Part II - TU Dresden...ray caustics statistics Wave scintillation Linearized shallow water wave equations in random bathymetry ß Ray equations 2 0 1 () 21 r c q q r r r E E E v H

Theory of branched flowsand the statistics of extreme waves in random media

Ragnar FleischmannMPI for Dynamics and Self-Organization

Göttingen

• Catastrophes, Caustics and the statistics of extreme waves

• Branching of Tsunami waves

• Teasers/outlook

Part II

Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation

Overview

Part I

• The phenomenon of branching

• In which systems can it be observed?

• Statistics of random caustics

• Characteristic transport length scale

• Wave intensity/height statistics

Part II



• Teasers/outlook


Overview

Part I






Part II



• Teasers/outlook


22

2

2

2

( )I

I Ix

I

variance of the intensity

mean intensity

Scintillation index

Propagation distance

2 ( )I x


Theory of intensity statistics


Predictions for the intensity distribution in branched flows:

• Log-normal distributionWolfson & Tomsovic JASA 2001

• Power-law tailsKaplan PRL 2002

2 ( )I x



saturated fluctuations=

random waves regime


• Saturated regime reached long before the mean free path!

2 ( )I x

strong fluctuations


Random waves intensity

2 2

2 2 2

1 2 1 2 with z and z Gaussian distributed

j j

j

j

j

j

j

j

j

j

a

I a z

i b

zb

1( )

I

IP I e

I

Rayleigh distributedP I

random wave intensities are exponentially distributed




random waves regime



2 ( )I x

strong fluctuations


Saturated fluctuations


Classical/ray-intensity

(0)y

(0)yv

( )y t

( )yv t

0 0

0

0 0

( , )

y y

y v

v v

y v

M t t

1

0

log-normally distributed?y

y

0

0

( (, ) 1, )n

n i i

i

M t tM t t

Michael A. Wolfson and Steven Tomsovic, J. Acoust. Soc. Am. 109, 2693 (2001).

0Tr ( , ) follows a log-normal-distributionM t t


Classical/ray-intensity


Intensity statistics


Intensity fluctuations governed by coherence

Metzger, et al, Phys. Rev. Lett. 111, 013901 (2013).




random waves regime



• Coherent waves show Rayleigh distribution

• Decoherent waves show log-normal distribution

• Theory for all degrees of coherence

J.J. Metzger, R. Fleischman, T. Geisel PRL 2013 2 ( )I x

strong fluctuations




Predictions for the intensity distribution in branched flows:

• Log-normal distributionWolfson & Tomsovic JASA 2001

Metzger, Fleischmann, Geisel PRL 2013

• Power-law tailsKaplan PRL 2002

2 ( )I x


random waves regime

strong fluctuations


( )1

cl

c

yy

Iy

Ray-Intensity at a Caustic

3( )p I I

2( ' ) ' ( ')I

P I I dI p I I


Classical/Ray-Intensity

P(I)

I

107 bins per lc

2I


Ray- and Wave-Intensity at a Caustic

1

2( ) ( )cl cy y yI

( ) ( )qm

yI y Ai

3( )p I I 2( ' ) ' ( ')I

P I I dI p I I




Power-law tails due to caustics only for extreme scale separation

2 ( )I x

strong fluctuations regime





2 ( )I x






610c

2 ( )I x





2 ( )I x

strong fluctuations regime 2I

I

P(I’>I)


Overview

Part I






Part II



• Teasers/outlook


Catastrophe theory

• Bifurcations from a different perspective.

• Studying singularities (discontinuities) of gradient maps

• Thom, Arnold & Zeemann

• Most important result: classification theorem (Thom’s Theorem)

Potential/Lyapunov-Function ( , )S C

0


The cusp catastrophe

Zeeman, E.C., 1976. Catastrophe Theory. Scientific American 65–83. doi:10.1038/scientificamerican0476-65


Zeeman's Catastrophe Machine in Flash by Daniel J. Crosshttp://lagrange.physics.drexel.edu/flash/zcm/

zcm.swf

zcm.swf


Classification

Zeeman, E.C., 1976. Catastrophe Theory. Scientific American 65–83.

Example: the cusp

Example: the fold

CuspCatastrophe.nb

FoldCatastrophe.nb


Catastrophe optics

Berry & Upstill 1980J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, IOP, 1999

C=(X,Y)

S=(x(s),y(s))

( , ) ( , , )s C s X Y

Potential

( , , )0

s X Y

s

raysC S

optical path length


Heller europhys.news 2005

“...nothing more catastrophic than the loss of two rays occurs. Nevertheless, the word catastrophe has been retained.” J.F. Nye in Natural focusing and the fine structure of light.

Caustics

Caustics are the catastrophes of ray optics with

Berry & Upstill, 1980

Example cusp

Example fold

2 3

2 30 ...

s s

Cusp.nb

Fold.nb


Wave amplitude near the fold caustic: diffraction

( ),

diffraction integral

( ) d i k x XJ X xe

31,

3

normal form of the fol

( )

d caustic

x X x X x

1/2

1/3 2/3 1/32( ) AiikRkX k e k X

iR

Fold at distance R from the initial wave front

R X

( ) ( )X J X


Wave amplitude near the fold caustic: diffraction

X0

1/2

1/3 2/3 1/32( ) AiikRkX k e k X

iR


Reminder: characteristics of the random medium

22 )1

( ) (V r V r

E

Random potential

E

( )V r

Kinetic energy of the particlesEnergy

c= correlation length,

i.e. typical length scalec

c


Reminder: characteristic length scale

Propagation distanceJ.J. Metzger, R. Fleischmann, and T. Geisel, Phys. Rev. Lett. 105 (2010).

