Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification

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1 Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification Andrew J Majda joint work with Di Qi Morse Professor of Arts and Science Department of Mathematics Founder of Center for Atmosphere Ocean Science (CAOS) Courant Institute, New York University and Principal Investigator, Center for Prototype Climate Modeling (CPCM) NYU Abu Dhabi (NYUAD) IPAM, January 9-3, 7 Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

2 Challenges for 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 random forcing or parameters. Turbulent dynamical systems are characterized by a large dimensional phase space and high degrees of internal instability. Instabilities through energy-conserving nonlinear interactions result in a statistical steady state that is usually non-gaussian. Accurate quantification for the statistical variability to general external perturbations is important in climate change sciences. Major Task of this work: Investigate a concise systematic framework for measuring and optimizing consistency and sensitivity of imperfect dynamical models. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

3 General framework for statistical modeling The system setup will be a finite-dimensional system of, u RN, with linear dynamics and an energy preserving quadratic part du dt = L [u] = (L + D) u + B (u, u) + F (t) + σ k (t) W k (t; ω), L = L; D ; () u B (u, u). EW STRATEGIES FOR REDUCED-ORDER MODELS FOR PREDICTING COMPLEX TURBULENT DYNAMICAL SYSTEMS Modeling Stages Math. & Computational Tools Model Seclection Ergodic Theory Statistical Measures in Equilibrium Model Calibration Empirical Information Theory Linear Response Theory Total Statistical Energy Equations Model Prediction Accurate and Efficient Schemes Numerical Stability Analysis Rigorous Math Theories Computational Methods Figure.. The general strategy for the development of reduced-order statistical models. Three sequential stages are required to carry out the reduced-order statistical model, and IPAM, rigorous mathandrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models January 9-3, 7 3 / 3

4 Related Works and Papers Recent new developments Majda, Introduction to turbulent dynamical systems in complex systems, Springer, 6. Qi and Majda, New strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, manuscript submitted to SIAM Review, 6. Statistical theories Majda, Statistical energy conservation principle for inhomogeneous turbulent dynamical systems. PNAS,. Majda and Gershgorin, Link between statistical equilibrium fidelity and forecasting skill for complex systems with model error. PNAS,. Majda and Wang, Linear response theory for statistical ensembles in complex systems with time-periodic forcing. CMS,. Improving imperfect model skill Majda, A and Qi, D, Improving prediction skill of imperfect turbulent models through statistical response and information theory, Journal of Nonlinear Science,. Qi, D and Majda, A, Low-dimensional reduced-order models for statistical response and uncertainty quantification: two-layer baroclinic turbulence, JAS, 6. Qi,D and Majda, A, Low-dimensional reduced-order models for statistical response and uncertainty quantification: barotropic turbulence with topography, Physica D, 6. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 4 / 3

5 Outline A two-layer quasi-geostrophic model for baroclinic turbulence Two-layer baroclinic turbulence in ocean and atmosphere regimes Reduced-order statistical models for general turbulent systems Formulation of the exact statistical moment dynamics A reduced-order statistical model with consistency and sensitivity Model calibration for optimal performance 3 Low-dimensional reduced-order models for the two-layer system Statistical equations for the two-layer model Tuning imperfect model parameters in the training phase Imperfect model prediction in various dynamical regimes Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

6 Outline A two-layer quasi-geostrophic model for baroclinic turbulence Two-layer baroclinic turbulence in ocean and atmosphere regimes Reduced-order statistical models for general turbulent systems Formulation of the exact statistical moment dynamics A reduced-order statistical model with consistency and sensitivity Model calibration for optimal performance 3 Low-dimensional reduced-order models for the two-layer system Statistical equations for the two-layer model Tuning imperfect model parameters in the training phase Imperfect model prediction in various dynamical regimes Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 6 / 3

7 The two-layer flow with forcing and dissipation The two-layer quasi-geostrophic model with baroclinic instability is one simple but fully nonlinear fluid model capable in capturing the essential physics in ocean and atmosphere science. Two-layer model q ψ t + ( ) J ψ,q ψ + J(τ,qτ ) + β ψ x + U τ x = κ (ψ τ) ν s q ψ + F ψ, q τ t + J(ψ,q τ) + J ( τ,q ψ ) + β τ x + U x ( ψ + k d ψ) = κ (τ ψ) ν s q τ + F τ. Barotropic and baroclinic modes: q ψ = ψ, ψ = (ψ + ψ ), q τ = τ k d τ, τ = (ψ ψ ). Normalized energy-consistent modes: p ψ,k = q ψ,k = k ψ k k, q τ,k p τ,k = = k + k k + kd d τ k. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 7 / 3

