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Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu

Some New Progresses in the Applications of Conditional Nonlinear Optimal Perturbations. Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG),

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Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu

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  1. Some New Progresses in the Applications of Conditional Nonlinear Optimal Perturbations Mu Mu F.F.Zhou,H.L.Wang and X.G.Wu State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics (IAP), Chinese Academy of Sciences (CAS) mumu@lasg.iap.ac.cn http://web.lasg.ac.cn/staff/mumu/

  2. Outline • 1. Concept of conditional nonlinear optimal • perturbation (CNOP) and the difference between • CNOP and LSV • 2.Adaptive observations (MM5 model) • 3.The sensitivity of ocean’s thermohaline circulation (THC) to the finite amplitude initial perturbations

  3. : (nonlinear) propagator of (1) Let be the solutions to (1) 1. Conditional Nonlinear Optimal Perturbation (1)

  4. Conditional Nonlinear Optimal Perturbation(CNOP) Constraint condition

  5. Physical meaning of CNOP • The initial error which has largest effect on the uncertainty at prediction time. • 2. The initial anomaly mode which will evolve into certain climate event most probably (ENSO) • 3. The most unstable (or sensitive ) initial mode of nonlinear model with the given finite time period

  6. is LSV if and only if , (2) : (linear) propagator of (1) where

  7. Reference [1] Mu Mu, Duan Wansuo, Wang Bin, 2003, Nonlinear Processes in Geophysics, 10, 493-501. [2] Duan Wansuo, Mu Mu, Wang Bin, 2004,. JGR Atmosphere, 109, D23105, doi:10.1029/2004JD004756. [3] Mu Mu, Sun Liang, D.A. Henk, 2004, J. Phys. Oceanogr., 34, 2305-2315. [4] Sun Liang, Mu Mu, Sun Dejun, Yin Xieyuan, 2005, JGR-Oceans, 110, C07025,doi: 10.1029/2005JC002897. [5] Mu Mu and Zhiyue Zhang,2006,J.Atmos.Sci.. [6] Mu Mu ,Hui Xu and Wansuo Dun(2007),GRL [7] Mu Mu ,Wansuo Duan and Bin Wang (2007),JGR [8] Mu Mu and Wang Bo,2007, Nonlinear Processes in Geophysics [9]Olivier Riviere et al,2008,JAS

  8. When nonlinearity is of importance , there exist distinct difference between CNOP and LSV represented by two facts: a. The initial patterns are different Note: LSV stands for the optimal growing direction , but CNOP the “pattern” b. Linear and nonlinear evolutions of CNOP and LSV are different. Mu Mu and Zhiyue Zhang,2006.J.Atmos.Sci.

  9. 2. Adaptive Observation • FASTEX (Snyder 1996) • NORPEX (Langland et al.1999) • WSR (Szunyogh et al. 2000,2002) • DOTSTAR (Wu et al.2005) • NATReC (Petersen et al. 2006 ) • THORPEX (in process)

  10. Methods used in Adaptive Observations • SV (Palmer et al.1998) • Adjoint Sensitivity (Ancell and Mass 2006) • ET (Bishop and Toth 1999) • EKF (Hamill and Snyder 2002) • ETKF (Bishop et al. 2001) • Quasi-inverse Linear Method (Pu et al.1997) • ADSSV (Wu et al. 2007)

  11. The sensitive areas identified by different methods may differ much. Which one is better is still in discussion (Majumdar et al.2006). • Conditional nonlinear optimal perturbation (CNOP), which is a natural extension of linear singular vector (SV) into the nonlinear regime, is in the advantage of considering nonlinearity(Mu et al, 2003; Mu and Zhang,2006).

