Interaction of a Large Amplitude Interfacial Solitary Wave of Depression with a Bottom Step - PowerPoint PPT Presentation

About This Presentation
Title:

Interaction of a Large Amplitude Interfacial Solitary Wave of Depression with a Bottom Step

Description:

Title: Interaction of a Large Amplitude Interfacial Solitary Wave of Depression with a Bottom Step Author: Tatiana Talipova Last modified by: TalipovaTG – PowerPoint PPT presentation

Number of Views:139
Avg rating:3.0/5.0
Slides: 48
Provided by: Tatia78
Category:

less

Transcript and Presenter's Notes

Title: Interaction of a Large Amplitude Interfacial Solitary Wave of Depression with a Bottom Step


1
The 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
2
Observations 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)
3
Theory 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
4
Cauchy Problem - Method of Inverse Scattering
5
Cauchy Problem
First Step t 0
Direct Spectral Problem
spectrum
Discrete spectrum solitons (real roots,
breathers (imaginary roots) Continuous spectrum
wave trains
6
Gardners 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
7
Positive and Negative Solitons
cubic, a1
Positive algebraic soliton
Negative algebraic soliton
quadratic a
Positive Solitons
Negative Solitons
Sign of the cubic term is principal!
8
Soliton interaction in KdV
9
Soliton interaction in Gardner, ?1 lt 0
10
Soliton interaction in Gardner, ?1 gt 0
11
Gardners 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
12
Gardner Breathers
?im? 0
?real gt ?im
?real lt ?im
13
Breathers positive cubic term
?1 gt 0
14
Breathers positive cubic term
b gt 0
15
Numerical (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
16
(No Transcript)
17
Envelopes and Breathers
Weak Nonlinear Groups
18
Nonlinear Schrodinger Equation
cubic, a1
cubic, ?
focusing
breathers
breathers
Envelope solitons
quadratic, a
defocusing
19
Transition Zone
(? ? 0)
Modified Schrodinger Equation
20
Modulation 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
21
Modulation 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.
22
X T diagram for internal rogue waves heights
exceeding level 1.2 for the initial maximal
amplitude 0.32
23
South China Sea
a
a1
There are large zones of positive cubic
coefficients !!!!
24
Quadratic nonlinearity, a, s-1
Arctic Ocean
Cubic nonlinearity, a1, m-1s-1
25
Horizontally variable background
H(x), N(z,x), U(z,x) 0 (input)
x

Q - amplification factor of linear long-wave
theory
Resulting model
26
WaveEvolutionon Malin Shelf
27
COMPARISON
Computing (with symbols) and Observed
2.2 km
5.2 km
6.1 km
28
Portuguese shelf
Blue line observation, black line - modelling
26.3 km
13.6 km
29
Section and coefficients
30
Focusing case
We put w 0.01 s-1
31
South China Sea
w 0.01
A 30m
32
(No Transcript)
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
Comparison with a1 0
a1 0
a1 gt 0
130 km
130 km
323 km
323 km
38
Baltic sea
Red zone is a1 gt 0
39
Focusing case
We put w 0.01 s-1
40
A0 6 m
41
(No Transcript)
42
(No Transcript)
43
No linear amplification Q 1
44
(No Transcript)
45
A0 8 m
46
Estimations 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
47
Conclusion
  • 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
Write a Comment
User Comments (0)
About PowerShow.com