Breathers of the Internal WavesBreathers of the Internal Waves

Tatiana TalipovaTatiana Talipovain collaboration within collaboration with

Roger Grimshaw, Efim Pelinovsky, Oxana Kurkina, Roger Grimshaw, Efim Pelinovsky, Oxana Kurkina, Katherina Terletska, Vladimir MaderichKatherina Terletska, Vladimir Maderich

Institute of Applied Physics RASInstitute of Applied Physics RASNizhny Novgorod, RussiaNizhny Novgorod, Russia

Institute of Mathematical Machine and System Problems, Kiev Ukraine

UK

Nizhny Novgoro Technical UniversityNizhny Novgoro Technical University

Do internal solitons exist in Do internal solitons exist in the ocean? the ocean?

Lev Ostrovsky, Yury Stepanyants, 1989

INTERNAL SOLITARY WAVE RECORDSINTERNAL SOLITARY WAVE RECORDS

Marshall H. Orr and

Peter C. Mignerey, South China sea

Nothern Oregon

J Small, T Sawyer, J.Scott,

SEASAMEMalin Shelf Edge

Internal waves in time-series in the South China Sea (Duda et al., 2004)Internal waves in time-series in the South China Sea (Duda et al., 2004)

Where internal solitons have been Where internal solitons have been reported (courtesy of Jackson)reported (courtesy of Jackson) The The horizontal horizontal ADCP ADCP velocities (Lee et al, 2006)velocities (Lee et al, 2006)

Observations of Internal Waves of Huge AmplitudesObservations of Internal Waves of Huge Amplitudes

Internal Solitary Waves on the Ocean Internal Solitary Waves on the Ocean ShelvesShelves

• 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

- 1 - 0 . 5 0 0 . 5 1

1 6 0

1 2 0

8 0

4 0

0

Mode structure

,0)())(( 22

zNdz

dzUc

dz

d

0)()0( H max 1

Eigenvalue problem for Eigenvalue problem for and and cc

(z)(z)

First modeFirst mode

Second modeSecond mode

0 0 .004 0 .008 0 .012 0 .016

N (z)

160

120

80

40

0

Brunt - VaisalaBrunt - Vaisala, , frequency, secfrequency, sec-1-1

Z, м

Theory for long waves of moderate amplitudes

03

32

1

x

u

x

uu

x

uu

t

u

•Full Integrable Model

Gardner equationGardner equation

Coefficients are the functions of the ocean stratification

Limited amplitude Limited amplitude aalimlim = =

< 0< 0

> 0> 0

sign ofsign of Gardner’s Solitons

aA

B

1

)),((cosh1),(

VtxB

Atxu

2

2

212

2

,6

1

,6

V

B

A

Two branches of solitons of bothTwo branches of solitons of both polarities, polarities, algebraic soliton algebraic soliton aalimlim == --//

I II

III IV

cubic, cubic, 11

quadratic quadratic αα

Positive SolitonsPositive SolitonsNegativeNegative SolitonsSolitons

Negative Negative algebraic algebraic solitonsoliton

Positive Positive algebraic algebraic solitonsoliton

Sign of the cubic term is principal!

Positive and Negative Solitons

Gardner’s Breathers cubic, cubic, > 0 > 0

)(ch)sin()sin()(sh

))sh(cos(-))cos(ch(atan2

κΦkΨl

κΦkΨl

xu

00 )(,)( κvtxlκwtxk

== 1 1, , = = 1212qq, , = = 66, , wherewhere qq is is arbitraryarbitrary))

2222 3,3 lkvlkw

andand are the phases of carrierare the phases of carrier wave and envelopewave and envelope

propagating with speedspropagating with speeds

There are 4 free parameters: There are 4 free parameters: 00 ,, 00 and two energetic parameters and two energetic parameters

q

ikliΨΦ

2tan 1

)(sh)(sin)(ch)(cos

)2(sh2222 ΨΦΨΦ

Ψqk

)(sh)(sin)(ch)(cos

)2sin(2222 ΨΦΨΦ

Φql

Pelinovsky D&GrimshawPelinovsky D&Grimshaw, 1997, 1997

Gardner Breathers

10 8 6 4 2 0 2 4 6 8 104

2

0

2

44

3.803

ui

1010 xi

imim→ 0→ 0 realrealimim

realrealimim

Breathers: Breathers: positive cubic termpositive cubic term11 > 0 > 0

Breathers: Breathers: positive cubic termpositive cubic term11 > 0 > 0

Numerical (Euler Equations) Numerical (Euler Equations) modeling of breathermodeling of breather

K. Lamb, O. Polukhina, T. Talipova, E. Pelinovsky, W. Xiao, A. K. Lamb, O. Polukhina, T. Talipova, E. Pelinovsky, W. Xiao, A. Kurkin.Kurkin.

Breather Generation in the Fully Nonlinear Models of a Stratified Breather Generation in the Fully Nonlinear Models of a Stratified Fluid. Fluid. Physical Rev. E. Physical Rev. E. 2007, 75, 4, 0463062007, 75, 4, 046306

Why IBW do not obserwed?Why IBW do not obserwed?

Do Internal Breathers Exist in Do Internal Breathers Exist in the Ocean?the Ocean?

11 > 0 Grimshaw, Pelinovsky, > 0 Grimshaw, Pelinovsky,

Talipova, NPG, 1997Talipova, NPG, 1997

South China SeaSouth China Sea

There are large zones of positive cubic coefficients !!!!