8 Flow in high-latitude homogeneous regimes regime N β kd U κ (kmin, kmax ) σmax (kx, ky )max ocean, high lat. 6 9 (., 4.6).4 (4, ). (.8, 6.78).99 (, ) 4 atmosphere, highmodels lat. with 6 4 flow. 4 Reduced-order homogeneous mean linear growth rate, ocean regime nonlinear flux, sum of eigenvalues meridional wavenumber meridional wavenumber nonlinear flux eigenvalues linear growth rate zonal wavenumber - wavenumber zonal wavenumber (a) high-latitude ocean regime linear growth rate, atmosphere regime zonal wavenumber 4 6 meridional wavenumber meridional wavenumber 6 nonlinear flux, sum of eigenvalues nonlinear flux eigenvalues linear growth rate zonal wavenumber wavenumber (b) high-latitude atmosphere regime Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 8 / 3

9 Flow in low/mid-latitude regimes with zonal jets Reduced-order models with inhomogeneous jet flow mean in relative vorticity mean in relative vorticity barotropic baroclinic barotropic baroclinic variance in relative vorticity variance in relative vorticity barotropic baroclinic k-. barotropic baroclinic -.6 k Andrew Majda and Di Qi (CIMS) statistical energy Low-Dimensional Reduced-Order Statistical Models statistical energy IPAM, January 9-3, 7 9 / 3

10 Outline A two-layer quasi-geostrophic model for baroclinic turbulence Two-layer baroclinic turbulence in ocean and atmosphere regimes Reduced-order statistical models for general turbulent systems Formulation of the exact statistical moment dynamics A reduced-order statistical model with consistency and sensitivity Model calibration for optimal performance 3 Low-dimensional reduced-order models for the two-layer system Statistical equations for the two-layer model Tuning imperfect model parameters in the training phase Imperfect model prediction in various dynamical regimes Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

11 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), () u(t ;ω) = u (ω). (3) 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 =. 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. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

12 Exact statistical moment equations 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 ijb ( v i,v j ) + F(t), dr dt = L vr + RL v + 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 m Z n Z j B(vm,v n ) v i + Z m Z n Z i B(v m,v n ) v j. note that energy is still conserved in this nonlinear interaction part Tr[Q F ] = Z m Z n Z i B(v m,v n ) v i = B(Z m v m,z n v n ) Z i v i = B ( u,u ) u =. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

13 Reduced-Order Statistical Energy Closure The true statistical model dū dt = (L + D)ū + B(ū,ū) + R ijb ( v i,v j ) + F(t), ū R N, dr dt = L v (ū)r + RL v (ū) + Q F + Q σ, R R N N. Q F,ij = Z m Z n Z j B(vm,v n ) v i + Z m Z n Z i B(v m,v n ) v j Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 3 / 3

14 Reduced-Order Statistical Energy Closure The reduced-order approximation ū M R M, M N dū M dt dr M dt = (L + D)ū M + B(ū M,ū M ) + R M,ij B(v i,v j ) + F, = L v R M + R M L v + Q M F + Q σ, A preferred approach for the nonlinear flux Q M F combining both the detailed model energy mechanism and control over model sensitivity is proposed [ ] Q M F = Q M, F + Q M,+ F = f (E)[ (N M,eq + d M I N )R M ] + f (E) Q + F,eq + Σ M. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 3 / 3

15 Reduced-Order Statistical Energy Closure A preferred approach for the nonlinear flux Q M F combining both the detailed model energy mechanism and control over model sensitivity is proposed [ ] Q M F = Q M, F + Q M,+ F = f (E)[ (N M,eq + d M I N )R M ] + f (E) Q + F,eq + Σ M. Higher-order corrections from equilibrium statistics: Q F,eq = Q F,eq + Q+ F,eq = L v (ū eq )R eq R eq L v (ū eq ) Q σ, N M,eq = Q F,eq R eq. Additional damping and noise to model nonlinear flux: Q add M = d MR M + Σ M. Statistical energy-consistent scaling to improve model sensitivity: ( ) / ( ) 3/ E E f (E) =, f (E) =. E eq E eq Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 3 / 3

16 Climate fidelity for equilibrium Equilibrium fidelity refers to the convergence to the same final unperturbed statistical equilibrium R eq in the reduced-order models R M in each resolved component. Specifically, it requires that the model nonlinear flux correction term Q M converges to the truth, Q M Q F,eq, when no external perturbation is added dr M,eq dt = = L v (ū eq )R M,eq + R M,eq L v (ū eq ) + Q M F,eq + Q σ R M,eq = R eq. the first component statistics. ( ) N M,eq,Q + F,eq comes from the true equilibrium climate consistency requires the second component correction makes no contribution in the unperturbed case Σ M = d MR eq, f (E eq ) =, f (E eq ) =. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 4 / 3

