Introduction to Turbulent Dynamical Systems in Complex Systems

Transcription

1 Introduction to Turbulent Dynamical Systems in Complex Systems Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 216 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

2 Lecture 1: Introduction to Turbulent Dynamical Systems in Complex Systems 1 General turbulent dynamical systems for complex systems Basic issues for prediction, uncertainty quantification, and state estimation Stochastic toolkit for UQ Detailed structure in complex systems Energy conservation principles 2 Prototype examples of complex turbulent dynamical systems Turbulent dynamical systems for complex geophysical flows: one-layer model The Lorenz 96 model as a turbulent dynamical system Statistical triad models, the building blocks of turbulent dynamical systems More rich examples of complex turbulent dynamical systems Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

3 Outline 1 General turbulent dynamical systems for complex systems Basic issues for prediction, uncertainty quantification, and state estimation Stochastic toolkit for UQ Detailed structure in complex systems Energy conservation principles 2 Prototype examples of complex turbulent dynamical systems Turbulent dynamical systems for complex geophysical flows: one-layer model The Lorenz 96 model as a turbulent dynamical system Statistical triad models, the building blocks of turbulent dynamical systems More rich examples of complex turbulent dynamical systems Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

4 General complex turbulent dynamical systems Consider a general dynamical system, perhaps with noise, written in the Ito sense as given by du = F (u, t) + σ (u, t) Ẇ (t). (1) dt General properties The turbulent dynamical systems live in a large dimensional phase space u R N, with N 1; σ is an N K noise matrix and W R K is K-dimensional white noise; The noise often represents degrees of freedom that are not explicitly modeled such as the small-scale surface wind on the ocean; Typically one thinks about the evolution of smooth probability density p (u, t) associated with (1). Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

5 Challenges in predicting turbulent systems du = F (u, t) + σ (u, t) Ẇ (t). dt The turbulent dynamical systems are characterized by a large dimensional phase space u R N, with N 1 The phase space contain a large dimensional of unstable directions measured by the number of positive Lyapunov exponents; or non-normal transient growth subspace. All these linear instabilities are mitigated by energy-conserving nonlinear interactions that transfer energy from linearly unstable modes to the stable ones Increase of energy in the linearly unstable modes is balanced by the nonlinear interactions; The linear stable modes dissipate the energy transferred from the unstable modes; A statistical steady state is reached for the complex turbulent system. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

6 Challenges for uncertainty quantification in turbulent dynamical systems Uncertainty quantification (UQ) deals with the probabilistic characterization of all the possible evolutions of a dynamical system given an initial set of possible states as well as the statistical characteristics of the random forcing or parameters. Sources of uncertainty Internal instabilities Initial and boundary conditions Approximations on the model Limited observations and data Computational challenges Non-Gaussian statistics Large dimensional phase-space Non-stationary dynamics Wide range of scales Major Goal: Obtain accurate statistical estimates such as change in the mean and variance for the key statistical quantities in the nonlinear response to changes in external forcing or uncertain initial data. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

7 Imperfect models with model error The efforts in UQ are hampered by the inevitable model errors and the curse of ensemble size for complex turbulent dynamical systems Perfect Model : Imperfect Model : du dt = F (u, t) + σ (u, t) Ẇ (t), u RN, du M = F M (u M, t) + σ M (u M, t) Ẇ (t), u M R M. dt Practical complex turbulent models often have a huge phase space with N = 1 6 to N = 1 1 ; Model errors typically arise from lack of resolution M N compared with the original perfect model. The perfect model is too expensive to simulate directly; The lack of physical understanding of certain physical effects; The noise σm is often non-zero and judicious chosen to mitigate model errors. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

8 Monte-Carlo ensemble prediction For chaotic turbulent dynamical systems, single predictions often have little statistical information and Monte-Carlo ensemble predictions are utilized L p L (u (x, t)) = p j δ (u u j (t)), where du j = F (u j, t) dt, j=1 with initial data p (u) L = p,j δ (u u,j (t)), u j t= = u,j. j=1 The curse of ensemble size arise for practical predictions since N is huge while L = O (1) is small by computational limitation; With model errors using lower resolution M, L can be increased but model errors can swamp this gain in statistical accuracy; It is a grand challenge to devise methods that make judicious model errors that lead to accurate predictions and UQ. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