  12. Applications of CNOP to Adaptive Observations • Rainstorms • Tropical cyclones

  13. Rainstorms • Case A: Rainfall during 0000 UTC 4 July~ 0000 UTC 5 July, 2003 on the Jianghuai drainage basin in China • Case B: Rainfall during 0000 UTC 5 Aug~ 0000 UTC 6 Aug, 1996 on the Huabei plain in China

  14. optimization algorithm • SPG2(Spectral projected gradient, • Birgin etal,2001) Characters: box or ball constraints linearity convergence high dimensions The constraint in this study is The optimization time interval is 24 hours.

  15. Experimental design • Model: MM5 and its Adjoint • Grid number: 51*61*10 Grid distance: 120km • Top level: 100hPa • Physical parameterizations: • dry-convective adjustment • grid-resolved large scale precipitation • high resolution PBL scheme • Anthes-Kuo cumulus parameterization scheme • Data: NCEP analysis • ECMWF reanalysis • routine observations

  16. Total dry energy is chosen as a metric： where, The integration extends the full horizontal domain D and the vertical direction .

  17. Nonlinear evolutions a (CNOP) b(CNOP) Case A Figure1. The temperature (shaded, unit:K) and wind (vector, unit: m/s) componentsof CNOP(a,b), FSV (c,d) and loc CNOP (e,f) on level at 0000 UTC 4 July (a,c,e) and their nonlinear evolutions at 0000 UTC 5 July (b,d,f). c (FSV) d (FSV) f (loc CNOP) e (loc CNOP)

  18. Table 1.Case A: The maxima (minima) of temperature (unit: K), zonal and meridional wind (unit: m/s) on level time type 0000 UTC 4 July, 2003 0000 UTC 5 July, 2003 Figure 2. Case A The evolution of the total dry energy on targeting area during the optimization time interval. CNOP (solid), local CNOP(dashed), FSV (dot) and -FSV (dashdotted). The TE showed is divided by the initial.

  19. Nonlinear evolutions b (CNOP) a (CNOP) c (FSV) d (FSV) Figure 3. Same as Fig.1(a,b,c,d), but for case B at 0000 UTC 5 Aug, 1996 (a,c) and at 0000 UTC 6 Aug, 1996 (b,d)

  20. Table 2. Same as table 1, but for case B time type 0000 UTC 5 Aug, 1996 0000 UTC 6 Aug, 1996 Figure 4 Same as Fig.2, but for case B

  21. Sensitivity experiments Case A Case B Figure 5. the variations of the cost function due to the reductions of CNOP (solid) or FSV (dashed) during the optimization time interval for case A and case B.

  22. Tropical Cyclones • Case C: • Mindulle, North-West Pacific Tropical cyclones • 0000 UTC 28 Jun ~ 0000 UTC 29 Jun, 2004 • Case D: • Matsa,North-West Pacific Tropical cyclones • 0000 UTC 5 Aug ~ 0000 UTC 6 Aug, 2005

  23. optimization algorithm • SPG2(Spectral projected gradient, • Birgin etal,2001) The constraints are for case C, and for case D. The optimization time intervals for these two cases are still 24 hours.

  24. Experimental design Model: MM5 and its Adjoint Grid number: 41*51*11(case C), 55*55*11(case D) Grid distance: 60km Top level: 100hPa Physical parameterizations: dry-convective adjustment grid-resolved large scale precipitation high resolution PBL scheme Anthes-Kuo cumulus parameterization scheme Data: NCEP reanalysis

  25. Metrics dynamic energy total dry energy where, The integration extends the full horizontal domain D and the vertical direction .

