A toy model of climatic variability with scaling behaviour - ITIA

Hydrofractals ’03 An international conference on fractals in hydrosciences Monte Verità, Ascona, Switzerland, 24-29 August 2003

A toy model of climatic variability with scaling behaviour Demetris Koutsoyiannis Department of Water Resources, School of Civil Engineering, National Technical University, Athens, Greece

Introduction: The notion of a toy model Definition: A model in which the features represented are kept to a minimum in order to show that some empirical phenomenon can or cannot be produced from primitive assumptions (adapted from Cox and Isham, 1998) Objectives „

„ „

Investigate whether simple mechanisms can produce a complex phenomenon Identify essentials and discard details in the system dynamics Identify sets of parameters for which the phenomenon occurs

Parameter issues „ „

A small number of parameters is involved Formal fitting may be irrelevant

Examples „ „ „

ENSO dynamics (Andrade et al., 1995) Biological evolution of species (Wandewalle & Ausloos, 1996) Attraction of parasites and predators (Freund & Grassberger, 1992) D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 2

The phenomenon studied: Simple scaling of climatic time series in discrete time Clarifications Scaling is meant here in terms of the behaviour of the time series aggregated (averaged) on different time scales Time scales are from annual to thousands of years Long time series are required for the study

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 3

A simple scaling process as a stochastic process A stochastic process at the annual scale The mean of Xi

Xi µ := E[X i]

The standard deviation of Xi The lag-j autocorrelation of Xi

σ := Var[Xi] ρj := Corr[Xi, Xi – j]

The aggregated stochastic process at scale k ≥ 1

(k) Zi

(k)

ik

∑ Xl

:=

l = (i – 1) k + 1 (k)

The mean of Zi

E[Z i ] = k µ (k)

The standard deviation of Zi

σ(k) :=

Definition of a simple scaling stochastic process or a simple scaling signal (SSS; also known as (a) stationary increments of self-similar process (b) Fractional Gaussian noise – FGN)

(k) (Z i

(k)

The standard deviation of an SSS Zi (a power law of scale k) (k)

The lag-j autocorrelation of an SSS Zi (a power law of lag j; independent of scale k)

(k)

Var [Zi ]

– kµ)

d  k H =  l   

(Z

(l) j

– lµ)

for any scales k and l and for a specified H (0 < H 0

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 4

Empirical basis of the study: (a) Nile data set Nilometer index

1500

The Nilometer series indicating the annual minimum water level of the Nile river for the years 622 to 1284 A.D. (663 years; Beran, 1994)

1400 1300 1200 1100 1000

900

Annual value

800 600

700

800

Average, 5 years 900

1000

Average, 25 years 1100

1200

1300

Nilometer index

Log(Standard deviation)

Year A.D. 3.2 3 2.8 2.6 2.4

H

2.2

=0

.85

1.8 0.4

1300 1200 1100 900

2 0.2

1400

1000

0.6

0.8

1

1.2

Log(Scale)

Annual value

Average, 5 years

Average, 25 years

800

Empirical White noise Modelled 0

1500

600

700

800

900

1000

1100

1200

1300 Year

1.4

A white noise series (for comparison) D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 5

Northern Hemisphere temperature anomalies in °C with reference to 1961– 1990 mean (992 years, Jones et al., 1998)

Temperature anomaly

(b) Jones data set 0.6 0.4 Annual 5-year average 25-year average 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

This series was constructed using temperature sensitive paleoclimatic multi-proxy data from 10 sites worldwide that include tree rings, ice cores, corals, and historical documents. Only four of the proxy data series go back before 1400 AD and, therefore, data prior to about 600 years ago are more uncertain.

Log(Standard deviation)

Year A.D.

1.5

Empirical Empirical, excl. 20th century Modelled White noise

1 0.5 0

H

=0

.88

-0.5 -1 0

0.5

1

1.5 2 Log(Scale)

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 6

10 400 year scale

o

The Vostok ice core deuterium data set going back to 422 766 years before present (Petit et al., 1999) Temperature difference with reference to the mean recent time value

Temperature anomaly ( C)

(c) Vostok data set 2000 year scale

10000 year scale

5 0 -5 -10 400000

300000

200000

100000 Years before present

o

=

0. 9

4

2

H

This temperature difference is calculated based on the deuterium content of the ice using a deuterium/temperature gradient of 9‰/°C, after accounting for the isotopic change of seawater. The temporal resolution ranges from 17 years (present time) to 631 years. Here the series was re-interpolated using a constant 400 year temporal resolution.

