2 1. HEURISTICS

The other component determining the absorbed radiative power Rin is a rough

and diﬃcult to model averaged surface albedo of the earth, i.e. the proportion of the

solar power absorbed. It is supposed to be just (average) temperature dependent.

For temperatures below T , for which the surface water on earth is supposed to have

turned into ice, and the surface is thus constantly bright, the albedo is assumed to

be constantly equal to a, for temperatures above T , for which all ice has melted, and

the surface constantly brown, it is assumed to be given by a constant a a. For

temperatures between T and T , the two constant values a and a are simply linearly

interpolated in the most basic model. The rough albedo function has therefore the

ramp function shape depicted in Figure 1.1.

a

a

1

0 T T

T

a(T )

Figure 1.1. The albedo function a = a(T ).

To have a simple model of Rout, the earth is assumed to behave approximately

as a black body radiator, for which the emitted power is described by the Stefan–

Boltzmann law. It is proportional to the fourth power of the body’s temperature

and is given by γ T

4(t),

with a constant γ proportional to the Stefan constant.

Hence the simple energy balance equation with periodic input Q on which the

model is built is given by

(1.1) c

d

dt

T (t) = Q(t) 1 − a(T (t)) − γ T

(t)4,

where the constant c describes a global thermal inertia. According to (1.1), (qua-

si-) stationary states of average temperature should be given by the solutions of

the equation

dT (t)

dt

= 0. If the model is good, they should reasonably well interpret

glacial period and warm age temperatures. Graphically, they are given by the

intersections of the curves of absorbed and emitted radiative power, see Figures 1.2

and 1.3.

As we shall more carefully explain below, the lower (T1(t)) and upper (T3(t))

quasi-equilibria are stable, while the middle one (T2(t)) is unstable. The equilibrium

T1(t) should represent an ice age temperature, T3(t) a warm age, while T2(t) is

not observed over noticeably long periods. In their dependence on t they should

describe small fluctuations due to the variations in the solar constant. But here

one encounters a serious problem with this purely deterministic model. If the

fluctuation amplitude of Q is small, then we will observe the two disjoint branches

of stable solutions T1 and T3 (Figure 1.4).