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Wave turbulence beyond spectra

This talk explores wave turbulence, a stochastic field of weakly interacting dispersive waves. It discusses the Kolmogorov-Zakharov cascades and the non-Gaussianity of wave probability density function (PDF) and intermittency. The talk also covers the effects of discreteness and the sanpile behavior of energy cascade. Other examples of wave turbulence are discussed, including sound waves, plasma waves, and waves in Bose-Einstein condensates.

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Wave turbulence beyond spectra

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  1. Wave turbulence beyond spectra Sergey Nazarenko, Warwick, UK • Collaborators: L. Biven, Y. Choi, Y. Lvov, A.C. Newell, M. Onorato, B. Pokorni, & V.E. Zakharov Nazarenko, Warwick Dec 8 2005

  2. Plan of the talk: • Statistical waves – “Wave Turbulence”. • Kolmogorov-Zakharov cascades. • Non-gaussianity of wave PDF; intermittency. • Discreteness effects; sanpile behaviour of energy cascade. Nazarenko, Warwick Dec 8 2005

  3. What is Wave Turbulence? WT describes a stochastic field of weakly interacting dispersive waves. Nazarenko, Warwick Dec 8 2005

  4. Other Examples of Wave Turbulence: • Sound waves, • Plasma waves, • Spin waves, • Waves in Bose-Einstein condensates, • Interstellar turbulence & solar wind, • Waves in Semi-conductor Lasers. Nazarenko, Warwick Dec 8 2005

  5. How can we describe WT? • Hamilitonian equations for the wave field. • Weak nonlinearity expansion. Separation of the linear and nonlinear timescales. • Statistical averaging, - closure. Nazarenko, Warwick Dec 8 2005

  6. Free surface motion r is a 2D vector in horizonal plane; z is the vertical coordinate Nazarenko, Warwick Dec 8 2005

  7. Zakharov equation Deep water waves, 2 =gk, Wis complicated (Krasitski 1992) Nazarenko, Warwick Dec 8 2005

  8. Frequency renormalization Nazarenko, Warwick Dec 8 2005

  9. Statistical variables in WT • Amplitude & phase: ak = Akk ; k = exp(ik). • Stationary distribution ofk – unsteady distribution ofk , • Randomk - correlatedk . Nazarenko, Warwick Dec 8 2005

  10. Statistical objects in WT • Spectrum nk = <Ak2> E.g. Kolmogorov-Zakharov spectrum • N-mode PDF: probability forAk2 to be in[sk, sk +dsk]and forkto be in[ξk, ξk+dξk] , P (N) {s,ξ} = <δ(s-A2) δ(ξ-)>; s={s1,s2,…,sN}; A={A1,A2,…,AN}; ξ={ξ1,ξ2,…,ξN}; ={1,2,…,N}. Nazarenko, Warwick Dec 8 2005

  11. Random Phase & Amplitude (RPA) wavefield: • All the amplitudes and the phase factors are independent random variables, • The phase factors are uniformly distributed on the unit circle in the complex plane. Nazarenko, Warwick Dec 8 2005

  12. RPA fields are not Gaussian. • Gaussian distribution means P(a)(s) ~ e-s/n. • RPA does not fix the amplitude PDF. Nazarenko, Warwick Dec 8 2005

  13. Weak nonlinearity expansion Choose T in between the linear and nonlinear timescales: Nazarenko, Warwick Dec 8 2005

  14. Iterations Nazarenko, Warwick Dec 8 2005

  15. Evolution of WT statistics • Substitute value of ak(T) into the PDF definition. • Apply RPA to ak(0) . • Replace [P(T)-P(0)]/T with ∂t P. Nazarenko, Warwick Dec 8 2005

  16. Equation for the N-mode PDF (Choi, Lvov & SN, 2004) Where Fj is the j-component of the flux, Nazarenko, Warwick Dec 8 2005

  17. Use of the N-mode PDF • Validation that RPA holds over the nonlinear evolution time • Non-Gaussian statistics of the wave amplitudes Nazarenko, Warwick Dec 8 2005

  18. Single-mode staitstics Eqn. for the 1-mode PDF (Choi, Lvov, SN, 2003): Nazarenko, Warwick Dec 8 2005

  19. Kinetic equation for spectrum Taking 1st moment of the 1-mode PDF eqn: Hasselmann, 1963 Nazarenko, Warwick Dec 8 2005

