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Frontiers in Nonlinear Waves University of Arizona March 26, 2010

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Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado – PowerPoint PPT presentation

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Title: Frontiers in Nonlinear Waves University of Arizona March 26, 2010


1
Frontiers in Nonlinear WavesUniversity of
Arizona March 26, 2010
  • The Modulational Instability
  • in water waves
  • Harvey Segur
  • University of Colorado

2
The modulational instability
  • Discovered by several people, in different
    scientific disciplines, in different countries,
    using different methods
  • Lighthill (1965), Whitham (1965, 1967),
    Zakharov (1967, 1968), Ostrovsky (1967), Benjamin
    Feir (1967), Benney Newell (1967),
  • See Zakharov Ostrovsky (2008) for a historical
    review of this remarkable period.

3
The modulational instability
  • A central concept in these discoveries
  • Nonlinear Schrödinger equation
  • For gravity-driven water waves
  • surface slow modulation fast oscillations
  • elevation

4
Modulational instability
  • Dispersive medium waves at different
    frequencies travel at different speeds
  • In a dispersive medium without dissipation, a
    uniform train of plane waves of finite amplitude
    is likely to be unstable

5
Modulational instability
  • Dispersive medium waves at different
    frequencies travel at different speeds
  • In a dispersive medium without dissipation, a
    uniform train of plane waves of finite amplitude
    is likely to be unstable
  • Maximum growth rate of perturbation

6
Experimental evidence of modulational instability
in deep water - Benjamin (1967)
  • near the wavemaker 60 m downstream
  • frequency 0.85 Hz, wavelength 2.2 m,
  • water depth 7.6 m

7
Experimental evidence of modulational instability
in an optical fiber
  • Hasegawa Kodama
  • Solitons in optical
    communications
  • (1995)

8
Experimental evidence of apparently stable wave
patterns in deep water -(www.math.psu.edu/dmh/
FRG)
  • 3 Hz wave 4 Hz wave
  • 17.3 cm wavelength 9.8 cm

9
More experimental results (www.math.psu.edu/dmh/F
RG)
  • 3 Hz wave 2 Hz wave
  • (old water) (new water)

10
Q Where did the modulational instability go?

11
Q Where did the modulational instability go?
  • The modulational (or Benjamin-Feir) instability
    is valid for waves on deep water without
    dissipation

12
Q Where did the modulational instability go?
  • The modulational (or Benjamin-Feir) instability
    is valid for waves on deep water without
    dissipation
  • But any amount of dissipation stabilizes the
    instability (Segur et al., 2005)

13
Q Where did the modulational instability go?
  • This dichotomy exists both for (1-D) plane waves
    and for 2-D wave patterns of (nearly) permanent
    form. The logic is nearly identical. (Carter,
    Henderson, Segur, JFM, to appear)
  • Controversial

14
Q How can small dissipation shut down the
instability?
  • Usual (linear) instability
  • Ordinarily, the (non-dissipative) growth rate
    must exceed the dissipation rate in order to see
    an instability. So very small dissipation does
    not stop an instability.

15
Q How can small dissipation shut down the
instability?
  • Set

16
Q How can small dissipation shut down the
instability?
  • Set
  • Recall maximum growth rate

17
Experimental verification of theory
  • 1-D tank at Penn State

18
Experimental wave records
  • x1
  • x8

19
Amplitudes of seeded sidebands(damping factored
out of data)
  • (with overall decay factored out)
  • ___ damped NLS theory
  • - - - Benjamin-Feir growth rate
  • ? ? ? experimental data

20
Q What about a higher order NLS model (like
Dysthe) ?
  • __, damped NLS ----, NLS - - -, Dysthe
  • ? ? ?, experimental data

21
Numerical simulations of full water wave
equations, plus damping
  • Wu, Liu Yue,
  • J Fluid Mech, 556,
  • 2006

22
Inferred validation
  • Dias, Dyachenko Zakharov (2008) derived the
    dissipative NLS equation from the equations of
    water waves
  • See also earlier work by Miles (1967)
  • Both papers provide analytic formulae for ??

23
How to measure ???
  • Integral quantities of interest
  • ,
  • ,

24
Dissipationin wavetankmeasuredafter waiting
a timeinterval
  • 15 min.
  • 45 min.
  • 60 min.
  • 80 min.
  • 120 min.
  • 1 day
  • 2 days
  • 6 days

25
Open questions
  • What is the correct boundary condition at the
    waters free surface?
  • Do we need a damping rate that varies over days?
  • If so, why?

26
Thank you for your attention
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