Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
Turbulence in 3D and 2D
                            The inverse energy cascade in 2D
Generation of large scale coherent st...
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Large scale coherent structures and turbulence in quasi-2D hydrodynamic models

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Large scale coherent structures and turbulence in quasi-2D hydrodynamic models

  1. 1. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Large scale coherent structures and turbulence in quasi-2D hydrodynamic models Colm Connaughton Centre for Complexity Science and Mathematics Institute University of Warwick United Kingdom Simulation Hierarchies for Climate Modelling IPAM, 3-7 May 2010 Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  2. 2. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Outline 1 Turbulence in 3D and 2D 2 The inverse energy cascade in 2D 3 Generation of large scale coherent structures by finite size effects 4 Conclusions Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  3. 3. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Large scale coherent structures in geophysical flows coexist with turbulence Simulation of Earths Zonal turbulence on oceans (Earth Simulator Jupiter (NASA) Center/JAMSTEC) Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  4. 4. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Turbulence in 3D ∂u 2 +u· u =− p+ν u+f ∂t ·u =0 LU Reynolds number R = ν . Energy injected into large eddies. Energy removed from small eddies at viscous scale. Energy transferred by interaction between scales. Concept of inertial range and energy cascade. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  5. 5. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Kolmogorov theory of 3D turbulence K41 : As R → ∞, small scale statistics depend only on the scale, k , and the energy flux, . Dimensionally : 2 5 E(k ) = c 3 k−3 Kolmogorov spectrum Fluctuations characterised by structure functions: n r Sn (r ) = (δuL )n = (u(x + r ) − u(x)) · . r Scaling form in stationary state: lim lim lim Sn (r ) = Cn ( r )ζn . r →0 ν→0 t→∞ Third order structure function is special. In inertial range: 4 S3 (r ) = − r ζ3 = 1 (exact). 5 Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  6. 6. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Turbulence in two dimensions Draining soap film (W. Electromagnetically driven Goldburg et al., Pitts- fluid layer (R. Ecke et al., burgh) Los Alamos) Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  7. 7. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Conservation Laws in two-dimensional hydrodynamics Streamfunction formulation of 2D NS : u = (∂y ψ, −∂x ψ). Vorticity : × u = ∆ψ ˆ ω= z d ∂ ∂ψ ∂ω ∂ψ ∂ω Advective derivative : = + − dt ∂t ∂y ∂x ∂x ∂y d N-S equation: (∆ψ) = ν∆ (∆ψ) − α∆ψ + f (x, t) dt There are two inviscid invariants : 1 1 Total energy : E= | ψ|2 dx = k 2 |ψk |2 dk 2 2 1 1 Total enstrophy : H= ω 2 dx = k 4 |ψk |2 dk 2 2 This profoundly affects the physics. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  8. 8. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Inverse Cascade in Two Dimensional Turbulence A K41–like phenomenological theory of 2-D turbulence was developed by Kraichnan, Leith and Batchelor in 1967-68. For a stationary state, E and H must cascade in opposite directions in k . H goes to smaller scales with E(k ) = cη η 2/3 k −3 . E goes to larger scales with E(k ) = c 2/3 k −5/3 . Inverse cascade is a new phenomenon. Analogue of 4/5-Law (Lindborg 1999): S3 (r ) = 2 r Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  9. 9. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions The Atmospheric Energy Spectrum on Earth Nastrom and Gage, 1985. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  10. 10. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions How should we interpret the spectra? We must at least establish the direction of the energy flux: S3 (r ) = (δ uL )3 Analogue of 4/5-Law (Lindborg 1999): S3 (r ) = 2 r Positive S3 (r ) corresponds to an inverse cascade. This is unclear from the atmospheric data: Lindborg 1999 : “S3 (r ) is negative”. Cho and Lindborg 2001 : “S3 (r ) is positive”. Falkovich et al. 2008 : “S3 (r ) is very sensitive to presence of coherent structures”. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  11. 11. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Development of the Inverse Cascade Peak of the energy spectrum moves to larger scales leaving k −5/3 spectrum behind it. Timescale slows down as cascade proceeds. Growth eventually saturates due to the dissipation of energy near the spectral peak due to friction. Vorticity movie Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  12. 12. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Statistics of Power Input Why is the direction of the energy flux not immediately obvious? Consider the local power injected into the system : P(x, t) = v(x, t) · f(x, t). Timeseries of P(t) PDF, Π(P) with Gaussian forcing Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  13. 13. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Strong intermittency of energy injection? Yes, but not for a very interesting reason: Distribution of products of normals If x1 and x2 are joint normally distributed random variables with 2 2 mean zero, variances σ1 and σ2 and correlation coefficient ρ, ρz e (1−ρ2 )σ1 σ2 |z| PXY (z = x1 x2 ) = K0 πσ1 σ2 1− ρ2 (1 − ρ2 )σ1 σ2 z = ρ σ1 σ2 must be equal to the total energy dissipation rate in the steady state. Degree of asymmetry is controlled by the dissipation rate. Deviations from the product distribution have dynamical significance, not the distribution itself. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  14. 14. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Comparison with numerical measurements 2D 3D Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  15. 15. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Gallavotti-Cohen fluctuation relation An amusing aside: the product distribution satisfies the fluctuation relation: Consider the sum of n samples from PXY : n 1 Mn = zi where zi ∼ PXY (z) n i=1 Large deviation principle: P(Mn > x) e−nI(x) . The rate function, I(x), is explicitly computable. P(Mn > X ) = e−n(I(X )−I(−X )) , (1) P(Mn < −X ) where: 2ρ I(x) − I(−x) = −Σx with Σ = . (1−ρ2 )σ1 σ2 Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  16. 16. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Gallavotti-Cohen fluctuation relation Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  17. 17. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Scale-by-Scale Energy Balance Inertial range energy flux is a third moment not a product. How do its statistics look compared to the injected flux? Filter streamfunction, ψ(x), at scale l: ¯ ψ(x) = dy ψ(y) Gl (x − y) ¯ 2 Large scale energy, El = ψ(x) , satisfies ∂El + · S = Πl ∂t S is the spatial energy flux and ¯ Πl = ψ · (u 2ψ −u 2 ψ) is the scale to scale energy transfer. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  18. 18. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions PDF of Πl from numerical data (inverse cascade) PDFs of Πl for l = 8lf , l = 16lf and l = 32lf where lf is the forcing scale. PDFs have been rescaled with their variances. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  19. 19. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Energy “Condensation”: coherent structures as a finite size effect In absence of friction, inverse cascade is blocked at scale L at t = t ∗ . After t ∗ , energy accumulates at large scales. (Smith & Yakhot 1991) Before t ∗ , E(k ) ∼ k −5/3 at large scales (Kraichnan). After t ∗ , observe a gradual cross-over to E(k ) ∼ k −3 at large scales. What is the x-space picture? Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  20. 20. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Real space view Large scale coherent vortices at the scale of the box. Organisation of smaller scale fluctuations Separation of time-scales. Stream function before Stream function after con- condensation densation Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  21. 21. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Laboratory experiments of Shats et al. Laboratory experiments on driven fluid layers by M. Shats’ group clearly demonstrate condensa- tion phenomenon. (Also numerics by Clercx et al.) Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  22. 22. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Extracting Coherent Flow using Wavelets Spectral representation is not useful for splitting the coherent flow from the fluctuations: use wavelets. Coherent component defined to contain 0.95 of the total energy. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  23. 23. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Decomposed spectrum Large scale k −3 spectrum is entirely the signature of the coherent structure. Nothing to do with cascades. Background turbulent fluctuations have a k −1 spectrum. Lesson: don’t over-emphasise spectral representation! Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  24. 24. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions What about confirming the direction of energy transfer? Computation of S3 (r ) is dominated by the presence of the coherent part. Time required to average out coherent flow is very long. Very sensitive to windowing. Conclude S3 (r ) is not an good diagnostic of the energy cascade. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  25. 25. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Scale-to-scale energy transfer in presence of coherent flow Measurement of the total scale-to-scale energy transfer in the inertial range shows strong signature of the coherent structure. Energy transfer l = 0.4. Quadrupolar stucture of the coherent energy transfer reflects the fact that a symmetric vortex cannot sustain a flux due to “depletion” of nonlinearity. Snapshot of Πl (x). Movie Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  26. 26. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions Conclusions The inverse cascade is a robust mechanism for the organisation of large scale structures from small scale fluctuations in two dimensional turbulence (and in more geophysically realistic models). Cartoon view of a smooth river of energy flowing through scales is misleading. Fluctuations are huge. The XY-product distribution is the benchmark distribution for understanding fluctuations in composite quantities like fluxes, not the normal distribution. Finite size effects may lead to strong organisation of the large scales (”condensation”). Diagnostics developed for h.i.t. (eg structure functions, spectra) do not work well in the presence of such coherent structures but others exist. Colm Connaughton, University of Warwick Inverse cascades and coherent structures
  27. 27. Turbulence in 3D and 2D The inverse energy cascade in 2D Generation of large scale coherent structures by finite size effects Conclusions References M. M. Bandi, S. G. Chumakov and C. Connaughton, "On the probability distribution of power fluctuations in turbulence", Phys. Rev E 79, 016309, 2009 M. M. Bandi and C. Connaughton, "Craig’s XY-distribution and the statistics of Lagrangian power in two-dimensional turbulence", Phys. Rev E 77, 036318, 2008 M. Chertkov, C. Connaughton, I. Kolokolov and V. Lebedev, "Dynamics of Energy Condensation in Two-Dimensional Turbulence", Phys. Rev. Lett. 99, 084501 (2007) Colm Connaughton, University of Warwick Inverse cascades and coherent structures

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