17 Statistical energy conservation principle Theorem (Statistical Energy Conservation Principle) Under the structural symmetries on the basis v i, for any turbulent dynamical systems in the form () the total statistical energy, E = Ē + E = ū ū + trr, satisfies de dt = ū Dū + ū F + tr(dr) + trq σ, where R satisfies the exact covariance equation. Corollary Under the assumption of the Theorem, assume D = di, with d >, then the turbulent dynamical system satisfies the closed statistical energy equation for E = ū ū + trr, de dt = de + ū F + trq σ. Majda, Statistical energy conservation principle for inhomogeneous turbulent dynamical systems, PNAS,. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

18 Model calibration blending statistical response and information theory Accurate modeling about the model sensitivity to various external perturbations requires the imperfect reduced-order models to correctly reflect the true system s memory about its previous states. the linear response operator can characterize the model sensitivity involving the nonlinear effects in the system regardless of the specific forms of the external perturbations. empirical information theory can be used as the distance between these two operators to calculate the unbiased and invariant measure for model distributions. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 6 / 3

19 Linear response operator R A (t) Linear response theory: The linear response of a system, u t = f(u), f δ = f + δ f (t), can be predicted by observing appropriate statistics of the system in equilibrium π eq E δ A(u) = E eq (A) + δ E A + O ( δ ), δ E A = t R A (t s)δ f (s)ds. without the need of applying any perturbations. kicked response: For δ small enough, the linear response operator R A (t) can be calculated by solving the unperturbed system with a perturbed initial distribution π t= = π eq (u δ u) = π eq δ u π eq + O ( δ ). δr A (t) δ u R A = A(u)δπ + O ( δ ). Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 7 / 3

20 Link between equilibrium fidelity and forecasting skill Given the optimal model for the unperturbed climate π eq, how can we assess the error in the climate change prediction P ( ) π δ,πδ M based on the unperturbed climate? Under assumptions with diagonal covariance matrices R = diag(r k ) and equilibrium model fidelity P ( π G,π M G ) = P ( ) π δ,πδ M = S ( ) π G,δ S (πδ ) + ( ) ( ) δ ūk δ ū M,k R k δ ūk δ ū M,k k + 4 k ( ) R k δ Rk δ R M,k + ( O δ 3). R k is the equilibrium variance in k-th component, and δ ū k and δ R k are the linear response operators for the mean and variance in k-th component. (Majda & Gershogorin, PNAS, ) Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 8 / 3

21 Optimization in the training phase To summarize, consider a class of imperfect models, M. The optimal model M M that ensures best information consistent responses is characterized with the smallest additional information in the linear response operator R A, such that ( ) P p δ,p M δ L([,T]) = min P ( ) p δ,p M L δ ([,T]), M M p M δ can be achieved through a kicked response procedure in the training phase compared with the actual observed data p δ in nature; the information distance between perturbed responses P ( ) p δ,p M δ can be calculated through the expansion formula; the information distance P ( p δ (t),p M δ (t)) is measured at each time instant, so the entire error is averaged under the L -norm inside a time window [, T]. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 9 / 3

22 Outline A two-layer quasi-geostrophic model for baroclinic turbulence Two-layer baroclinic turbulence in ocean and atmosphere regimes Reduced-order statistical models for general turbulent systems Formulation of the exact statistical moment dynamics A reduced-order statistical model with consistency and sensitivity Model calibration for optimal performance 3 Low-dimensional reduced-order models for the two-layer system Statistical equations for the two-layer model Tuning imperfect model parameters in the training phase Imperfect model prediction in various dynamical regimes Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

23 Exact statistical moment equations for the two-layer model The rescaled set of equations of () can be summarized in the abstract form dp k dt = B k (p k,p k ) + (L k D k )p k + F k, p k = ( ) T p ψ,k,p τ,k, p k B k (p k,p k ), k where the normalized state variable p k = ( p ψ,k,p τ,k ) T is in barotropic and baroclinic mode, the linear operator is decomposed into non-symmetric part L k involving β -effect and shear flow U and dissipation part D k, together with the forcing F k combining deterministic component and stochastic component. ik xβ k L k = ik x U (k d/ k ) [ Bψ,k B k (p k,p k ) = B τ,k ik xβ +(k d / k ) k +k d ] = ik xu +(k d / k ), D k = κ +(k d / k ) +(k d / k ) +(k d / k ) ( ) m n n m+n=k k m p n ψ,mp ψ,n + +k d m p +k d τ,m p τ,n ( m+n=k m n k +k d n +k d m, ) n. p ψ,m p τ,n + p τ,m p ψ,n m +k d Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

