1

Fluctuations, Response, and Prediction in Geophysical Fluid

DynamicsValerio Lucarini

valerio.lucarini@uni-hamburg.de

Meteorologisches Institut, Universität HamburgDept. 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

Response theory 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

Perturbed chaotic flow as:

Change in expectation value of Φ:

nth order perturbation:

Ruelle (’98) Response Theory

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

*)1()1( )]([)( 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

0t tttt

Fourier Transform We end up with the linear susceptibility... Let’s rewrite he equation:

So: difference between 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

Correlations Power Spectra

2122

,

21

22

,, AAPAP

15

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

Fxxxxx iiiii 211 Nii xxNi ,...,1

16

17

Spectroscopy – Im [χ(1)(ω)]

Rigorous extrapolation

LW HF

L. and Sarno 2011

18

Using stochastic forcing… Squared modulus of Blue: Using stoch pert; Black: deter

forcing ... And many many many less integrations

)1(e

L. 2012

We choose observable A, forcing e Let’s perform an ensemble of

experiments Linear response:

Fantastic, we estimate

…and we obtain:

…we can predict

Broadband forcing

Broadband forcing G(1)

(t)

20 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

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:

Time scale of prediction

23

(Non-)Differentiability of the measure for the climate system

CO2 S*

Boschi et al. 2013

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:

A Climate Change experiment

Model Starter and

Graphic User Interface

Spectral Atmospheremoist primitive equations

on σ levels

Sea-Icethermodynamic

Terrestrial Surface: five layer soil

plus snow

Vegetations(Simba, V-code,

Koeppen)

Oceans:LSG, mixed layer,or climatol. SST

PlaSim: Planet Simulator

Key features• portable• fast• open source• parallel

• modular• easy to use• documented• compatible

28

What we get – CO2 doubling

29

G(1)(t)Climate Prediction - TS

CLIMATE SENSITIVITY

30

Change in Vertical Stratification

TS-T500

31

Meridional TS gradient

32

Meridional T500 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 Δ expectation 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)