Tatiana TalipovaTatiana Talipova

in collaboration within collaboration withEfim Pelinovsky, Oxana Kurkina, Roger Efim Pelinovsky, Oxana Kurkina, Roger Grimshaw, Anna Sergeeva, Kevin Lamb Grimshaw, Anna Sergeeva, Kevin Lamb

Institute of Applied Physics, Institute of Applied Physics, Nizhny Novgorod, RussiaNizhny Novgorod, Russia

The modulational The modulational instability of long instability of long

internal wavesinternal waves

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

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

Alfred OsbornAlfred Osborn““Nonlinear Ocean

Waves & the Inverse Scattering Transform”,

2010

Theory for long waves of Theory for long waves of moderate amplitudesmoderate amplitudes

03

32

1

x

u

x

uu

x

uu

t

u

•Full Integrable Model

Reference systemReference systemOne mode (mainly the first)One mode (mainly the first)

Gardner equationGardner equation

Coefficients are the functions of the ocean stratification

03

32

1

x

u

x

uu

x

uu

t

u

LAALALt

L

][

),(,/ txuxL

),(,/ txuxAt

xu

uxL

2

12

1

Cauchy Problem - Method of Inverse Scattering

First Step: t = 0

L

)0,(xu spectrumspectrum

DiscreteDiscrete spectrum – spectrum – solitons (real solitons (real roots, breathers (imaginary roots)roots, breathers (imaginary roots)

ContinuousContinuous spectrum – spectrum – wave trainswave trains

Direct Spectral ProblemDirect Spectral Problem

Cauchy Problem

xu

uxL

2

12

1

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

Soliton interaction in KdV

Soliton interaction in Gardner, 1 < 0

Soliton interaction in Gardner, 1 > 0

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

ikltgiΨΦ

21

)(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 term> 0> 0

Breathers: Breathers: positive cubic termpositive cubic term > 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

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,

0focusingfocusing

defocusingdefocusing

Envelope solitonsEnvelope solitons

breathersbreathers breathersbreathers

2

2

1 6 k

cubic,cubic, kL 3

Transition ZoneTransition Zone (( 0) 0)

Modified Schrodinger EquationModified Schrodinger Equation

k3 23

2

k

3

22

21

9

2

k

AA

61

*

1211

411

212

12

1 ||||A

AiAAAAAA

i

Modulation Instability only for positiveModulation Instability only for positive cubic,

focusingfocusing0

breathersbreathersbreathersbreathers

Wave groupWave groupof large amplitudesof large amplitudes

Wave groupWave groupof large amplitudesof large amplitudes

Wave groupWave groupof weak amplitudesof weak amplitudes

31

2

4

||

crAA

cubic,cubic,

quadratic, quadratic,

Modulation instability of internal wave packets (mKdV model)

Formation of IW of large amplitudesFormation of IW of large amplitudesGrimshaw 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 X – T diagram for internal rogue waves heights exceeding level 1.2 for the initial maximal amplitude exceeding level 1.2 for the initial maximal amplitude

0.320.32

South China SeaSouth China Sea

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

Cubic nonlinearity, Cubic nonlinearity, mm-1-1ss-1-1

Quadratic nonlinearity, Quadratic nonlinearity, ss-1-1

Arctic OceanArctic Ocean

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

WaveWaveEvolutionEvolutionon Malin on Malin

ShelfShelf

COMPARISONComputing (with symbols) and

Observed

4 6 8 1 0

tim e , h r

-3 0

-2 0

-1 0

0

1 0

2 0

elev

atio

n, m

4 6 8 1 0

tim e , h r

-4 0

-2 0

0

2 0

elev

atio

n, m

4 5 6 7 8 9

tim e , h r

-4 0

-2 0

0

2 0

elev

atio

n, m

2.2 km 5.2 km

6.1 km

Portuguese shelfPortuguese shelf

1 2 3 4

tim e , h r

-4 0

-3 0

-2 0

-1 0

0

1 0

elev

atio

n, m

0 1 2 3 4

tim e , h r

-3 0

-2 0

-1 0

0

1 0

elev

atio

n, m

Blue line – observation, black line - modelling

13.6 km 26.3 km

Section and coefficients

0 100 200 300x,км

1000

500

глу

би

на,

м

0

1

2

3

c,

м/с

-0.01

0

0.01

,

сек

-1

0

100000

200000

, м

3 /с

0.00004

0.00008

1,

м-1

с-1

0

1

2

Q

16 crkk

16

ccr

Focusing caseFocusing case

We put

== s s-1-1

0 100 200 300 400x , k m

0

0.001

0.002

0.003

cr sec -1

South China SeaSouth China Sea

0 100 200 300x

1000

500

dep

th, m

0

1

2

3

c, m

/s

0 .01

0

-0.01

,

s-1

0

100000

200000

, m

3/s

0.00004

0.00008

1,

m-1

s-1

0

1

2

Q

0 4 8 12

t, hour

-80

-40

0

40

80, m

A = 30m 0.01

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 100 200 300x

1000

500

dep

th, m

0

1

2

3c,

m/s

0 .01

0

-0.01

,

s-1

0

100000

200000

, m

3/s

0.00004

0.00008

1,

m-1

s-1

0

1

2

Q

0 100 200 300x

1000

500

dep

th, m

0

1

2

3c,

m/s

0 .01

0

-0.01

,

s-1

0

100000

200000

, m

3/s

0.00004

0.00008

1,

m-1

s-1

0

1

2

Q

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 100 200 300x

1000

500

dep

th, m

0

1

2

3

c, m

/s

0 .01

0

-0.01

,

s-1

0

100000

200000

, m

3/s

0.00004

0.00008

1,

m-1

s-1

0

1

2

Q

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 100 200 300x

1000

500

dep

th, m

0

1

2

3

c, m

/s

0 .01

0

-0.01

,

s-1

0

100000

200000

, m

3/s

0.00004

0.00008

1,

m-1

s-1

0

1

2

Q

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 100 200 300x

1000

500

dep

th, m

0

1

2

3

c, m

/s

0 .01

0

-0.01

,

s-1

0

100000

200000

, m

3/s

0.00004

0.00008

1,

m-1

s-1

0

1

2

Q

Comparison with Comparison with = 0= 0

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 4 8 12

t, hour

-80

-40

0

40

80, m

130 km

= 0= 0 > 0> 0

130 km

0 4 8 12

t, hour

-80

-40

0

40

80, m

0 4 8 12

t, hour

-80

-40

0

40

80, m

323 km323 km

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

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

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

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

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

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

0 40 80 120 160x0

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

AA00 = 8 m = 8 m

0 2 4 6

t, час

-40

-20

0

20

40, м

72 км

Estimations of instability lengthEstimations of instability length

2

22~

A

cLins

2

2

1 6

c

South China SeaSouth China Sea

LLinsins~ 0.6 km~ 0.6 kmLLinsins~ 60 km~ 60 kmStart pointStart point Last pointLast point

Baltic SeaBaltic SeaCentral pointCentral point Last pointLast point

LLinsins~ 5 km~ 5 km LLinsins~ 600 km~ 600 km

Conclusion:Conclusion:

• Modulational instability is possible for Modulational instability is possible for Long Long Sea Internal Waves Sea Internal Waves on “shallow” water.on “shallow” water.• Modulational instability may take place Modulational instability may take place when the background stratification leads to when the background stratification leads to the the positive cubic nonlinear termpositive cubic nonlinear term..• Modulational instability of large-amplitude Modulational instability of large-amplitude wave packets results in wave packets results in rogue waverogue wave formationsformations