Title: Interaction of a Large Amplitude Interfacial Solitary Wave of Depression with a Bottom Step
1The modulational instability of long internal
waves
Tatiana Talipova in collaboration with Efim
Pelinovsky, Oxana Kurkina, Roger Grimshaw, Anna
Sergeeva, Kevin Lamb Institute of Applied
Physics, Nizhny Novgorod, Russia
2Observations of Internal Waves of Huge Amplitudes
Alfred Osborn Nonlinear Ocean Waves the
Inverse Scattering Transform, 2010
Internal waves in time-series in the South China
Sea (Duda et al., 2004)
The horizontal ADCP velocities (Lee et al, 2006)
3Theory for long waves of moderate amplitudes
Gardner equation
- Full Integrable Model
- Reference system
- One mode (mainly the first)
Coefficients are the functions of the ocean
stratification
4Cauchy Problem - Method of Inverse Scattering
5Cauchy Problem
First Step t 0
Direct Spectral Problem
spectrum
Discrete spectrum solitons (real roots,
breathers (imaginary roots) Continuous spectrum
wave trains
6Gardners Solitons
sign of a1
a1 lt 0
Limited amplitude alim -a/a1
a1 gt 0
Two branches of solitons of both polarities,
algebraic soliton alim -2 a/a1
7Positive and Negative Solitons
cubic, a1
Positive algebraic soliton
Negative algebraic soliton
quadratic a
Positive Solitons
Negative Solitons
Sign of the cubic term is principal!
8Soliton interaction in KdV
9Soliton interaction in Gardner, ?1 lt 0
10Soliton interaction in Gardner, ?1 gt 0
11Gardners Breathers
cubic, a1 gt 0
b 1, ? 12q, a1 6, where q is arbitrary)
? and ? are the phases of carrier wave and
envelope
propagating with speeds
There are 4 free parameters ?0 , ?0 and two
energetic parameters
Pelinovsky D. Grimshaw, 1997
12Gardner Breathers
?im? 0
?real gt ?im
?real lt ?im
13Breathers positive cubic term
?1 gt 0
14Breathers positive cubic term
b gt 0
15Numerical (Euler Equations) modeling of breather
K. Lamb, O. Polukhina, T. Talipova, E.
Pelinovsky, W. Xiao, A. Kurkin. Breather
Generation in the Fully Nonlinear Models of a
Stratified Fluid. Physical Rev. E. 2007, 75, 4,
046306
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17Envelopes and Breathers
Weak Nonlinear Groups
18Nonlinear Schrodinger Equation
cubic, a1
cubic, ?
focusing
breathers
breathers
Envelope solitons
quadratic, a
defocusing
19Transition Zone
(? ? 0)
Modified Schrodinger Equation
20Modulation Instability only for positive b
cubic, ?
cubic, b
focusing
breathers
breathers
Wave group of large amplitudes
Wave group of large amplitudes
Wave group of weak amplitudes
quadratic, a
21Modulation instability of internal wave packets
(mKdV model)
Formation of IW of large amplitudes
Grimshaw R., Pelinovsky E., Talipova T., Ruderman
M., Erdely R., Short-living large-amplitude
pulses in the nonlinear long-wave models
described by the modified Korteweg de Vries
equation. Studied of Applied Mathematics 2005,
114, 2, 189.
22X T diagram for internal rogue waves heights
exceeding level 1.2 for the initial maximal
amplitude 0.32
23South China Sea
a
a1
There are large zones of positive cubic
coefficients !!!!
24Quadratic nonlinearity, a, s-1
Arctic Ocean
Cubic nonlinearity, a1, m-1s-1
25Horizontally variable background
H(x), N(z,x), U(z,x) 0 (input)
x
Q - amplification factor of linear long-wave
theory
Resulting model
26WaveEvolutionon Malin Shelf
27COMPARISON
Computing (with symbols) and Observed
2.2 km
5.2 km
6.1 km
28Portuguese shelf
Blue line observation, black line - modelling
26.3 km
13.6 km
29Section and coefficients
30Focusing case
We put w 0.01 s-1
31South China Sea
w 0.01
A 30m
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37Comparison with a1 0
a1 0
a1 gt 0
130 km
130 km
323 km
323 km
38Baltic sea
Red zone is a1 gt 0
39Focusing case
We put w 0.01 s-1
40A0 6 m
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43No linear amplification Q 1
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45A0 8 m
46Estimations of instability length
South China Sea
Last point
Start point
Lins 0.6 km
Lins 60 km
Baltic Sea
Last point
Central point
Lins 5 km
Lins 600 km
47Conclusion
- Modulational instability is possible for Long
Sea Internal Waves on shallow water. - Modulational instability may take place when the
background stratification leads to the positive
cubic nonlinear term. - Modulational instability of large-amplitude wave
packets results in rogue wave formations