Introduction to (weak) wave turbulence

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Review lecture given at Winter School on Wave Turbulence

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Introduction to (weak) wave turbulence

  1. 1. An introduction to (weak) wave turbulence Colm Connaughton Mathematics Institute and Centre for Complexity Science, University of Warwick, UK Turbulence d’ondes - Wave turbulence Ecole de Physique des Houches 26 March 2012 http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  2. 2. Motivation from natural sciences Waves are ubiqitous in the physical sciences. Most are dispersive. All become nonlinear at finite amplitude. In most situations, waves are excited and damped by “external" processes. This lecture: An introduction to wave turbulence. Mostly focusing on concepts and avoiding detailed calculations. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  3. 3. What is wave turbulence? Working definition Wave turbulence is the non-equilibrium statistical dynamics of ensembles of interacting dispersive waves. non-equilibrium : forcing and dissipation are key so equipartition of energy has limited relevance. statistical : many degrees of freedom are active. interacting : nonlinearity cannot be neglected. dispersive : non-dispersive waves are much tougher theoretically. Applications Interfacial waves in fluids (gravity, capillary). Waves in plasmas (Alfvén, drift, sound). Nonlinear optics and BECs (NLS). Geophysical waves (Rossby, inertial). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  4. 4. Linear waves and complex coordinates A linear wave equation for h(x, t): α ∂2h ∂2 = −c 2 − 2 h. ∂t 2 ∂x Such an equation is more natural in Fourier space: h(x, t) = hk (t)ei k x dk . Each (complex-valued) Fourier mode, hk (t), executes simple harmonic motion, d 2 hk = −c 2 k 2α hk ≡ −ωk hk . 2 dt 2 with frequency ωk = c k α . At the linear level, different types of waves differ only in their dispersion relation. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  5. 5. Linear waves and complex coordinates The simple harmonic motion equation, d 2 hk 2 = −ωk hk , dt 2 can be represented as a pair of first order equations for (xk , yk ) = (hk , ∂hk ): ∂t dxk = yk dt dyk 2 = −ωk xk . dt These are equivalent to a single first order equation for the complex coordinate ak = yk + i ωk xk : dak = i ωk ak . dt http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  6. 6. Linear waves and complex coordinates The wave equation in complex coordinates, dak = i ωk ak , dt is probably the simplest example of a Hamiltonian system: ∂ak δH2 ∗ =i ∗ H2 = dk ωk ak ak , (1) ∂t δak where we have reverted to treating k as a continuous variable. Hamiltonian structure is not necessary but it is convenient. Nonlinearities manifest themselves as higher order powers of ak in Eqs. (1): H = H2 + H3 + H4 + . . . . In this lecture we will stop at H3 . (H4 in Nazarenko lecture). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  7. 7. Mode coupling in nonlinear wave equations A simple model of H3 is ∗ ∗ ∗ H3 = Vk1 k2 k3 (ak1 ak2 ak3 +ak1 ak2 ak3 )δ(k1 −k2 −k3 )dk1 dk2 dk3 . Gives equations of motion: ∂ak = iωk ak + Vkk2 k3 ak2 ak3 δ(k − k2 − k3 )dk2 dk3 ∂t ∗ + 2 Vk2 kk3 ak2 ak3 δ(k2 − k − k3 )dk2 dk3 RHS couples all modes together in groups of 3 (triads). Projection of RHS onto a single triad, k3 = k1 + k2 , which is resonant (ωk3 = ωk1 + ωk2 ) gives ODEs: dB1 ∗ dB2 ∗ dB3 = B2 B3 = B1 B3 = −B1 B2 . dt dt dt (see talks by Bustamante and Kartashova). