Fourier Spectrum of Riemann Waves
description
Transcript of Fourier Spectrum of Riemann Waves
![Page 1: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/1.jpg)
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
![Page 2: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/2.jpg)
26 May 1983Japan Sea
(Shuto, 1983)
Tsunami Wave Tsunami Wave ShapesShapes
at at Japanese CoastJapanese Coast
MotivationMotivation
![Page 3: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/3.jpg)
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
![Page 4: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/4.jpg)
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
![Page 5: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/5.jpg)
)sin()( kxAxF Initial Sine Disturbance
Breaking point location
> 0 < 0
A
CQ Cubic/quadratic nonlinear ratio
![Page 6: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/6.jpg)
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
![Page 7: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/7.jpg)
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
![Page 8: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/8.jpg)
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
![Page 9: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/9.jpg)
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
![Page 10: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/10.jpg)
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
![Page 11: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/11.jpg)
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
![Page 12: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/12.jpg)
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:
![Page 13: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/13.jpg)
)()( FtV )()( tFtx
Riemann Wave Solution
)( brbrbr TFx Breaking coordinate
maxmax )/(
1
)/(
1
ddFdxdVT
where br is extreme of ddF /
see, breaking time
![Page 14: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/14.jpg)
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
![Page 15: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/15.jpg)
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)
![Page 16: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/16.jpg)
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.
![Page 17: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/17.jpg)
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
![Page 18: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/18.jpg)
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
![Page 19: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/19.jpg)
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.
![Page 20: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/20.jpg)
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
![Page 21: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/21.jpg)
Denys Dutykh BreakingRiemannWave_KdV.avi
![Page 22: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/22.jpg)
Burgers Equation
Shock wave formation
k-4/3
k-1
![Page 23: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/23.jpg)
Strongly nonlinear Riemann Waves in Water Channels
2004 Indian Ocean Tsunami
![Page 24: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/24.jpg)
2004 Indian Ocean Tsunami
![Page 25: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/25.jpg)
![Page 26: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/26.jpg)
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
![Page 27: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/27.jpg)
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
![Page 28: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/28.jpg)
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.
![Page 29: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/29.jpg)
Shock Wave Formation A/h = 0.2
Computation with CLAWPACK
![Page 30: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/30.jpg)
Shock Wave Formation A/h = 0.6
![Page 31: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/31.jpg)
Shock Wave Formation A/h = 0.9
![Page 32: Fourier Spectrum of Riemann Waves](https://reader036.fdocuments.net/reader036/viewer/2022062519/56814e21550346895dbb8522/html5/thumbnails/32.jpg)
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.