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

2. Reinhard Hagenbrock, Working Group on Climate Dynamics2/19 June 18., 2004 Outline Introduction A zero-dimensional energy balance model A stochastic energy balance model Some aspects of Stochastic Resonance

3. Reinhard Hagenbrock, Working Group on Climate Dynamics3/19 June 18., 2004 Introduction Observed climate variability: Strong spectral peak at periods of 10 5 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)

4. Reinhard Hagenbrock, Working Group on Climate Dynamics4/19 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.

5. Reinhard Hagenbrock, Working Group on Climate Dynamics5/19 June 18., 2004 Introduction http://www.umbrars.com/sr/biblio.htm

6. Reinhard Hagenbrock, Working Group on Climate Dynamics6/19 June 18., 2004 A zero-dimensional energy balance model The Budyko-Sellers model:

7. Reinhard Hagenbrock, Working Group on Climate Dynamics7/19 June 18., 2004 A zero-dimensional energy balance model Solutions of F(T)= 0 represent steady or equilibrium states i.e. climates). To investigate the stability properties of climates, introduce the pseudo-potential

8. Reinhard Hagenbrock, Working Group on Climate Dynamics8/19 June 18., 2004 A zero-dimensional energy balance model T 0  T  T 1 T2T2 T 1 <T<T 2 : F<0  T  T 1 T 2 0  T  T 3 T 3 <T: F<0  T  T 3 T 1, T 3 stable “climates” T 2 unstable “climate”

9. Reinhard Hagenbrock, Working Group on Climate Dynamics9/19 June 18., 2004 A stochastic energy balance model Extend the simple energy balance model by a stochastic forcing: Resulting power spectrum:

10. Reinhard Hagenbrock, Working Group on Climate Dynamics10/19 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=T 1, glaciation) to another climate state (i.e. T=T 3, interglaciation), but at no preferred residence time in one potential well  Adding noise to the model can by itself not explain the observed 10 5 -year cycle.

11. Reinhard Hagenbrock, Working Group on Climate Dynamics11/19 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

12. Reinhard Hagenbrock, Working Group on Climate Dynamics12/19 June 18., 2004 A stochastic energy balance model

13. Reinhard Hagenbrock, Working Group on Climate Dynamics13/19 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

14. Reinhard Hagenbrock, Working Group on Climate Dynamics14/19 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.

15. Reinhard Hagenbrock, Working Group on Climate Dynamics15/19 June 18., 2004 Some aspects of Stochastic Resonance Behaviour of the system is dependent on the set of parameters used…

16. Reinhard Hagenbrock, Working Group on Climate Dynamics16/19 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

17. Reinhard Hagenbrock, Working Group on Climate Dynamics17/19 June 18., 2004 Some aspects of Stochastic Resonance For climate change investigations, stochastic resonance based models predict abrupt jumps of the climate…

18. Reinhard Hagenbrock, Working Group on Climate Dynamics18/19 June 18., 2004 Some aspects of Stochastic Resonance

19. Reinhard Hagenbrock, Working Group on Climate Dynamics19/19 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… …