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  8. 8. Models of nonlinear wave interactions ∗ H2 = dk ωk ak ak ∗ ∗ ∗ H3 = Vk1 k2 k3 (ak1 ak2 ak3 + ak1 ak2 ak3 )δ(k1 − k2 − k3 )dk1 dk2 dk3 . What should we take for the nonlinear interaction coefficient, Vk1 k2 k3 and the linear frequency ωk ? Many interesting cases possess scale invariance : ωhk = hα ωk Vhk1 hk2 hk3 = hβ Vk1 k2 k3 . Detailed formulae for Vk1 k2 k3 can be quite complex. 3 9 Example: capillary waves: d = 2, α = 2 and β = 4 . Model systems (cf MMT) are often studied, especially numerically. For example: ωk = c k α (2) β Vk1 k2 k3 = λ (k1 k2 k3 ) 3 (3) http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  9. 9. Conservation laws in turbulence In turbulent systems, quantities which are conserved by the nonlinear terms in the equations of motion are very important. Navier-Stokes turbulence ∂v + (v · )v = − p + ν∆v + f ∂t ·v=0 1 Nonlinear terms conserve kinetic energy, 2 v2 . Energy is added by forcing and removed by the viscous terms. Wave turbulence ∂ak δH = i ∗ + fk + D [ak ] ∂t δak In WT, H, is conserved by nonlinearity - not quadratic. Possibly other conserved quantities (cf Nazarenko lecture). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  10. 10. Kolmogorov phenomology of Navier-Stokes turbulence LU Reynolds number R = ν . 1 Energy, 2 v2 , injected at large scales and dissipated at small scales. Conservative energy transfer in between due to interaction between scales. Concept of inertial range and energy cascade. K41 : As R → ∞, small scale statistics depend only on the scale, k , and the energy flux, J. Dimensionally : 2 5 E(k ) = c J 3 k − 3 Kolmogorov spectrum Similar phenomenology for wave turbulence. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  11. 11. What can we learn about WT by dimensional analysis? For the purposes of power counting: 2 H = c k α ak dk + λ k β ak δ(k) (dk)3 3 2 nk δ(k) = ak Dimensions: [c] = T −1 K −d 1 1 d [ak ] = H 2 T 2 K − 2 1 1 d [λ] = H 2 T 2 K − 2 [nk ] = HT One more dimensional parameter, the flux: [J] = HT −1 K d . Flux is "H per unit volume per unit time". http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  12. 12. What can we learn about WT by dimensional analysis? We have enough dimensional parameters to get any spectum: nk = c w J x λy k −z Matching powers of H, T and K gives: 2(β + d − z) w = 2α − β 2β − 2α + d − z x = − 2α − β 2α + d − z y = − . 2α − β There are special scalings: (w = 0) : z = β + d (KZ) (x = 0) : z = 2β − 2α + d (Generalised Phillips) (y = 0) : z = 2α + d (???). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  13. 13. What do we want from a statistical description? Ideally want full joint PDF, P(ak1 (t1 ), ak2 (t2 ), . . .)! Start with (single-time) correlation functions: ∗ C2 (k1 , k2 ) = ak1 ak2 ∗ C3 (k1 , k2 , k3 ) = ak1 ak2 ak3 . is an ensemble average with respect to the statistics of the forcing (forced/steady turbulence) or initial condition (unforced/decaying turbulence). We almost always assume ergodicity in practice. For steady turbulence, this allows ensemble averages to be replaced with time averages. We also typically assume statistical spatial homogeneity. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  14. 