Geoscience Reference
In-Depth Information
-
is the deviation of the global temperature with regard to the
x
=
T
−
T
G
E
equilibrium;
-
G
x
is the infrared flow radiated, supposing that it is proportional to
small temperature deviations.
λ
The energy stored by the planet is the product
I
G
.T
G
, and its variation is
proportional to the balance of flows, hence
dT
. The equilibrium
G
I
=
Φ
G
R
dt
d
temperature
T
E
,
which being constant by definition, gives
T
=
0
, and
dt
results from the equation for the model:
d
I
x
=
−
λ
x
+
α
u
G
G
dt
At any time
t
, we can consider that the variable
x
(
t
) constitutes the
state
of this system, and the expression above is known as a
state space equation
.
Here, it is a first order equation. The process defined thus is a linear time
invariant system (LTI).
To an equation of this kind, we can associate a transmittance (or transfer
function)
1
. This can be done by replacing the differential operator
d
/
with
the symbol
s
(the Laplace variable). The equation above then becomes
u
dt
, that can be processed like an ordinary algebraic equation.
I
s
x
=
−
λ
x
+
α
G
G
After factoring,
, it finally becomes:
(
I
s
+
λ
)
x
=
α
u
G
G
⎛
α
⎞
⎜
⎜
⎝
⎟
⎟
⎠
x
=
u
I
s
+
λ
G
G
α
+
The expression
is the transmittance associated to the
G
(
s
)
=
I
G
s
λ
G
initial equation of state. It is a rational fraction in
s
. It symbolically
represents the process of transmission from a causal input
u
to the output
x
.
1 A presentation of these concepts, relevant to this work, is given in Chapter 3 entitled “LTI
systems and transmittances” of the topic
Automatique Appliquée
by Philippe de Larminat
[DEL 09].
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