Nonlinear Local Lyapunov Vectors and Their Applications to Ensemble Predictions

1. Nonlinear Local Lyapunov Vectors and Their Applications to Ensemble Predictions Ruiqiang Ding, Jianping Li, Jie Feng, and Zhaolu Hou LASG, IAP, Chinese Academy of Sciences GCESS, Beijing Normal University November 23, 2017

2. 1 2 3 5 4 Outline Limitations of linear Lyapunov vector Nonlinear local Lyapunov vector (NLLV) The barotropic quasi-geostrophic model Performance of the NLLVs in ensemble prediction Summary

3. Linear Lyapunov vectors (LVs) In dynamical system theory, Lyapunov vectors describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial perturbations for ensemble forecasting in numerical weather prediction (Kalnay2007). (From http://en.wikipedia.org/wiki/Lyapunov_vector)

4. Leading Lyapunov vector (LLV) Calculation of Lyapunov vectorsfrom differential equations For initial orthonormal vectors In a chaotic system, each vector tends to fall along the local direction of most rapid growth. The Gram- Schmidt reorthonormalization (GSR) procedure: Lyapunov vectors and exponent spectrum from the fastest to slowest directions: Orthogonalized,then normalized

5. The limitations of linear Lyapunov theory N-dimensional dynamical system Error evolution equation Linear approximation equation Nonlinear term Linear propagator The largest Lyapunov exponent (LE)

6. linear nonlinear Linear----- Nonlinear —— Lorenz system Here

7. Noninear local Lyapunov exponents and vectors Nonlinear propagator Nonlinear local Lyapunov exponent (NLLE) and vectors (NLLVs) GSR procedure to obtain the NLLVs

8. Differences between the LVs and NLLVs Nonlinear Model Linear Model Linear LEs NLLEs The LVs are based on the linear error model; applicable for sufficiently small error. NLLVs: nonlinear generalizations of LVs; applicable for finite amplitude error.

9. The NLLE spectrum in the Lorenz63 model Linear phase Nonlinear phase LE spectrum Fastest NLLE spectrum Slowest The NLLE spectrum may realistically reflect the time-varying characteristics of error growth rates along different directions from the linear to nonlinear phases, which is an improvement over the traditional LE spectrum.

10. The NLLE spectrum in the Lorenz96 40-variable model The NLLE method can effectively separate the slowly and rapidly growing perturbations, which is very important for studies of the predictability and error growth dynamics.

11. Intaseasonal oscillation (ISO) Monthly and seasonal mean GPH Monthly mean SST (单位:月) (month) Some predictability studies by using the NLLE method • Predictability of toy chaotic models(Ding and Li 2009, 2012) • Relationships between the Limit of predictability and initial error; • Influences of initial and parameter errors on predictability. • Predictability of weather and climate(Li and Ding 2008, 2009, 2011; Ding and Li 2010, 2011, 2012)

12. Ensemble forecasting The atmosphere is a chaotic system, and its predictability is severely limited by the uncertainties related to the initial value and the model. Ensemble forecasting offers a feasible approach to improve a single forecast. (ECMWF, 2009) The basic idea behind an ensemble generation scheme is to sample the uncertainties related to errors in the initial conditions.

13. Some methods producing ensemble perturbations • Dynamical sampling in growing sub-space: Breeding vector(BV; Toth and Kalnay 1993) • Advantage: nonlinear, time saving (the adjoint model unnecessary), flow dependence • Limitation: perturbations not orthogonal (dependence among perturbations) Singular vector(SV; Buizza and Palmer 1995) • Advantage: orthogonal, flow dependent • Limitation: linear, time consuming (the adjoint model necessary) • EnKF with perturbed observations (Houtekamer and Mitchell 1998) • Ensemble-based data assimilation: ETKF (Wang and Bishop 2001) The breeding method is used extensively as an ensemble generation technique for its simple concept and cheap computing.