14. The wave spectrum, nk The second order correlation function is of particular interest: ∗ ak1 ak2 = a(x1 ) a∗ (x2 ) e−i k1 ·x1 +i k2 ·x2 dx1 d x2 = f (x1 − x2 ) e−i (k1 −k2 )·x1 e−i k2 ·(x1 −x2 ) dx1 d x2 = f (r) e−i k2 ·r dr e−i (k1 −k2 )·x1 dx1 r = x1 − x2 = n(k2 ) δ(k1 − k2 ). Equations of motion lead to closure problem: ∂n(k1 ) ∗ = 2 Vk1 k2 k3 Im ak1 ak2 ak3 δ(k1 − k2 − k3 )dk2 dk3 ∂t ∗ − 2 Vk2 k3 k1 Im ak2 ak3 ak1 δ(k2 − k3 − k1 )dk2 dk3 ∗ − 2 Vk3 k1 k2 Im ak3 ak1 ak2 δ(k3 − k1 − k2 )dk2 dk3 http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  15. 15. Weakly nonlinear wave kinetics Interaction Hamiltonian is small com- pared to the linear Hamiltonian. H3 /H2 = 1 provides an expansion parameter. Strategy (cf lecture by Newell): 1 Write moment hierarchy in terms of cumulants and solve perturbatively in . 2 Higher order cumulants decay as t → ∞ (asymptotic closure) but resonant triads lead to secular terms: 2 (2) (2) n(k, t) = n(0) (k, t)+ n(1) (k, t)+ t nsec (k, t) + nnon−sec (k, t) +. . . 3 Allow slow variation of lower order terms: n(0) (k, t, 2 t). 4 Choose dependence on 2t to cancel secular terms. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  16. 16. The wave kinetic equation for 3-wave systems At leading order, this procedure tells us that n(0) (k, t), satisfies the following equation: The 3-wave kinetic equation: 2 ∂t nk = (Rkk1 k2 − Rk1 k2 k − Rk2 kk1 )dk1 dk2 2 Rkk1 k2 = Vkk1 k2 (nk1 nk2 − nk nk2 − nk nk1 )δ(k − k1 − k2 ) δ(ωk − ωk1 − ωk2 ) Closed kinetic equation. Interactions are restricted to resonant triads. Quadratic energy, H2 = ωk nk dk is conserved (to leading order). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  17. 17. The wave kinetic equation in frequency space For isotropic systems, it is helpful to write the KE in frequency space using the change of variables 1 d−α k = ωα k d−1 dk = ω α dω. We define the angle averaged frequency spectrum, Nω , by Ωd d−α nk dk = nk k d−1 dΩd dk = nω ω α dω = Nω dω. α Translation between nk and Nω Nω ∼ w −z ⇐⇒ nk ∼ k −αz+α−d . Nω satisfies the kinetic equation ∂t Nω = (Rωω1 ω2 − Rω1 ω2 ω − Rω2 ωω1 )dω1 dω2 Rωω1 ω2 = Lω1 ω2 (Nω1 Nω2 − Nω Nω2 − Nω Nω1 ) δ(ω − ω1 − ω2 ) http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  18. 18. The wave kinetic equation in frequency space Details are hidden in the (very messy) kernel L(ω1 , ω2 ). Scaling: 2β − α L1 (ω1 , ω2 ) ∼ ω ζ , ζ= α The frequency-space kinetic equation can be written: ∂Nω = S1 [Nω ] + S2 [Nω ] + S3 [Nω ]. ∂t The RHS has been split into forward-transfer terms (S1 [Nω ]) and backscatter terms (S2 [Nω ] and S3 [Nω ]). Each conserves energy independently. S1 [Nω ] describes the kinetics of cluster aggregation. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  19. 19. How does the solution of the kinetic equation look? Illustrative numerical simulations for L(ω1 , ω2 ) = 1. Unforced turbulence Forced turbulence http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  20. 20. The stationary Kolmogorov–Zakharov spectrum Look for stationary solutions of 3WKE Nω = C ω −x Zakharov Transformations: ωω1 ω 2 (ω1 , ω2 ) → ( , ) ω2 ω2 ω 2 ωω2 (ω1 , ω2 ) → ( , ). ω1 ω1 ∞ 0 = C2 dω1 dω2 L(ω1 , ω2 ) (ωω1 ω2 )−x ω 2−ζ−2x δ(ω − ω1 − ω2 ) 0 2x−ζ−2 2x−ζ−2 (ω x − ω1 − ω2 ) ω 2x−ζ−2 − ω1 x x − ω2 ζ+3 Stationary solutions: x = 1 (thermodynamic) and x = 2 (constant flux). ζ+3 (Translation: Nω ∼ ω − 2 gives nk ∼ k −β−d . Nω ∼ ω −ζ gives nk ∼ k −2β+2α−d .) http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  21. 21. Calculation of the Kolmogorov constant Amplitude C can be calculated exactly. Product kernel: ζ L(ω1 , ω2 ) = (ω1 ω2 ) 2 . Sum kernel: 1 L(ω1 , ω2 ) = ω ζ + ω2 . ζ 2 1 −1/2 √ dI C= 2J where dx x= ζ+3 2 1 1 I(x) = L(y , 1 − y ) (y (1 − y ))−x (1 − y x − (1 − y )x ) 2 0 (1 − y 2x−ζ−2 − (1 − y )2x−ζ−2 ) dy . http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  22. 