Nonlinear Internal Waves From the Nonlinear Internal Waves From the Luzon StraitLuzon Strait

Eos, Vol. 87, No. 42, 17 October 2006

Russian ArcticRussian 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., 2006Lee, Lozovatsky et al., 2006

Alfred OsbornAlfred 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 decreasesModulation instability of internal wave groupTransformation of the solitary wave of the second mode through the bottom step

MechanizmsMechanizms

= + 1

= 0

= - 0 .6

= - 1

-15

0

15

-15

0

15

225 250 275 300x

-15

0

15

-15

0

15

11 = 0.2 = 0.2

2 3 0 2 4 0 2 5 0 2 6 0 2 7 0 2 8 0 2 9 0

x

Breather formation at the Breather formation at the end of transient zoneend of transient zone

Quadratic nonlinear coefficient changes the signQuadratic nonlinear coefficient changes the sign

Grimshaw, Pelinovsky, Talipova Physica D, 1999

Horizontally variable backgroundHorizontally variable background H(x), N(z,x), U(z,x)

0 (input)x

xxxc

dxt ,

)(

)(

),(),(

xQ

xx

Q - amplification factor of linear long-wave theory

dzdzdUcc

dzdzdUccQ

22

2000

20

)/)((

)/)((

Resulting model

03

3

42

2

21

2

τ

ξ

c

β

τ

ξ)ξ

c

Qαξ

c

αQ(

x

ξ

Model parameters on the North West Model parameters on the North West Australian shelfAustralian shelf

0 40 80 x, km

500

0

H, m

0

2

4

Q

0

1

2

c, m

/s

0

6000

12000

, m3

/s

-0.012

0

0.012

, s-1

-0.0008

-0.0004

0

0.0004

1, m

-1s-1

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

Internal soliton transformation on the Internal soliton transformation on the North West Australian shelfNorth West Australian shelf

Modulation Instability of Modulation Instability of Long IWLong IW

Grimshaw, D Pelinovsky, E. Pelinovsky, Talipova, Physica D, 2001

Weak Nonlinear GroupsWeak Nonlinear Groups

..),()2exp(),(

)exp(),(),(

022 ccAiA

iAtxu

tkx )(2 tcx gr

t2 1ε

Envelopes and BreathersEnvelopes and Breathers

222 6A

kA

2

20 3A

kA

Nonlinear Schrodinger EquationNonlinear Schrodinger Equation

AAkA

kA

i 22

2

||3

cubic,

quadratic, quadratic,

0focusinfocusingg

2

2

1 6 k

cubic,cubic,

Wave groupWave groupof weak amplitudesof weak amplitudesWave groupWave group

of large amplitudesof large amplitudesWave groupWave group

of large amplitudesof large amplitudes

31

2

4

||

crAA

Bendjamin- Feir instability in the Bendjamin- Feir instability in the mKdV modelmKdV model

x= a(1+mcosKx)coskx 1 1 > 0> 0

Twenty satellitesTwenty satellitesTwenty satellites just fulls the condition for a narrow initial spectrum. The evolution of the wave field with AAmaxmax = 0.5 = 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 = 0, t = t = 404000

R. Grimshaw, E. Pelinovsky, T. Taipova, and A. Sergeeva, European Physical Journal, 2010

AAmaxmax = = 1.21.2

t = 0t = 0 t = t = 151500

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.

аб

SAR Images of IW on the Baltic Sea

Baltic seaBaltic seaRed zone is Red zone is > 0> 0

0 40 80 120 160x

60

30

0

dep

th, m

0

0.5

1c,

m/s

0

0.005

0.01

,

s-1

0

500

1000

, m

3/s

-0.00040

0.00000

0.00040

1,

m-1

s-1

0

1

2

Q

Focusing caseFocusing caseWe put

== s s-1-1

0 40 80 120 160x , k m

0

0.004

0.008

0.012

0.016

cr sec -1

0 40 80 120 160x

60

30

0

dep

th, m

0

0.5

1c,

m/s

0

0.005

0.01

,

s-1

0

500

1000

, m

3/s

-0.00040

0.00000

0.00040

1,

m-1

s-1

0

1

2

Q

0 2 4 6

t, hour

-20

-10

0

10

20, m

AA00 = 6 m = 6 m

0 40 80 120 160x

60

30

0

dep

th, m

0

0.5

1

c, m

/s

0

0.005

0.01

,

s-1

0

500

1000

, m

3/s

-0.00040

0.00000

0.00040

1,

m-1

s-1

0

1

2

Q

0 40 80 120 160x

60

30

0d

epth

, m0

0.5

1

c, m

/s

0

0.005

0.01

,

s-1

0

500

1000

, m

3/s

-0.00040

0.00000

0.00040

1,

m-1

s-1

0

1

2

Q

0 2 4 6

t, hour

-20

-10

0

10

20, m

0 40 80 120 160x

60

30

0

dep

th, m

0

0.5

1

c, m

/s

0

0.005

0.01

,

s-1

0

500

1000

, m

3/s

-0.00040

0.00000

0.00040

1,

m-1

s-1

0

1

2

Q

0 2 4 6

t, hour

-20

-10

0

10

20, m

No linear amplification Q ~ 1No linear amplification Q ~ 1

Interaction of interfacial Interaction of interfacial solitary wave of the second solitary wave of the second

mode with bottom stepmode with bottom step

Terletska, Talipova, Maderich, Grimshaw, Pelinovsky

In Progress

Numerical tankNumerical tank

Breaking parameter hh2+2+/|/|aaii | |

b = 2.17b = 2.17

= 12 cm, H = 23 cm= 12 cm, H = 23 cm

Slow soliton and Slow soliton and some breathers some breathers of the of the first modefirst mode plus plus intensive solitary intensive solitary wave of the wave of the second second modemode are formed are formed after the stepafter the step

CONCLUSIONSCONCLUSIONS

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