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Wave Turbulence: Foundations and Challenges

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Wave Turbulence: Foundations and Challenges Wave Turbulence: foundations and challenges Sergey Nazarenko University of Warwick, UK March 25, 2015 Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 1 / 30 Recent Wave Turbulence books Book with focus on experiment Book with focus on experiment Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 2 / 30 Recent Wave Turbulence books Book with focus on theory and new applications Book with focus on theory and new applications Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 3 / 30 Recent Wave Turbulence books Book on foundations of fluid dynamics Foundations of fluid dynamics, inc. waves and turbulence Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 4 / 30 Well, anyway ... What is Wave Turbulence? Wave Turbulence is a non-equilibrium statistical system of many randomly interacting waves. Kinetic equations of Wave Turbulence describe evolution of the wave energy in Fourier space. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 5 / 30 Examples of Wave Turbulence Ocean waves Wave Turbulence on ocean surface Kinetic equations of Wave Turbulence are used for the wave weather forecasts e.g. at ECMWF. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 6 / 30 Examples of Wave Turbulence Water waves in laboratory Waves in laboratory flumes Gravity and capillary waves are the most actively studied in laboratory for validating the Wave Turbulence predictions. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 7 / 30 Examples of Wave Turbulence Magneto-Hydrodynamic waves Alfv´enWave Turbulence in solar wind Alfv´enWave Turbulence is at the heart of MHD turbulence theory. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 8 / 30 Examples of Wave Turbulence Drift Wave Turbulence Waves in fusion plasmas Drift Wave Turbulence is the cause of the anomalous energy and particle losses. Its interaction with zonal flows leads to Low-to-High transitions. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 9 / 30 Examples of Wave Turbulence Quantum Wave Turbulence Superfluid Turbulence Kolmogorov cascade is taken over by Kelvin Wave Turbulence at the classical-quantum crossover scale. Bottleneck effect. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 10 / 30 Examples of Wave Turbulence \Liquid light" systems Optical Wave Turbulence Optical Wave Turbulence: interacting waves and solitons Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 11 / 30 Challenges in Wave Turbulence Challenges WT assumes small nonlinearity, which may not be uniformly valid across scales. How can we prove that what we see in experiment is WT? What is the role of interaction with strongly turbulent scales, coherent structures? Does the phase-amplitude randomness survive over nonlinear time? Controlling dissipation in experiment. To validate steps of the WT theory, one has to deal with joint PDF of wave modes. Tricky infinite-box limit. Unclear relation between the solutions of the kinetic equations, like Kolmogorov-Zakharov (KZ), and solutions for joint PDF. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 12 / 30 Schematic structure of Wave Turbulence Wave Turbulence maze 2. Fourier space. Interaction representation. 1. Nonlinear wave equation 3. Wave amplitude at intermediate time. Weak nonlinearity expansion. 4. Substitute the weak−nonlinearity expansion into the N−mode generating function and average assuming that initial field is RP. 8.Take initial field to be RPA. 5. Take large−box limit followed by weak nonlinearity limit. 6. Evolution equation for the N−mode generating function & N−mode PDF. 9. Check that RPA survives over the nonlinear time. 7. Check that RP survives over the nonlinear time. This validates use of the equation for the N−mode 10. Derive equation for reduced objects, e.g. generating function over the nonlinear time, 1−mode PDF, 1−mode moments, spectrum. i.e. not only near t=0. 12. Cascade states. 13. Non−Gaussian scalings 11. Study multi−mode statistics KZ spectra. and intermittency. when amplitudes are correlated. 14. Advance mathematical description of WT. 15. Including coherent structures and wavebreaking into the WT description. WT Life Cycle. