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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

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

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Page 1: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Frontiers in Nonlinear WavesUniversity of Arizona

March 26, 2010

The Modulational Instability in water waves

Harvey SegurUniversity of Colorado

Page 2: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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.

Page 3: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

The modulational instability

A central concept in these discoveries:

Nonlinear Schrödinger equation

For gravity-driven water waves:

surface slow modulation fast oscillations

elevation

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A = 0

η(X,Y,T;ε) ~ ε[A(ε(X − cgT),εY,ε2X) ⋅e iθ + A*e−iθ ]+O(ε 2)

Page 4: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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

Page 5: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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:

Ω∝| A0 |2

Page 6: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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

Page 7: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Experimental evidence of modulational instability in an optical fiber

Hasegawa & Kodama

“Solitons in optical communications”

(1995)

Page 8: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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

QuickTime™ and aMotion JPEG OpenDML decompressor

are needed to see this picture.

Page 9: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

More experimental results (www.math.psu.edu/dmh/FRG)

3 Hz wave 2 Hz wave

(old water) (new water)

Page 10: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Q: Where did the modulational instability go?

Page 11: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Q: Where did the modulational instability go?

• The modulational (or Benjamin-Feir) instability is valid for waves on deep water without dissipation

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A = 0

Page 12: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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)

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A = 0

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A + iδA = 0

Page 13: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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

Page 14: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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.

Ωobserved =Ω predicted −δdissipation

Page 15: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Q: How can small dissipation shut down the instability?

Set

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A + iδA = 0

A(x,y, t) = e−δ ⋅tΒ(x,y, t)

i∂tB +α∂x2B + β∂y

2B + γe−2δ ⋅t |B |2 B = 0

Page 16: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Q: How can small dissipation shut down the instability?

Set

Recall maximum growth rate:

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A + iδA = 0

A(x,y, t) = e−δ ⋅tΒ(x,y, t)

i∂tB +α∂x2B + β∂y

2B + γe−2δ ⋅t |B |2 B = 0

Ω∝| A0 |2 → Ω∝ e−2δ ⋅t | A0 |

2

Page 17: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Experimental verification of theory

1-D tank at Penn State

Page 18: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Experimental wave records

x1

x8

Page 19: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Amplitudes of seeded sidebands(damping factored out of data)

(with overall decay factored out)

___ damped NLS theory

- - - Benjamin-Feir growth rate

experimental data

Page 20: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Q: What about a higher order NLS model (like Dysthe) ?

__, damped NLS ----, NLS - - -, Dysthe

, experimental data

Page 21: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Numerical simulations of full water wave equations, plus damping

Wu, Liu & Yue, J Fluid Mech, 556, 2006

Page 22: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

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 €

i∂tA +α∂x2A + β∂y

2A + γ | A |2 A + iδA = 0

Page 23: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

How to measure

Integral quantities of interest:

,

,

i∂tA +α∂x2A + γ | A |2 A + iδA = 0

M(t) = | A(x, t) |2∫ dx

M(t) = M(0) ⋅e−2δ ⋅t

P(t) = i [A∂xA*−A*∂x∫ A]dx

P(t) = P(0) ⋅e−2δ ⋅t

Page 24: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Dissipationin wave

tankmeasured

after waiting a timeinterval

15 min.

45 min.

60 min.

80 min.

120 min.

1 day

2 days

6 days

Page 25: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Open questions

What is the correct boundary condition at the water’s free surface?

Do we need a damping rate that varies over days?

If so, why?

Page 26: Frontiers in Nonlinear Waves University of Arizona March 26, 2010 The Modulational Instability in water waves Harvey Segur University of Colorado

Thank you for your attention