  26. Simulation of case C (Mindulle) a a: model domain b: target area Figure 6. Simulation track from MM5 (red) and the observation track (blue) from CMA b

  27. Simulation of case D (Matsa) a a: model domain b: target area a Figure 7. Simulation track from MM5 (red) and the observation track (blue) from CMA b b

  28. Results CNOP Mindulle FSV dynamic energy, 24-h at 0000 UTC 28 Jun FSV CNOP at 0000 UTC 29 Jun Nonlinear evolutions

  29. Mindulle CNOP dry energy, 24-h FSV at 0000 UTC 28 Jun CNOP FSV at 0000 UTC 29 Jun Nonlinear evolutions

  30. Case C (Mindulle) The evolutions of the dynamic energies (KE) and total dry energies (TE) of CNOP (blue) and FSV (red) on targeting area during the optimization time interval. Unit: J/kg

  31. Matsa CNOP dynamic energy, 24-h FSV at 0000 UTC 5 Aug CNOP FSV at 0000 UTC 6 Aug Nonlinear evolutions

  32. Matsa FSV CNOP dry energy, 24-h at 0000 UTC 5 Aug CNOP FSV at 0000 UTC 6 Aug Nonlinear evolutions

  33. Case D (Matsa) The evolutions of the dynamic energies (KE) and total dry energies (TE) of CNOP (blue) and FSV (red) on targeting area during the optimization time interval. Unit: J/kg

  34. Sensitivity experiments Define: Where is the projection operator, is a constant less than one. Benefits obtained from the reductions of CNOP or FSV are evaluated by:

  35. Benefits obtained from the reductions of CNOP or FSV Case C Mindulle Case D Matsa

  36. Conclusions The pattern of CNOP may differ from that of FSV, and its nonlinear evolutions are larger than those of FSV, as well as the loc CNOP and –FSV. The forecasts are more sensitive to the CNOP kind errors than the FSV kind. It is indicated that reduction of the CNOP kind errors benefits more than reduction of the FSV kind errors.

  37. Discussions • The determination of the sensitivity area according to CNOP • Comparisons with other methods • Choice of the constraints • Optimization algorithm: L-BFGS, no constraint • Evaluations of the effectiveness of adaptive observation • Feasibility and the time limitation

  38. 3.The sensitivity of ocean’s thermohaline circulation (THC) to finite amplitude initial perturbations and decadal variability Mu Mu, Sun Liang, D.A. Henk, 2004, J. Phys.Oceanogr., 34, 2305-2315 Sun Liang, Mu Mu, Sun Dejun, Yin Xieyuan, 2005, JGR-Oceans, 110,C07025,doi:10.1029/2005JC002897. Wu Xiaogang,Mu Mu, 2008,J.P.O. in review

  39. Sensitivity and stability study of THC

  40. The day after tomorrow?

  41. Floods & Impacts to New York

  42. Stommel box Model Strength of the thermal forcing Strength of the freshwater forcing Ratio of the relaxation time of T and S to surface forcing

  43. One disadvantage of S-model The ignoring the effect of wind-stress To consider the impact of small- and meso-scale motions of wind-driven ocean gyres (WDOG) of THC, Longworth et al (2005,J.of Climate) introduce a diffusion term to represent the effect of WDOG.

  44. Longworth’s model (2a) (2b) : the diffusion coefficient

  45. Steady state Thermally-driven, TH Salinity-driven, SA Perturbation Norm

  46. The effect of WDOG on the existence of multi-equilibrium Figure 1. The bifurcation diagram of box models for ， as a plot of versus . The curves from left to right : 0.0, 0.01, 0.05, 0.09 and 0.17. Circles in the figure represent the bifurcation points, which separate the linearly stable equilibrium TH-states and unstable ones. Besides, negative corresponds to the linearly stable SA-state.

  47. when ,we have , Fig.1 shows ,hence (numerical result)

  48. nonlinear stability analysis Figure 2. The evolution of (a) (c) cost function J and (b) (d) overturning function versus t computed with CNOPs superposed on the equilibrium state as initial conditions for , . (a) (b) : the TH-state with , and (c) (d) : SA-state with . Solid (dashed) curve is for L (S) model.

  49. Fig.2 a, b WDOG stabilizes the TH-state Fig.2 c, d WDOG destabilizes the SA-state

  50. Understanding nonlinear stable regime The smallest magnitude of a finite perturbation which induces a transition from TH state to SA state and vise versa.

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