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Tatiana Talipova in collaboration with

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. Observations of Internal Waves of Huge Amplitudes.

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Tatiana Talipova in collaboration with

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  1. The modulational instability of long internal waves Tatiana Talipova in collaboration with EfimPelinovsky, OxanaKurkina, 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) Thehorizontal 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. sign ofa1 Gardner’s Solitons a1 < 0 Limited amplitude alim = -a/a1 a1 > 0 Two branches of solitons of bothpolarities, algebraic soliton alim=-2 a/a1

  7. cubic, a1 Positive and Negative Solitons Positive algebraic soliton Negative algebraic soliton quadratic α NegativeSolitons Positive Solitons Sign of the cubic term is principal!

  8. Soliton interaction in KdV

  9. Soliton interaction in Gardner, 1 < 0

  10. Soliton interaction in Gardner, 1 > 0

  11. cubic, a1 > 0 Gardner’s Breathers b = 1, = 12q, a1 = 6, whereq isarbitrary) andare the phases of carrierwave 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> im real< im

  13. Breathers: positive cubic term 1> 0

  14. Breathers: positive cubic term b > 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. Envelopes and Breathers Weak Nonlinear Groups

  17. Nonlinear Schrodinger Equation cubic,a1 cubic,  focusing breathers breathers Envelope solitons quadratic, a defocusing

  18. Transition Zone ( 0) Modified Schrodinger Equation

  19. Modulation Instability only for positiveb cubic,  cubic,b focusing breathers breathers Wave group of large amplitudes Wave group of large amplitudes Wave group of weak amplitudes quadratic, a

  20. 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.

  21. X – T diagram for internal rogue waves heights exceeding level 1.2 for the initial maximal amplitude 0.32

  22. South China Sea a a1 There are large zones of positive cubic coefficients !!!!

  23. Quadratic nonlinearity, a, s-1 Arctic Ocean Cubic nonlinearity, a1, m-1s-1

  24. Horizontally variable background H(x), N(z,x), U(z,x) 0 (input) x Q- amplification factor of linear long-wave theory Resulting model

  25. WaveEvolutionon Malin Shelf

  26. COMPARISON Computing (with symbols) and Observed 2.2 km 5.2 km 6.1 km

  27. Portuguese shelf Blue line – observation, black line - modelling 26.3 km 13.6 km

  28. Section and coefficients

  29. Focusing case We put w= 0.01 s-1

  30. South China Sea w = 0.01 A = 30m

  31. Comparison with a1= 0 a1 = 0 a1> 0 130 km 130 km 323 km 323 km

  32. Red zone is a1 > 0 Baltic sea

  33. Focusing case We put w= 0.01 s-1

  34. A0 = 6 m

  35. No linear amplification Q ~ 1

  36. A0 = 8 m

  37. 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

  38. 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

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