Holger Hennig,
Department of Physics, Harvard
University, Cambridge, MA 02138, USA and Broad Institute of MIT and
Harvard, Cambridge, MA 02142, USA
Published
March 24, 2014
|
Physics 7, 31 (2014)
|
DOI:
10.1103/Physics.7.31
Nonlinear freak waves in water can be
generated experimentally by exploiting the time-reversal symmetry of the
equations that govern their propagation.
Freak or rogue waves are giant waves in the ocean, up to tens of
meters high, that seem to form out of the blue, posing a major threat
for even the largest ships. It is no wonder that freak waves have often
been described as a mysterious phenomenon. How can they form and why do
they occur orders of magnitude more often than what is expected from
commonly used models, such as a “random-plane-wave model” based on the
linear superposition of plane waves? One possible mechanism for their
formation is via so-called “breathers,” oscillating localized and
high-amplitude waves that can occur in nonlinear media.
Breathers are solutions of the nonlinear Schrödinger
equation (NLS), which describes the dynamics of a large variety of
nonlinear media including water, optical fibers, and clouds of
Bose-Einstein condensates. As reported in
Physical Review Letters,
Amin Chabchoub at the Swinburne University of Technology in Melbourne,
Australia, and Mathias Fink at the Institut Langevin of the French
National Center for Scientific Research (CNRS) have demonstrated the
experimental generation of breathers in water [
1]
based on a particular mathematical property of the NLS: its
time-reversal symmetry. This property allows the authors, for the first
time, to “refocus” breathers: after they are generated, propagated, and
are recorded at some distance from the origin, the decayed nonlinear
wave profile is time-reversed and re-emitted in such a way that their
energy focuses back, as though a movie of the propagating waves had been
played backwards. The effect, which occurs despite the inevitable
presence of damping and noise, may be exploited for the generation of
breathers in a variety of media or to study the role of nonlinearity in
the formation of rogue waves.
Historically, the simplest approach to model freak waves
has been the random-plane-wave model, in which the linear superposition
of waves leads to a statistical (Rayleigh) distribution of wave
intensities. Yet, as confirmed by recent research on freak waves in
microwave resonators [
2],
it has been known for many years that this underestimates by orders of
magnitude the probability for the occurrence of freak waves. This puzzle
remains unsolved even when the model is refined by including multiple
wave-scattering events. But two other physical effects could be
responsible for the unexpectedly high frequency of freak waves observed
in the real world: (i) nonlinearity, i.e., the formation of breathers
and (ii) caustics, i.e., the focusing of waves to high intensity due to
purely linear wave propagation in random media.
Caustics form naturally in random media and act like
lenses for waves. They determine, for instance, the changing light
patterns that you can see on the bottom of a swimming pool on sunny
days. While the contribution of caustics to freak waves has recently
been uncovered [
3],
the role of breathers in the formation of freak waves in the ocean is
still foggy. Therefore, a way to experimentally control freak waves
formed via breathers would be highly desirable.
Chabchoub and Fink tackle the nonlinear contribution to freak wave dynamics experimentally. They use a
15-meter-long unidirectional water flume with a water depth of
1
meter (m)—basically a water tank carefully optimized to control wave
propagation in one dimension. To minimize dissipation and its influence
on breather propagation in the tank, they cleaned the walls of the flume
and used filtered water before performing the experiments. The
researchers refocused breathers using the following protocol: (i)
Building on previous work [
4],
they provided the initial conditions to generate a breather by moving a
single wave-forming paddle. Such initial conditions, e.g., the
amplitude profile of the breather, can be calculated analytically. (ii)
They measured the amplitude profile of the decaying breather at a
9-m distance. (iii) They calculated and generated the time-reversed amplitude profile of the breather.
The time-reversal symmetry of the NLS would imply that a
state at a given time could be back-propagated to yield the initial
conditions that generated it. The time-reversed signal would then lead
to the re-formation of a breather after propagating for
9m.
