tatiana talipova in collaboration with efim pelinovsky, oxana kurkina, roger grimshaw, anna...
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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
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