Mathematical Theories of Turbulent Dynamical Systems

Transcription

1 Mathematical Theories of Turbulent Dynamical Systems Di Qi, and Andrew J. Majda Courant Institute of Mathematical Sciences Fall 2016 Advanced Topics in Applied Math Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

2 Lecture 11 & 12: Mathematical Theories of Turbulent Dynamical Systems 1 Nontrivial turbulent dynamical systems with a Gaussian invariant measure 2 Exact equations for the mean and covariance of the fluctuations Turbulent dynamical systems with nontrivial third-order moments Statistical dynamics in the L-96 model and statistical energy conservation 3 A statistical energy conservation principle for turbulent dynamical systems Details about deterministic triad energy conservation symmetry A generalized statistical energy identity Enhanced dissipation of the statistical mean energy, the statistical energy principle, and Eddy viscosity Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

3 Outline 1 Nontrivial turbulent dynamical systems with a Gaussian invariant measure 2 Exact equations for the mean and covariance of the fluctuations Turbulent dynamical systems with nontrivial third-order moments Statistical dynamics in the L-96 model and statistical energy conservation 3 A statistical energy conservation principle for turbulent dynamical systems Details about deterministic triad energy conservation symmetry A generalized statistical energy identity Enhanced dissipation of the statistical mean energy, the statistical energy principle, and Eddy viscosity Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

4 Turbulent systems with special forms of damping and noise The system setup will be a finite-dimensional system of, u R N, with linear dynamics, an energy preserving quadratic part, and random white noise forcing du dt = L [u] = B (u, u) + Lu dλu + Λ1 /2 σẇ (t; ω). Properties in the operators u B (u, u) = 0, Energy Conservation, u Lu = 0, Skew Symmetry for L, div u (B (u, u)) = 0, Liouville Property. The scalar d and σ satisfy σ 2 eq = σ 2 /2d, and Λ 0 (positive-definite) controls the linear damping rate and the noise amplitude. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

5 Turbulent systems with special forms of damping and noise The system setup will be a finite-dimensional system of, u R N, with linear dynamics, an energy preserving quadratic part, and random white noise forcing du dt = L [u] = B (u, u) + Lu dλu + Λ1 /2 σẇ (t; ω). Properties in the operators u B (u, u) = 0, Energy Conservation, u Lu = 0, Skew Symmetry for L, div u (B (u, u)) = 0, Liouville Property. The scalar d and σ satisfy σ 2 eq = σ 2 /2d, and Λ 0 (positive-definite) controls the linear damping rate and the noise amplitude. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

6 General setup of turbulent systems Examples: du dt = L [u] = B (u, u) + Lu dλu + Λ1 /2 σẇ (t; ω). General 3 3 triad system model; Truncated Burgers-Hopf (TBH) equation, and inviscid L-96; Quasi-geostrophic flow with topography, mean flow in pseudo-energy metric; Inviscid Boussinesq or modified shallow water equation. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

7 Gaussian invariant measure Proposition du dt = L [u] = B (u, u) + Lu dλu + Λ1 /2 σẇ (t; ω). (1) Associated with SDE in (1), with the structural properties, then (1) has the Gaussian invariant measure defined as in p eq ; The invariant measure has equipartition of energy in each component of u, p eq = C 1 N exp ( 1 2 σ 2 eq u u ), with σeq 2 = σ 2 /2d. Proof. The Fokker-Planck Equation for (1), using the properties, is given by ( dp Λσ 2 ) = div u [(B (u, u) + Lu) p] + div u (dλup) + div u dt 2 p ( ) = (B (u, u) + Lu) p + div u dλup + Λσ2 2 p. Insert p eq dλupeq + Λσ2 Λσ2 upeq = dλupeq 2 2 σ 2 eq up eq 0. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