9 State estimation (data assimilation, or filtering) 2. Basic idea of filtering. In analysis step, Bayesian formula is utilized: p(u m+1 v m+1 ) p(u m+1 )p(v m+1 u m+1 ) In linear and Gaussian case, the optimal filter is the so-called Kalman filter. (See Majda-Harlim book for more details.) 46 / 48 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

10 Examples for complex dynamical systems with uncertainty Large or dimensional systems with uncertainties Examples of continuous systems Examples of continuous systems Thermal transport in heterogeneous media Plasma flow Geophysical fluid flows Geophysical fluid flows Examples of discrete systems Examples of discrete Systems biology Water waves spectrum evolution High Reynolds numbers systems Structural systems US power network 2 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

11 Distribution and density function Distribution and density functions Monday, September 19, 211 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38 7

12 Moments of random variables 4. Moments. Consider 1-D case for simplicity. The n-th moment and central moment of X are defined respectively by Z 1 µ n = E[X n ]= x n p(x)dx, n 1, 1 Z 1 µ n = E[(X µ) n ]= (x µ) n p(x)dx, n 2. 1 Statistics Description Moment Formula Mean the central tendency 1-st moment µ Variance spreading out 2-nd central moment 2 = µ 2 Skewness the asymmetry 3-rd normalized central moment µ 3 / 3 Kurtosis the peakedness 4-th normalized central moment µ 4 / 4 PDF with different skewness (linear scale).6.5 Skewness = Skewness = Skewness =.78 Skewness = Logarithm scale PDF with different kurtosis (linear scale).5 Kurtosis = 9.4 Kurtosis = 7.3 Kurtosis = 5.2 Kurtosis = 3.1 Kurtosis = Logarithm scale 1 2 fat tails sub Gaussian 5 / 48 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

13 Stochastic processes Ito Stochastic differential equations Definitions: Stochastic process X satisfies the Ito SDE t>t >, X(t )=X if for all t, t Z t Z t Definition: X(t) [Wiener =X(tprocess )+ A(X(s),s)ds (Brownian motion) + B(X(s),s)dW ] (s) t t Real-valued stochastic process W(t) such that + (with probability one) X nonanticipating Stieltjes integral Ito integral conditions (Lipschitz and growth cnds.) on the functions A and B for existence and uniqueness of path-wise solutions (see, e.g., Oksendal 2) W i W (t i ) W (t i 1 ) Definition: [Ito stochastic integral] Z t nx E( Wi 2 )= t i B(X(s),s)dW (s) :=m.s. lim B(X(t j 1 ),t j 1 )(W (t j ) W (t j 1 )) t n!1 j=1 Sample paths of the Wiener process are, with probability one, nowhere differentiable where the limit is taken in the mean-square sense. Monday, September 19, 211 m.s. lim n!1 X n = X ) lim n!1 h(x n X) 2 i = 9 Monday, September 19, 211 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38 1

14 Basic properties of Ito calculus Basic properties of Ito calculus Ito formula Assume that dx = a(x, t)dt + b(x, t)dw (t) Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

15 The Fokker-Planck equation The Fokker Planck equation Describes evolution of probability density associated with a stochastic process which satisfies an SDE Scalar case Consider a C 2 function f. Then the mean evolution Integrate by parts, note that f arbitrary and by definition Fokker-Planck equation Multivariate case Monday, September 19, Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

16 Stationary solutions of scalar Fokker Fokker-Planck equation equation L FP p eq (x) = Consider linear SDE with Fokker-Planck equation Thus, implies Gaussian density Monday, September 19, 211 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38 13

17 Linear Linear (unforced) Langevin Langevin equation (scalar equation unforced case) with Path-wise solutions Second-order statistics Equilibrium statistics Equilibrium density hui eq = Var eq (u) = 2 2 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

