Formulating an Energy Balance Model - Climate Change: Identification and Projections - page 51

If the final value is reduced to the unit, the response above defines the
step response (the response at a unit level) of the transmittance ( G . Having

arrived here via a GCM, its order is a priori very high. Fluctuations resulting
from the appearance of chaos are inherent to natural turbulent flows and
their numerical simulations. The underlying response obeys the general
behavior of heat propagation: very fast initial rise, and ultimate transitory
phases which return to their asymptote extremely slowly, typical of thermal
exchange processes.

)

Nevertheless, its modeling with an idealized (fractal) equation of heat
diffusion is not required (see Van Hateren, [VAN 13]), since it is no more
compliant than any other to the complex physical reality of the climatic
machine.

Ordinary tools for identification usually allow transmittances in the form
of a ratio of any two polynomials in s , which gives too much latitude to the
identified models: instabilities, resonant modes or non-minimum phases.
Moreover, optimizations required by identification techniques often appear
to be poorly conditioned in relation to denominator coefficients. We
therefore suggest defining G ( s ) as the sum of a priori fixed time constants T i ,
where only the positive multiplicative coefficients c i are free:

1 1 1

G ( s ) = c + c + " + c

1 2

n

1 + s T 1 + s T 1 + s T

1 2 n

Thanks to a sufficient amount of time constants T i , it is possible to
approximate any aperiodic system. In practice, three or four time constants
in geometric progression are enough to represent a system with multiple
scale systems. Therefore, the approximation given on Figure 4.2 corresponds
to the transmittance response , where:

8 . 3 × G ( s )

0.2 0.6 0.2

(where ) Gs () = + + c + c + c = 1

1 2 3

1 + s 1 + 10 s 1 + 100 s

The climate would be likely to keep a memory of its past through even
greater time constants, of around a millennium or more, and we had planned
to complete the series by a term in . However, it then becomes

practically impossible to distinguish the unknown equilibrium temperature
T E from the memory effect associated to this time constant. Here, we prefer

1 /( 1 + 1000 s )