8 Gaussian invariant measure Proposition du dt = L [u] = B (u, u) + Lu dλu + Λ1 /2 σẇ (t; ω). (1) Associated with SDE in (1), with the structural properties, then (1) has the Gaussian invariant measure defined as in p eq ; The invariant measure has equipartition of energy in each component of u, p eq = C 1 N exp ( 1 2 σ 2 eq u u ), with σeq 2 = σ 2 /2d. Proof. The Fokker-Planck Equation for (1), using the properties, is given by ( dp Λσ 2 ) = div u [(B (u, u) + Lu) p] + div u (dλup) + div u dt 2 p ( ) = (B (u, u) + Lu) p + div u dλup + Λσ2 2 p. Insert p eq dλupeq + Λσ2 Λσ2 upeq = dλupeq 2 2 σ 2 eq up eq 0. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

9 Example: a simplified 57-mode testing model 12 In the numerical experiments, the truncation k 2 Λ, with Λ = 17 are utilized with 57 degrees of freedom d ˆω k dt dv dt = = ( ) ψ q k ) + ik 1 (β/ k 2 V ˆω k ik 1 ĥ k V ( dλ k ω k + σ 1 + µ/ k 2) 1/2 1/2 λ k Ẇ k, ik 1 k 2 ĥ k ˆω k dλ 0V + σµ 1/2 λ 1/2 0 Ẇ 0. 1 k Λ The pseudo-energy metric induces p eq (V, ω) = C 1 exp µ 2 σ 2 eq ( V V ) σ 1 eq ( ) 1 + µ k 2 (ω k ω k ) 2. 1 k 2 Λ So the steady state large and small scale variables reach the equilibrium mean and variance V = β µ, ψ k = h k µ + k 2, V 2 = σ2 eq µ, ψ 2 = σeq 2 ( ). k 2 k 2 + µ 1 Majda, Timofeyev, and Vanden-Eijnden, JAS, Majda, Franzke, Fischer, and Crommelin, PNAS, 2006 Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

10 Equilibrium steady state data V V 2 ψ (1,0) ψ 2 (1,0) theory i numerics i ψ 2 (0,1) ψ 2 (2,0) ψ 2 (0,2) V ψ (1,0) theory numerics ( i) 10 3 Table: Unperturbed equilibrium statistics of the 57-mode model with damping and forcing term. The parameters µ = 2, σ 2 eq = 1, β = 1, H = 3 2/4, d = 0.5 are used in the model. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

11 Outline 1 Nontrivial turbulent dynamical systems with a Gaussian invariant measure 2 Exact equations for the mean and covariance of the fluctuations Turbulent dynamical systems with nontrivial third-order moments Statistical dynamics in the L-96 model and statistical energy conservation 3 A statistical energy conservation principle for turbulent dynamical systems Details about deterministic triad energy conservation symmetry A generalized statistical energy identity Enhanced dissipation of the statistical mean energy, the statistical energy principle, and Eddy viscosity Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

12 General setup of turbulent systems with quadratic nonlinearities The system setup will be a finite-dimensional system with linear dynamics and an energy preserving quadratic part with u R N du dt = L [u (t; ω) ; ω] = (L + D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω), (3) u (t 0 ; ω) = u 0 (ω). (4) 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 a quadratic operator which conserves the energy by itself so that it satisfies B (u, u) u = 0. 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 with white noise characteristics. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

13 General setup of turbulent systems with quadratic nonlinearities The system setup will be a finite-dimensional system with linear dynamics and an energy preserving quadratic part with u R N du dt = L [u (t; ω) ; ω] = (L + D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω), (3) u (t 0 ; ω) = u 0 (ω). (4) 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 a quadratic operator which conserves the energy by itself so that it satisfies B (u, u) u = 0. 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 with white noise characteristics. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

14 Ideas about Reduced Order Methods (ROMs) Approach: Project dynamics onto a finite-dimensional representation of the stochastic field consisting of fixed-in-time, N-dimensional, orthonormal basis {v i } N i=1 u (t) = ū (t) + u (t), u = N Φ i (t; ω) v i i=1 s Φ i (t; ω) v i ; i=1 Modeling is restricted to subspace V s = span {v i } s i=1 ; Aim is to capture as much of the dynamics as possible with few modes; Attractive for reduced computational cost. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