18 Detailed structure in complex systems The system setup will be a finite-dimensional system of, u R N, with linear dynamics and an energy preserving quadratic part du dt = L [u] = (L D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω). (2) L being a skew-symmetric linear operator L = L, representing the β-effect of Earth s curvature, topography etc. D being a negative definite symmetric operator D = D, representing dissipative processes such as surface drag, radiative damping, viscosity etc. B (u, u) being the quadratic operator which conserves the energy by itself so that it satisfies B (u, u) u =. F (t) + σ k (t) Ẇ k (t; ω) being the effects of external forcing, i.e. solar forcing, seasonal cycle, which can be split into a mean component F (t) and a stochastic component. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

19 Energy conserving principle Consider the energy E = 1 2 u 2 ; The skew-symmetric operator makes no contribution u Lu = ; The symmetric operator D represents strict dissipation so that u Du d u 2, d >. de dt = d ( ) 1 dt 2 u 2 = (Du u) + F u d 2 u d F 2 = de + 1 2d F 2. The elementary inequality a b a b 2 2 has been used, the Gronwall inequality guarantees the global existence of bound smooth solutions. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

20 Local existence and uniqueness theorem Definition Let O R N be an open set. A function F : O R N is said to be Lipschitz on O if there exists a constant K such that for all X, Y O. Theorem Consider the initial value problem F (Y ) F (X ) K Y X, Ẋ = F (X ), X () = X where X R N. Suppose that F : O R N is Lipschitz. Then there exists a > and a unique solution X : ( a, a) R N of this differential equation satisfying the initial condition. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

21 Extending solutions Definition Let O R N be open, and let F : O R N be Lipschitz. Let Y (t) be a solution of Ẋ = F (X ) defined on a maximal open interval J = (α, β) R with β <. Then given any compact set C O, there is some t (α, β) with Y (t ) / C. This theorem says that if a solution Y (t) cannot be extended to a larger time interval, then this solution leaves any compact set in O. This implies that, as t β, Y (t) accumulates on the boundary of O. Using the energy equation, Gronwall s inequality implies de dt de + 1 2d F 2 E (t) E () e dt + 1 2d ˆ t e d(t s) F (s) 2 ds < C. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

22 Outline 1 General turbulent dynamical systems for complex systems Basic issues for prediction, uncertainty quantification, and state estimation Stochastic toolkit for UQ Detailed structure in complex systems Energy conservation principles 2 Prototype examples of complex turbulent dynamical systems Turbulent dynamical systems for complex geophysical flows: one-layer model The Lorenz 96 model as a turbulent dynamical system Statistical triad models, the building blocks of turbulent dynamical systems More rich examples of complex turbulent dynamical systems Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

23 One-layer model The one-layer geophysical flow on a periodic domain T 2 = [ π, π] 2 is given by dq dt + ψ q = D ( ) ψ + f (x) + Ẇ t, q = ψ F 2 ψ + h (x) + βy. q is potential vorticity, ψ is stream function, and the flow by u = ψ = ( y ψ, x ψ); The operator D ( ) = l j=1 ( 1)j γ j j stands for a general dissipation operator; f (x) is the external deterministic forcing, the random forcing W t is a Gaussian random field; βy is the β-plane approximation of the Coriolis effect, and h (x) is the periodic topography; The constant F = L 1 R, where L R = gh /f is the Rossby radius which measures the relative strength of rotation to stratification. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

24 One-layer model The idealized geophysical flow can produce a remarkable number of realistic phenomena: the formation of coherent jets and vortices; direct and inverse turbulent cascades; statistical bifurcations between jets and vortices. Numerical experiments and statistical approximations are only possible or valid for large finite times; Recent paper 1 proves with full mathematical rigor a unique smooth invariant measure which attracts all statistical initial data at exponential rate. 1 Majda & Tong, Ergodicity of truncated stochastic Navier-Stokes with deterministic forcing and dissipation, JNLS, 215 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

25 Lorenz 96 system L-96 du i dt = u i 1 (u i+1 u i 2 ) u i + F i, i =, 1,..., J 1. usually take J = 4 to simulate mid-latitude turbulence; energy conservation term B (u, u) u = (nonlinear advection); negative definite term D = I (dissipation); system is invariant under translations, steady state statistics become spatially homogeneous, thus Fourier basis a natural choice; external noise is, so uncertainty builds from unstable modes of linearized dynamics L v. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

