Fourier Spectrum of Riemann Waves

1. Efim Pelinovsky Elena Tobish (Kartashova) Tatiana Talipova Dmitry Pelinovsky Fourier Spectrum of Riemann Waves 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

2. Motivation Tsunami Wave Shapes at Japanese Coast 26 May 1983 Japan Sea (Shuto, 1983)

3. 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

4. Weakly Nonlinear Riemann Waves Coefficients can have either sign RiemannWave Wave Steepness First Wave Breaking

5. Initial Sine Disturbance Breaking point location  < 0  > 0 Cubic/quadratic nonlinear ratio

6. Breaking Time ( = 0) ( = 0) Cubic nonlinearity reduces breaking time

7. Time evolution of the wave shape Cubic nonlinearity CQ = 0.4

8. Fourier spectrum of a nonlinearly deformed wave Implicit formula Change of variable F(y) = A sin(ky) Explicit formula Final formula

9. Quadratic Nonlinearity Bessel-Fubinni Series (Nonlinear Acoustics) Power asymptotics at breaking time S(k) ~ k-4/3 from Tbr E(k) ~ k-8/3 E = (S/A)2

10. Cubic Nonlinearity AGAIN Power asymptotics at breaking time S(k) ~ k-4/3 E(k) ~ k-8/3

11. Quadratic – cubic nonlinearity CQ = 1 CQ = 0.2 Power asymptotics at breaking time S(k) ~ k-4/3 CQ = 5 E(k) ~ k-8/3

12. Universal Spectrum Asymptotics S(k) ~ k-4/3 means existence of singularity in wave shape Proof: Or in equivalent form

13. Riemann Wave Solution Breaking coordinate where is extreme of see, breaking time

14. Decomposition Taylor series of Riemann wave in the vicinity of breaking = 0 = -1/T Finally

15. So Similar Taylor series for function, V In breaking point the wave shape has a singularity This singularity leads to power spectrum The same for η due to V(η)

16. Rigorous Results for Shape Singularity in Riemann Wave • 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.

17. Korteweg - de Vries (α1 = 0)or Gardner equation The dispersion leads to solitary waves formation at the front of breaking wave

18. Energy spectrum in KdV computations before solitons tends to k-8/3 k-8/3 = k/k0

19. 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.

20. Solitary wave formation in spectrum

21. Denys Dutykh BreakingRiemannWave_KdV.avi

22. Burgers Equation Shock wave formation k-4/3 k-1

23. Strongly nonlinear Riemann Waves in Water Channels 2004 Indian Ocean Tsunami

24. 2004 Indian Ocean Tsunami

26. Nonlinear Shallow Water Theory 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

27. Riemann Wave Particle velocity Local speed “right” deformation Critical Depth when V = 0 trough crest “left” deformation

28. Wave amplitude Location of the breaking point in trough on the shallow water wave • 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.

29. Shock Wave Formation A/h = 0.2 Computation with CLAWPACK

30. Shock Wave Formation A/h = 0.6

31. Shock Wave Formation A/h = 0.9

32. 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.