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Linear and non linear persistence in climate and its effect on the extremes

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Linear and non linear persistence in climate

and its effect on the extremes

Armin Bunde, Sabine Lennartz, Mikhail Bogachev

Justus-Liebig Universität Giessen

In cooperation with:

E. Koscielny-Bunde (Giessen), H.J. Schellnhuber (PIK),

S. Havlin (Tel Aviv), D. Rybski (Giessen, PIK)

H. v. Storch (GKSS), J. Eichner (Giessen, Re Munich)

I. Linear long-term correlations in climate

white noise

1/f noise

non stationary

i

i

Seasonal mean

Climate records:

Analysis problems:Finite Size Effects, Trends

Seasonal standard deviation

Alternative: Fluctuation analysis

Advantage: Modifications (DFA1, DFA2, ...Wavelet Methods) allow to detect long-term correlations in the presence of trends, with reduced finite size effects

For the inverse problem of trend detection in the presence of long-term memory, with application to anthropogenic global warming,see talk by Sabine Lennartz on Thursday

Summary of the fluctuation exponents: (a) Observational data

J.Eichner et al, 2003, D. Rybski et al, 2004, 2006, E. Koscielny-Bunde et al, 1996, 1998, 2004

(b) Model temperature data (1 000y): Erik the Red (Hamburg),

D. Rybski, A. Bunde, H. v. Storch, 2008, see also Fraedrich + Blender, 2006

II Extreme events

Q

Q

Q

threshold Q

return intervals ri

Result for long-term correlated records with correlation exponent :

The return intervals are

(a) long-term correlated with the same

(b) and their probability density scales as

A. Bunde, J. Eichner, S. Havlin, J. Kantelhardt, 2005

Comparison with paleo-climate data

A. Bunde, J. Eichner, S. Havlin, J. Kantelhardt, 2005

t

∆t

III Risk estimation: Hazard functionAssume: Last Q-exceeding event occured t time units ago. We are interested in the probability that within the next time units at least one event occurs:

trivial prediction

linear long-term correlations

strong nonlinear correlations

A. B., J. Eichner, J.Kantelhardt, S. Havlin, 2005; M. Bogachev, A.B., 2007, 2010

IV Precipitation and river run-offs

Cascade model:

days

days

Precipitation

To obtain the proper α-value, we shift

the multifractal spectrum by H´

River run-offs

V Non linear correlations: Multifractality

Generalized fluctuation function depends on q: Multifractality

See also: Schertzer, Lovejoy et al, Kantelhardt et al, Koscielny- Bunde et al, 2000-2006

VI PDF of the return intervals

Pronounced power law behavior independent of α, result of strong nonlinear memory

Weak deviations from exponential: result of weak linear and nonlinear memory .

Instrumental recordHistorical runControl runHistorical run (biannual)

Instrumental recordHistorical run

Historical run

Reconstructed record (Kaplan)Historical runControl runHistorical run (biannual)

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