CHAPTER 1

Heuristics of noise induced transitions

1.1. Energy balance models of climate dynamics

The simple concept of energy balance models stimulated research not only in

the area of conceptual climate models, but was at the cradle of a research direc-

tion in physics which subsequently took important examples from various domains

of biology, chemistry and neurology: it was one of the first examples for which

the phenomenon of stochastic resonance was used to explain the transition dynam-

ics between different stable states of physical systems. For a good overview see

Gammaitoni et al. [43] or Jung [62].

In the end of the 70’s, Nicolis [83] and Benzi et al. [5] almost simultaneously

tried stochastic resonance as a rough and qualitative explanation for the glaciation

cycles in earth’s history. They were looking for a simple mathematical model appro-

priate to explain experimental findings from deep sea core measurements according

to which the earth has seen ten glacial periods during the last million years, alter-

nating with warm ages rather regularly in periods of about 100 000 years. Mean

temperature shifts between warm age and glacial period are reported to be of the

order of 10 K, and relaxation times, i.e. transition times between two relatively

stable mean temperatures as rather short, of the order of only 100 years. Math-

ematically, their explanation was based on an equation stating the global energy

balance in terms of the average temperature T (t), where the global average is taken

meridionally (i.e. over all latitudes), zonally (i.e. over all longitudes), and annually

around time t. The global radiative power change at time t is equated to the differ-

ence between incoming solar (short wave) radiative power Rin and outgoing (long

wave) radiative power Rout.

The power Rin is proportional to the global average of the solar constant Q(t)

at time t. To model the periodicity in the glaciation cycles, one assumes that Q

undergoes periodic variations due to one of the so-called Milankovich cycles, based

on periodic perturbations of the earth’s orbit around the sun. Two of the most

prominent cycles are due to a small periodic variation between 22.1 and 24.5 degrees

of the angle of inclination (obliquity) of the earth’s rotation axis with respect to

its plane of rotation, and a very small periodic change of only about 0.1 percent of

the eccentricity, i.e. the deviation from a circular shape, of the earth’s trajectory

around the sun. The obliquity cycle has a duration of about 41 000 years, while

the eccentricity cycle corresponds to the 100 000 years observed in the temperature

proxies from deep sea core measurements mentioned above. They are caused by

gravitational influences of other planets of our solar system. In formulas, Q was

assumed to be of the form

Q(t) = Q0 + b sin ωt,

with some constants Q0,b and a frequency ω =

10−5[ 1

y

].

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http://dx.doi.org/10.1090/surv/194/01 http://dx.doi.org/10.1090/surv/194/01