Log(standard deviation in C)

3

1 Empirical White noise Modelled 0 2

3

4 5 Log(scale in years) D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 7

0

Vostok data series of temperature difference Identification of periodicity Plot of the principal harmonic roughly corresponding to the period of orbital stretch

10 5

-5 -10 400000

1

H

=

94 . 0

0 2

3

4 5 Log(scale in years)

300000

200000

Equation of harmonic

100000

0

Years before present

͠xt = 3.917 cos (2π t/τ +

o

Temperature anomaly ( C)

o

Log(standard deviation in C)

2

2000 year scale Harmonic

0

0.2146) – 4.856 τ = 103 598 years Explains 25% of variance

3 Empirical White noise Modelled

400 year scale 10000 year scale

o

Temperature anomaly ( C)

(c2) Vostok data set adapted

10

Subtraction of the harmonic

5 0 -5 400 year scale

2000 year scale

10000 year scale

-10 400000

300000

200000

100000 Years before present

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 8

0

Major climate change processes and feedbacks Milankovitch cycles (in scales of thousands of years) Ice-albedo feedback (positive) Water vapour feedback (positive) Cloud feedback (negative)

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 9

Synthesis of positive and negative feedbacks

I = xt – 1– x* +

+

Negative feedback |1 – f2(xt – 1)| ≥ 1

1 System state at time t (x t )

O = xt – x*

0.8 0.6

f2(xt - 1)

f1(xt - 1) Positive feedback |1 – f1(xt – 1)| ≤ 1

Positive feedback: x t = x * + (x t - 1 - x *) / [1 - f 1(x t

- 1)]

Negative feedback: x t = x * + (x t - 1 - x *) / [1 - f 2(x t

- 1)]

Stationary point x * = f 1 (x * ) = f 2 (x *)

0.4

Synthesis of both feedbacks: x t = x * + (x t - 1 - x *) / [1 - f 1(x t

0.2

- 1)f 2 (x t - 1 )]

Equality line: xt =xt -1

0 0

0.2

0.4

0.6

0.8

System state at time t - 1 (x

1 t - 1) D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 10

Simplified modelling of compound positive and negative feedbacks Starting point: The generalised tent map

0 -20 = α -20

-3

0.8

α =2

Example usage: The map approximates the relation between successive maxima in the variable x(t) from the Lorenz equations that describe climatic dynamics (Lasota and Mackey, 1994, p. 150)

1.8

0.6

1

=0

with 0 ≤ xt ≤ 1, α < 2

1

α

(2 – α) min (xt – 1, 1 – xt – 1) = ――――――――――― 1 – α min (xt – 1, 1 – xt – 1)

State at time t

xt = g(xt – 1; α) =

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

State at time t - 1

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 11

Simplified modelling of compound positive and negative feedbacks (2) xt = gn(xt – 1; α) = g(g(…(g(xt – 1; α)…); α); α)

123 n

1 0.8 0.6

Equivalent definition of gn( ) xt = yn t with yn t = g(yn t – 1; α), y0 = x0, t = 0, 1, 2, …

g4

0.4

g

1

=g

0.2

where the intermediate terms y(n – 1)t , … yn t – 1 are regarded as hidden terms

State at time t

More complex maps resulting from the generalised tent map

0 0

0.2

0.4

0.6

0.8

1

State at time t - 1 D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 12

Resulting time series Synthetic series Synthetic series, moving average of 50 values 1 xi

A time series generated by the transformation g4(x; α) with α = 0.317

0.8 0.6 0.4 0.2 0 0

200

400

600

800

1000 Time, i

Standard deviation for time scale k

• Random appearance at the basic scale • Stable behaviour at larger scales • Hurst coefficient = 0.5

