Fluctuations, Response, and Prediction in Geophysical Fluid Dynamics

1. Fluctuations, Response, and Prediction in Geophysical Fluid Dynamics Valerio Lucarini valerio.lucarini@uni-hamburg.de MeteorologischesInstitut, Universität Hamburg Dept. of Mathematics and Statistics, University of Reading F. Lunkeit, F. Ragone, S. Sarno Cambridge,,November 1st 2013

2. Motivations and Goals • What makes it so difficult to model the geophysical fluids? • Some gross mistakes in our models • Some conceptual/epistemological issues • What is a response? • Examples and open problems • Recent results of the perturbation theory for non-equilibrium statistical mechanics • Deterministic & Stochastic Perturbations • Spectroscopy/Noise/Broadband analysis • Applications on systems of GFD interest • Climate Change prediction

3. IPCC scenarios

4. Models’ Response

5. Climate ResponseIPCC scenario 1% increase p.y.

6. Responsetheory • The response theory is a Gedankenexperiment: • a system, a measuring device, a clock, turnable knobs. • Changes of the statistical properties of a system in terms of the unperturbed system • Divergence in the response tipping points • Suitable environment for a climate change theory • “Blind” use of several CM experiments • We struggle with climate sensitivity and climate response • Deriving parametrizations!

7. Axiom A systems • Axiom A dynamical systems are very special • hyperbolic on the attractor • SRB invariant measure • Smooth on unstable (and neutral) manifold • Singular on stable directions (contraction!) • When we perform numerical simulations, we implicitly set ourselves in these hypotheses • Chaotic hypothesis by Gallavotti& Cohen (1995, 1996): systems with many d.o.f. can be treated as if Axiom A • These are, in some sense, good physical models!!! • Response theory is expected to apply in more general dynamical systems AT LEAST FOR SOME observables

8. Ruelle (’98) Response Theory • Perturbed chaotic flow as: • Change in expectation value of Φ: • nthorderperturbation:

9. This is a perturbative theory… • with a causal Green function: • Expectation value of an operator evaluated over the unperturbed invariant measure ρSRB(dx) • where: and • Linear term: • Linear Green: • Linear suscept:

10. Applicability of FDT • If measure is singular, FDT has a boundary term • Forced and Free fluctuations non equivalent • Recent studies (Cooper, Alexeev, Branstator….): FDT approximately works • In fact, coarse graining sorts out the problem • Parametrization by Wouters and L. 2012 has noise • The choice of the observable is crucial • Gaussian approximation may be dangerous

11. Simpler and simpler forms of FDT • Various degrees of approximation

12. Kramers-Kronig relations • FDT or not, in-phase and out-of-phase responses are connected by Kramers-Kronig relations: • Measurements of the real (imaginary) part of the susceptibility K-K  imaginary (real) part • Every causal linear model obeys these constraints • K-K exist also for nonlinear susceptibilities with Kramers, 1926; Kronig, 1927

13. Linear (and nonlinear) Spectroscopy of L63 • Resonances have to do with UPOs L. 2009

14. Stochastic forcing • , • Therefore, and • We obtain: • The linear correction vanishes; only even orders of perturbations give a contribution • No time-dependence • Convergence to unperturbed measure

15. Correlations  Power Spectra • Fourier Transform • We end up with the linear susceptibility... • Let’s rewrite he equation: • So: differencebetween the power spectra • → square modulus of linear susceptibility • Stoch forcing enhances the Power Spectrum • Can be extended to general (very) noise • KK  linear susceptibility  Green function

16. Lorenz 96 model • Excellent toy model of the atmosphere • Advection, Dissipation, Forcing • Test Bed for Data assimilation schemes • Popular within statistical physicists • Evolution Equations • Spatially extended, 2 Parameters: N & F • Properties are intensive

17. Spectroscopy –Im [χ(1)(ω)] LW HF L. and Sarno 2011 Rigorous extrapolation

18. Using stochastic forcing… • Squared modulus of • Blue: Using stoch pert; Black: deter forcing • ... And many many many less integrations L. 2012

19. Broadband forcing • We choose observable A, forcing e • Let’s perform an ensemble of experiments • Linear response: • Fantastic, we estimate • …and we obtain: • …we can predict

20. Broadband forcing G(1)(t) • Inverse FT of the susceptibility • Response to any forcing with the same spatial pattern but with general time pattern

21. Climate Prediction: convolution with T(t)=sin(2πt)

22. Time scale of prediction • Noise due to finite length L of integrations and of number of ensemble members N • We assume • We can make predictions for timescales: • Or for frequencies:

23. (Non-)Differentiability of the measure for the climate system Boschi et al. 2013 CO2 S*

24. A Climate Change experiment • Observable: globally averaged TS • Forcing: increase of CO2 concentration • Linear response: • Let’s perform an ensemble of experiments • Concentration at t=0 • Fantastic, we estimate • …and we predict:

26. PlaSim: Planet Simulator Vegetations (Simba, V-code, Koeppen) Terrestrial Surface: five layer soil plus snow Oceans: LSG, mixed layer, or climatol. SST Sea-Ice thermodynamic Spectral Atmosphere moist primitive equationson σ levels • Key features • portable • fast • open source • parallel • modular • easy to use • documented • compatible Model Starter andGraphic User Interface

28. What we get – CO2 doubling

29. G(1)(t)Climate Prediction - TS CLIMATE SENSITIVITY

30. Change in Vertical StratificationTS-T500

31. Meridional TS gradient

32. MeridionalT500 gradient

33. Conclusions • Impact of deterministic and stochastic forcings to non-equilibrium statistical mechanical systems • Frequency-dependent response obeys strong constraints • We can reconstruct the Green function – Spectroscopy/Broadband • Δexpectation of observable ≈variance of the noise • SRB measure is robust with respect to noise • Δ power spectral density ≈ l linear susceptibility |2 • More general case: Δ power spectral density >0 • We can predict climate change given the scenario of forcing and some baseline experiments • Limits to prediction • Decadal time scales • Now working on IPCC/Climateprediction.net data

34. References • D. Ruelle, Phys. Lett. 245, 220 (1997) • D. Ruelle, Nonlinearity 11, 5-18 (1998) • C. H. Reich, Phys. Rev. E 66, 036103 (2002) • R. Abramov and A. Majda, Nonlinearity 20, 2793 (2007) • U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Phys. Rep. 461, 111 (2008) • D. Ruelle, Nonlinearity 22 855 (2009) • V. Lucarini, J.J. Saarinen, K.-E. Peiponen, E. Vartiainen: Kramers-Kronig Relations in Optical Materials Research, Springer, Heidelberg, 2005 • V. Lucarini, J. Stat. Phys. 131, 543-558 (2008) • V. Lucarini, J. Stat. Phys. 134, 381-400 (2009) • V. Lucarini and S. Sarno, Nonlin. Proc. Geophys. 18, 7-27 (2011) • V. Lucarini, J. Stat. Phys. 146, 774 (2012) • J. Wouters and V. Lucarini, J. Stat. Mech. (2012) • J. Wouters and V. Lucarini, J Stat Phys. 2013 (2013) • V. Lucarini, R. Blender, C. Herbert, S. Pascale, J. Wouters,Mathematical Ideas for Climate Science, in preparation(2013)