breathers of the internal waves
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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