QUANTITATIVE METHODS FOR PROBLEMS IN

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QUANTITATIVE METHODS FOR PROBLEMS IN ENGINEERING
SCIENCE
FRANCESCO FEDELE Goddard Earth Sciences and
Technology center Global Modeling Assimilation
Office NASA Goddard Space Flight Center ,
Maryland, USA
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Unifying theme advanced quantitative methods as
Adjoint methods, numerical methods and advanced
stochastic nonlinear theory
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EXTREME WAVES IN NONLINEAR RANDOM SEAS
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A natural beauty.
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Which can be anomalous! .. so-called FREAK
WAVES
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DRAUPNER EVENT JANUARY 1995
Hmax25.6 m ! 1 in 200,000 waves
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LINEAR WAVES GAUSSIAN SEAS
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TYPICAL WAVE SPECTRA OF THE MEDITERRANEAN SEA
Time covariance
Spectrum
from Boccotti P. Wave Mechanics 2000 Elsevier
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OCCURRENCE OF A HIGH WAVE IN GAUSSIAN SEAS
Theory of quasi-determinism, Boccotti P. Wave
Mechanics 2000 Elsevier
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SINGLE WAVE GROUP EVOLUTION
What happens in the neighborhood of a point x0
if a large crest followed by large trough are
recorded in time at x0 ?
Boccotti P. Wave Mechanics 2000 Elsevier
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SUCCESSIVE WAVE CRESTS IN TIME
(h02h12)1/2
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STOCHASTIC WAVE GROUP
What happens in the neighborhood of a point x0
if two large successive wave crests are
recorded in time at x0 ?
amplitude of the wave group is a random variable
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NONLINEAR RANDOM SEAS
Third order effects FOUR-WAVE RESONANCE
Second order effects BOUND WAVES
Crest-trough symmetry kurtosisgt3 Modulation
instability Effects on slow time scale gtgt wave
period DOMINANT ONLY IN UNIDIRECTIONAL
NARROW-BAND SEAS !
Cresttrough asymmetry skewnessgt0 Tayfun
distribution Effects on Short time scale wave
period
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NONLINEAR EVOLUTION OF A STOCHASTIC WAVE GROUP
t0
t-t0
(linear wave group)
h
hNLgth
x
wave action and wave momentum Always conserved
identities
x0
Hamiltonian invariant
Symmetric third order effects
HmaxhNLa (hNL)2f(h) a f(h)2
Asymmetric second order effects
h Rayleigh distributed
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THE PROBABILITY OF EXCEEDING A FIXED WAVE
AMPLITUDE
Wave tank experiments unidirectional
narrow-band seas ( Onorato et all. 2005) THIRD
ORDER MODULATION SECOND ORDER EFFECTS
P
P
H
h
Wave crest
Wave height
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OPTIMAL PERTURBATIONS IN QUASI-GEOSTROPHIC
ATMOSPHERE
SELF-SUSTAINING PROCESSESNONLINEAR TRAVELLING
WAVES
NON-MODAL ENERGY GROWTH MECHANISMS
OPTIMAL PERTURBATIONS
FEM AND BEM SOLUTION
Unifying theme adjoint methods
COMMON QUANTITATIVE METHOD USED ADJOINT
TECHNIQUES !
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OPTIMAL PERTURBATIONS IN PULSATILE PIPE FLOWS
t0
t1
t2
t3
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OPTIMAL PERTURBATIONS IN QUASI-GEOSTROPHIC
ATMOSPHERE
data measurements attained at time T
t
True trajectory Of State of atmosphere Wind,
humidity, Temperature
Kalman filtering
Unknown state at T
Known State at T0
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a)
b)
OPTICAL TOMOGRAPHY
FEM solution 7,000 FEM nodes reconstruction
time 10 minutes
BEM solution For photon light intensity
FEM mesh 6956 nodes
BEM mesh 708 nodes
d)
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(No Transcript)
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Questions ?
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RESEARCH PLANS

• Extreme wave statistics ( Shell , ONR YIP )
• Coastal processes (Skidaway Inst. of
  Oceanography)
• Turbulence
• Weak wave turbulence
• self sustaining processes
• in quasi-geostrophic atmosphere ( NASA )
• Atmospheric Data assimilation
• Reduced models and ensemble techniques ( NASA ,
  NSF YIP )
• Optimal tomography
• Boundary element method ( NIH YIP )