accurately referred to as ‘climate simulation’ where the
primary objective is to achieve a maximum of realism, at the
expense of an in-depth understanding of how the system
operates. In the most sophisticated climate models, there are
several million degrees of freedom thus making it difficult to
understand and analyze the detail of the sequence of processes involved in the simulations carried out by these
models. A typical strategy is therefore to multiply the
number of simulations, as described later, by carrying out
sensitivity experiments. Conversely, much simpler models,
which may produce less realistic results, can provide insight
into the root causes of certain mechanisms that underlie the
phenomena being represented. If the aim of modeling is
summarized in the maxim ‘understand so as to better simulate’, it is obvious that a whole spectrum of models of
varying complexities is necessary in order to tackle the
different aspects of a problem.
Before further describing climate modeling in general and
the problems encountered in paleoclimatology in particular,
we will revisit Eq. (25.1) in more detail in order to explain
some concepts that are widely used either implicitly or
explicitly. The vector X(t) which describes the whole system
is also called the prognostic variable set of the model. This
refers to all quantities X i (t) in the system (25.1) possessing
an equation of evolution. Moreover, it is often useful to
include additional variables to represent the physical quantities used in the equations, quantities which depend directly
on the prognostic variables X i (t) without recourse to an
associated evolution equation. For example, the quantity y
(t) = X 1 (t) + X 2 (t) is deduced from the quantities X i (t) and so
the evolution equation for the derivative dy(t)/dt is redundant
in the system of Eq. (25.1). These additional variables are
called the diagnostic variables of the model, because they
are mainly used to provide a better understanding of the
model in terms of the customary physical values. Thus,
typically, in an atmospheric circulation model, the only
prognostic variables at each point of the grid of the model
are temperature, humidity, and wind velocity on the horizontal plane, with evolution equations representing the
conservation of energy and water (transport equations) and
the conservation of momentum (i.e. the Navier-Stokes
equation) on the horizontal plane. All other values (vertical
velocities, energy fluxes, precipitation, clouds etc.) are
deduced more or less directly. These are merely diagnostic
or secondary variables, but they are nevertheless very useful
at all the stages of modeling, from the design of the model to
the analysis of the results. These diagnostic variables, often
more numerous than the prognostic variables, do not mean
additional degrees of freedom.
Moreover, the notation of the system of Eq. (25.1) always
involves values which are established at the outset, deeming
these to be either physical values external to the model under
consideration, or more or less well defined constants. These
values are the model parameters. When these parameters are
spatialized, i.e., dependent on their geographical location,
they are then considered to be boundary conditions. When
the parameters are time dependent, they may be referred to as
model forcings. For example, for an atmospheric model, the
surface temperature of the oceans is a boundary condition
(and also a forcing, if it depends on time), and the atmospheric concentration of CO 2 is a parameter (and also a
forcing, if it depends on time). For a coupled oceanatmosphere model, this same sea surface temperature is a
prognostic variable while pCO 2 remains a parameter. For a
climate-carbon coupled model, pCO 2 is explicitly calculated
and thus becomes a model variable as well. It is often interesting to explore how the model outcomes change when the
values of certain parameters change. These are called sensitivity experiments, because the objective is not to perform
realistic climate simulations, but to see how sensitive the
model is to certain parameters (examples of experiments of
this type are shown in Sections “General Circulation Models,
Complex Models of the Earth System” and “Examples of
Long-Term Simulations and Studies of Sensitivity to Forcings”). When this type of study systematically includes many
parameter values and many parameters, this is called the
exploration of the parameter space of the model, and is
sometimes imprecisely referred to as the exploration of ‘the
phase space’ (although, strictly speaking, it is the space of the
prognostic variables and not of the parameters).
Dynamic Systems
It is also important to briefly outline the general results that
can be obtained from an equation system such as system
25.1. First, the choice of functions f(X, t) must be restricted
to cases likely to have a physical meaning. Instead of starting
from a single initial condition X 0 , we start with a set of
proximate initial conditions, which fill an initial volume V 0
in the phase space. For the ‘physical’ cases, the second
principle of thermodynamics implies that, at time t, the
corresponding states X(t) fill a volume V(t) which decreases
with time (in the case of dissipative systems) or remains
constant (in the case of conservative systems). While the
conservative systems retain the memory of the initial condition, since the volume V(t) remains constant, this information is gradually lost in dissipative systems. Indeed, in
general, this volume tends towards zero as time t approaches
infinity. Climate (like many other physical systems) is a
dissipative system. Figure 25.1 gives examples of typical
behaviors of a system for two different initial conditions.