15 Finite-dimensional representation of the stochastic field Use a finite-dimensional representation of the stochastic field consisting of fixed-in-time, N-dimensional (u R N ), orthonormal basis {v i } u (t) = ū (t) + u (t) = ū (t) + N Z i (t; ω) v i, ū (t) = u (t), u = 0. Constructing the random field v i from equilibrium statistics i=1 C uu (x, y) = E ω [ (u (x; Geometrical ω) u (x; ω)) picture (u (y; ω) of KL u expansion (y; ω)) ] Solve the eigenvalue problem For a stochastic Cfield uu (x, u( y) x ;ωv) we i dx have = λ 2 i v ( i, (y) ). ( ( ; ) ( ; ))( ( ; ) ( ; T C E )) ω uu x y = u x ω u x ω u y ω u y ω Provides principal directions v i spanning Provides principal directions the phase space over which probability is distributed Solution of the eigenvalue problem ui x of the infinite-dimensional space over which probability is distributed For each principal direction For each v i principal the direction ui ( x) the associated eigenvalue λ associated eigenvalue λdefines i defines the spread theof spread i the probability of the probability ( ) ui 2 ( x) 2 C ( x, y) ui( x) dx= λi ui( y) uu ui 1 ( x) Probability measure For infinite-dimensional probability measures (case of stochastic fields) we have Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories Countable of Turbulent infinity Systems of principal directions Nov. 17, / 39 u i 3 ( x)

16 Finite-dimensional representation of the stochastic field Use a finite-dimensional representation of the stochastic field consisting of fixed-in-time, N-dimensional (u R N ), orthonormal basis {v i } u (t) = ū (t) + u (t) = ū (t) + N Z i (t; ω) v i, ū (t) = u (t), u = 0. Constructing the random field v i from equilibrium statistics i=1 C uu (x, y) = E ω [ (u (x; Geometrical ω) u (x; ω)) picture (u (y; ω) of KL u expansion (y; ω)) ] Solve the eigenvalue problem For a stochastic Cfield uu (x, u( y) x ;ωv) we i dx have = λ 2 i v ( i, (y) ). ( ( ; ) ( ; ))( ( ; ) ( ; T C E )) ω uu x y = u x ω u x ω u y ω u y ω Provides principal directions v i spanning Provides principal directions the phase space over which probability is distributed Solution of the eigenvalue problem ui x of the infinite-dimensional space over which probability is distributed For each principal direction For each v i principal the direction ui ( x) the associated eigenvalue λ associated eigenvalue λdefines i defines the spread theof spread i the probability of the probability ( ) ui 2 ( x) 2 C ( x, y) ui( x) dx= λi ui( y) uu ui 1 ( x) Probability measure For infinite-dimensional probability measures (case of stochastic fields) we have Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories Countable of Turbulent infinity Systems of principal directions Nov. 17, / 39 u i 3 ( x)

17 Development of statistical moment dynamics u (t) = ū (t) + u (t) = ū (t) + N Z i (t; ω) v i, u = 0. i=1 Take expectation on both sides of the equations du dt = (L + D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω) (5) Equation for the mean field ū dū dt = (L + D) ū + B (ū, ū) + R ijb (v i, v j ) + F (t), (6) where R ij = Z i Zj, B (u, u) = B (ū + u, ū + u ) = B (ū, ū) + B (ū, u ) + B (u, ū) + B (u, u ), B (u, u ) = B (Z i v i, Z j v j ) = Z i Zj B (vi, v j ). Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

18 Equation for the fluctuation component du dt = (L + D) u + B (ū, u ) + B (u, ū) + B (u, u ) R jk B (v j, v k ) + σ k (t) Ẇ k (t; ω). (7) Equations for the stochastic coefficients dz i dt = Z j [(L + D) v j + B (ū, v j ) + B (v j, ū)] v i + [B (u, u ) R jk B (v j, v k )] v i + Ẇ k σ k v i. (8) Equation for the covariance matrix R ij = Z i Zj dr dt = L v (ū) R + RL v (ū) + Q F + Q σ. (9) Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