10

1

0.50 = H

0.1 1

10

Time scale, k

100

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 13

The toy model Make parameter of the tent transformation time dependent using the same (tent) transformation zt = G(zt – 1; κ, λ) = g(zt – 1; κ αt – 1) with αt = g(αt – 1; λ) Extend the tent transformation by adding hidden terms zt = Gn(zt – 1; κ, λ) defined by zt = yn t with yn t = G(yn t – 1; κ, λ), y0 = z0, t = 0, 1, 2, … Apply a rescaling the transformation to shift from [0, 1] to [0, ∞) xt = b + c tan (π zt / 2)d The final model for xt „ is two dimensional (involves two degrees of freedom corresponding to α0 and z0) „ contains five parameters (κ, λ, b, c, d) D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 14

Model behaviour with respect to parameters κ and λ

Runaway area (H ≈ 1)

2

κ 1.5

1

Scaling area (0.5 < H < 1)

0.5

0 -10

-8

-6

-4

-2

0 λ

Non-scaling area (H = 0.5)

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 15

Parameter fitting Aim of parameter fitting to the example data sets: Generation of a synthetic series that resembles „ downward and upward trends „ statistical properties of historical series Criteria for parameter fitting: Large correlation with historical series „ for time scale of 1 time step „ for time scale of 50 time steps Note: In all examples, n = 4 Fitted parameters

Initial values

κ

λ

b

c

d

z0

α0

Nilometer

1.871

0.477

-26871.1

28130.5

0.0013

0.030

0.335

Jones

1.765

0.317

73.3

-73.8

0.0013

0.797

0.325

Vostok

1.810

0.332

624.8

-628.6

0.0011

0.988

0.327

Data set

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 16

Nilometer index1

1400 1300 1200 1100 1000

Historical series, annual scale Historical series, 50-year moving average

900

Nilometer index1

800 600 1500

700

800

900

1000

1100

1400

1200 1300 Year A. D.

1300 1200 1100 1000 900

Synthetic series, annual scale Synthetic series, 50-year moving average

800 Nilometer index1

Data set: Nilometer Generated series

1500

1300600

700

800

900

1000

1100

1200 1300 Year A.D.

1200 1100 1000 Historical series, 50-year moving average Synthetic series, 50-year moving average

900 800 600

700

800

900

1000

1100

1200 1300 Year A.D.

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 17

Non exceedence frequency1

Data set: Nilometer Comparison of statistical properties between historical and generated series

1 0.8 0.6 0.4 0.2 0 800

10000

1000

Historical

1000

H

=

1200

1400

Nilometer index

Synthetic

Autocorrelation1

Standard deviation for time scale k

Historical Synthetic

5 0.8

100

1 0.75

Historical Synthetic

0.5 0.25 0 -0.25 -0.5

10 1

10

100 Time scale, k (years)

0

100

200

Lag

300

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 18

o

o

o

Temperature anomaly ( C) Temperature anomaly ( C) Temperature anomaly ( C)

Data set: Jones Generated series

0.6

Historical data series, annual scale Historical data series, 50-year moving average

0 -0.6 -1.2 1000 0.6

1200

1400

1600

1800

2000

Synthetic data series, annual scale Year A.D. Synthetic data series, 50-year moving average

0 -0.6 -1.2 1000 0.6

1200

1400

1600

1800

2000

Historical data series, 50-year movingYear average A.D. Synthetic data series, 50-year moving average

0 -0.6 -1.2 1000

1200

1400

1600

1800

2000

Year A.D. D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 19

Non exceedence frequency1

Data set: Jones Comparison of statistical properties between historical and generated series

1 Historical Synthetic

0.8 0.6 0.4 0.2 0 -1.2

Historical

-0.6

0

0.6 o

Temperature anomaly ( C)

Synthetic

H

1

.8 =0

8 Autocorrelation1

Standard deviation for time scale k

10

1 0.8

Historical Synthetic

0.6 0.4 0.2 0

0.1

-0.2 1

10

100 Time scale, k (years)

0

100

200

Lag

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 20

300

o

Temperature anomaly ( C)

o

Temperature anomaly ( C)

o

Temperature anomaly ( C)