24 Exact statistical moment equations Statistical energy in each spectral mode pψ,k R k = p p k p k = ψ,k p τ,k, p p ψ,k p τ,k pτ,k,k p,k = p,k p,k + p,k p,k. R k combines the variability in both mean and variance. The true statistical dynamical equations form a system about R k C dr k dt = (L k D k )R k + Q F,k + Q σ,k + c.c., k N, Q F,k = p k B k (p k,p k ) = [ p ψ,k B ψ,k p τ,k B ψ,k p ψ,k B τ,k p τ,k B τ,k ], trq F,k. k Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

25 Statistical energy conservation principle The total statistical energy dynamical equation concerns the evolution of the total variability in mean and variance in response to external perturbations E = ( ) k ψ k + k + kd τ k = pψ,k + pτ,k. k N k N The exact dynamics for the statistical energy can be derived as de dt + H f = κe + κ F νh + Q σ. H f is the meridional heat flux due to baroclinic instability, F is the additional damping effects due to the non-symmetry in Ekman drag only applied on the bottom layer H f = k d U ψ x τ = k d U ik x ψ k τ k, F = k d τ k + k Reψ k τ k. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 3 / 3

26 Set-up for the numerical problem The true statistics is calculated by a pseudo-spectra code with 8 spectral modes zonally and meridionally, corresponding to 6 6 grid points in total. In the reduced-order methods, only the large-scale modes k are resolved, which is about.% of the full model resolution. External forcing in stochastic and deterministic component: The amplitude of the stochastic forcing σ k Ẇ k is introduced according to the equilibrium energy so that σψ,k = δσ qψ,k, σ eq τ,k = δσ qτ,k eq. The deterministic forcing is introduced through a perturbation in the background shear U δ = U + δ U ( δ f ψ,k = δ Uik x k ) ) τ k, δ f τ,k = δ Uik x ( k + k d ψ k. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 4 / 3

27 educed-order models with homogeneous mean flow Tuning parameters in the training phase tuning parameters with energy correction. tuning parameters without energy correction d τ.. d τ d ψ d ψ. (a) tuning with energy scaling (b) tuning without energy scaling baroclinic energy unperturbed spectrum perturbed response w/ energy correction w/o energy correction wavenumber w/ energy correction w/o energy correction (c) prediction and info. errors information error wavenumber. 4.: Figure: Tuning Tuning imperfect imperfect model parameters model parameters in the training in phase. the training The information phase. errors The information with varying model p rameters, d M = d y,d t, are plotted for stochastic barotropic perturbation case. The errors using tot errors energy with as varying scalar factor model from parameters, the statistical equation d M = ( ) d and ψ,d method τ, are without plotted the for scaling stochastic factor are compare barotropic The prediction perturbation skill and case. information error with and without using the total energy correction are compare in the last row for a typical test case of perturbing the barotropic mode. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 / 3

28 High-latitude: Mean shear flow perturbation The model is perturbed by changing the background zonal flow strength U; The entire spectral is perturbed due to the mean flow advection in each spectral mode. Reduced-order models with inhomogeneous jet flow Reduced-order models withinhomogeneous jet flow barotropic energy 6 barotropic energy barotropic energy 6 barotropic energy baroclinic energy baroclinic energy baroclinic energy baroclinic energy heat flux heat flux heat flux heat flux wavenumber wavenumber wavenumber wavenumber (a) U =. (b) U =.9 (a) U =. (b) U =.9 (a) ocean regime δ U = ±., U = (b) atmos. regime δ U = ±., U =. Fig. 4.4: Reduced-order model predictions to mean shear flow perturbation du = ±. Fig. (that 4.: is, Reduced-order % of the original model predictions to mean shear flow perturbation du =.,. (that i value U) in the ocean regime. The spectra for the resolved modes apple k < are compared. originalblack valuelines U) in the atmosphere regime. The spectra for the resolved modes apple k < with circles show the perturbed model responses in the normalized barotropic energy, baroclinic Black lines energy, with andcircles show the perturbed model responses in barotropic energy, baroc heat flux. The dashed black lines are the unperturbed statistics. And the reduced order heat model flux. predictions The dashed black lines are the unperturbed statistics. And the reduced order m are in red lines. are in red lines. Model responses to the perturbed mean shear du regime N b kd U k n s (kmin,kmax) In checking the model responses to deterministic forcing, we introduce the forcing perturbation ocean by changing regime, low/mid the background Andrew jet strength Majda Uand as in Di(4.). Qi (CIMS) The same perturbation is Low-Dimensional tested in [4] for a more Reduced-Order complicated atmosphere reduced-order Statistical regime, modified low/mid Modelslat. 6. IPAM, 4.January. 9-3, 7 4 (.,7.6) 6 / lat (7.4,.63) 3