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 13 / 30 Master example: Petviashvilli equation Master example: Petviashvilli equation Petviashvilli equation descrives drift waves in inhomoheneous plasmas in the limit kρs 1 in presence of scalar (thermal) nonlinearity: 2 @t = r @x − @x ; (1) where = (x; y; t) is a real function of two space coordinates (x; y) and time t. Petviashvilli equation conserves "energy": 1 Z E = 2 dx = const; 2 and "potential enstrophy": 1 Z 1 Ω = (r )2 + 3 dx = const: 2 3 Note: Hamiltonian is actually given by Ω, not E: δΩ @ = −@ : t x δ Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 14 / 30 From the physical space to Fourier Fourier space 2 Let us consider a double-periodic system, x 2 T , with period L in both directions Using Fourier coefficients Z 1 −ik·x ak(t) = 2 (x:t)e dx ; (2) L Box rewrite Petviashvilli equation as 2 X k a_ k + ikx k ak + i a1a2k2x δ12 = 0; (3) 1;2 k where dot is for time derivative, a1;2 ≡ ak1;2 , and δ12 = δ(k − k1 − k2). In the linear limit, −i!k t ak = Ak e ; (4) where Ak 2 C is a time-independent amplitude of the wave with wavevector k, and the frequency of this wave is 2 !k = kx k : (5) Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 15 / 30 From the physical space to Fourier Interaction Representation Let us separate the time scales by introducing interaction representation variables as a e−i!k t b = k ; (6) k 1=2 jkx j where we introduced parameter > 0 for easier nonlinearity power counting. We now have: X k _ k i!12t ibk = sign(kx ) V12k b1b2δ12e ; (7) 1;2 where we have introduced the interaction coefficient 1 V = pjk k k j : (8) 12k 2 x 1x 2x k and !12 = !k − !1 − !2. Note that the linear term !k ak is gone, and that there is explicit time-dependence in the nonlinear term. We have not made any approximations yet. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 16 / 30 Statistical objects. Discrete k-space 2 Since we consider a periodic system, x 2 T , the wavenumbers are 2 discrete, k 2 Z (hereafter for simplicity L = 2π). Let the total number of modes be finite and bounded by some kmax (eg. a dissipation cutoff at high wavenumbers). Denote by BN the set of all wavenumbers kl inside 2 the k-space box of volume (2kmax) : 2 Figure: Set of active wave modes, BN ⊂ Z . Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 17 / 30 Statistical objects. Amplitude-Phase representation We write the wave function in terms of its amplitude and phase p a(k; t) = Jkφk ; + 1 where Jk 2 R is the intensity and φk 2 S is the phase factor of the 1 mode k. By S we mean the unit circle in the complex plane, i.e. i' φk = e k : Re d ξl φl ξl +d ξl ξl 0 Im Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 18 / 30 Statistical objects. Probability Density Functions. Denote the set of all Jk and φk with k such that k 2 BN as fJ; φg. + The probability of finding Jk inside (sk; sk + dsk) ⊂ R and finding φk on 1 the arch (ξk; ξk + dξk) ⊂ S (see Figure) is given in terms of the joint PDF P(N)fs; ξg as (N) Y P fs; ξg dskjdξkj : (9) k2BN M-mode joint PDF (M < N): 0 1 Z 1 I (M) Y P = ds jdξ j P(N)fs; ξg : (10) k1;k2;:::;kM @ k k A 0 1 k6=k1;k2;:::;kM S N-mode amplitude-only PDF obtained by integrating out all the phases, 0 1 I (N;a) Y (N) P fsg = @ jdξkjA P fs; ξg : (11) 1 k2BN S Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 19 / 30 Shades of randomness RP fields. (1;a) (a) Single-mode amplitude PDF, P ≡ Pk (sk), is obtained via integrated out all phases ξ, and all amplitudes s but one, sk. Definition. Random phase (RP) field: all φ are independent random 1 variables (i.r.v.) each uniformly distributed on S . Thus for a RP field 1 P(N)fs; ξg = P(N;a)fsg : (2π)N Note: RP (in addition to the weak nonlinearity) is enough for the lowest level WT closure leading to an equation for the N-point amplitude-only PDF. However, it is not sufficient for the one-point WT closure, in particular the wave kinetic equation, and we need to assume something about the amplitudes too. Sergey Nazarenko University of Warwick, UK Wave Turbulence: foundations and challenges 20 / 30 Shades of randomness Random Phase and Amplitude (RPA) field definition 1 All the amplitudes and all the phases are i.r.v., 1 2 All the phases are uniformly distributed on S , 3 For RPA fields, the PDF has a product-factorized form, 1 Y (a) P(N)fs; ξg = P (s ): (12) (2π)N j j kl 2BN We have changed the standard meaning of RPA which usually stands for \Random phase approximation". In our definition of RPA: 1 The amplitudes are random, not only the phases. 2 RPA is defined as a property of the field, not an approximation. (a) RPA does not mean Gaussianity because it does not specify Pj (sj ).
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