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 Institute of Mathematical Machine and System Problems, Kiev Ukraine UK Nizhny Novgoro Technical University

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

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

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) Observations of Internal Waves of Huge Amplitudes

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

Mode structure Eigenvalue problem for  and c  (z) First mode Second mode Brunt - Vaisala, frequency, sec -1 Z, м

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

Limited amplitude a lim =     < 0   > 0 sign of   Gardner’s Solitons Two branches of solitons of both polarities, algebraic soliton a lim = -   /  

cubic,  1 quadratic α Positive Solitons Negative Solitons Negativealgebraicsoliton Positivealgebraicsoliton Sign of the cubic term is principal! Positive and Negative Solitons

Gardner’s Breathers cubic,   > 0  = 1,  = 12q,   = 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

Gardner Breathers im → 0 im → 0 real    im real    im real    im real    im

Breathers: positive cubic term  1 > 0

Breathers: positive cubic term 1 >01 >01 >01 >0

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,

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

  South China Sea There are large zones of positive cubic coefficients !!!!

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

Russian Arctic   Positive values for the cubic nonlinearity are not too exotic on the ocean shelves Sign variability for quadratic nonlinearity is ordinary occurance on the ocean shelves

Lee, Lozovatsky et al., 2006

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

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 Mechanizms

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

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

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), Grimshaw, R., Pelinovsky, E., and Talipova, T. Modeling Internal solitary waves in the coastal ocean. Survey in Geophysics, 2007, 28, 2, 273

Internal soliton transformation on the North West Australian shelf

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

Weak Nonlinear Groups Envelopes and Breathers

Nonlinear Schrodinger Equation cubic, quadratic,  focusing cubic,   Wave group of weak amplitudes Wave group of large amplitudes Wave group of large amplitudes

Bendjamin- Feir instability in the mKdV model  x  = a(1+mcosKx)coskx   1 > 0

Twenty satellites A max = 0.5 Twenty satellites just fulls the condition for a narrow initial spectrum. The evolution of the wave field with A max = 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

A max = 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.

абаб SAR Images of IW on the Baltic Sea

Baltic sea Red zone is   > 0

Focusing case We put   =  s -1

A 0 = 6 m

No linear amplification Q ~ 1

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

Numerical tank h 2+ /|a i | Breaking parameter h 2+ /|a i |

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

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