1. Sensitivity of the climate system to small perturbations of external forcing V.P. Dymnikov, E.M. Volodin, V.Ya. Galin, A.S. Gritsoun, A.V. Glazunov, N.A. Diansky, V.N. Lykosov Institute of Numerical Mathematics RAS, Moscow

2. The climate system - the system consisting of atmosphere, hydrosphere, cryosphere, land and biota. The climate - the ensemble of states the climate system passes through during a sufficiently long time interval. Characteristics of the climate system as a physical object: quasi-two-dimensionality impossibility of purposeful physical experiments

3. The central direction of the climate sensitivity studies: mathematical modeling Problems: 1. The identification of models by sensitivity. 2. Is it possible to determine the sensitivity of the climate system to small external forcing using single trajectory?

4. The climate model sensitivity to the increasing of CO 2 CMIP - Coupled Model Intercomparison Project http://www-pcmdi.llnl.gov/cmip CMIP collects output from global coupled ocean-atmosphere general circulation models (about 30 coupled GCMs). Among other usage, such models are employed both to detect anthropogenic effects in the climate record of the past century and to project future climatic changes due to human production of greenhouse gases and aerosols.

5. Climate simulations and investigation of the climate sensitivity to the increase of CO2 with coupled atmosphere - ocean GCM

8. Response to the increasing of CO2 CMIP models (averaged) INM model

9. Global warming in CMIP models in CO2 run and parameterization of lower inversion clouds. T - global warming (K), LC - parameterization of lower inversion clouds (+ parameterization was included, - no parameterization, ? - model description is not available). Models are ordered by reduction of global warming.

10. 1. Formulation of model equations 2. Proof of the existence and uniqueness theorems 3. Attractor existence theorem, dimension estimate 4. Stability of the attractor (as set) and measure on it 5. Finite-dimensional approximations and correspondent convergence theorems Mathematical theory of climate

11. (sensitivity) 6. Construction of the response operator for measure and its moments (“optimal perturbation”, inverse problems,….) 7. Methods of approximation for 8. Numerical experiments

12. Response operator for 1 st moment (linear theory) Linear model for the low-frequency variability of the original system: ( A is stable, is white noise in time) Perturbed system

13. Stationary response For covariance matrix we have and response operator M could be obtained as

14. Response operator for 1 st moment (nonlinear theory) Nonlinear model for system dynamics: ( is the white noise in time) Perturbed system

15. Stationary response Fokker-Plank equation for the density of invariant measure has unique stationary solution.

16. To the first order in Consequently, In the case of normal distribution we arrive at

17. Numerical experiments Construction of the approximate response operator (A.S.Gritsoun,G.Branstator, V.P.Dymnikov, R.J.Numer.Analysis&M.Model, (2002), v.17,p. 399)

18. Reconstruction of the CCM0 response to the sinusoidal heating anomaly

19. FieldCorrelationAmplitude ratio | Nonlin.|/|Lin.| 0.881.090.13 0.871.130.10 0.851.160.17 0.760.920.15 average0.841.080.14 FieldCorrelationAmplitude ratio | Nonlin.|/|Lin.| 0.691.060.21 0.631.050.17 0.780.910.27 0.720.820.17 average0.710.960.20 Reconstruction of the equatorial low-level heating anomalies. Average values of correlations, amplitude ratios are shown (for 24 different heating positions). Reconstruction of the equatorial sinusoidal heating anomalies. Average values of correlations, amplitude ratios are shown (for 24 different heating positions). Reconstruction of the CCM0 response (continued)

20. Reconstruction of the low-level heating anomalies using the inverse response operator

21. Construction of the optimal heating forcing for the excitation of the AO (using NCEP/NCAR data and output of AGCM of INM RAS) AO (1EOF of surface pressure) calculated using DJF NCEP/NCAR data AO (1EOF of surface pressure) calculated using output of AGCM of INM RAS

22. Procedure for the construction of the approximate response operator is analogues to (A.S.Gritsoun, G.Branstator, V.P.Dymnikov, R.J.Numer. Analysis&M.Model, (2002), v.17,p. 399) Optimal perturbation for AO (1EOF of PS) calculated using NCEP/NCAR reanalysis data (zonal average) Optimal perturbation for AO (1EOF of PS) calculated using output of AGCM of INM RAS (zonal average)

23. Global warming in CMIP models in CO2 run and parameterization of lower inversion clouds. T - global warming (K), LC - parameterization of lower inversion clouds (+ parameterization was included, - no parameterization, ? - model description is not available). Models are ordered by reduction of global warming.

24. Acknowledgments Our studies are supported by: Russian Ministry for Industry, Sciences and Technology Russian Academy of Sciences Russian Foundation for Basic Research INTAS