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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
€
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)
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:
€
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
€
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.
€
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
€
P(t) = i [A∂xA*−A*∂x∫ A]dx
€
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