1. What makes it so difficult to model the geophysical fluids? › Some gross mistakes in our...
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Transcript of 1. What makes it so difficult to model the geophysical fluids? › Some gross mistakes in our...
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Fluctuations, Response, and Prediction in Geophysical Fluid
Dynamics
Valerio [email protected]
Meteorologisches Institut, Universität HamburgDept. of Mathematics and Statistics, University of
Reading
F. Lunkeit, F. Ragone, S. Sarno
Cambridge,,November 1st 2013
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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
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IPCC scenarios
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Models’ Response
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Climate ResponseIPCC scenario 1% increase p.y.
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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!
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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
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Perturbed chaotic flow as:
Change in expectation value of Φ:
nth order perturbation:
Ruelle (’98) Response Theory
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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:
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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
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Simpler and simpler forms of FDT
Various degrees of approximation
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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
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Linear (and nonlinear) Spectroscopy of L63
Resonances have to do with UPOs
L. 2009
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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
,
2
122
,, AAPAP
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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
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Spectroscopy – Im [χ(1)(ω)]
Rigorous extrapolation
LW HF
L. and Sarno 2011
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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
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Climate Prediction: convolution with T(t)=sin(2πt)
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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
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(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
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What we get – CO2 doubling
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G(1)(t)Climate Prediction - TS
CLIMATE SENSITIVITY
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Change in Vertical Stratification
TS-T500
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Meridional TS gradient
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Meridional T500 gradient
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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
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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)