19 Covariance Equation dr dt = L v (ū) R + RL v (ū) + Q F + Q σ the linear dynamics operator L v expressing energy transfers between the mean field and the stochastic modes (B), as well as energy dissipation (D), and non-normal dynamics (L) {L v } ij = [ (L + D) v j + B ( ū, v j ) + B ( vj, ū )] v i. the positive definite operator Q σ expressing energy transfer due to external stochastic forcing {Q σ} ij = v i σ k σ kv j. the third-order moments expressing the energy flux between different modes due to non-linear terms Q F = Z mz nz j B (vm, v n) v i + Z mz nz i B (v m, v n) v j. note that energy is still conserved in this nonlinear interaction part Tr [Q F ] = 2 Z mz nz i B (v m, v n) v i = 2 B (Z mv m, Z nv n) Z i v i = 2 B ( u, u ) u = 0. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

20 Energy flow in the quadratic systems Statistical mean and covariance dynamics, u = ū + Z i v i, R ij = Z i Z j, dū dt = (L + D) ū + B (ū, ū) + R ij B ( v i, v j ) + F (t), dr dt = Lv (ū) R + RL v (ū) + Q F + Q σ. Q F = Z mz nz j B (vm, v n) v i + Z mz nz i B (v m, v n) v j (a) perfect model (b) imperfect closure Figure 3: Energy flow in the L-96 system. Energy flows from the mean Figure field to 4: the Energy linearly flow unstable in the MQG UQ scheme. Energy flows from the mean field to the linearly modes and then redistributed through nonlinear, conservative termsunstable to the stable modesmodes. and then Redredistributed through empirical, conservative fluxes to the stable modes. arrows denotes dissipation, while the dashed box indicates terms that Red conserve arrows energy. denotes dissipation, while the dashed box indicates terms that conserve energy. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

21 Turbulent dynamical systems with non-gaussian statistical steady Turbulent states Dynamical Systems with Non-Gaussian Statistical Steady States and Nontrivial Third Moments Consider a turbulent dynamical system without noise, s 0, and assume it has a statistical steady state so that ū eq and R eq are time independent. Since d dt R eq = 0, R eq necessarily satisfies the steady covariance equation (3.6) Lūeq R eq + R eq L ū eq = Q F,eq, (3.11) where Q F,eq are the third moments from (3.9) evaluated at the statistical steady state. Thus a necessary and sufficient condition for a non-gaussian statistical steady state is that the first and second moments satisfy the obvious requirement that the matrix Lūeq R eq + R eq L ū eq 6= 0, (3.12) so the above matrix has some non-zero entries. This elementary remark can be So viewed the above as a sweeping matrix has generalization some non-zero for turbulent entries. dynamical The non-trivial systems third of the moments Karmanplay Howarth a crucial equation dynamical for therole Navier-Stokes in the L-96 equation model and (see forchapter two-layer 6 of ocean [47]). turbulence. The nontrivial third moments play a crucial dynamical role in the L-96 model and for twolayer ocean turbulence [158, 157], and is discussed in later chapters. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

22 Turbulent dynamical systems with non-gaussian statistical steady states In the system with a Gaussian invariant measure du dt = B (u, u) + Lu dλu + Λ1 /2 σẇ (t; ω). There ū eq = 0, R eq = σeqi 2, and Lūeq = L with L skew-symmetric, so LR eq + R eq L 0; The damping with matrix D = dλ exactly balances the stochastic forcing variance σ 2 Λ; For a strictly positive covariance R eq, there is a whitening linear transformation, so that TR eq T 1 = I. Nontrivial third moments are satisfied if TLūeq T 1, is non-zero. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

