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
Institute of Mathematical Machine and
System Problems, Kiev Ukraine
UK
Do internal solitons exist in
the ocean?
Lev Ostrovsky, Yury Stepanyants, 1989
INTERNAL SOLITARY WAVE RECORDS
Marshall H. Orr and
Peter C. Mignerey,
South China sea
J Small, T Sawyer, J.Scott,
SEASAME
Malin Shelf Edge
Nothern Oregon
Observations of Internal Waves of Huge Amplitudes
Internal waves in time-series in the South China Sea (Duda et al., 2004)
Where internal solitons have been
reported (courtesy of Jackson)
The horizontal ADCP velocities (Lee et al, 2006)
Internal Solitary Waves on the Ocean
Shelves
• 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
Mode structure
Brunt - Vaisala, frequency, sec-1
0
0.004
0.008
0.012
0
0.016
-1
-0.5
0
0.5
1
(z)
0
N(z)
40
40
Second mode
80
80
120
120
Z,
160 м
d
dz
First mode
160
Eigenvalue problem for  and c

2 d 
2
(
c

U
(
z
))

N
(
z
)
  0,


dz 

(0)  ( H )  0
 max  1
Theory for long waves of moderate
amplitudes
u
u
u
2 u
 u
 1u
 3 0
t
x
x
x
3
Gardner equation
•Full Integrable Model
Coefficients are the functions of the ocean stratification
Gardner’s Solitons
A
u ( x, t ) 
1  Bcosh( ( x  Vt )),
sign of 1
2
6 
A
,

B  1
2
1
<0
Limited amplitude
alim = /1
1
>0
V  
61 

2
2
2
A
a
1 B
Two branches of solitons of both polarities,
algebraic soliton alim = -2 /1
,
cubic, 1
Positive and Negative
I
Solitons
1
II
Positive
algebraic
soliton
Negative
algebraic
soliton

quadratic
α
III
IV
Negative Solitons
Positive Solitons
Sign of the cubic term is principal!
Gardner’s Breathers
cubic, 1 > 0
 = 1,  = 12q, 1 = 6, where q is arbitrary)

lch(Ψ )cos( ) - kcos(Φ)sh(κ )
u2
at an
x
lsh (Ψ ) sin( )  k sin(Φ)ch(κ )
 and  are the phases of carrier wave and envelope
  k ( x  wt )   0 , κ  l ( x  vt)  κ 0
propagating with speeds
w  k 2  3l 2 , v  3k 2  l 2
There are 4 free parameters: 0 , 0 and two energetic parameters
Pelinovsky D&Grimshaw, 1997
 l  ik 
Φ  iΨ  tan 

2
q


1
sh(2Ψ )
sin(2Φ)
k q 2
lq 2
2
2
2
2
2
2
cos (Φ)ch (Ψ )  sin (Φ)sh (Ψ )
cos (Φ)ch (Ψ )  sin (Φ)sh (Ψ )
Gardner Breathers
im→ 0
4
real > im
4
2
ui
0
real < im
2
 3.803
4
10
 10
8
6
4
2
0
xi
2
4
6
8
10
10
Breathers: positive cubic term
1 > 0
Breathers: positive cubic term
1 >
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, 046306
Do Internal Breathers Exist in
the Ocean?
Why IBW do not obserwed?
1 > 0 Grimshaw, Pelinovsky, Talipova,
NPG, 1997
South China Sea
1

There are large zones of positive cubic coefficients !!!!
Nonlinear Internal Waves From the
Luzon Strait
Eos, Vol. 87, No. 42, 17 October 2006
Russian Arctic
Sign variability for
quadratic nonlinearity is
ordinary occurance on the
ocean shelves

