysthe equation does
not Ablowitz,
conserve momentum,
we have
obtained qualitatively
M.J.,
Hammack,
J.,
Henderson,
D., Schober,
Ablowitz,
M.J.,
J.,
Henderson,
Schober, C.M.
C.M.
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Periodic
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irreversible
downshifting
also occurs for all values of β,
Rev.
Lett.
84:887–890.
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84:887–890.
although
on
a
slightly
longer
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Rogue waves
do not
Ablowitz,
M.J.,
Hammack,
J.,
D.,
Ablowitz,
M.J.,
Hammack,
J., Henderson,
Henderson,
D., Schober,
Schober, C.M.
C.M.
occur
in
the
time
series
after
the
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is
permanent.
(2001).
Long
time
dynamics
of
the
modulational
damped
(2001).
Long time dynamics of the modulational instability
instability of
of
Similarly,
δcutoff
generally
decreases
as the strength of the
deep
water
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D
152-153:416–433.
deep
water
waves.
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increases
suggesting
the heightened
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of ScatterAblowitz,
M.J.,
H.
(1981).
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and
Inverse
Ablowitz,
M.J., Segur,
Segur,
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the
Inverse
proximity
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obtaining
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(1996). Mel’nikov
Mel’nikov analysis
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in the
the
Acknowledgements.
This work equation.
was partially
supported
by NSF
nonlinear
Schrödinger
Physica
D
89:227–260.
nonlinear Schrödinger equation. Physica D 89:227–260.
Grant
# NSF-DMS0608693.
Calini,
Calini, A.,
A., Schober,
Schober, C.M.
C.M. (2002).
(2002). Homoclinic
Homoclinic chaos
chaos increases
increases the
the
likelihood
of
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waves.
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A
298:335–349.
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systems. Journal of Computational Physics 176:430–455.
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for stiff
systems. Journal of Computational Physics 176:430–455.
A. Islas and C. M. Schober: Rogue waves and downshifting
Edited by: C. Kharif
Reviewed by: two anonymous referees
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