  20. Kolmogorov-Zakharov spectra • Power-law spectra describing a down-scale energy cascade and an up-scale wave-action cascade • WT may break at a large or small scale Nazarenko, Warwick Dec 8 2005

  21. Breakdown of WT • Water surface: wavebreaking means there is no amplitudes higher than critical • Hard breakdown, n~s*, Biven, Newell, SN, 2001 • Weak breakdown, n<<s*.Choi, Lvov, SN, 2003 Nazarenko, Warwick Dec 8 2005

  22. Steady state PDF Choi, Lvov, SN, 2003 • Gaussian core, non-gaussian tail Nazarenko, Warwick Dec 8 2005

  23. Direct Numerical Simulations • Truncated (at the 3rd order in amplitude) Euler equations for the free water surface. • Pseudo-spectral method 256X256. Nazarenko, Warwick Dec 8 2005

  24. Energy spectrum Onorato et al’ 02, Dyachenko et al’03, Nakoyama’04, Lvov et al’05 Nazarenko, Warwick Dec 8 2005

  25. One-mode PDF • Anomalously high amplitude of large waves – Freak Waves Nazarenko, Warwick Dec 8 2005

  26. Correlation of ’s. • In agreement with WT, ’s are decorrelated from A’s and among themselves Nazarenko, Warwick Dec 8 2005

  27. ’s are correlated! • Correct theory is based on random ’s and not random ’s. Nazarenko, Warwick Dec 8 2005

  28. Exact 4-wave resonances • Collinear (Dyachenko et al’94): all 4 k’s parallel to each other (unimportant – null interaction). • Symmetric (Kartashova’98): |k1| = |k3|, |k2| = |k4| or |k1| = |k4|, |k2| = |k3| . • Tridents (Lvov et al’05): k1 anti-parallel k3, k2 is mirror-symmetric with k4 with respect to k1 -k3axis. Nazarenko, Warwick Dec 8 2005

  29. Tridents Parametrisation (SN’05): Nazarenko, Warwick Dec 8 2005

  30. Cascade on quasi-resonances • Cascade starts at resonance broadening << k-grid spacing • It is anisotropic and supported by small fraction of k’s. Nazarenko, Warwick Dec 8 2005

  31. Frequency peaks at fixed k. • 2nd peak is contribution of k/2 mode • Weak turbulence: 1st peak << 2nd peak • Sometimes 2nd peak gets > 1st peak • Diagnostics of nonlinear activity Nazarenko, Warwick Dec 8 2005

  32. Phase runs • Phase runs – diagnostics of nonlinear activity Nazarenko, Warwick Dec 8 2005

  33. “Sandpile avalanches” • Nonlinear activity at k1and k2 are correlated (k2>k1), • It is time delayed and amplified at k2 with respect k1. • “sandpile avalanches”. Nazarenko, Warwick Dec 8 2005

  34. Cycle of discrete turbulence • Weak turbulence at forcing scale – no resonance, no cascade. • Energy accumulation, growth of nonlinearity. • Nonlinear resonance broadening, cascade activation – “avalanche”. • Avalanche drains energy from the forcing scale, -> beginning of the cycle. Nazarenko, Warwick Dec 8 2005

  35. Summary • Generalised WT description: PDF. • Kolmogorov-Zakharov spectrum. • RPA validation. Correlations of phases. • Anomalous distribution of waves with high amplitudes • Discreteness effects: exact and quasi-resonances, sanpile behavior of cascade. Nazarenko, Warwick Dec 8 2005

  36. Phases vs Phase factors • Illustration through an example : • Phases are correlated, because • Phase factors are statistically independent, Nazarenko, Warwick Dec 8 2005

  37. Mean phase • Expression for phase • Evolution eqn. the mean value of the phase is steadily changing over the nonlinear time and it would be incorrect to assume that phases remains uniformly distributed in [-,]. Nazarenko, Warwick Dec 8 2005

  38. Dispersion of the phase is always positive and the phase fluctuations experience ultimate growth (linear in steady state) Nazarenko, Warwick Dec 8 2005

  39. Essentially RPA fields • The amplitude variables are almost independent is a sense that for each M<<N modes the M-mode amplitude PDF is equal to the product of the one-mode PDF’s up to and corrections. Nazarenko, Warwick Dec 8 2005

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