14. BVs vsNLLVs • Nonlinear Local Lyapunov vectors： • Extension of leading LVs into nonlinear environment; • Orthogonalization (GSR; each NLLV was orthogonalized); • Orthogonal. • Bred Vector: • Extension of leading LVs into nonlinear environment; • Natural breeding (each BV generated by independent breeding cycle); • Quasi-orthogonal. RO Schematic of BV (Toth and Kalnay, 1997) Schematic of NLLV

15. BVs NLLVs span the fast-growing perturbation subspace efficiently, and thus may grasp more components in analysis errors than the BVs.

16. Quasi-geostrophic (QG) model is the streamfunction,is the potential vorticity,is the planetary Froude number. topography initial state

17. Comparison of local dimension of BVs and NLLVs (Good for increasing ensemble spread and instabilities) • The NLLVs have more independent local structure (thus more effectively capture the unstable subpace) than the BVs.

18. The spatial patterns of BV and NLLV respective five perturbations The local perturbations of NLLVs identify the baroclinic instabilities in a more diverse way than BVs.

19. Comparison of the error growth structure EnKF ensemble perturbations Corr(a, b)=0.27 Corr(a, c)=0.52 NLLVs can better sample the analysis error than BVs.

20. 92

21. Error growth structure projected on the subspace of BVs and NLLVs The variances of projected on the subspace of BV and NLLV absolute perturbations: (least square method to build the regression model of the analysis error onto BVs or NLLVs) (explained variance by regression) If the larger projection of error growth structure on the subspace of BVs or NLLVs, the better the analysis error could be captured by the unstable mode constructed by BVs or NLLVs.

22. Correlation = 0.86 NLLVs have greater projected variance than BVs.

23. Comparison of different initialization schemes in a barotropic model) Explained variances of the eigenvectors of ensemble perturbation covariance matrix Variances of the analysis errors explained by the ensemble perturbations NLLVs can better sample the analysis error than BVs, and comparable to ETKF. NLLVs/ETKF are more independent than BVs.

24. Comparison of ensemble prediction skill RMSE PAC Brier score

25. Comparison of ensemble schemes based on the QG model Prediction Skill: NLLVs~ETKF >BVs > RP Computational Time: NLLVs~1/3 ETKF The NLLV scheme involves a simpler algorithm and less computation time than the ETKF scheme, and therefore it has potential for ensemble forecasting.

26. Performance of NLLVs in the ZC model • Atmospheric model:a Gill-type steady-state linear atmospheric model; • Oceanic model: a reduced-gravity oceanic model with a rectangular ocean basin that extends from 124°E to 80°W and from 29°N to 29°S on a 2° longitude × 0.5° latitude grid; • The circulation is forced by a heating anomaly that depends partly on local heating that is associated with SST anomalies (SSTAs) and on low-level moisture convergence. Zebiak-Cane model (an intermediate coupled ENSO model; Zebiak and Cane, 1987)

27. Evolution of SSTA structures • NLLVs are dependent on the background. • NLLVs can well represent the instability structure associated with ENSO. SSTA structures (contour) and first three NLLVs (shaded)

28. Subspace of NLLVs and BVs against the analysis error • The subspace consisting of NLLVs can better and more effectively capture the analysis error than that of an equal number of BVs.

29. Comparison of local dimension of BVs and NLLVs • The local dimensions of NLLVs are higher than those of BVs. NLLV BV

30. Ensemble prediction experiment • Ensemble perturbations：NLLV, BV and Random Vectors • Ensemble members：20 (positive–negative pairs) • The ensemble skill of the NLLVs is higher than that of BVs. RMSE Correlation

31. Summary • The NLLVs are generated by dynamically evolving the perturbations with full nonlinear models and capture the orthogonal basis to span the most unstable perturbation subspace. • The NLLVs have overall higher but correlated local dimensions compared to the BVs which may be beneficial for the former to increase the ensemble spread and capture the instabilities as well. • The NLLV scheme has the potential for operational application. Next step we will use the more complex models (WRF for hurricane and CESM for ENSO) to test the performance of the NLLVs in ensemble forecasting.

32. Thanks for your attention! Ding R. Q., J. P. Li, and B. S. Li, Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system. Adv. Atmos. Sci., 10.1007/s00376-017-7011-8.