1
Positive values for the
cubic nonlinearity are
not too exotic on the
ocean shelves
Lee, Lozovatsky et al., 2006
Alfred Osborn
“Nonlinear Ocean Waves & the Inverse Scattering
Transform”, 2010
Mechanizms
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
decreases
Modulation instability of internal wave group
Transformation of the solitary wave of the second mode
through the bottom step
Quadratic nonlinear coefficient changes the sign
Breather formation at the
end of transient zone
1 = 0.2
Grimshaw, Pelinovsky, Talipova
Physica D, 1999
=+1
15
0
-15
15
=0
0
-15
15
 = - 0.6
0
-15
15
=-1
0
-15
225
250
275
x
300
230
240
250
260
x
270
280
290
Horizontally variable background
H(x), N(z,x), U(z,x)
x
0 (input)
 ( , x) 
dx
 t
, xx
c( x)
 ( , x)
Q( x)
Q - amplification factor of
linear long-wave theory
Resulting model
Q
c02  (c0  U 0 )( d 0 / dz ) 2 dz
c 2  (c  U )( d / dz ) 2 dz
ξ
αQ
α1Q 2 2 ξ
β  3ξ
( 2 ξ  2 ξ )
 4
0
3
x
c
c
τ c τ
c, m/s
Model parameters on the North West
Australian shelf
2
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.
1
, m3 /s
0
12000
6000
0
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
-0.012
0.0004
1 , m s
-1 -1
-1
, s
0
0.012
0
-0.0004
Q
-0.0008
4
2
H,m
0
0
500
0
40
80
x, km
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
North West Australian shelf
Modulation Instability of
Long IW
Grimshaw, D Pelinovsky, E. Pelinovsky,
Talipova, Physica D, 2001
Envelopes and Breathers
Weak Nonlinear Groups
u( x, t )  A( , ) exp(i ) 
 A2 ( , ) exp(2i )  A0 ( , )  c.c.
2
  kx  t
   ( x  cgr t )
2
  t
2
ε << 1


2
2
A2 
A
A0  
A
2
2
6k
3k
Nonlinear Schrodinger Equation
A
 A
2
i
 3k 2  k | A | A


2
 0

  1 
2
6k
2
cubic,
cubic, 1
focusing
Wave group
of large amplitudes
Wave group
of weak amplitudes
| | 2
A > Acr 
3
41
Wave group
of large amplitudes
quadratic, 
Bendjamin- Feir instability in the
mKdV model
 0
1 > 0
(x,0) = a(1+mcosKx)coskx
Twenty satellites
Twenty satellites just fulls the condition for a narrow initial spectrum. The
evolution of the wave field with Amax = 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 = 400
R. Grimshaw, E. Pelinovsky, T. Taipova, and A. Sergeeva, European
Physical Journal, 2010
Amax = 1.2
t=0
t = 150
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.
SAR Images of IW on the Baltic Sea
а
б
Red zone is 1 > 0
1, m-1s-1
Baltic sea
0.00040
0.00000
-0.00040
, s-1
0.01
0.005
Q
0
2
1
0
, m3/s
1000
500
0
c, m/s
1
0.5
depth, m
0
0
30
60
0
40
80
120
160
x
1, m-1s-1
Focusing case
We put
 = 0.01 s-1
0.00000
-0.00040
0.01
, s-1
cr, sec
-1
0.00040
0.016
0.005
Q
0
2
1
0.012
0
, m3/s
1000
0.008
500
0
c, m/s
1
0.004
0.5
depth, m
0
0
0
40
80
x, km
120
160
0
30
60
0
40
80
120
160
x
0.00040
0.00000
-0.00040
0.01
, m
, s-1
20
1, m-1s-1
A0 = 6 m
0.005
Q
0
2
10
1
0
, m3/s
1000
0
500
0
c, m/s
1
-10
0.5
depth, m
0
-20
0
2
4
t, hour
6
0
30
60
0
40
80
120
160
x
1, m-1s-1
0.00000
-0.00040
0.01
, m
, s-1
20
0.00040
0.005
Q
0
2
10
1
0
, m3/s
1000
0
500
0
c, m/s
1
-10
0.5
-20
0
2
4
t, hour
6
depth, m
0
0
30
60
0
40
80
120
160
x
1, m-1s-1
No linear amplification Q ~ 1
0.00000
-0.00040
0.01
, s-1
20
, m
0.00040
0.005
0
2
Q
10
1
0
, m3/s
1000
0
500
0
c, m/s
1
-10
0.5
-20
0
2
4
t, hour
6
depth, m
0
0
30
60
0
40
80
120
160
x
Interaction of interfacial
solitary wave of the second
mode with bottom step
Terletska, Talipova, Maderich, Grimshaw,
Pelinovsky
In Progress
Numerical tank
Breaking parameter h2+/|ai |
 = 12 cm, H = 23 cm
b = 2.17
Slow soliton and
some breathers of the
first mode plus
intensive solitary
wave of the second
mode are formed
after the step
CONCLUSIONS
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