Data set: Vostok Generated series

10 Historical data series, 400-year scale Historical data series, 20000-year average

5 0 -5 -10 10

400000

300000

200000

100000

0

Synthetic data series, 400-year scale present Years before Synthetic data series, 20000-year average

5 0 -5 -10 10

400000

300000

200000

100000

0

Historical data series, 20000-year average Years before present Synthetic data series, 20000-year average

5 0 -5 -10 400000

300000

200000

100000 Years before present

0

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 21

Non exceedence frequency1

Data set: Vostok Comparison of statistical properties between historical and generated series

0.8 0.6 0.4 Historical Synthetic

0.2 0

1000

-10

Historical Synthetic

-5

0

5

10

Temperature anomaly (oC)

10

Autocorrelation1

=

0. 9

4

100

H

Standard deviation for time scale k

1

1

Historical Note: 1 time step Synthetic corresponds to

0.75 0.5

400 years

0.25 0

-0.25 -0.5

1 100

1000

10000 100000 Time scale, k (years)

0

100

200

Lag

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 22

300

o

Temperature anomaly ( C)

o

Temperature anomaly ( C)

o

Temperature anomaly ( C)

Data set: Vostok minus harmonic Generated series

10 Historical data series, 400-year scale Historical data series, 20000-year average

5 0 -5 -10 10

400000

300000

200000

100000

0

Synthetic data series, 400-year scale present Years before Synthetic data series, 20000-year average

5 0 -5 -10 10

400000

300000

200000

100000

0

Years before present Historical data series, 20000-year average Synthetic data series, 20000-year average

5 0 -5 -10 400000

300000

200000

100000

0

Years before present D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 23

Non exceedence frequency1

Data set: Vostok minus harmonic Comparison of statistical properties between historical and generated series

Historical Synthetic

0.8 0.6 0.4 0.2 0

1000

-10

Historical Synthetic

-5

0

5

10

Temperature anomaly (oC)

=

0. 9

4

Autocorrelation1

100

H

Standard deviation for time scale k

1

10

1 Historical Synthetic

0.75 0.5

Note: 1 time step corresponds to 400 years

0.25 0

-0.25 1

-0.5 100

1000

10000 100000 Time scale, k (years)

0

100

200

Lag

300

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 24

The evolution of “climate” is totally different but the statistical characteristics (especially the Hurst exponent) remain the same

o

0.4 0.2

Historical data series, 50-year moving average Synthetic data series 1, 50-year moving average Synthetic data series 2, 50-year moving average

0 -0.2 -0.4 -0.6 1000

1200

1400

1600

1800 2000 Year A.D.

10 Standard deviation for time scale k

Data set: Jones Series 2 was generated with same parameters as series 1, with same initial value z0, but with initial value of parameter α0 greater by 0.01%

Temperature anomaly ( C)

The role of initial values

Historical Synthetic 2

1

H

.8 0 =

8

0.1 1

10

100 Time scale, k (years)

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 25

0.4 0.2 0 -0.2 -0.4 -0.6 2000

4000

6000

8000

10000

12000

ʺYearsʺ

Data set: Jones Series 1 extended to 12 000 “years” (plotted is the 50-”year” moving average) The statistical characteristics (especially the Hurst exponent) do not depend seriously on length or location within time series

Standard deviation for time scale k 1

0

Standard deviation for time scale k 1

0.6

o

Temperature anomaly ( C)

The role of series length

100 ʺYearsʺ 1-1000 ʺYearsʺ 1-2000 ʺYearsʺ 1-6000 ʺYearsʺ 1-12000

10

1

0.1 100 1

10 100 k Time scale, (ʺyearsʺ) ʺYearsʺ 1-1000 ʺYearsʺ 6001-7000 ʺYearsʺ 11001-12000

10

1

0.1 1

10 100 Time scale, k (ʺyearsʺ)

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 26

Data set: Jones Correlation sum C2 as a function of scale length ε and embedding dimension m for the series 1 with length N = 12 000 The slope of correlation integral d2(ε, m) increases with embedding dimension m That is, the standard algorithm fails to capture the low dimensional determinism (D = 2) in the produced series (for N = 12 000) and deems it as a random series