But proving that the mathematically predicted refocusing could work as
planned under real-world experimental conditions was the greatest
challenge for the researchers. Refocusing has been demonstrated in the
linear regime for single giant peaks, also called hot spots [
5].
Can nonlinearity-based refocusing also succeed, despite the inevitable
damping that breaks the time-reversal symmetry of the NLS? Chabchoub and
Fink indeed report a successful outcome: the experimental refocusing of
a first-order (so-called Peregrine) breather. Furthermore, they were
able to reverse in time an even higher (second) order
(Akhmediev-Peregrine) breather. The amplitude of the wave envelope is
amplified by a factor of
5 in the case of the second-order breather. Thus, if a ship were to sail across
2-m random waves in the ocean, encountering a second-order breather would translate to facing waves up to
10m high.
It is worth noting that time reversal of breathers as
proposed by the authors may provide insight into a broader class of
situations beyond freak waves. Moving breathers (more widely known as
solitons) manifest themselves in rivers as part of tidal bores, in the
form of a wave front followed by a train of solitons [
6]
(see Fig. 1). This phenomenon is well known among surfers: the tidal
bore in the Severn River in the UK can carry surfers several miles
upstream. A five star bore—the highest category—attracted surfers and
“breather watching” spectators alike
in February 2014.
The authors’ results may also have implications in other
domains of physics. The NLS describes a variety of physical systems,
including Bose-Einstein condensates (BECs) confined to cigar-shaped
traps—a system that can be controlled experimentally with a very high
precision. In the presence of an optical lattice potential, the NLS
turns into a lattice equation: the discrete NLS. Generating breathers in
BECs is appealing, as breathers are stable and localized matter waves
that are coherent on long time scales [
7],
properties that could be used, for instance, in quantum computing.
Experiments with BECs may address whether time reversal of decayed
breathers is possible. However, quantum effects may pose an obstacle:
classically instable motion (including the decay of a breather) leads to
decoherence [
8], which goes beyond the validity of the NLS.
While the scheme studied by the authors is one
dimensional (like solitons in narrow rivers), future research may
investigate the simulation of breathers in two dimensions, providing a
more realistic approach to understand possible mechanisms for the
formation and decay of freak waves on the ocean surface. Furthermore,
disentangling nonlinear contributions (i.e., breathers) from linear
contributions (i.e., caustics) to freak-wave formation remains an
outstanding challenge for researchers in a wide range of fields,
including microwave chaos, nonlinear optics, and water waves.
References
- Amin Chabchoub and Mathias Fink, “Time-Reversal Generation of Rogue Waves,” Phys. Rev. Lett. 112, 124101 (2014).
- H.-J. Stöckmann, in Chaos,
edited by B. Duplantier, S. Nonnenmacher, and V. Rivasseau Progress in
Mathematical Physics, Vol 66 (Birkhäuser, Boston, 2013)[Amazon][WorldCat].
- R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak Waves in the Linear Regime: A Microwave Study,” Phys. Rev. Lett. 104, 093901 (2010).
- A. Chabchoub, N. Hoffmann, M. Onorato, and N.
Akhmediev, “Super Rogue Waves: Observation of a Higher-Order Breather in
Water Waves,” Phys. Rev. X 2, 011015 (2012).
- A. Przadka, S. Feat, P. Petitjeans, V. Pagneux, A. Maurel, and M. Fink, “Time Reversal of Water Waves,” Phys. Rev. Lett. 109, 064501 (2012).
- M. A. Porter, N. J. Zabusky, B. Hu, and D. K. Campbell, “Fermi, Pasta, Ulam and the Birth of Experimental Mathematics,” American Scientist 97, 214 (2009).
- H. Hennig, D. Witthaut, and D. Campbell,
“Global Phase Space of Coherence and Entanglement in a Double-Well
Bose-Einstein Condensate,” Phys. Rev. A 86, 051604 (2012).
- Y. Castin and R. Dum, “Instability and Depletion of an Excited Bose-Einstein Condensate in a Trap,” Phys. Rev. Lett. 79, 3553 (1997).
