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Günter Steinmeyer1, Simon Birkholz1, Carsten Brée2, Ayhan Demircan3
1 Max Born Institute, Berlin, Germany
2 Weierstrass Institute, Berlin, Germany 3 Institut für Quantenoptik, University of Hannover
Nonlinear correlation analysis in rogue wave data
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Big Data & Real-time Analytics in Photonics UCLA, Los Angeles, CA
March 27, 2015
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Outline
• Ocean rogue waves may cause serious damage to ships
• Qualitatively similar behavior observed in optical physics
• Exact reasons for ocean rogue waves unknown
• Predictability of rogue events?
• Tracing the reasons: looking for determinism in measured data
• Nonlinear correlation analysis
• Numerical complexity O(N2): N~105: 1h on Desktop PC N~107: 1yr on desktop PC
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What Exactly is a Rogue Wave?
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Rogue Waves – Towards a Unifying Concept? Eur. Phys. J. Special Topics 185 (2010).
Criteria generally agreed upon:
1. Sparsity: rogue events are extremely rare
2. Extremeness: rogue events surpass any prediction derived from observation of previous events
3. Statistical unlikelihood: rogue waves defy regular Gaussian statistics
4. Unpredictability: rogue waves appear w/o warning and disappear w/o a trace
Difficult to quantify
Only criterion so far
New criterion
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Optical Fiber Rogue Waves
Solli et al., Nature 450, 1054 (2007).
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Is this a Rogue Wave?
J.H.E. Cartwright & H. Nakamura, What kind of a wave is Hokusai's Great wave off Kanagawa ? Notes Rec. R. Soc. 63, 119-135 (2009).
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Are Mountains Actually Rogue Waves?
Source: Geo Elevation Data Mathematica, Wolfram Research
Earth surface sampled at 0.1O steps 6,400,000 data points
Sea level
This is where rogue starts
This is where we are 125 m = 400 ft
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Multifilament Rogue Scenario
xenon cell (2 bar)
Ti:sapphire laser 1 kHz, 5 mJ, 40 fs
1 kHz linescan or 100 Hz 2D camera
dynamic beam profile
f =0.75m
Ref.: S. Birkholz et al., Phys. Rev. Lett. 111, 243903 (2013)
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Spatio-temporal isolation
Ref.: S. Birkholz et al., Phys. Rev. Lett. 111, 243903 (2013)
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The „Birkholz“ event
Draupner oil rig, North Sea January 1, 1995, 3:20pm
MBI labs, Berlin April 2012
Ref.: Dysthe et al. “Oceanic Rogue Waves” Annu. Rev. Fluid Mech. 40, 287 (2008).
Birkholz
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Statistical Analysis Laser Single
Filament Multi
Filament
Events at 10x significant wave height F1/3
Ref.: S. Birkholz et al., Phys. Rev. Lett. 111, 243903 (2013)
W. Weibull, J. Appl. Mech.-Trans. 18, (1951)
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Predictability of rogue events
1. Select segment Si,i+L 2. Compute ||Si,i+L-Sj,j+L||2 for all i,j
P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).
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Histogram representation
||Si,i+L-Sj,j+L||2
Cou
nts
most segments are fairly dissimilar
S. Birkholz et al., submitted to PRL
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Surrogates
Time
Am
plitu
de
Refs.: D. Prichard & J. Theiler PRL 73, 951 (1994). T. Schreiber & A. Schmitz PRL 77, 635 (1996) True random numbers from www.random.org
1. Maintain histogram
2. Maintain spectrum
3. Conserve linear correlation function
4. Use „true“ random numbers
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Determinism of multifilament data
surrogates
original
several thousand “déjà vus“ in the original
200 ms segment length = 15x lin. correlation time
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Analysis for original Draupner event
20s segment length
Data is clearly deterministic !
Original data of Draupner event courtesy M. Olagnon IFREMER, Brest, France and Statoil ASA, Norway
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Analysis for fiber Rogue waves
Original data courtesy D. Solli, UCLA D. Solli et al., Nature 450, 1054 (2007)
Original data more random than surrogates
Positively no determinism !!!
1 µs segment length
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Can we actually predict rogue events?
Total data 60,000 camera lines
400 pxls each
Contains 920 spatially
and temporally isolated
rogue events
1 min
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A new type of surrogate analysis
1. Select 100ms long segment before each rogue wave
2. Select identical number of random segments across data set
3. Redo surrogate 100x
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Predictability of rogue waves?
surrogates
originals
Specific dynamics herald the
impact of rogue waves
100ms look ahead time (~10s equivalent for Draupner)
Distance
Cou
nt
J. Martinerie et al., “Epileptic seizures can be anticipated by non-linear analysis,“
Nature Medicine 4, 1173 (1998).
