Transcript segur - University of Arizona

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.
The modulational instability
A central concept in these discoveries:
Nonlinear Schrödinger equation
it A   A   A   | A | A  0
2
x
2
y
2
For gravity-driven water waves:
(X,Y,T;) ~ [A((X  c gT),Y, 2 X)  e i  A*ei ]  O( 2 )
surface
elevation
slow modulation
fast oscillations
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:
 | 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)
QuickTime™ and a
Motion JPEG OpenDML decompressor
are needed to see this picture.
3 Hz wave
17.3 cm
wavelength
4 Hz wave
9.8 cm
More experimental results
(www.math.psu.edu/dmh/FRG)
3 Hz wave
(old water)
2 Hz wave
(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t A   A   A   | A | A  0
2
x
2
y
2

Q: Where did the modulational
instability go?
• The modulational (or Benjamin-Feir)
instability is valid for waves on deep water
without dissipation
it A   A   A   | A | A  0
2
x
2
y
2
• But any amount of dissipation stabilizes the
instability (Segur et al., 2005)
it A   A   A   | A | A  iA  0
2
x
2
y
2
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:
observed   predicted  dissipation
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.
Q: How can small dissipation
shut down the instability?
it A   A   A   | A | A  iA  0
2
x
Set
2
y
2
A(x, y,t)  e  t (x, y,t)
it B  x2 B  y2 B  e2 t | B |2 B  0


Q: How can small dissipation
shut down the instability?
it A   A   A   | A | A  iA  0
2
x
Set
2
y
2
A(x, y,t)  e  t (x, y,t)
it B  x2 B  y2 B  e2 t | B |2 B  0
 Recall maximum growth rate:

 | A0 |
2
 e
2  t
2
| A0 |
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
  , experimental data
----, NLS
- - -, Dysthe
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
it A   A   A   | A | A  iA  0
2
x
2
y
2
See also earlier work by Miles (1967)
Both papers provide analytic formulae for 
How to measure 
i t A   A   | A | A  iA  0
2
x
2
Integral quantities of interest:
M(t)   | A(x,t) |2dx ,
M(t)  M(0)  e2 t

P(t)  i  [Ax A*A* x A]dx,


2 t
P(t)  P(0)  e
15 min.
Dissipation
in wave
tank
measured
after
waiting
a time
interval
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