29 Flow in low-latitude regimes with zonal jets regime N β kd U κ (kmin, kmax ) σmax (kx, ky )max ocean, high lat. 6 (7.4,.63).4 (, 8) atmosphere, high lat (., 7.6).3 (3, ) Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 7 / 3

30 Low-latitude: Stochastic perturbation with δ σ =. autocorrelation functions and probability density functions mode (6,3) - 3 mode (6,) - mode (6,) 3 mode (6,) - mode (6,) mode (,) mode (,).. mode (,) mode (,) mode (,).4 mode (,). - mode (6,3).4 (a) ocean (b) atmosphere Fig..: Autocorrelation functions and the probability distribution functions in low/mid-latitude ocean6 and atmo6 Summary 6 Summary sphere regime. The first three most energetic baroclinic modes are displayed. In the autocorrelations, the solid lines show the real part while the dashed lines are the imaginary part of the functions. In the pdfs, the corresponding Gaussian distributions with the same variance are also plotted in dashed black lines. barotropic energy total barotropic energy baroclinic energy 3 total baroclinic energy baroclinic energy total baroclinic energy. 3-3 heat flux 3 total heat flux 6 - truth model 4 total barotropic energy unperturbed truth model truth model 3 barotropic energy unperturbed truth model heat flux wavenumber time (a) spectra for the energy and heat flux (b) time-series for total responses 3 -. total heat flux 3. 4 wavenumber (a) spectra for the energy and heat flux time (b) time-series for total responses Fig..6: Model responses in low/mid-latitude forcing perturbation sin. (whileregime no atmosphere regime with ran Fig..7: Model responses low/mid-latitude (c) ocean regimeocean regime with random (d) atmosphere = stochastic forcing for the unperturbed Low-Dimensional case). The left panel shows the spectra formodels theforcing barotropic baroclinic stochastic s and =. for the unperturbed case). 8 The Andrew Majda and Di Qi (CIMS) Reduced-Order Statistical IPAM, January 9-3, 7 / 3firs

31 Current and future work Reduced-order stochastic modeling strategies to capture passive scalar intermittency 34 ; Design of a mitigation control strategy by using novel low-order statistical models; More detailed consideration about the model nonlinearity. Majda & Tong, Intermittency in turbulent diffusion models with a mean gradient, Nonlinearity, 3 Majda & Gershgorin, Elementary models for turbulent diffusion with complex physical features: eddy diffusivity, spectrum and intermittency, Phil. Trans. R. Soc. A, 3 4 Qi & Majda, Predicting fat-tailed intermittent probability distributions in passive scalar turbulence with imperfect models through empirical information theory, Comm. Math. Sci,. Qi & Majda, Predicting Extreme Events for Passive Scalar Turbulence in Two-Layer Baroclinic Flows through Reduced-Order Stochastic Models, in preparation. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 9 / 3

32 Related Works and Papers Recent new developments Majda, Introduction to turbulent dynamical systems in complex systems, Springer, 6. Qi and Majda, New strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems, manuscript submitted to SIAM Review, 6. Statistical theories Majda, Statistical energy conservation principle for inhomogeneous turbulent dynamical systems. PNAS,. Majda and Gershgorin, Link between statistical equilibrium fidelity and forecasting skill for complex systems with model error. PNAS,. Majda and Wang, Linear response theory for statistical ensembles in complex systems with time-periodic forcing. CMS,. Improving imperfect model skill Majda, A and Qi, D, Improving prediction skill of imperfect turbulent models through statistical response and information theory, Journal of Nonlinear Science,. Qi, D and Majda, A, Low-dimensional reduced-order models for statistical response and uncertainty quantification: two-layer baroclinic turbulence, JAS, 6. Qi,D and Majda, A, Low-dimensional reduced-order models for statistical response and uncertainty quantification: barotropic turbulence with topography, Physica D, 6. Andrew Majda and Di Qi (CIMS) Low-Dimensional Reduced-Order Statistical Models IPAM, January 9-3, 7 3 / 3