Fourier Spectrum of Riemann Waves
Efim Pelinovsky
Elena Tobish (Kartashova)
Tatiana Talipova
Dmitry Pelinovsky
Institute for Analysis
Institute of Applied Physics, Nizhny Novgorod, Russia
State Technical University, Nizhny Novgorod, Russia
Wave Interactions WIN-2014, Linz, Austria, 23-36 April 2014
Motivation
Tsunami Wave
Shapes
at
Japanese Coast
26 May 1983
Japan Sea
(Shuto, 1983)
Internal Wave Observations
Marshall H. Orr and Peter
C. Mignerey,
South
China sea
J Small, T Sawyer, J.Scott,
SEASAME
Malin Shelf Edge
Nothern Oregon
Weakly Nonlinear Riemann Waves

t
 V ( )

x
0
V (  )    
2
Coefficients can have either sign
 ( x , t )  F [ x  V (  )t ]
Wave Steepness

x

dF / dx
1
dV
dx
t
Riemann Wave
First Wave Breaking
T 
1
(  dV / dx ) max
Initial Sine Disturbance
F ( x )  A sin( kx )
Breaking point location
>0
CQ 
A

<0
Cubic/quadratic nonlinear ratio
Breaking Time
|  | kA
( = 0)
T3 
( = 0)
1
|  | kA
2
1
0 .8
Cubic nonlinearity
reduces
breaking time
0 .6
T /T 2
T2 
1
0 .4
0 .2
0
0
2
4
6
|C Q |
8
10
Time evolution of the wave shape
1
c u b ic
t= 0
t= T
1
q u a d ra tic
t= 0
t= T
0 .5
 /A
 /A
0 .5
0
0
-0 .5
-0 .5
-1
-1
0
0
0 .5
k x /2 
0 .4
0 .5
1
k x /2 
2
0 .3
0 .4
0 .5
1
n e g a tiv e
p o s itiv e
1
0 .2


Cubic
nonlinearity
0
0
-1
CQ = 0.4
-0 .2
-2
-0 .4
0
2
4
kx
6
0
2
4
kx
6
Fourier spectrum of a nonlinearly deformed wave
a 0 (t )
( x, t ) 


2
Change of variable
S n (t ) 
S n (t ) 
2 / k

n
iA
n
0
2
dF
dy
n
( t ) cos( nkx )  b n ( t ) sin( nkx ) 
n 1
S n ( t )  a n  ib n 
i
 a
2 / k
k

  ( x , t ) exp inkx dx
0
y  x  V ( ) t
F(y) = A sin(ky)
exp ink  y  V ( F ) t dy

Explicit formula
 cos x exp in x   kA sin x   kA sin
0
Implicit formula
2
2

x t dx
Final formula
Quadratic Nonlinearity
S n  (  1)
n 1
2 iA
n  kAt
J n ( n  kAt )
Bessel-Fubinni Series
(Nonlinear Acoustics)
0
-8 /3
Power asymptotics
at breaking time
lo g (E )
-1 0
Q u a d ra tic
1 /4
1 /2
3 /4
1
-2 0
S(k) ~ k-4/3
from Tbr
-3 0
1
10
n
E = (S/A)2
100
E(k) ~ k-8/3
Cubic Nonlinearity
Sm 
iA
2m  1
q ( t )   ktA / 2
2
exp[ i ( 2 m  1) q ]J m ([ 2 m  1] q )  iJ m  1 ([ 2 m  1] q )
0
AGAIN
Power asymptotics
at breaking time
-8 /3
lo g (E n )
-1 0
S(k) ~ k-4/3
C u b ic
1 /4
1 /2
3 /4
1
-2 0
E(k) ~ k-8/3
-3 0
1
10
n
100
Quadratic – cubic nonlinearity
0
0
C u b ic -q u a d ra tic
B re a k in g
C u b ic -q u a d ra tic
B re a k in g
-2
lo g (E )
lo g (E )
-2
-4
CQ = 1
-4
CQ = 0.2
-6
-6
1
10
100
1
10
100
n
n
Power asymptotics
at breaking time
0
C u b ic -q u a d ra tic
B re a k in g
-2
lo g (E )
S(k) ~ k-4/3
-4
CQ = 5
E(k) ~ k-8/3
-6
1
10
n
100
Universal Spectrum Asymptotics
S(k) ~ k-4/3
means existence of singularity in wave shape
 ( x , t ) ~ ( x  x br )
1/ 3
Proof:

t
 V ( )

x
V
0
Or in equivalent form
t
dx
dt
V
V
dV
dt
V
x
0
0
Riemann Wave Solution
V ( t )  F ( )
x ( t )    tF (  )
Breaking coordinate
x br   br  TF ( br )
where  br is extreme of dF / d 
see, breaking time
T 
1
(  dV / dx ) max

1
(  dF / d  ) max
Decomposition
   br  
  0
Taylor series of Riemann wave in the vicinity of breaking
x   br   TF ( br   )   br   
2
2
3
3


dF
 d F
 d F
 T  F ( br )  
( br ) 
( br ) 
( br )  ... 
2
3
d
2 d
6 d


Finally
x  x br
 3 d 3F

T
( br ) 
3
 6 d

So


6
  3
3 

d
/
F
Td


1/ 3
( x  x br )
1/ 3
Similar Taylor series for function, V
V ( x , T )  F ( br   )  F ( br )  
dF
d
( br )  ...  F ( br ) 

