Stochastic Resonance in Climate Research Reinhard Hagenbrock Working Group on Climate Dynamics, June 18., 2004.

Stochastic Resonance in Climate Research

Reinhard Hagenbrock

Working Group on Climate Dynamics, June 18., 2004

Reinhard Hagenbrock,Working Group on Climate Dynamics 2/19 June 18., 2004

Outline

Introduction A zero-dimensional energy balance model A stochastic energy balance model Some aspects of Stochastic Resonance

Reinhard Hagenbrock,Working Group on Climate Dynamics 3/19 June 18., 2004

Introduction

Observed climate variability:

•Strong spectral peak at periods of 105 years

•Coincides with external periodic forcing (Milankovich cycle)

•External forcing (variation of the solar constant by ~0.1% is to week to explain strong (and abrupt) climate shifts (i.e. between glacial and interglacial)

Reinhard Hagenbrock,Working Group on Climate Dynamics 4/19 June 18., 2004

Introduction

Nonlinear interaction is believed to magnify the impact of the relatively weak external forcing

“Dynamical approach”: The dynamics of the thermohaline circulation (THC) are investigated

“Stochastical approach”: Stochastic Resonance is investigated (mostly using simple energy balance models)

Stochastic Resonance is a Paradigm which does not replace dynamical considerations, but rather builds a framework for them.

The idea of Stochastic Resonance came up in connection to investigations on climate variability and change, but found applications in many areas of physics.

Reinhard Hagenbrock,Working Group on Climate Dynamics 5/19 June 18., 2004

Introduction

http://www.umbrars.com/sr/biblio.htm

Reinhard Hagenbrock,Working Group on Climate Dynamics 6/19 June 18., 2004

A zero-dimensional energy balance model

inR

4)()()(

)(

)()(

TTRTTR

QTR

TRTRdt

dTC

inout

solarin

outin

outR

The Budyko-Sellers model:

Reinhard Hagenbrock,Working Group on Climate Dynamics 7/19 June 18., 2004

A zero-dimensional energy balance model

)(TFdt

dT

Solutions of F(T)= 0 represent steady or equilibrium states i.e. climates).

To investigate the stability properties of climates, introduce the pseudo-potential

dTTF )(

Reinhard Hagenbrock,Working Group on Climate Dynamics 8/19 June 18., 2004

A zero-dimensional energy balance model

TTF

dt

dT

)(

T<T1: F>0

T T1

T2

T1<T<T2: F<0

T T1

T2<T<T2: F>0

T T3

T3<T: F<0

T T3

T1, T3 stable “climates”

T2 unstable “climate”

Reinhard Hagenbrock,Working Group on Climate Dynamics 9/19 June 18., 2004

A stochastic energy balance model

)()( tsTFdt

dT

Extend the simple energy balance model by a stochastic forcing:

Resulting power spectrum:

processWienernormalized:)(t

Reinhard Hagenbrock,Working Group on Climate Dynamics 10/19 June 18., 2004

A stochastic energy balance model

Temperature spectrum decays exponentially No spectral peak is found System changes from one climate state (i.e.

T=T1, glaciation) to another climate state (i.e. T=T3, interglaciation), but at no preferred residence time in one potential well

Adding noise to the model can by itself not explain the observed 105-year cycle.

Reinhard Hagenbrock,Working Group on Climate Dynamics 11/19 June 18., 2004

A stochastic energy balance model

Modify the model so far by adding the orbital forcing:

i.e. F (and therefore Φ) change over time

years10/2

)cos0005.01()(),(~

5

tTFtTF

Reinhard Hagenbrock,Working Group on Climate Dynamics 12/19 June 18., 2004

A stochastic energy balance model

Reinhard Hagenbrock,Working Group on Climate Dynamics 13/19 June 18., 2004

A stochastic energy balance model

Residence time is strongly dependent on the depth of the potential well.

When the potential well is shallow, the climate system will almost certainly switch to the other equilibrium state.

The observed variability shows a peak at the frequency of the external forcing

Stochastic Resonance

Reinhard Hagenbrock,Working Group on Climate Dynamics 14/19 June 18., 2004

Some aspects of Stochastic Resonance

to sum up: Orbital forcing of a simple energy balance model results

in the right spectrum, but the amplitude is to small. Noise added to a simple model with prescribed stable

equilibrium states results in the right amplitude, but the spectrum shows no peak.

Combination of both is able to explain both amplitude and frequency of observed climate shifts.

Reinhard Hagenbrock,Working Group on Climate Dynamics 15/19 June 18., 2004

Some aspects of Stochastic Resonance

Behaviour of the system is dependent on the set of parameters used…

Reinhard Hagenbrock,Working Group on Climate Dynamics 16/19 June 18., 2004

Some aspects of Stochastic Resonance

• Correlation between jumping time and external forcing is only observed if the noise level is well tuned!

therefore the term “resonance”

• Parameters of the model (distance between equilibrium temperatures, depth of the potential, variance of stochastic forcing) estimated from climate records and model studies

Reinhard Hagenbrock,Working Group on Climate Dynamics 17/19 June 18., 2004

Some aspects of Stochastic Resonance

For climate change investigations, stochastic resonance based models predict abrupt jumps of the climate…

Reinhard Hagenbrock,Working Group on Climate Dynamics 18/19 June 18., 2004

Some aspects of Stochastic Resonance

Reinhard Hagenbrock,Working Group on Climate Dynamics 19/19 June 18., 2004

Some aspects of Stochastic Resonance

Other investigated aspects include:

• application to other systems with stable equilibrium states (such as blocked/zonal flow), possibly with asymmetric potential wells

• solution of the associated Fokker-Planck equation (numerically and analytically)

• fluctuation-dissipation relations (FDR): relate the deterministic and stochastic components of a system…

• …