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Conclusions Filament rogue waves behave like ocean rogue
waves concerning several aspects Heavytail-distributions (actually more extreme
than the ocean) Deterministic behavior → predictability Main driver: atmospheric turbulence (as in the
ocean) Is it really true that rogue waves appear w/o any
warning?
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Acknowledgments Simon Birkholz, MBI Rogue wave experiments
Erik Nibbering, MBI 15 year experience w/ filaments
Carsten Brée, WIAS Numerical Simulations
Stefan Skupin, Univ. Bordeaux Propagation Code
Ayhan Demircan, WIAS Theory support
Goëry Genty, TUT, Finland Original idea
Fedor Mitschke, Rostock Grassberger & Procaccia, PhD thesis advisor
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Microscopic origin of rogue events • What is the origin of this sudden intensity/fluence spikes
in the filaments?
• Clamping intensity ???
• We are not measuring inside the cell, beams were allowed to diffract
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Some thoughts on the role of mergers • Can we control these mergers?
• Is it possible to combine two or more filament strings?
• No violation of clamping intensity
• Increased energy / fluence in combined filament
• Control prior to filament formation futile!
• Control during multifilament propagation
• Maybe such experiments work in solid or liquid medium?
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lateral dimension 0 6.5 mm
Freq
uenc
y
The Input Profile – Calm Waters
Gaussian
rogue
average intensity
peak intensity
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Single Filamentation – Choppy Seas
lateral dimension 0 10 mm
peak intensity ≈ 2x average intensity
peak intensity ≈ average intensity
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Multiple Filamentation
lateral dimension 0 10 mm
peak intensity > 10x average intensity
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Numerical Simulations
x
y
0
30 cm 200 µm
2D+1 numerical solution of the NLSE code developed by S. Skupin
0
π
2π Phase
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The Role of Breathers
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Rogue waves arise from mergers C
ount
s
Intensity
all filaments
mergers only
L. Cavaleri et al., “Rogue waves in crossing seas: The Louis Majesty accident,” J. Geophys. Research: Oceans 117, C5 (2012).
V. P. Ruban, “Giant waves in weakly crossing sea states,” J. Exp. Theo. Phys. 110, 529 (2010).
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Isolated breather event
Input: 2 filament strings w/ identical phase
Filaments attract due to XPM
z
Reseparation in orthogonal plane
Further breathing after collapse
• Incidental interference of 2 waves
• Phase plays dice here
• No rogues w/o nonlinearity
• Similarity w/ 1D breathers
• The universal mechanism found?
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Where does the determinism come from?
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Macroscopic origin of rogue waves Experiments with different parameters: • No rogue waves with water as a medium
--- despite higher nonlinearity • 10x reduction of repetition rates with chopper
→ no rogue waves • xenon and SF6 work (2 bar pressure, 1 kHz rep. rate)
Thermal convection mechanism
inside cell Y.-H. Cheng et al. "The effect of long timescale gas
dynamics on femtosecond filamentation," Opt. Express 21, 4740 (2013)
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Turbulent thermal convection
K. Noto et al. J. Thermophys. Heat Transfer 13, 82 (1999). S. Grafsrønningen et al., Int. J. Heat Mass Transfer 57, 519 (2013).
Local Grashof Number
Natural convection Irregular horizontal cylinder array
Transition from laminar to turbulent heat exchange
2x108 < Gry,Q < 2x109 Transition regime Possible to meet for xenon
(low heat conductivity and viscosity) Water requires 1000x higher powers
y
z
Nonlinear correlation analysis in rogue wave dataOutlineWhat Exactly is a Rogue Wave?Optical Fiber Rogue WavesIs this a Rogue Wave?Are Mountains Actually Rogue Waves?Multifilament Rogue ScenarioSpatio-temporal isolationThe „Birkholz“ eventStatistical AnalysisPredictability of rogue eventsHistogram representationSurrogatesDeterminism of multifilament dataAnalysis for original Draupner eventAnalysis for fiber Rogue wavesCan we actually predict rogue events?A new type of surrogate analysisPredictability of rogue waves?ConclusionsAcknowledgmentsMicroscopic origin of rogue eventsSome thoughts on the role of mergersThe Input Profile – Calm WatersSingle Filamentation – Choppy SeasMultiple FilamentationNumerical SimulationsThe Role of BreathersRogue waves arise from mergersIsolated breather eventWhere does the determinism �come from?Macroscopic origin of rogue wavesTurbulent thermal convection