Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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
• Catastrophes, Caustics and the statistics of extreme waves
• Branching of Tsunami waves
• Teasers/outlook
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
• Catastrophes, Caustics and the statistics of extreme waves
• Branching of Tsunami waves
• Teasers/outlook
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
22
2
2
2
( )I
I Ix
I
variance of the intensity
mean intensity
Scintillation index
Propagation distance
2 ( )I x
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
Predictions for the intensity distribution in branched flows:
• Log-normal distributionWolfson & Tomsovic JASA 2001
• Power-law tailsKaplan PRL 2002
2 ( )I x
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
saturated fluctuations=
random waves regime
Propagation distance
• Saturated regime reached long before the mean free path!
2 ( )I x
strong fluctuations
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
saturated fluctuations=
random waves regime
Propagation distance
• Saturated regime reached long before the mean free path!
2 ( )I x
strong fluctuations
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Saturated fluctuations
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Classical/ray-intensity
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Intensity statistics
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Intensity fluctuations governed by coherence
Metzger, et al, Phys. Rev. Lett. 111, 013901 (2013).
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
saturated fluctuations=
random waves regime
Propagation distance
• Saturated regime reached long before the mean free path!
• 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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
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
saturated fluctuations=
random waves regime
strong fluctuations
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
( )1
cl
c
yy
Iy
Ray-Intensity at a Caustic
3( )p I I
2( ' ) ' ( ')I
P I I dI p I I
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Classical/Ray-Intensity
P(I)
I
107 bins per lc
2I
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
Power-law tails due to caustics only for extreme scale separation
2 ( )I x
strong fluctuations regime
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
Power-law tails due to caustics only for extreme scale separation
2 ( )I x
strong fluctuations regime
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
Power-law tails due to caustics only for extreme scale separation
610c
2 ( )I x
strong fluctuations regime
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
2 ( )I x
strong fluctuations regime 2I
I
P(I’>I)
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
• Catastrophes, Caustics and the statistics of extreme waves
• Branching of Tsunami waves
• Teasers/outlook
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The cusp catastrophe
Zeeman, E.C., 1976. Catastrophe Theory. Scientific American 65–83. doi:10.1038/scientificamerican0476-65
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Zeeman's Catastrophe Machine in Flash by Daniel J. Crosshttp://lagrange.physics.drexel.edu/flash/zcm/
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Classification
Zeeman, E.C., 1976. Catastrophe Theory. Scientific American 65–83.
Example: the cusp
Example: the fold
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Wave amplitude near the fold caustic: diffraction
X0
1/2
1/3 2/3 1/32( ) AiikRkX k e k X
iR
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Theory of intensity statistics
Propagation distance
• 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
strong fluctuations regime
J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112, 203903 (2014).
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
main propagation direction
L
J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112 (2014)
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
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
J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112 (2014)
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
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
J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112 (2014)
12 2/3
b
bR
0
2/3
0b c cv t v
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
main propagation direction
L
J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112 (2014)
1/3 2/9scale with ( / )max cI
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
The statistics of extreme (rogue) waves
[( )/ ]1/3 1/3 2/9( / [ ] )a
maxI m s
c maxP I I e
J. J. Metzger, R.Fleischmann, and T. Geisel, PRL 112 (2014)
• extreme value statistics
• Stochastic nonlinear dynamics
• diffraction
• interference
Distribution of highest waves
wavelength
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
11 March 2011
Do tsunami waves show branching?
NOAA 2011
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Do tsunamis show branching?
1500km
1500km
Standard deviation of the ocean depth < 7%
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Do tsunamis show branching?
Standard deviation of ocean depth /mean ocean depth ≈ 7%
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Do tsunamis show branching?
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Tsunami created by a fault source
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
11 March 2011
Only of academic interest?
NOAA 2011
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
Model of the ocean floor
http://www.gebco.net/general_interest/faq/
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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%
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
In collaboration with...
Jakob Metzger
Henri Degueldre
Theo Geisel
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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
Ragnar Fleischmann Max-Planck-Institut für Dynamik und Selbstorganisation
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)