23 weakly chaotic Lorenz 96 system strongly chaotic fully turbulent L-96Table 2.1 Dynamical properties of L-96 model for weakly chaotic regime (F = 6), strongly chaotic regime (F = 8)and fully turbulent regime (F = 16). Here, l1 denotes the largest Lyapunov exponent, N + denotes the dimension of the expanding subspace of the attractor, KS denotes the dui Kolmogorov-Sinai and Tcorr denotes the decorrelation time ofi energy-rescaled corre= ui 1 (uentropy, = 0, 1,..., Jtime 1, J i+1 ui 2 ) di (t) ui + Fi (t), lation dt function. = 40. uj = u0. F=6 time 20 F= space F = space space Fig. 2.1 Space-time of numerical solutions of L-96 model for weakly chaotic (F = 6), strongly Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

24 Statistical formulation of the L-96 system We use a Fourier basis v k = { } J 1 1 J e 2πık j J because they diagonalize translation invariant systems with spatial homogeneity. Here are the statistical dynamics for L-96 model: j=0, dū (t) dt dr k (t) dt = d (t) ū (t) + 1 J J/2 k= J /2+1 r k (t) Γ k + F (t) =2 [ Γ k ū (t) d (t)] r k (t) + Q F,kk, k = 0, 1,..., J /2. (10) Here we denote Γ k = cos 4πk J cos 2πk J, r k = Z k Z k = Zk Zk = r k, and the nonlinear flux Q F for the third moments becomes diagonal Q F,kk = 2 J with energy conservation trq F = 0. m { ( 2m+k 2πi Re Z m Z m k Z k e J e )} m+2k 2πi J δ kk, Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

25 Statistical energy conservation principle for L-96 model By multiplying ū on both sides of the mean equation and by summing up all the modes in the variance equation in (10), we get dū 2 dt dtrr dt = 2dū 2 + 2ūF + 2 J ( ) = 2 k Γ k r k ū Γ k r k ū k 2dtrR + trq F. It is convenient to define the statistical energy including both mean and total variance as E (t) = J 2 ū2 + 1 trr. (11) 2 With this definition the corresponding dynamical equation for the statistical energy E of the true system can be easily derived as de dt = 2dE + JF ū trq F = 2dE + JF ū, (12) with symmetry of nonlinear energy conservation, trq F = 0, assumed. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

26 Nontrivial third moments for statistical steady state of L-96 model Consider the original L-96 model with constant forcing F and damping d = 1 with the statistical steady states ū eq, R eq, with R eq = {r eq,i δ ij } diagonal. For the L-96 model with homogeneous statistics, { } the linear operator is also block diagonal with the symmetric part L s ū eq = L s ū δ eq,i ij and L s ū eq,i = Γ i ū eq 1, i = 0,, J/2. The third moments are non-zero and the statistical steady state is non-gaussian provided L s ū eq,ir eq,i 0, for some i with 0 i J 2, in which case 2L s ū eq,ir eq,i = Q F,eq,i, 0 i J 2. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

27 Nontrivial third moments for statistical steady state of L-96 mode 14 Predicting Fat-Tailed Intermittent Probability Distributions in Passive Scalar Turbulence k 3 x 3 third order moments in spectral domain, F = State of the 16 Mean u l third order moments in spectal domain, F = x Fig. 2.5: 0 Model prediction of 0the 0 mean 10 state 20 ū30 40 x 1 u M (left upper row) and averaged energy trr M /J (or 1- i K = 120 K = 19 K = 015 K = k k 2 u k time (a) third order moments1-point between variance modes in spectral domain K = third order moments in physical domain, F = 8 third order moments in physical domain, F = 8 K = K = K = time u j (a) Model prediction for the mean and total variance Di Qi and Andrew J. Majda mode on each grid point, left lower row) from Galerkin truncated L-96 models in regime F =8 with numb (b) third order moments between state variables at different physical grid points Fourier modes K =19,15,10, compared with the true model with total number of modes K =20 Fig. Di state Qi, 2.6: andthird spectra Andreworder J. achieved Majda moments (CIMS) through of the flowthese Mathematical variable truncated u in Theories steady models of state Turbulent inare both Systems compared spectral and with physical the state domain Nov. variables in 17, regime 2016 û k 23 from / Energy Spectra N=20 N=19 N=15 N=10 (b) Model prediction for energy spectra