/ bx

2/3

b c

x

1

2

mfp c≪ mean free pathbranching length

#Caustics( )

2 Lengthbr xN

c

wavelength

𝑁𝑏𝑟(x

)/𝜎3

𝜎2




• power-law tails only in semi-classical limit (not realistic)

• intensity distribution still unknown but

Analytic result for the distribution of extreme (rogue) waves

2 ( )I x


J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112, 203903 (2014).


The statistics of extreme (rogue) waves





main propagation direction

L

J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112 (2014)



Y0y

Mapping branched flow on catastrophe optics

(see e.g. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, IOP Publishing, 1999).

3

0 0 0

1,

3( )Y Yyy y


23( )

2 3

yf y y

R

2

1

R

2 2v t

2/3

2/3

22

1c

c

c

t

bR

1/2

1/3 2/3 1/32( ) AiikRkX k e k X

iR

normal form of the fold caustic



Y0yMapping branched flow on catastrophe optics

(see e.g. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, IOP Publishing, 1999).

2 1/3 2/9

fold fold ( , , ) ( / )c cI k


12 2/3

b

bR

0

2/3

0b c cv t v



main propagation direction

L


1/3 2/9scale with ( / )max cI



[( )/ ]1/3 1/3 2/9( / [ ] )a

maxI m s

c maxP I I e


• extreme value statistics

• Stochastic nonlinear dynamics

• diffraction

• interference

Distribution of highest waves

wavelength


11 March 2011

Do tsunami waves show branching?

NOAA 2011


Do tsunamis show branching?

1500km

1500km

Standard deviation of the ocean depth < 7%



Standard deviation of ocean depth /mean ocean depth ≈ 7%




Tsunami created by a fault source


Degueldre

Scaling of shallow water waves height fluctuations

2 2

0

2 2

, 1 (

0

) ,

t r t c r r t

ray caustics statistics

Wave scintillation

Linearized shallow water wave equations in random bathymetry ß

Ray equations

2

0 1 ( )

( )

2 1 ( )

r c q

q

r

r

r

2/3

Degueldre, Metzger, Geisel, Fleischmann, submitted


11 March 2011

Only of academic interest?

NOAA 2011


Degueldre

Scaling of shallow water waves height fluctuations

2 2

0

2 2

, 1 (

0

) ,

t r t c r r t

ray caustics statistics

Wave scintillation

Linearized shallow water wave equations in random bathymetry ß

Ray equations

2

0 1 ( )

( )

2 1 ( )

r c q

q

r

r

r

2/3

Degueldre, Metzger, Geisel, Fleischmann, submitted


Model of the ocean floor

http://www.gebco.net/general_interest/faq/


Model of the ocean floor

Sandwell, D.T., Gille, S.T., and W.H.F.Smith, eds., Bathymetry from Space: Oceanography, Geophysics, andClimate, Geoscience Professional Services, Bethesda, Maryland, June 2002

“Topography [...] derived from sparse ship soundings and satellite altimeter measurements reveals the large-scale structures created by seafloor spreading (ridges and transforms) but the horizontal resolution (15 km) and vertical accuracy (250 m) is poor.”

What is the “State-of-the-Art“?

250𝑚

4 𝑘𝑚> 6%


Add fluctuations to the recorded depth profile < error

A precise knowledge of the ocean's bathymetry, beyond todays accuracy, is absolutely indispensable for reliable tsunami forecasts


In collaboration with...

Jakob Metzger

Henri Degueldre

Theo Geisel


External Collaborators

Theory

Eric Heller

Scot Shaw

(Harvard Univ.)

Experiment

Marc Topinka

Brian LeRoy

Robert Westervelt

(Harvard Univ.)

Denis MaryenkoJürgen SmetKlaus von Klitzing(MPI-FKF Stuttgart)

Sonja BarkhofenUlrich KuhlHans-Jürgen Stöckmann(Univs. Marburg & Nice)

DFG

Forschergruppe FOR760


Conclusions

• General phenomenon of wave propagation in weakly scattering media

• Universal statistics of branches

• Fundamental length scale of transport in disordered systems

• Scaling of peak position in the scintillation index

• Experimental observation ofscaling behavior

• Branching leads to heavy-tailed intensity distribution: rogue waves

• Theory of extreme waves

• Branching of tsunami waves

• Tsunami forecasts

J.J. Metzger, R. Fleischmann, and T. Geisel, Phys. Rev. Lett. 105, 020601 (2010).

D. Maryenko, F. Ospald, K. v. Klitzing, J. H. Smet, J. J. Metzger, R. Fleischmann, T. Geisel, and V. Umansky, Phys. Rev. B 85, 195329 (2012).

J. J. Metzger, R. Fleischmann, and T. Geisel,Phys. Rev. Lett. 111, 013901 (2013).

S. Barkhofen, J. J. Metzger, R. Fleischmann, U. Kuhl, and H.-J. Stöckmann, Phys. Rev. Lett. 111, 183902 (2013).

J. J. Metzger, R. Fleischmann, and T. Geisel, Phys. Rev. Lett. 112, 203903 (2014).

H. Dequeldre, J. J. Metzger, T. Geisel, and R. Fleischmann, to appear Nature physics (2015)

Documents

Part II - TU Dresden...ray caustics statistics Wave scintillation Linearized shallow water wave equations in random bathymetry ß Ray equations 2 0 1 () 21 r c q q r r r E E E v H