Enhanced sensitivity of persistent events to weak forcing in dynamical and stochastic systems: Implications for climate change
Questions What are the characteristics of the unforced Lorenz system? What do the terms state space, fixed point, regime, and stability mean in the context of dynamical systems? Does the forcing of the model change the mean persistence time, PDF maxima, and the PDF of persistence? Why or why not? Can this model be used to help predict ENSO? How would anthropogenically forced climate changes be seen in ENSO?
Prior Work Anthropogenic forcing of the climate system IPCC (1996) focused on the mean Palmer (1999) & Corti et al. (1999) both used chaotic models with small forcings Khatiwala et al. used a chaotic model as well, but focused on how the persistence of events changed
Lorenz System Two nonlinear terms @ xy and xz Fixed Points Always fixed - (0,0,0) (for non-forced system) Symmetric Points C + and C - Depends on r, b, σ Strogatz, Steven H., Nonlinear dynamics and chaos; with applications to physics, biology, chemistry, and engineering (1994)
Stability Stability Globally attracting Liapunov stable all trajectories that start close to x* remain close Neutrally Liapunov stable, but not attracting Asymptotically Both Liapunov stable and attracting Unstable Neither Liapunov stable nor attracting Strogatz, Steven H., Nonlinear dynamics and chaos; with applications to physics, biology, chemistry, and engineering (1994)
Lorenz System Strange Attractors attractor which has a sensitive dependence on initial conditions Strogatz, Steven H., Nonlinear dynamics and chaos; with applications to physics, biology, chemistry, and engineering (1994)
Lorenz System The system is characterized by chaotic oscillations around two unstable fixed points Two regimes Fluctuations between regimes Aperiodic never repeats exactly Strogatz, Steven H., Nonlinear dynamics and chaos; with applications to physics, biology, chemistry, and engineering (1994)
Forcing No forcing regimes are equal With forcing one regime is favored PDF maxima, which nearly coincide with the location of the fixed points, do not change position in phase space
Palmer (1999) Anthropogenically forced changes in climate would project largely onto modes of natural climate variability The effect of anthropogenic forcing would manifest itself through changes in the frequency of occurrence of natural patterns of variability
Forced Lorenz System Distinct difference between the persistence time of the two regions Expect that the distance from the fixed point increases until it reaches a critical value when the system flips to the other regime
Forced Lorenz System Most events have short persistence times Small difference at short T p Large difference at large T p
Forced Lorenz System Most events have short persistence times Short persistence times dominate the mean
Forced Lorenz System Enhancement in the frequency of persistent events is not due to an increase in the probability of a closer approach to the fixed points It is due to a change in stability of the regimes
Forced Lorenz System The slope of the PDF of T p is directly related to the linear stability at the fixed points Slope gets shallower as stability becomes smaller There can be orders of magnitude enhancement of the probability of extremely persistent events, even though the mean changes only by a small amount
Comparison to a Stochastic Process Particle in a double well Stable equilibria at x=±1 Unstable equil. at x=0 Addition of forcing allows jumping from one regime to the other
Comparison to a Stochastic Process PDF of T p as f 0 is varied is similar to Lorenz system Dominant effect in both is to change the frequency of occurrence of extremely persistent events
Comparison to a Stochastic Process Dominant effect is the change in frequency of occurrence of extremely persistent events Slopes of PDF s are insensitive to how the two regimes are defined It is the stability of the regime that sets the persistence of long-lived events
Implications Dramatic change in the frequency of occurrence of extremely long events Climate example: forcing is global warming Overall reduction in rainfall more droughts Increase in prolonged periods of drought
ENSO Is the frequency of persistent events changing? http://www.cdc.noaa.gov/people/klaus.wolter/mei/mei.html
ENSO Is the frequency of persistent events changing? http://www.pmel.noaa.gov/~kessler/enso/soi-1950-98.gif
PDO 0.6 0.4 0.2 0-0.2-0.4-0.6 1600 1650 1700 1750 1800 1850 1900 1950 2000 PDO with 9-yr running average Is the frequency of persistent events changing? Evans (2001)
Questions What are the characteristics of the unforced Lorenz system? What do the terms state space, fixed point, regime, and stability mean in the context of dynamical systems? Does the forcing of the model change the mean persistence time, PDF maxima, and the PDF of persistence? Why or why not? Can this model be used to help predict ENSO? How would anthropogenically forced climate changes be seen in ENSO?