Normal-form Based Analysisof Climate Time Series

Jan Sieber

Department of Mathematics

in collaboration with J.M.T. Thompson, FRS,School of Engineering, Aberdeen

Outline

É Time series analysisÉ Saddle-node induced tippingÉ Estimate of normal form parameters from time

series

Tipping in palaeoclimate time series

−8

−6

−4

−2

0

2

Temperature

170

180

190

200

Grayscale

−5 −4 −3 −2 −1 0

x 104

−1.3 −1.25 −1.2 −1.15

x 104

time (yrs)time (yrs)

End of younger DryasEnd of last glaciation

Background — time series analysis

u

y

input

outputy

Model

(t) = ƒ (t, , p)(t0) = 0

Experiment

Background — time series analysis

u

y

input

outputy

Model

(t) = ƒ (t, , p)(t0) = 0

Experiment

−8

−6

−4

−2

0

2

−4 −2 0×104time (yrs)

Observation

Background — time series analysis

Low-dim chaos& small noise

High-dim chaosor large noise

embedding, recurrence

[Kantz & Schreiber, 2000]

Questions

Assumption quasi-stationary

 finite t

 qualitativechange of attractor

Background — time series analysis

Low-dim chaos& small noise

High-dim chaosor large noise

embedding, recurrence

[Kantz & Schreiber, 2000]

Questions

Assumption quasi-stationary

 finite t

 qualitativechange of attractor

Tipping — Mechanical caricature of

positive feedback

É squishy beam, clamped and loaded withgradually increasing mass m

Tipping — Mechanical caricature of

positive feedback

V()

mass m

=pμ−pμ

+ noise

μ ∼mcriticl−m

d

dt = −V′() = μ− 2

d

dtμ = −ϵ

saddle-node normal formwith drift and noise

Tipping — Mechanical caricature of

positive feedback

V()

μ ∼ 1μ > 0μ ≈ 0 μ ≈ 0

μ < 0

Estimate from time series

time

Approach to Saddle-node = ƒ (, μ) + σηt

stable eq

unstableunstable

parameter μ = −ϵ

Estimate from time series

time

Approach to Saddle-node = ƒ (, μ) + σηt

stable eq

unstableunstable

parameter μ = −ϵ

(t)− eq

approx eq

zero order = −(?)[− eq(ϵt)] + ?

Estimate from time series

time

Approach to Saddle-node = ƒ (, μ) + σηt

stable eq

unstableunstable

parameter μ = −ϵ

fitfit

AR(1)DFAVar

κ

?a.u.

0

first order = − κ(ϵt)[− eq(ϵt)] + ?

(t)− eq

approx eq

zero order = −(?)[− eq(ϵt)] + ?

Estimate of linear decay rate

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

Estimate of linear decay rate

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

AR(1) (tn+1) = α(tn) ⇒ fit α ⇒ α = exp(−κΔt)

Estimate of linear decay rate

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

AR(1) (tn+1) = α(tn) ⇒ fit α ⇒ α = exp(−κΔt)

Variance ⇒ Var =σ2

κ

stationary distribution normal

When linear is not enough

d

Need to knowκ and d

escape prob.

μ 6∼ ϵt

κ assuming saddle-node

κ = 0

time

Back to estimates

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

AR(1) (tn+1) = α(tn) ⇒ fit α ⇒ α = exp(−κΔt)

Variance ⇒ Var =σ2

κ

stationary distribution normal

Back to estimates

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

AR(1) (tn+1) = α(tn) ⇒ fit α ⇒ α = exp(−κΔt)

Variance ⇒ Var =σ2

κ

stationary distribution normal

= − κ(ϵt)+Nx2 + σηt

Back to estimates

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

AR(1) (tn+1) = α(tn) ⇒ fit α ⇒ α = exp(−κΔt)

Variance ⇒ Var =σ2

κ

stationary distribution normal

= − κ(ϵt)+Nx2 + σηt

generalisation poor

Back to estimates

fitfit

κ

?a.u.

0

(t) noisefirst order = − κ(ϵt)+ σηt

AR(1) (Held&Kleinen’04)DFA (Livina&Lenton’07)Variance

AR(1) (tn+1) = α(tn) ⇒ fit α ⇒ α = exp(−κΔt)

Variance ⇒ Var =σ2

κ

stationary distribution normal

= − κ(ϵt)+Nx2 + σηt

generalisation poor

stationary distribution non-normal

GuttalLivina,Kwasniok,Lenton’10

Estimates for nonlinear parts

Fokker-Planck equation Density p of

= ƒ (, μ) + σηt

satisfies

∂tp =σ2

2∂p− ∂ [ƒ (, μ)p]

Stationary density p()

1

2∂p() = σ−2ƒ (, μ)p() + c

Estimates for nonlinear parts

Fokker-Planck equation Density p of

= ƒ (, μ) + σηt

satisfies

∂tp =σ2

2∂p− ∂ [ƒ (, μ)p]

Stationary density p()

1

2∂p() = σ−2ƒ (, μ)p() + c

empirical

Estimates for nonlinear parts

Fokker-Planck equation Density p of

= ƒ (, μ) + σηt

satisfies

∂tp =σ2

2∂p− ∂ [ƒ (, μ)p]

Stationary density p()

1

2∂p() = σ−2ƒ (, μ)p() + c

empirical

c

linear estimate −κx

Paleo-climate recordsEnd of last glaciation End of Younger Dryas

time−1.7× 104

time−11.6× 103

temperature greyscale

pot. well pot. well

2.8% 10.8%

fitted c fitted cempirical skewness empirical skewness

18% 41%

Summary

É accuracy of estimateszero order > first order > second order

É but second order term necesssary to estimatetipping time/probability

É estimate tipping time/probability based onsaddle-node normal form

[JMTT,JS on arxiv]