28 Outline 1 Nontrivial turbulent dynamical systems with a Gaussian invariant measure 2 Exact equations for the mean and covariance of the fluctuations Turbulent dynamical systems with nontrivial third-order moments Statistical dynamics in the L-96 model and statistical energy conservation 3 A statistical energy conservation principle for turbulent dynamical systems Details about deterministic triad energy conservation symmetry A generalized statistical energy identity Enhanced dissipation of the statistical mean energy, the statistical energy principle, and Eddy viscosity Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

29 A general statistical energy principle In L-96 system with homogeneous statistics, we have the statistical energy dynamics for the total statistical energy E (t) = J 2 ū trr de dt = 2dE + JF ū. A fundamental issue in UQ is that complex turbulent dynamical systems are highly anisotropic with inhomogeneous forcing and the statistical mean can exchange energy with the fluctuations. There is suitable statistical symmetry so that the energy of the mean plus the tracer of the covariance matrix satisfies an energy conservation principle for the general conservative nonlinear dynamics du dt = (L + D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω), where exact mean statistical field equation and the covariance equation can be calculated as dū dt = (L + D) ū + B (ū, ū) + R ijb (e i, e j ) + F, dr dt =Lv R + RL v + Q F + Q σ. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

30 Detailed symmetry in triad energy conservation Proposition Consider the three dimensional Galerkin projected dynamics spanned by the triad e i, e j, e k for 1 i, j, k N for the pure nonlinear model Assume the following: A) The self interactions vanish, (u Λ ) t = P Λ B (u Λ, u Λ ). (13) B (e i, e i ) 0, 1 i N; (14) B) The dyad interaction coefficients vanish through the symmetry, e i [B (e l, e i ) + B (e i, e l )] = 0, for any i, l. (15) Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

31 Detailed symmetry in triad energy conservation Proposition (continue) Then the three dimensional Galerkin truncation becomes the triad interaction equations for u = (u i, u j, u k ) = (u Λ e i, u Λ e j, u Λ e k ) du i dt =A ijku j u k, with coefficient satisfying du j dt = A jkiu k u i, du k dt = A kij u i u j. (16) A ijk + A jik + A kji = 0, (17) which is the detailed triad energy conservation symmetry, since A ijk + A jki + A kij e i [B (e j, e k ) + B (e k, e j )] + = 0. e j [B (e k, e i ) + B (e i, e k )] + e k [B (e i, e j ) + B (e j, e i )] Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

32 The projected energy conservation law Define Λ as the index set for the resolved modes under the orthonormal basis {e i } N i=1 of full dimensionality N in the truncation model u Λ = P Λ u = i Λ u i e i. The projected energy conservation law for truncated energy E Λ = 1 2 u Λ u Λ is satisfied depending on the proper conserved quantity and the induced inner product de Λ = u Λ P Λ B (u Λ, u Λ ) = 0. (18) dt The second equality holds since u Λ P Λ B (u Λ, u Λ ) = u Λ [e i B (u Λ, u Λ )] e i = j Λ i Λ u j e j N [e i B (u Λ, u Λ )] e i i=1 = u Λ B (u Λ, u Λ ), Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

33 Proof of the Proposition Now consider the triad truncated system about state variable u Λ in a three dimensional subspace. We take the index set with resolved modes as Λ = {i, j, k}. the explicit expressions can thus be calculated along each mode e m, m = i, j, k by applying assumption A) and B) in the Proposition du m dt = e m P Λ B (u Λ, u Λ ) = e m B ( u n e n, n Λ l Λ = u n u l e m B (e n, e l ) = n,l Λ n l Λ {m} u l e l ) u n u l e m B (e n, e l ) = u n u l e m [B (e n, e l ) + B (e l, e n )], n l m. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