26 F Linear Analysis Mean Statistics Mixing k begin k end Reω ū E (ū) E p λ 1 N + T corr F = 5.12 F = 8.8 F = 16 Percentage of Energy % Percentage of Energy % Percentage of Energy % wavenumber wavenumber wavenumber 15 F = F = F = time 1.63 time time space space space Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

27 Triad system: building block of turbulence The three-dimensional triad system with a quadratic part that is both divergence free and energy preserving, u = (u 1, u 2, u 3 ) T L = du dt = (L + D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω). λ 12 λ 13 λ 12 λ 23, B (u, v) = B 1u 3 v 2 B 2 u 1 v 3, D = γ 1 γ 2 λ 13 λ 23 B 3 u 2 v 1 γ 3 energy conservation : B 1 + B 2 + B 3 =. Condition for Gaussian invariant measure σ 2 1 2γ 1 = σ2 2 2γ 2 = σ2 3 2γ 3 = E, Strong nonlinear regime with energy cascade, Σ = ( p eq = C 1 exp 2E 1 u 2). σ 1 σ 2 σ 3 λ 12 =, λ 13 =, λ 23 =, B 1 = 2, B 2 = 1, B 3 = 1, γ 1 =.1, γ 2 = 1, γ 3 = 1, σ1 2 = 1.41, σ2 2 =.4, σ2 3 =.4. The first component is weakly damped and strongly forced by noise while the other two components are strongly damped and weakly forced. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

28 Figure: Figure Strongly 4: Strongly nonlinear nonlinear regime regime with with energy energy cascade: full-system Full-systemstatistics predicted with direct direct Monte-Carlo Monte-Carlo in the in the original original system. In In the the right right plots plots the stready the steady state conditional state conditional probability pdfs of density functions of p u1u2u3 are shown as well as 2D scatter plots. p u1 u 2 u 3. In Figure 4 we present the statistics of the system as computed by the direct Monte-Carlo This instructive example illustrates very clearly that the small magnitude of the method. Thus, in the statistical steady state regime we have one mode carrying most of the uncertainty system energy in specific and twodirections low-energy may modesnot absorbing always energy be an fromefficient the rst one. criterion This energy to neglect transfer dynamics. property This manifested is especially by the the third-order case formoment turbulent hu 1 u 2 flows u 3 i (shown where in Figure energy4) transfers whose negative have to value indicates the energy transfer from the rst mode to the other two. This ow of energy is also be modeled even if the modes are associated with weak energy. illustrated by the nearly two-dimensional character of the joint probability density function (see [15] for a rigorous connection of the energy transfer properties and the dimensionality of the probability Di Qi, measure). and Andrew J. This Majda strongly (CIMS) nonlinear Introduction regime to with Turbulent energy Dynamical cascade Systems has a Fokker-Planck equation Nov. 1, with / 38

29 The elementary analysis in (2.5) suggests that we can expect a flow or cascade of energy from u 1 to u 2 and u 3 where it is dissipated provided the interaction coefficient B 1 has the opposite sign from B 2 and B 3. Di Qi, We and Andrew illustrate J. Majda this (CIMS) intuitionintroduction a simple to Turbulent numerical Dynamical Systems experiment in a nonlinear Nov. 1, regime / 38 turbulent dynamical system that are not resolved in the three dimensional subspace [121, 122, 123]. Stochastic triad models are qualitative models for a wide variety Intuition about energy transfer of turbulent phenomena regarding energy exchange and cascades and supply important intuition for such effects. They also provide elementary test models with subtle features for prediction, UQ, and state estimation [49, 15, 51, 156, 157]. Elementary intuition about energy transfer in such models can be gained by looking at the special situation with L = D = F = s so that there are only the nonlinear interactions in (2.4). We examine the linear stability of the fixed point, ū =(ū 1,,) T. Elementary calculations show that the perturbation du 1 satisfies = while the perturbations du 2,du 3 satisfy the second-order equation ddu 1 dt so that d 2 dt 2 du 2 = B 2 B 3 ū 2 1du 2, d 2 dt 2 du 3 = B 2 B 3 ū 2 1du 3, there is instability with B 2 B 3 > and the energy of du 2,du 3 grows provided B 1 has (2.5) the opposite sign of B 2 and B 3 with B 1 + B 2 + B 3 =.

