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)

)(S

I. Linear long-term correlations in climate

i

Climate records: Analysis problems: Finite Size Effects, Trends

i i

1,

2

)2)(1()( ssC sii

:0

:1

:1 white noise 1/f noise

non stationary

i

iii

xx

Seasonal mean

Seasonal standard deviation

Alternative: Fluctuation analysis

)2(

1~

22/1

)(2

h

ii ssF

s

2/)1(

2/1)2(

h

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

2/12/)1(,~)( ssF

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

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

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

II Extreme events

threshold Q

return intervals ri

QQQ

ix~

QQ

QQQ RrRr

RrRrrP

,)/(ln)1(

,)/(~))(ln(

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

Comparison with paleo-climate data

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

III Risk estimation: Hazard function

t

Q

tt

t

Q

Q

drrP

drrP

ttW ;

Q

QQ

Q R

tWe

RrP QRr

1

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:

t

t ∆t

???

t

tQWrrP QQ

Q 1~

trivial prediction

strong nonlinear correlations

1

~~QQ

QQ

Q R

t

R

tWe

dr

drP

Rr linear long-term correlations

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

IV Precipitation and river run-offs

daysdays

i

i

Precipitation

River run-offs

To obtain the proper α-value, we shift

the multifractal spectrum by H´

Cascade model:

V Non linear correlations: Multifractality

Generalized fluctuation function depends on q: Multifractality

)2(h qhssqFs

ii

qq

~1

/1

)(

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

VI PDF of the return intervals

Weak deviations from exponential: result of weak

linear and nonlinear memory .

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

memory

End of the talk

Instrumental record

Historical run

Instrumental recordHistorical runControl runHistorical run (biannual)

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

Historical run

(b) Temperature, precipitation and run-off records

22,~)(~)( ssCssF