1E-01 1E-02

1 5

2 6

3 7

4 8

1E-03 1E-04 1E-05 1E-06 1E-07 1E-08 1E-09 1E-10 1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1E+00 ε

d 2(ε , m )

C 2 (ε , m )

Tracing of determinism in a generated series

1E+00

8 7 6 5

1 5

2 6

3 7

4 8

4 3 2 1 0 1E-06

1E-05

1E-04

1E-03

1E-02

1E-01

1E+00 ε

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 27

Synopsis Long climatic time series reveal irregular changes (upward and downward fluctuations) on all time scales These comply with the fact that “Climate changes irregularly, for unknown reasons, on all timescales” (National Research Council, 1991, p. 21).

The irregular changes on all scales are equivalent to a scaling behaviour of climatic series The scaling behaviour is quantified through a Hurst exponent greater than 0.5 Synthetic time series with scaling behaviour are typically generated by appropriate stochastic models Even a simple two-dimensional deterministic toy model can reproduce the scaling behaviour of climatic processes The simplicity of the deterministic toy model (in comparison with stochastic models which are more complex) enables easy implementation and convenient experimentation D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 28

Synopsis (2) This toy model is based on the “chaotic tent map”, which may represent the compound result of a positive and a negative feedback mechanism Application of the toy model gives traces that can resemble historical climatic time series; in particular, exhibit scaling behaviour with a Hurst exponent greater than 0.5 Moreover, application demonstrates that large-scale synthetic “climatic” fluctuations can emerge without any specific reason and their evolution is unpredictable, even when they are generated by this simple fully deterministic model with only two degrees of freedom Obviously, the fact that such a simple model can generate time series that are realistic surrogates of real climatic series does not mean that the real climatic system involves that simple dynamics

D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 29

Conclusion A simple two-dimensional deterministic dynamical system can produce series that resemble climatic series, especially their scaling behaviour with Hurst exponent > 0.5 This simple toy model illustrates the great uncertainty and unpredictability of the climate system, showing that they can emerge even from caricature, purely deterministic, dynamics with only two degrees of freedom Obviously, the dynamics of the real climate system is greatly more complex than this simple toy model D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 30

This presentation is available on line at http://www.itia.ntua.gr/e/docinfo/585/ References Andrade, J.S. Jr., I. Wainer, J. Mendes Filho andJ.E. Moreira, Self-organized criticality in the El Niño Southern Oscillation, Physica A, 215, 331-338, 1995. Beran, J., Statistics for Long-Memory Processes, vol. 61 of Monographs on Statistics and Applied Probability. Chapman & Hall, New York, USA, 1994. Cox, D. R., and V. Isham, Stochastic spatial-temporal models for rain, in Stochastic Methods in Hydrology: Rain, Landforms and Floods, edited by O.E. Barndorff-Nielsen, V.K. Gupta, V. Pérez-Abreu and E. Waymire, pp. 1-24, World Scientific, Singapore, 1998. Freund, H., and P. Grassberger, The Red Queen’s walk, Physica A, 190, 218-237, 1992. Jones, P. D., Briffa, K. R., Barnett, T. P. & Tett, S. F. B., High-resolution paleoclimatic records for the last millennium: interpretation, integration and comparison with General Circulation Model control-run temperatures. Holocene 8(4), 455–471, 1998. Lasota, A., and M.C. Mackey, Chaos, Fractals and Noise, Stochastic Aspects of Dynamics, SpringerVerlag, 1994. National Research Council (1991) Opportunities in the Hydrologic Sciences, National Academy Press, Washington DC, USA. Petit J.R., Jouzel J., Raynaud D., Barkov N.I., Barnola J.M., Basile I., Bender M., Chappellaz J., Davis J., Delaygue G., Delmotte M., Kotlyakov V.M., Legrand M., Lipenkov V., Lorius C., Pépin L., Ritz C., Saltzman E., Stievenard M., Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica, Nature, 399, 429-436, 1999. Wandewalle, N., and M. Ausloos, A toy model for life at the “edge of chaos”, Comput. & Graphics, 20(6), 921-923, 1996. D. Koutsoyiannis, A toy model of climatic variability with scaling behaviour 31