The Benjamin-Feir instability-a popular and
debated issue. ( Dysthe and Trulsen)
• The beginning, selfaction effects in various
waves.
• Initial growth of side bands.
• Recurrence, and the NLS.
• Frequency downshift.
• Longtime evolution.
• How to make exstreme waves.
• BF instability of stokastic waves.
• 3D development
Self-action of waves
The LASER-induced revolution in optics led to
experimental demonstration of self-action phenomena:
self-focusing Lallemand & Bloembergen (1965)
and
self-trapping Garmire et.al. (1966).
Self-modulation
of gravity waves demonstrated at roughly the same time (Benjamin
& Feir 1967), was not induced by advances in technology. One
wanders why it was not realized before?
There were, however, corroborating theoretical results obtained at
the same time :
Lighthill (1965), Whitham (1967), Ostrovsky (1963, 1967), Zakharov
(1967) and Benney & Newell (1967).
Early history of self-modulation
• It is a remakable fact that the opening of this
field of reseach happened at the same time in
US, England and USSR.
• The observations were in a number of physical
situations: water waves, plasma waves, laser
beams and electromagnetic transmission lines.
• For an excellent overview of the early
development, see:
”Modulation instability: The beginning”
Zakharov & Ostrovsky: Physica D 238, 2009
Benjamin &Feir (1967)
• A short glimpse of their
experiment.
• There must have been
previous experimenters
seeing the same, when
trying to produce regular
Stokes waves.
• They probably conceived
it as a nuisance (possibly
caused by a poor
wavemaker??) rather
than a scientific
challenge.
Benjamin and Feir saw the selfmodulation as a nonlinear
interaction between 4 waves, mediated by the second
harmonic ”virtual wave”.
 (1   )


2
 (1   )
First major experimental verification of the
BF instability (10 years later).
Lake, Yuen, Rungaldier
and Ferguson:
J.Fluid Mech. 1977
Growth of upper (open symbols) and
lower sidebands (solid symbols)
• Growth of the most
unstable sidebands,
Lake et.al. 1977
• The good agrement with
B&F’s theoretical growth
rate is due to the fact that
they used a nominal
steepness, only 80% of
the measured one.
• If instead the measured
steepness is used, their
growth rate agree with
later measurement (e.g.
Waseda & Tulin 1999)
Waseda & Tulin (1999)
New opening for the theory of dispersive waves
at the end of the 1960th.
• The development of the NLS equation has proved to be
of great value, because:
• It predicts qualitatively correct many of the new and
surprising phenomena observed for nonlinear dispersive
waves, such as the BF instability.
• Its 2D-version can be solved by the inverse scattering
method (Zakharov & Shabat 1971) like the KdV
equation, and many explicite solutions are known.
Long-time evolution of the BF instability
according to the NLS equation
(Yuen & Lake 1980)
The ”breather” family of solutions
of the NLS equation
Asymptotic state of the BF
instability according to the NLS
equation.
• Ma (1979) used the inverse scattering method on
the NLS equation. He found that :
• the asymptotic state of a Stokes wave that is
perturbed over a finite domain, is a series of
Kuznetsov breathers and dispersive waves.
Simulation of a periodically perturbed
wavetrain with the NLS equation.
Simulation of a periodically perturbed wavetrain
with higher order MNLS equations.
First glimpse of a ”downshift”
A closer look at the B-F evolution from start to a ”quasi- recurrence ”.
(Lake et.al.1977)
• Counting the number of waves
at 5ft (13), and 30ft (10), a
frequency downshift is seen to
occur. Not mentioned by the
authors.
• However, in a later paper by
Yuen and Lake: (Ann.Rev.of
Fluid Mech. 1980), it was
pointed out.
• Melville (1982) reported that
this phenomenon is happening
when the BF modulation is
deep enough to produce
breaking.
Frequency ”separation”. Experiment at Marintek,
Trulsen and Stansberg (2001).
• Bichromatic
waves evolve
• 10 m from
wavemaker
frequency:
1  2
2
• 80 m from
wavemaker,
”separation”
1 and 2
have separated
0.3
probe 1,