34 Proof of the Proposition The interaction coefficients therefore can be defined as A ijk = e i [B (e j, e k ) + B (e k, e j )], with symmetry and vanishing property A ijk = A ikj, A ijk = 0, if two of the index i, j, k coincident. With this explicit definition of the coefficients A ijk, the detailed triad energy conservation symmetry is just direct application of the above formulas, that is for any u i, u j, u k with i j k de Λ dt du i = u i dt + u du j j dt + u du k k dt = (A ijk + A jki + A kij ) u i u k u j 0. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

35 Dynamics for the mean and fluctuation energy Proposition The change of mean energy Ē = 1 2 (ū ū) satisfies ( ) d 1 dt 2 ū 2 = ū Dū + ū F R ijū [B (e i, e j ) + B (e j, e i )]. (19) The last term represents the effect of the fluctuation on the mean, ū. Proposition Under the structure assumption in (14) and (15) on the basis e i, the fluctuating energy, E = 1 2trR, for any turbulent dynamical system satisfies, de dt = 1 2 tr ( DR + R D ) trq σ 1 2 R ijū [B (e i, e j ) + B (e j, e i )], (20) where R satisfies the exact covariance equation. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

36 Statistical energy conservation principle 3 Theorem ( Statistical Energy Conservation Principle) Under the structural assumption (14), (15) on the basis e i, for any turbulent dynamical systems in the form du dt = (L + D) u + B (u, u) + F (t) + σ k (t) Ẇ k (t; ω), the total statistical energy, E = Ē + E = 2ū 1 ū + 1 2trR, satisfies de ( ) dt = ū Dū + ū F + tr DR trq σ, (21) where R satisfies the exact covariance equation. 3 Majda, Statistical energy conservation principle for inhomogeneous turbulent dynamical systems, PNAS, Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

37 Illustration about the proof ˆR is the tensor representation of the covariance matrix with u = Z i e i, ˆR = u u = i,j R ij e i e j ; ( ) the matrix B ū, ˆR is defined as the componentwise interaction with each column of ˆR. Expand ū by ū = ū M e M, and use tr (a b) = a b, we have ( ) ) ] 1 [B tr ū, ˆR + B (ˆR, ū + transpose part 2 = 1 2 R ijū M [e i B (e M, e j ) + B (e i, e M ) e j + e j B (e M, e i ) + B (e j, e M ) e i ]. Now use the detailed triad conservation structure, the formula is given by e i [B (e j, e M ) + B (e M, e j )]+e j [B (e M, e i ) + B (e i, e M )] = e M [B (e i, e j ) + B (e j, e i )]. Now, use ū = ū M e M and get that the sum in (17) for this contribution to the trace is exactly 1 2 R ijū [B (e i, e j ) + B (e j, e i )]. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

38 Illustrative general examples and applications Corollary Under the assumption of the Theorem, assume D = di, with d > 0, then the turbulent dynamical system satisfies the closed statistical energy equation for E = 1 2ū ū trr, de dt = 2dE + ū F trq σ. (22) In particular, if the external forcing vanishes so that F 0, Q σ 0, for random initial conditions, the statistical energy decays exponentially in time and satisfies E (t) = exp ( 2dt) E 0. Assume the symmetric dissipation matrix, D, satisfies the upper and lower bounds, d + u 2 u Du d u 2, with d, d + > 0. Typical general dissipation matrices D are diagonal in basis with Fourier modes or spherical harmonics, any positive symmetric matrix R 0 we have the a priori bounds, ( ) DR + R D d +trr tr d trr. 2 Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

39 Illustrative general examples and applications Corollary Under the assumption of the Theorem, assume D = di, with d > 0, then the turbulent dynamical system satisfies the closed statistical energy equation for E = 1 2ū ū trr, de dt = 2dE + ū F trq σ. (22) In particular, if the external forcing vanishes so that F 0, Q σ 0, for random initial conditions, the statistical energy decays exponentially in time and satisfies E (t) = exp ( 2dt) E 0. Assume the symmetric dissipation matrix, D, satisfies the upper and lower bounds, d + u 2 u Du d u 2, with d, d + > 0. Typical general dissipation matrices D are diagonal in basis with Fourier modes or spherical harmonics, any positive symmetric matrix R 0 we have the a priori bounds, ( ) DR + R D d +trr tr d trr. 2 Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

