1 / 44

Breathers of the Internal Waves

Breathers of the Internal Waves. Tatiana Talipova. in collaboration with Roger Grimshaw , Efim Pelinovsky , Oxana Kurkina , Katherina Terletska , Vladimir Maderich. Institute of Applied Physics RAS Nizhny Novgorod, Russia. Nizhny Novgoro Technical University.

jamar
Download Presentation

Breathers of the Internal Waves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Breathers of the Internal Waves Tatiana Talipova in collaboration with Roger Grimshaw, EfimPelinovsky, OxanaKurkina, KatherinaTerletska, Vladimir Maderich Institute of Applied Physics RAS Nizhny Novgorod, Russia Nizhny Novgoro Technical University Institute of Mathematical Machine and System Problems, Kiev Ukraine UK

  2. Do internal solitons exist in the ocean? Lev Ostrovsky, Yury Stepanyants, 1989

  3. INTERNAL SOLITARY WAVE RECORDS Marshall H. Orr and Peter C. Mignerey, South China sea J Small, T Sawyer, J.Scott, SEASAME Malin Shelf Edge Nothern Oregon

  4. Observations of Internal Waves of Huge Amplitudes Internal waves in time-series in the South China Sea (Duda et al., 2004) Where internal solitons have been reported (courtesy of Jackson) The horizontal ADCP velocities (Lee et al, 2006)

  5. Internal Solitary Waves on the Ocean Shelves • Most intensive IW had been observed on the ocean shelves • Shallow water, long IW, vertical mode structure • There is no the Garrett-Munk spectrum • There is 90% of presence of the first mode

  6. Mode structure Brunt - Vaisala, frequency, sec-1 F(z) Second mode First mode Z, м Eigenvalue problem for Fand c

  7. Theory for long waves of moderate amplitudes Gardner equation • Full Integrable Model Coefficients are the functions of the ocean stratification

  8. 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

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

  10. 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 PelinovskyD&Grimshaw, 1997

  11. Gardner Breathers im→ 0 real> im real< im

  12. Breathers: positive cubic term 1 > 0

  13. Breathers: positive cubic term 1 > 0

  14. 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

  15. Do Internal Breathers Exist in the Ocean? Why IBW do not obserwed? 1 > 0 Grimshaw, Pelinovsky, Talipova, NPG, 1997

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

  17. Nonlinear Internal Waves From the Luzon Strait Eos, Vol. 87, No. 42, 17 October 2006

  18. Russian Arctic Sign variability for quadratic nonlinearity is ordinary occurance on the ocean shelves a a1 Positive values for the cubic nonlinearity are not too exotic on the ocean shelves

  19. Lee, Lozovatsky et al., 2006

  20. Alfred Osborn “Nonlinear Ocean Waves & the Inverse Scattering Transform”, 2010

  21. Mechanizms • Solitary wave transformation through the critical points • Breather as the secondary wave is formed from solitary wave of opposite polarity when the quadratic nonlinear coefficient changes the sign • Breather is formed from solitary wave of opposite polarity when the positive cubic nonlinear coefficient decreases • Modulation instability of internal wave group • Transformation of the solitary wave of the second mode through the bottom step

  22. Quadratic nonlinear coefficient changes the sign Breather formation at the end of transient zone 1 = 0.2 Grimshaw, Pelinovsky, Talipova Physica D, 1999

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

  24. Model parameters on the North West Australian shelf Holloway P., Pelinovsky E., Talipova T., Barnes B. A Nonlinear Model of Internal Tide Transformation on the Australian North West Shelf, J. Physical Oceanography, 1997, 27, 6, 871. Holloway P, Pelinovsky E., Talipova T. A Generalized Korteweg - de Vries Model of Internal Tide Transformation in the Coastal Zone, 1999, J. Geophys. Res.,104(C8), 18333 Grimshaw, R., Pelinovsky, E., and Talipova, T. Modeling Internal solitary waves in the coastal ocean. Survey in Geophysics, 2007, 28, 2, 273

  25. Internal soliton transformation on the North West Australian shelf

  26. Modulation Instability of Long IW Grimshaw, D Pelinovsky, E. Pelinovsky, Talipova, Physica D, 2001

  27. Envelopes and Breathers Weak Nonlinear Groups

  28. Nonlinear Schrodinger Equation cubic,  cubic,1 focusing Wave group of weak amplitudes Wave group of large amplitudes Wave group of large amplitudes quadratic, a

  29. Bendjamin- Feir instability in the mKdV model a = 0a1 > 0 h(x,0) = a(1+mcosKx)coskx

  30. Twenty satellites Twenty satellites just fulls the condition for a narrow initial spectrum. The evolution of the wave field with Amax = 0.5 is displayed below. The initial wave field consists of eight modulated groups of different amplitudes and each group contains 9-15 individual waves. t = 0, t = 400 R. Grimshaw, E. Pelinovsky, T. Taipova, and A. Sergeeva, European Physical Journal, 2010

  31. Amax = 1.2 t = 0t = 150 An increase of the initial amplitude leads to more complicated wave dynamics. The breathers formed here are narrower than in the previous case (3 - 5 individual waves). The largest waves here are two individual waves, and are not a wave group.

  32. SAR Images of IW on the Baltic Sea а б

  33. Red zone is a1 > 0 Baltic sea

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

  35. A0 = 6 m

  36. No linear amplification Q ~ 1

  37. Interaction of interfacial solitary wave of the second mode with bottom step Terletska, Talipova, Maderich, Grimshaw, Pelinovsky In Progress

  38. Numerical tank Breaking parameter h2+/|ai |

  39. l = 12 cm, H = 23 cm b = 2.17 Slow soliton and some breathers of the first mode plus intensive solitary wave of the second mode are formed after the step

  40. CONCLUSIONS • Mechanisms of surface rogue wave • formation can be applied for internal • rogue wave formation • Dynamics of internal waves is more • various than dynamics of surface waves • Additional mechanisms of internal rogue • wave formation connected with variable • water stratification are exists

More Related