2
k
0
x = 0
0.2
0.1
0
-0.1
-0.2
130
0.3
0.2
0.1
0
-0.1
-0.2
135
probe 4,
140
145
2
150
 k 0 x = 0.89
155
160
165
170
before breaking
after breaking.
Hwung et.al.: ”Observation of the evolution of wave
modulation” Proc. R. Soc 2007
Why dissipation stabilize the BF in the strict
mathematical sence (Segur et.al.2005)
(A pedestrian approach)
• As the amplitude decreases
due to dissipation, the domain
of instability for a sideband
perturbation shrinks.
• Consequently the growth of
that sideband is limited in
time.
• Therefore its amplitude can in
principle stay arbitrarily small,
provided that its initial value
is sufficiently small.
• Thus the damped Stokes wave
is modulationally stabel (in the
strict mathematical sence).
SCALED
GROWTH-RATE

 ka0
The pursuite of extreme waves
• Starting et the end of the
19-hundreds. Extreme
waves were much in the
news and scientists soon
got very interrested.
• The main question: is
there a new and
overlooked physical
mechanism responsible
for the ”freak-”or ”rogue”
waves?
Henderson, Peregrine and Dold (1999).
Long time history of the BF instability and
Steep Wave Events (SWE)
• Development of BF
instability. Full
nonlinear equations.
• The maximum surface
elevation of the
wavetrain as a function
of time.
• The SWE’s were found
to have a conspicuous
likeness to the breather
solution of the NLS.
Particularly to the
Kuznetsov breather.
~3a
4a~
”Explosion” of a group.
2D simulation with fully nonlinear equations.
• Initial condition in the shape of
a NLS-soliton with maximum
steepnes ka=0.14
• After a couple of dozen
periods this happens!
Zakharov,Dyachenko and Prokofiev (2005)
• Here a little less spectacular.
Max initial steepness ka=0.09
Clamond et.al.(2006)
MNLS animation 1
MNLS animation 3
MNLS animation 2
Alber (1978) found that even an irregular
(stokastic) wave-train may be unstable if the
spectrum is sufficiently narrow.
• The condition for instability is that the
relative half-width Δω/ωp of the frequency
spectrum at the spectral peak ωp , is less
than the wave steepness s i.e. when :
s > Δω/ωp
s  kp a
2 1/ 2
 H sT p
2
Development of narrow wave spectra due to
the Benjamin-Feir instability
(simulations, Dysthe et.al. 2003)
Development of the angular averaged spectrum.
 4
Experimental development of an irregular
wavetrain with BF-index ~1, Onorato et.al.(2005)
• The wavemaker produces
32min. irregular
wavetrains having a
JONSWAP spectrum
with random phases.
• The BF index is 0.9 with
BFI= s/ where  is the
spectral half width at half
the peak value and s the
steepness.
Wave statistics by large scale simulation by
the MNLS equations. Socquet-Juglard et.al.(2005)
• Initial conditions: JONSWAP spectrum with BFI >1.
Random phases. Three different angular distributions.
Average steepness s=0.1
• The computational domain contains appr. 10.000 typical
waves at any time.
Steepness
• Scatter-diagram of peak
period Tp and significant
wave height Hs .
• Data from the northern
North Sea collected from
5 platforms, 1973-2001.
(Haver, Eik)
• Each point extracted from
20 min. timeseries.
14H S
s
2
gTP
s  k p a2
 H sT p
1/ 2
2
Angular distribution
The probability of a crest height
exceeding x standard deviations
Does the B-F instability increase the
population of rogue waves??
• 2D: Yes, when Δω/ωp<s . This has been verified in a
wave flume (Marintek, Trondheim) Onorato et.al. (2004,
2006).
• 3D: Simulations indicate that it does not work when the
average crest length is less than appr. 10 wave lengths.
Socquet-Juglard et.al. (2005), Gramstad & Trulsen (2007).
• Independent experiments in two wave basins (at
Marintek, Trondheim and University of Tokyo) have
shown these conclutions to be at least qualitatively
correct.
• Onorato et.al. (2009)
Development of Zig-Zak patterns. 3D
BF effect).
Ruban (PRL 2007)
•
Ruban used a fully nonlinear and
”weakly 3D” numerical code to
analyse the late stages of a plane
Stokes wave. The phase, on
entering the computational area
has been randomly perturbed.
A generalized Stokesdevelopment :
1
2
    ( Aei  kz  A2 e 2i  2 kz  A3e3i 3kz  c.c.)  O( 4 )
and
1
2
    ( Bei  B2 e 2i  B3e3i  c.c.)  O( 4 )
where   k p x   p t .
T hecomplexamplitudesA, A2 , A3 , and B, B2 , B3 havesmall (order 1/2 )
rat esof variationin space and time.If theyare all expressedby B and its
derivatives we get theevolutionequat ions:
B
1 2
3i 2 B i 2 B * 
i
 LB  B B  B
 B

B
t
2
2
x 4
x x
and
 1  2

B
, z0
z 2 x
2  0 , z  0
L is thelinear operator:

L  (1   x ) 2   2y

1/ 4

1