34. Definition of nonlinear local Lyapunov vectors • BVs sample the fastest-growing perturbations, but may be strongly correlated regionally and not effectively span the fast-growing subspace. • LVs are corresponding to orthogonal directions with different growth rates, but they are linear constructs. Based on the breeding scheme, we extend linear LVs to nonlinear framework and introduce the orthogonal nonlinear local Lyapunov vectors (NLLVs). Algorithm of Gram-Schmidt orthogonalization Schematic of the generation of NLLVs

35. Zebiak-Cane model • Atmospheric model:a Gill-type steady-state linear atmospheric model; • Oceanic model: a reduced-gravity oceanic model with a rectangular ocean basin that extends from 124°E to 80°W and from 29°N to 29°S on a 2° longitude × 0.5° latitude grid; • The circulation is forced by a heating anomaly that depends partly on local heating that is associated with SST anomalies (SSTAs) and on low-level moisture convergence. (Cane et al., 1986; Zebiak and Cane, 1987; Mu, M., H. Xu, and W. Duan, 2007; Duan and Zhao, 2015).

36. 1 Calculation of NLLVs in the ZC model

37. Configuration of the reference trajectory • All experiments in the present paper assume a perfect model under which the output of a long-period integration from the model is regarded as the true trajectory • The reference trajectory isfrom the EnKF assimilation method assimilation variable：SST； observation：The total number of observation locations is 24 assimilation of parameters：inflation ratio-0.05，members-10 assimilation frequency：monthly RMSE of The SST analysis field:0.19℃

38. Reference breeding parameters for NLLV

39. How many breeding cycles • A sufficient number of breeding cycles ensures that the initial random perturbation can be fully developed to the fast growth directions. • The growth rate of different NLLVs can reach the statistically stable stage in 12 cycles • The growth rate of the SSTA field of the NLLV1 • The growth rate of the first ten NLLVs through 12 breeding cycles

40. NLLVs under two different initial random perturbations • Compared with NLLV1, the subsequent NLLVs are more sensitive, which is one of the instantaneous features of the NLLVs. The nonlinearity of the dynamical system contributes to the sensitivity of the NLLVs to the initial random perturbation seeds. 60.7% 2.7% 14.7% The pattern correlations between the SSTA fields of NLLVs using two different initial random perturbations for every breeding process. 59.3% 58.0% 60.1%

41. Statistical features of the NLLVs to different breeding parameter sets • The statistical features of the NLLVs can be captured by their empirical orthogonal functions (EOFs). Different breeding variable sets： • the reference configuration experiment using TO, UO, VO, H1; • using just TO; • using TO, H1, oceanic depth averaged zonal current (U1) and oceanic depth averaged meridional current (V1); • TO, UO, VO, H1, U1, V1. The NLLVs are not sensitive in a statistical sense to the choice of breeding variable group.

42. Relationship between the subspaces of NLLVs and the analysis errors Statistically, the NLLVs from different breeding parameter sets have great similarity in terms of the structure and explained variance of the EOFs.

43. 2 Growth rates and modes of NLLVs

44. The average growth rate of the first three NLLVs • The subsequent NLLVs besides NLLV1 have the probability to be the fastest growth direction during some phase. • It is necessary to consider the subsequent NLLVs to describe the multidimensional error growth space of the ZC model.

45. Evolution of SSTA structures • NLLVs are dependent on the background. • The subsequent NLLVs besides NLLV1 can also represent the instability structure associated with ENSO and contain finer scale information. SSTA structures (contour) and first three NLLVs (shaded)

46. 3 Advantages of NLLVs over BVs

47. The subspace of NLLVs and BVs against the analysis error • The orthogonalization of the breeding process ensures the different NLLVs adequately resolve the physics. • The subspace consisting of NLLVs can better and more effectively capture the analysis error than that of an equal number of BVs.

48. Local dimensionality • The local dimensions of NLLVs are higher than those of BVs, and this is not sensitive to the grid size or the number of vectors k. NLLV BV

49. 4 Application of NLLVs in ensemble prediction

50. Ensemble prediction experaiment • Ensemble perturbations：NLLV, BV and Random Vectors • Ensemble members：10 (positive–negative pairs) • The ensemble skill of the NLLVs is higher than that of BVs, which is due to the diversity of the NLLVs. RMSE Correlation