Frontiers in Nonlinear WavesUniversity of Arizona

March 26, 2010

The Modulational Instability in water waves

Harvey SegurUniversity 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.

The modulational instability

A central concept in these discoveries:

Nonlinear Schrödinger equation

For gravity-driven water waves:

surface slow modulation fast oscillations

elevation

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i∂tA +α∂x2A + β∂y

2A + γ | A |2 A = 0

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

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

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:

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Ω∝| A0 |2

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

Experimental evidence of modulational instability in an optical fiber

Hasegawa & Kodama

“Solitons in optical communications”

(1995)

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.

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

3 Hz wave 2 Hz wave

(old water) (new water)

Q: Where did the modulational instability go?

Q: Where did the modulational instability go?

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

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i∂tA +α∂x2A + β∂y

2A + γ | A |2 A = 0

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

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

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.

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Ωobserved =Ω predicted −δdissipation

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

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

Experimental verification of theory

1-D tank at Penn State

Experimental wave records

x1

x8

Amplitudes of seeded sidebands(damping factored out of data)

(with overall decay factored out)

___ damped NLS theory

- - - Benjamin-Feir growth rate

experimental data

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

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

, experimental data

Numerical simulations of full water wave equations, plus damping

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

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

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

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P(t) = i [A∂xA*−A*∂x∫ A]dx

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P(t) = P(0) ⋅e−2δ ⋅t

Dissipationin wave

tankmeasured

after waiting a timeinterval

15 min.

45 min.

60 min.

80 min.

120 min.

1 day

2 days

6 days

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?

Thank you for your attention