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
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