40 Illustrative general examples and applications Corollary Assume D is diagonal and satisfies the upper and lower bounds, then the statistical energy in (21) in the Theorem, E (t), satisfies the upper and lower bounds E + (t) E (t) E (t) where E ± (t) satisfy the differential equality in Corollary 1 with d d ±. In particular, the statistical energy is a statistical Lyapunov function for the turbulent dynamical system. Also, if the external forcings F, Q σ vanish, the statistical energy decays exponentially with these upper and lower bounds. Consider the Gaussian approximation to the one point statistics; recall that u = ū + Z i e i so at the location x, the mean and variance are given by ū (x) = ū M e M (x), var (u (x)) = Z j Z k e j (x) e k (x). We have control over the variance of the average over the domain, denoted by E x because E xe j (x) e k (x) = δ jk I ; thus, the average of the single point variance is bounded by trr which is controlled by E. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

41 Illustrative general examples and applications Corollary Assume D is diagonal and satisfies the upper and lower bounds, then the statistical energy in (21) in the Theorem, E (t), satisfies the upper and lower bounds E + (t) E (t) E (t) where E ± (t) satisfy the differential equality in Corollary 1 with d d ±. In particular, the statistical energy is a statistical Lyapunov function for the turbulent dynamical system. Also, if the external forcings F, Q σ vanish, the statistical energy decays exponentially with these upper and lower bounds. Consider the Gaussian approximation to the one point statistics; recall that u = ū + Z i e i so at the location x, the mean and variance are given by ū (x) = ū M e M (x), var (u (x)) = Z j Z k e j (x) e k (x). We have control over the variance of the average over the domain, denoted by E x because E xe j (x) e k (x) = δ jk I ; thus, the average of the single point variance is bounded by trr which is controlled by E. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

42 Example with turbulent flow The nonlinear terms define B with dynamics alone ω Λ t = P Λ ( ψ Λ ω Λ ), ψλ = ω Λ, which satisfy both the conservation of Energy, E Λ, and Enstrophy, E Λ with E Λ = 1 ˆ ψ Λ ω Λ dx, and E Λ = 1 ˆ ω Λdx. It is well-known that in a Fourier basis, the symmetry conditions of Proposition are satisfied for either the L 2 inner product for E Λ or with a rescaled version of E Λ ˆ ˆ u 1, u 2 E ω 1 ω 2 dx = ψ 1 ψ 2 dx, u 1, u 2 E ˆ ψ 1 ω 2 dx = ˆ ψ 1 ψ 2 dx, Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

43 Example with a dyad model To check the generalized energy principle, we consider the simplest dyad interaction equation u 1 t u 2 t = γ 1 u 1 u 2 + γ 2 u 2 2, = γ 1 u 2 1 γ 2 u 1 u 2. Take the two-dimensional natural basis e 1 = (1, 0) T and e 2 = (0, 1) T, and the inner product is defined as the standard Euclidean inner product. Then the conservative quadratic interaction becomes B (e 1, e 1 ) = (0, γ 1 ) T, ( 1 B (e 1, e 2 ) = B (e 2, e 1 ) = 2 γ 1, 1 ) T 2 γ 2, B (e 2, e 2 ) = (γ 2, 0) T. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

44 The energy conservation u B (u, u) = 0 is satisfied in this model. Obviously the assumptions in (14) and (15) become non-zero B (e i, e i ) 0, e i [B (e l, e i ) + B (e i, e l )] 0. On the other hand, the dyad interaction balance is satisfied so that e 1 B (e 1, e 2 ) + e 1 B (e 2, e 1 ) + e 2 B (e 1, e 1 ) = γ 1 γ 1 = 0. Therefore the statistical energy conservation law is still valid for this dyad system [ d 1 (ū2 dt ū2) 2 1 ( ) ] + u u 2 2 = 0. Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39

45 Questions & Discussions Di Qi, and Andrew J. Majda (CIMS) Mathematical Theories of Turbulent Systems Nov. 17, / 39