T
In breaking point the wave shape has a singularity
V ( x , t )  V br 
3
x  x br
This singularity leads to power spectrum
The same for η due to V(η)
k
4 / 3
Rigorous Results for Shape Singularity in Riemann Wave
1. Sulem, C. Sulem, P.-L., Frisch H. Tracing complex singularities with spectral
methods. J. Computational Physics, 1983, v. 50, 138-161.
2. Dubrovin B. On Hamiltonian perturbations of hyperbolic systems of conservation laws,
II: Universality of critical behaviour. Commun. Math. Phys., 2006, vol. 267, 117-139.
3. Pomeau Y., Jamin T., Le Bars M., Le Gal P., and Audoly B. Law of spreading of the
crest of a breaking wave. Proc. Royal Society London, 2008, vol. 464, 1851-1866.
4. Pomeau Y., Le Berre M. Gyuenne P., Grilli S. Wave-breaking and generic singularities
of nonlinear hyperbolic equations. Nonlinearity, 2008, vol. 21, T61-T79.
5. Mailybaev A.A. Renormalization and universality of blowup in hydrodynamic flow.
Physical Review E, 2012, vol. 85, 066317.
6. Kartashova E., Pelinovsky E., and Talipova T. Fourier spectrum and shape evolution
of an internal Riemann wave of moderate amplitude. Nonlinear Processes in Geophysics,
2013, vol. 20, 571-580.
7. Pelinovsky D., Pelinovsky E., Kartashova E., Talipova T., and Giniyatullin A.
Universal power law for the energy spectrum of breaking Riemann waves. JETP Letters,
2013, vol. 98, No. 4, 237-241.
Korteweg - de Vries (α1 = 0) or
Gardner equation
u
t
 u
u
x
  1u
2
u
x
 u
3

x
3
The dispersion leads to solitary waves
formation at the front of breaking wave
0
Energy spectrum in KdV computations
before solitons tends to k-8/3
k-8/3
= k/k0
Solitary wave formation
Mark J. Ablowitz, Douglas E. Baldwin.
Interactions and asymptotics of dispersive
shock waves – Korteweg–de Vries equation.
Physics Letters A, 2013, vol. 377, 5550559.
0.8
0.4
0
0.8
-0.4
0
40
80
120
160
t = 40
200
x
elevation
elevation
t = 27
0.4
0
-0.4
0
40
80
120
160
200
Solitary wave formation in spectrum
1
0.1
spectrum, S/S0
0.01
0.001
0.0001
1E-005
1E-006
40
1E-007
t = 24
1E-008
1E-009
1E-010
1
10
wave number, k
100
Denys Dutykh
BreakingRiemannWave_KdV.avi
Burgers Equation
Shock wave formation
k-4/3
k-1
Strongly nonlinear Riemann Waves in Water Channels
2004 Indian Ocean Tsunami
2004 Indian Ocean Tsunami
Nonlinear Shallow Water Theory

t
u
t


x
u
h   u   0
u
x
g

0
x
 is the water level displacement,
u is the horizontal velocity of water flow,
g is a gravity acceleration and
h is unperturbed water depth assumed to be constant
Riemann Wave
u()
Riemann Wave
u 2


t
 V ( )
g (h   ) 
V  3 g (h   )  2
gh


x
0
Particle velocity
Local speed
gh
5
4
1
0
-1
crest
9
h
2
trough
H cr 
V (H )/c
Critical Depth when V = 0
4
“right” deformation
3
“left” deformation
-2
0
1
2
3
Ãë óá è í à , H /h
4
5
a m p litu d e , a /h
0
0 .2
0 .4
 (t)
0 .6
-2 
a
-0 .4
2

h
-1 .6
b rea k in g p o in t
h
9 1   2
  = asin (  )
Wave amplitude
-0 .8
-1 .2
2  t
0
 ( 3 1    2 )  2 ( 3 1    2 )
 

A
 
  arcsin 
Location of the breaking point in
trough on the shallow water wave
a m p litu d e , a /h
d is p la c e m e n t a t b re a k in g p o in t,  /h
p h a s e o f b re a k in g p o in t,  t
0
0
0 .2
0 .4
0 .6
0
 
*
h
 *  a sin  
-0 .2
-0 .4
.6
•Zahibo, N., Slunyaev, A., Talipova, T., Pelinovsky, E., Kurkin,-0A.,
and Polukhina, O.
Strongly nonlinear steepening of long interfacial waves. Nonlinear Processes in Geophysics,
2007, vol. 14, No. 3, 247-256.
•Zahibo, N., Didenkulova, I., Kurkin, A., and Pelinovsky, E. Steepness and spectrum of
nonlinear deformed shallow water wave. Ocean Engineering. 2008, vol. 35, No. 1., 47-52.
•Pelinovsky, E.N., and Rodin, A.A. Nonlinear deformation of a large-amplitude wave on
shallow water. Doklady Physics, 2011, vol. 56, No. 5, 305-308.
Shock Wave Formation A/h = 0.2
Computation with CLAWPACK
Shock Wave Formation A/h = 0.6
Shock Wave Formation A/h = 0.9
Conclusions
• The time for breaking to occur depends only on the absolute values
of the coefficients of the quadratic and cubic nonlinear terms but not
on their signs and it decreases with increasing wave amplitude. The
shock appears on the face- or back-slope depending on the signs and
ratio of the quadratic and cubic nonlinear terms.
• Using the dispersionless Gardner equation, the spectrum evolution
of an initially sinusoidal wave has been analyzed and an explicit
formula for the Fourier spectrum in terms of Bessel functions
obtained. The asymptotic behavior of the Fourier spectrum has been
studied in detail.
• The energy spectrum of the Riemann wave at the point of breaking
is universal for any kind of nonlinearity and described by a power
law with a slope close to -8/3.
• The spectrum can be described by an exponential law for small
times and has a power asymptotic describing the form of the
singularity in the wave shape at the point where the wave breaks at
the time of breaking.