30 2.4 More Rich Examples of Complex Turbulent Dynamical Systems 11 Quantitative models Quantitative Models A) The truncated turbulent Navier-Stokes equations in two or three space dimensions with shear and periodic or channel geometry [143]. B) Two-layer or even multi-layer stratified flows with topography and shears in periodic, channel geometry or on the sphere [13, 171, 94]. These models include more physics like baroclinic instability for transfer of heat and generalize the one-layer model discussed in Section 2.1. There has been promising novel multiscale methods in two-layer models for the ocean which overcome the curse of ensemble size for statistical dynamics and state estimation called s- tochastic superparameterization. See [11] for a survey and for the applications [65, 66, 67, 68, 64, 63] for state estimation and filtering. The numerical dynamics of these stochastic algorithms is a fruitful and important research topic. The end of Chapter 1 of [13] contains the formal relationship of these more complex models to the one-layer model in Section 2.1. C) The rotating and stratified Boussinesq equations with both gravity waves and vortices [124, 171, 94]. There are even more models with clouds and moisture which could be listed. Next is the list of qualitative models with insight on the central issues for complex turbulent dynamical systems. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

31 Two-layer baroclinic model We consider the Phillips model in a barotropic-baroclinic mode formulation with periodic boundary condition given by dq dt = L (q) + B (q, q), with q = (q ψ, q τ ) T = ( 2 ψ, 2 τ λ 2 τ ) T, and the quadratic operator ( B (q 1, q 2 ) = J (ψ 1, q 2,ψ ) + J (τ 1, q 2,τ ) J (ψ 1, q 2,τ ) + J (τ 1, q 2,ψ ) + ξj (τ 1, q 2,τ ) as well as the linear operator ( (1 δ) r ( L (q) = 2 ψ a 1 τ ) U x 2 τ β ψ ) ( x δ (1 δ)r 2 ψ a 1 τ ) β τ x U ( x 2 ψ + λ 2 ψ + ξ 2 τ ) with U = δ (1 δ) (U 1 U 2 ), a = (1 δ) δ 1, ξ = (1 2δ) / δ(1 δ). ) Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

32 y y y y With 128 modes meridionally and zonally, corresponding to grid points, more than 125, state variables are needed for numerical integration of this turbulent system! mean barotropic stream function <ψ> low latitude.15 mean baroclinic stream function <τ> low latitude x x.1 mean barotropic stream function <ψ> high latitude mean baroclinic stream function <τ> high latitude x x Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

33 72 Chapter 2. Effects of Rotation and Stratification Boussinesq equation with gravity waves and vortices Summary of Boussinesq Equations The simple Boussinesq equations are, for an inviscid fluid: momentum equations: Dv + f v = φ + bk, Dt (B.1) mass conservation: v =, (B.2) buoyancy equation: Db Dt = ḃ. A more general form replaces the buoyancy equation by: (B.3) thermodynamic equation: Dθ Dt = θ, (B.4) salinity equation: DS Dt = Ṡ, (B.5) equation of state: b = b(θ,s,φ). (B.6) Energy conservation is only assured if b = b(θ,s,z). Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

34 Qualitative models Qualitative Models A) The truncated Burgers-Hopf (TBH) model: Galerkin truncation of the inviscid Burgers equation with remarkable turbulent dynamics with features predicted by simple statistical theory [119, 125, 12]. The models mimic stochastic backscatter in a deterministic chaotic system [2]. B) The MMT models of dispersive wave turbulence: One-dimensional models of wave turbulence with coherent structure, wave radiation, and direct and inverse turbulent cascades [116, 23]. Recent applications to multi-scale stochastic superparameterization [66], a novel multi-scale algorithm for state estimation [62], and extreme event prediction [35] are developed. C) Conceptual dynamical models for turbulence: There are low-dimensional models capturing key features of complex turbulent systems such as non-gaussian intermittency through energy conserving dyad interactions between the mean and fluctuations in a short self-contained paper [115]. Applications as a test model for non-gaussian multi-scale filtering algorithms for state estimation and prediction [91] will be discussed in Section 5.4. It is very interesting and accessible to develop a rigorous analysis of these models and also the above algorithms. Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

