Fourier Spectrum of Riemann WavesFourier Spectrum of Riemann Waves

Efim PelinovskyEfim PelinovskyElena Tobish (Kartashova) Elena Tobish (Kartashova)

Tatiana TalipovaTatiana TalipovaDmitry PelinovskyDmitry 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

26 May 1983Japan Sea

(Shuto, 1983)

Tsunami Wave Tsunami Wave ShapesShapes

at at Japanese CoastJapanese Coast

MotivationMotivation

Internal Wave ObservationsInternal Wave Observations

Marshall H. Orr and Peter

C. Mignerey, South China sea

Nothern Oregon

J Small, T Sawyer, J.Scott,

SEASAMEMalin Shelf Edge

Weakly Nonlinear Riemann Waves

0)(

xV

t

2)( V

Coefficients can have either sign

])([),( tVxFtx Riemann Wave

tdx

dVdxdF

x

1

/

max)/(

1

dxdVT

Wave Steepness First Wave Breaking

)sin()( kxAxF Initial Sine Disturbance

Breaking point location

> 0 < 0

A

CQ Cubic/quadratic nonlinear ratio

Breaking Time

kAT

||

12 23 ||

1

kAT

0 2 4 6 8 10|CQ|

0

0.2

0.4

0.6

0.8

1

T/T

2

( = 0) ( = 0)

Cubic nonlinearity reduces

breaking time

Time evolution of the wave shape

0 0.5 1

kx/2- 1

- 0 . 5

0

0 . 5

1

/A

quadratict=0t = T

0 0.5 1

kx/2- 1

- 0 . 5

0

0 . 5

1

/A

cubict=0t = T

0 2 4 6kx

-0.4

-0.2

0

0.2

0.4

0 . 30 . 40 . 5

CQ = 0.4

0 2 4 6kx

-2

-1

0

1

2

negativepositive

Cubicnonlinearity

Fourier spectrum of a nonlinearly deformed wave

1

0 )sin()()cos()(2

)(),(

nnn nkxtbnkxta

tatx

k

nnn dxinkxtxk

ibatS/2

0

exp),()(

tVxy )(Change of variable

k

n dytFVyinkdy

dF

n

itS

/2

0

)(exp)(

Explicit formulaF(y) = A sin(ky)

Implicit formula

2

0

22 sinsinexpcos)( dxtxkAxkAxinxn

iAtSn Final formula

Quadratic Nonlinearity

)(2

)1( 1 kAtnJkAtn

iAS n

nn

Bessel-Fubinni Series (Nonlinear Acoustics)

1 10 100n

-30

-20

-10

0

log

(E)

Quadratic1/41 / 23 / 41

-8/3

E = (S/A)2

from Tbr

Power asymptoticsat breaking time

S(k) ~ k-4/3

E(k) ~ k-8/3

Cubic Nonlinearity

)]12([)]12([])12(exp[12 1 qmiJqmJqmi

m

iAS mmm

2/)( 2ktAtq

1 10 100n

-30

-20

-10

0

log

(En)

Cubic1/41 / 23 / 41

-8/3 AGAINPower asymptotics

at breaking time

S(k) ~ k-4/3

E(k) ~ k-8/3

Quadratic – cubic nonlinearity

1 10 100n

-6

-4

-2

0

log

(E)

Cubic-quadraticBreaking

1 10 100n

-6

-4

-2

0

log

(E)

Cubic-quadraticBreaking

1 10 100n

-6

-4

-2

0

log

(E)

Cubic-quadraticBreaking

CQ = 0.2 CQ = 1

CQ = 5

Power asymptoticsat breaking time

S(k) ~ k-4/3

E(k) ~ k-8/3

Universal Spectrum AsymptoticsUniversal Spectrum Asymptotics

0

x

VV

t

V0)(

xV

t

Vdt

dx 0

dt

dVOr in equivalent form

S(k) ~ k-4/3

means existence of singularity in wave shape3/1)(~),( brxxtx

Proof:

)()( FtV )()( tFtx

Riemann Wave Solution

)( brbrbr TFx Breaking coordinate

maxmax )/(

1

)/(

1

ddFdxdVT

where br is extreme of ddF /

see, breaking time

Decomposition

br 0

...)(6

)(2

)()(

)(

3

33

2

22

brbrbrbr

brbrbr

d

Fd

d

Fd

d

dFFT

TFx

Taylor series of Riemann wave in the vicinity of breaking

Finally

)(

6 3

33

brbr d

FdTxx

= 0= -1/T

So 3/1

3/1

33)(

/

6brxx

dFTd

TF

d

dFFFTxV brbrbrbr

)(...)()()(),(

Similar Taylor series for function, V

3),( brbr xxVtxV

In breaking point the wave shape has a singularity

This singularity leads to power spectrum 3/4kThe same for η due to V(η)

Rigorous Results for Shape Singularity in Riemann Wave1. 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 (Korteweg - de Vries (αα11 = 0) = 0) or or

Gardner equationGardner equation

03

32

1

x

u

x

uu

x

uu

t

u

The dispersion leads to solitary waves The dispersion leads to solitary waves formation at the front of breaking waveformation at the front of breaking wave

EnergyEnergy spectrum in KdV computations spectrum in KdV computations before solitons tends to before solitons tends to k-8/3

k-8/3

= = k/kk/k00

Solitary wave formationSolitary wave formation

0 40 80 120 160 200

x

-0.4

0

0.4

0.8el

eva

tio

nt = 400 40 80 120 160 200

x

-0.4

0

0.4

0.8

elev

ati

on

t = 27

Mark J. Ablowitz, Douglas E. Baldwin. Interactions and asymptotics of dispersive shock waves – Korteweg–de Vries equation. Physics Letters A, 2013, vol. 377, 5550559.

Solitary wave formation in spectrumSolitary wave formation in spectrum

1 10 100

wave num ber, k

1E-010

1E-009

1E-008

1E-007

1E-006

1E-005

0.0001

0.001

0.01

0.1

1

spec

tru

m,

S/S

0

t = 24

40

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 TheoryNonlinear Shallow Water Theory

0

uhxt

0

xg

x

uu

t

u

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

u()Riemann Wave

0)(

xV

t

ghhgV 2)(3

Riemann Wave

ghhgu )(2

0 1 2 3 4 5Ãë óá è í à, H/h

-2

-1

0

1

2

3

4

5

V(H

)/c

Particle velocity

Local speed

“right” deformation

“left” deformation

cres

t

tro

ug

hhH cr 9

4

Critical Depth when V = 0

t0 2-2breaking point

=asin()

(t)

h

0 0.2 0.4 0.6

amplitude, a/h

-0.6

-0.4

-0.2

0

dis

pla

cem

ent

at b

reak

ing

po

int,

/h

0 0.2 0.4 0.6

amplitude, a/h

-1.6

-1.2

-0.8

-0.4

0

ph

ase

of

bre

aki

ng

po

int,

t

A

arcsin

Location of the breaking point in trough on the shallow water wave

219

)213(2)213(2

h

a

Wave amplitude

sin** ah

•Zahibo, N., Slunyaev, A., Talipova, T., Pelinovsky, E., Kurkin, A., 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

ConclusionsConclusions• 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.