Introduction - Noise-induced Phenomena in the Environmental Sciences - page 4

where φ ( t ) is the state variable characterizing the state of the system, t is time, and

) is a deterministic algebraic function of the state variable. A spatially extended
version of Eq. ( 1.1 ) would include also a term representing the effects on the dynamics
of the values of

f ( φ

in the neighboring sites; spatially extended systems are considered
in the second half of this topic, but the spatial coupling is neglected here to speculate
more easily on the role of noise in the dynamics of

φ

φ

.
The environment from which dynamical system ( 1.1 ) is extracted is generally
much bigger than the dynamical system itself and is also called external environment
to stress the fact that it is external to the dynamical system. Because the external
environment is often too “large,” complex, and also partially unknown to be modeled
deterministically, its action on the dynamical system is represented as a stochastic pro-
cess. Therefore we account for the randomness of environmental conditions through
a stochastic forcing, which is modeled as noise ξ ( t ). Thus the dynamics of the state

variable read

d d t =

f ( φ ) + g ( φ ) ξ ( t ) , (1.2)

where the (linear or nonlinear) algebraic function g ( ) accounts for the possibility

that the effect of the random forcing on the system is modulated by the state of the
system itself. The noise is additive when g (

φ

φ ) = const, whereas it is multiplicative

otherwise.
Because the dynamical system is much smaller than the external environment, it is
generally unable to affect its environmental drivers. However, in some dynamics the
impact of the system on its environment can be important. In these cases a feedback
exists between the state of the system and environmental conditions. This topic dis-
cusses some examples of feedbacks relevant to the biogeosciences. Feedbacks with
random environmental drivers are typically expressed either through the multiplicative
function g (

)].
One of the most crucial issues in the representation of these stochastic dynamics
arises from the modeling of the random forcing. To address this point, we need to
consider two time scales underlying the dynamics, namely, the time scale

φ ) or through a state dependency in the stochastic forcing [i.e., ξ ( t ,φ

τ s of the

deterministic dynamical system and the time scale τ n of the random forcing. The

former describes the response time of the deterministic system after a displacement
from its steady state(s) τ s expresses how slowly or quickly the

systemwill converge to its stable state(s). For example,

φ s . In other words,

τ s can be expressed as inversely

f (

proportional to the first-order derivative of f ( φ ) calculated in φ s , τ s 1 / | φ s ) | .

τ n of the random forcing is a function of its autocorrelation, which
expresses the interrelations existing within the noise signal, i.e., how the values of
random forcing at different times depend on their temporal separation (a formal
definition is provided in Chapter 2). The time scale

The time scale

n can be expressed as the integral
of the autocorrelation function, which represents the (linear) temporal memory of

τ