35 Need Statistically Accurate Inexpensive Forecast Models to Beat the Curse of Ensemble Size for MMT modelprediction, State Estimation and UQ The MMT equation The MMT equation (Majda, McLaughlin and Tabak, 1997; Cai and M.M.T., Phys. D 21) iu t x 1 2 u + u 2 u iau + F. Here we consider the case with the focusing nonlinearity, = 1, which induces spatially coherent solitonic excitations at random spatial locations. I The instability of collapsing solitons radiate energy to large scales producing direct and inverse turbulent cascades. I In geophysical applications energy oftern flows from small scales to large scales (inverse cascade) creating a challenge for reduced modelling. I Fractional dispersion are crucial with completely different behavior from NLS equation! 39 / 51 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

36 Visualization of (x, t) from simulation with F =.163; darker colors indicate higher amplitudes. Here the number of Fourier modes are From Cai etal, Physica D / 51 Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38

37 ū ū u u Conceptual dynamical models for turbulence variables u A hallmark of turbulence is that the large scales can destabilize k,k =1, 2,...,5 have essentially zero means with the smaller scales in fluctuations variances.446,.174,.49,.14, and.5 respec- impact intermittently and this increased small scale energy cantively with the correlation the large time for scales u approximately (Majda 34 while & Lee, those for u k,k =1, 2,...,5 are decreasing with k and approximately 29,16,6,4, and 3 respectively. These are all the features PNAS, 214) of anisotropic turbulence required from (A)-(D) and demonstrated in the conceptual dynamical models; furthermore all dū K dt = dū ( ) + γ u 2 of these k ᾱū 3 conditions occur in a robust fashion for F increasing in magnitude with F apple.55 and.55 apple F apple.1. All + F, of the detailed data discussed above can be found in Tables utral stability ability matrix [1] if and only if rs for the nutualmodelin scenarios with oth cases, the ow while for = [11] =.5. e two critical d =.67 hile the critto u perturs a Lyapunov nverge to eihe marginally t behavior in as the forcing lues, F with s for the case with a single le to pertur- R), u CR 6=, 1-3 of the supplementary material. There is an evident role k=1 for the unstable damping of the large scales, d =.1 toincrease the variance of u with its mean near the marginally du k = d k u k dt k + σ critical value u so that the instability mechanism from [5] kẇk, operates1vigorously k in the K. model and creates more variance in u k,k =1, 2,...,5. Thus, we expect the system with stable all simulations the initial value is u =1.5 withu k = for damping and the same values of F with F =.55 to have k =1, 2,...,5. less variance. Negative large scale damping d =.1 u 2 u 3 u 4 u 5 u u 1.5 ū.5.5 u u u u 4 that the instability mechanism elucidated in [5] is operating on all modes and creating intermittency. The total energy of the mean flow u exceeds that of the fluctuations, u k. The Positive large scale damping d = time Di Qi, and Andrew J. Majda (CIMS) u Introduction 5 to Turbulent Dynamical Systems.1.2 u 5 Nov. 1, / time u 1 u 2 u 3 u 4 u u ū.5.5 u u u u in Fig. 2. Th tal energy with very close to th intermittent ins again. Both th similar fashion discussed above model with pos dynamical mod tures in (A)-(D) F with.55 no fat tails for as F =.1 expected from forcing with F apple three less varia damping case co case. Document tensive tables in cross correlation negligible in the less than the 5% In the above 5 to mimic the turbulence and a mathematical lowing: what is intermittency a Versions of the intermittency in (A)-(D) for bot the supplement with K =1al Gaussian behav level is varied in Concluding D Conceptual dyn been introduced key features of v tual dynamical scale mean flow on variety of s wave-mean flow stochastic forci a transparent m bilize the k-th analysis yields strong large sca aclimatological ble in the sense the mean u ofte

38 Questions & Discussions Di Qi, and Andrew J. Majda (CIMS) Introduction to Turbulent Dynamical Systems Nov. 1, / 38