22. Locality of the KZ spectrum ζ+3 The integral I(x) can vanish or diverge when x = 2 and the KZ spectrum then runs into trouble. For a kernel having asymptotics: ν µ L(ω1 , ω2 ) ∼ ω1 ω2 for ω1 ω2 , the integral is finite and non-zero provided: |ν − µ| < 3 (4) xKZ > xT . (5) Such cascades are referred to as “local”. Introduce a cut-off, Ω, and study what happens as Ω → ∞. In systems for which Eqs. (4) and (5) are not satisfied the spectrum in the inertial range continues to depend on Ω in this limit. Hence the term “non-local”. Not much known about nonlocal cascades (see Connaughton talk). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  23. 23. Breakdown of the KZ spectrum and the generalisedPhillips (critical balance) spectrum Weak turbulence requires the linear timescale, τL , associated with the waves to be much faster than the nonlinear timescale, τNL , associated with resonant energy transfer between waves. Linear timescale: τL ∼ ω −1 . Nonlinear timescale: −1 1 τNL ∼ Nω ∂Nω . ∂t If Nω ∼ ω −x , the ratio τL /τNL is: τL ∼ ω ζ−x . τNL If x = ζ this ratio is uniform in ω (Phillips spectrum). If x = ζ+3 this ratio becomes 2 τL ζ−3 ∼ω 2 (6) τNL Conclusion: breakdown criterion KZ spectrum breaks down as ω → ∞ if ζ > 3 (or β > 2α). http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  24. 24. The concept of capacity √ ζ+3 Nω = C J ω− 2 . Total quadratic energy contained in the spectrum: √ Ω ζ+1 E =C J dω ω − 2 . 1 E diverges as Ω → ∞ if ζ ≤ 1 (β < α): Infinite Capacity . E finite as Ω → ∞ if ζ > 1 (β > α): Finite Capacity . Transition occurs at ζ = 1. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  25. 25. Dissipative anomaly in finite capacity systems Finite capacity systems exhibit a dissipative anomaly as the dissipation scale → ∞, infinite capacity systems do not: ζ = 3/4 ζ = 3/2 http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  26. 26. “Snakes in the grass” Despite the elegance of the theory which I have described, experimental observations of clean KZ scaling, are rare. There are many possible complications: Insufficient inertial range. Crossover between different scaling regimes. More than one cascade leading to mixing. KZ and equilibium scaling exponents coincide. Spectrally broad forcing can contaminate the inertial range. KZ spectrum can be nonlocal. KZ spectrum can break down at scales of interest. Presence of coherent structures with their own scaling. Finite basin can invalidate the continuum description of resonances. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence
  27. 27. Summary (before we get on to the interesting stuff) This lecture has introduced most of the basic concepts of wave turbulence theory in the context of a single cascade of energy in a 3-wave system. Many properties are in common with hydrodynamic turbulence but some important differences. For weakly nonlinear waves, the statistical dynamics has a natural closure which leads to the wave kinetic equation for the wave spectrum. For isotropic systems it is much neater to study the kinetic equation in frequency space. Under certain conditions it is possible to find the stationary Kolmogorov-Zakharov solution of the kinetic equation exactly. Concepts of locality, capacity and breakdown were introduced. http://www.slideshare.net/connaughtonc An introduction to (weak) wave turbulence

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