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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),

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

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
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
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
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
(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
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


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

Comparison with paleo-climate data

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


Iii risk estimation hazard function

???

t

∆t

III Risk estimation: Hazard function

Assume: 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
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


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

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


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

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 .



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

Instrumental recordHistorical runControl runHistorical run (biannual)

Instrumental recordHistorical run

Historical run

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