Tatiana Talipova in collaboration with Efim Pelinovsky, Oxana Kurkina, Roger Grimshaw, Anna Sergeeva, Kevin Lamb Institute of Applied Physics, Nizhny Novgorod, Russia The modulational instability of long internal waves

Internal waves in time-series in the South China Sea (Duda et al., 2004) The horizontal ADCP velocities (Lee et al, 2006) Observations of Internal Waves of Huge Amplitudes Alfred Osborn “ “Nonlinear Ocean Waves & the Inverse Scattering Transform”, 2010

Theory for long waves of moderate amplitudes Full Integrable Model Reference system One mode (mainly the first) Gardner equation Coefficients are the functions of the ocean stratification

Cauchy Problem - Method of Inverse Scattering

First Step: t = 0 spectrum Discrete spectrum – solitons (real roots, breathers (imaginary roots) Continuous spectrum – wave trains Direct Spectral Problem Cauchy Problem

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

Soliton interaction in KdV

Soliton interaction in Gardner,  1 < 0

Soliton interaction in Gardner,  1 > 0

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    > 0

Breathers: positive cubic term  > 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,

Weak Nonlinear Groups Envelopes and Breathers

Nonlinear Schrodinger Equation cubic, quadratic,  focusing defocusing Envelope solitons breathers breathers cubic,  

Transition Zone (   0) Modified Schrodinger Equation

Modulation Instability only for positive  cubic, focusing breathers breathers Wave group of large amplitudes Wave group of large amplitudes Wave group of weak amplitudes cubic,  quadratic, 

Modulation instability of internal wave packets (mKdV model) Formation of IW of large amplitudes Grimshaw R., Pelinovsky E., Talipova T., Ruderman M., Erdely R., 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.

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

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

Cubic nonlinearity,    m -1 s -1 Quadratic nonlinearity,  s -1 Arctic Ocean

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

Wave Evolution on Malin Shelf

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

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

Section and coefficients

Focusing case We put   =  s -1

South China Sea A = 30m  0.01

Comparison with   = km   = 0  >0 >0 >0 >0 130 km 323 km

Baltic sea Red zone is   > 0

Focusing case We put   =  s -1

A 0 = 6 m

No linear amplification Q ~ 1

A 0 = 8 m

Estimations of instability length South China Sea L ins ~ 0.6 km L ins ~ 60 km Start point Last point Baltic Sea Central point Last point L ins ~ 5 km L ins ~ 600 km

Conclusion: Modulational instability is possible for Long Sea Internal Waves on “shallow” water. 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 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 Modulational instability of large-amplitude wave packets results in rogue wave formations