As dissipative systems gradually forget their initial condition, this may turn out to be positive: because this initial
information is in any case lost after a certain time, this
information is not relevant to the long-term behavior of the
25 Modeling and Paleoclimatology
321
primary objective is to achieve a maximum of realism, at the
expense of an in-depth understanding of how the system
operates. In the most sophisticated climate models, there are
several million degrees of freedom thus making it difficult to
understand and analyze the detail of the sequence of processes involved in the simulations carried out by these
models. A typical strategy is therefore to multiply the
number of simulations, as described later, by carrying out
sensitivity experiments. Conversely, much simpler models,
which may produce less realistic results, can provide insight
into the root causes of certain mechanisms that underlie the
phenomena being represented. If the aim of modeling is
summarized in the maxim ‘understand so as to better simulate’, it is obvious that a whole spectrum of models of
varying complexities is necessary in order to tackle the
different aspects of a problem.
Before further describing climate modeling in general and
the problems encountered in paleoclimatology in particular,
we will revisit Eq. (25.1) in more detail in order to explain
some concepts that are widely used either implicitly or
explicitly. The vector X(t) which describes the whole system
is also called the prognostic variable set of the model. This
refers to all quantities X i (t) in the system (25.1) possessing
an equation of evolution. Moreover, it is often useful to
include additional variables to represent the physical quantities used in the equations, quantities which depend directly
on the prognostic variables X i (t) without recourse to an
associated evolution equation. For example, the quantity y
(t) = X 1 (t) + X 2 (t) is deduced from the quantities X i (t) and so
the evolution equation for the derivative dy(t)/dt is redundant
in the system of Eq. (25.1). These additional variables are
called the diagnostic variables of the model, because they
are mainly used to provide a better understanding of the
model in terms of the customary physical values. Thus,
typically, in an atmospheric circulation model, the only
prognostic variables at each point of the grid of the model
are temperature, humidity, and wind velocity on the horizontal plane, with evolution equations representing the
conservation of energy and water (transport equations) and
the conservation of momentum (i.e. the Navier-Stokes
equation) on the horizontal plane. All other values (vertical
velocities, energy fluxes, precipitation, clouds etc.) are
deduced more or less directly. These are merely diagnostic
or secondary variables, but they are nevertheless very useful
at all the stages of modeling, from the design of the model to
the analysis of the results. These diagnostic variables, often
more numerous than the prognostic variables, do not mean
additional degrees of freedom.
Moreover, the notation of the system of Eq. (25.1) always
involves values which are established at the outset, deeming
these to be either physical values external to the model under
consideration, or more or less well defined constants. These
values are the model parameters. When these parameters are
spatialized, i.e., dependent on their geographical location,
they are then considered to be boundary conditions. When
the parameters are time dependent, they may be referred to as
model forcings. For example, for an atmospheric model, the
surface temperature of the oceans is a boundary condition
(and also a forcing, if it depends on time), and the atmospheric concentration of CO 2 is a parameter (and also a
forcing, if it depends on time). For a coupled oceanatmosphere model, this same sea surface temperature is a
prognostic variable while pCO 2 remains a parameter. For a
climate-carbon coupled model, pCO 2 is explicitly calculated
and thus becomes a model variable as well. It is often interesting to explore how the model outcomes change when the
values of certain parameters change. These are called sensitivity experiments, because the objective is not to perform
realistic climate simulations, but to see how sensitive the
model is to certain parameters (examples of experiments of
this type are shown in Sections “General Circulation Models,
Complex Models of the Earth System” and “Examples of
Long-Term Simulations and Studies of Sensitivity to Forcings”). When this type of study systematically includes many
parameter values and many parameters, this is called the
exploration of the parameter space of the model, and is
sometimes imprecisely referred to as the exploration of ‘the
phase space’ (although, strictly speaking, it is the space of the
prognostic variables and not of the parameters).
Dynamic Systems
It is also important to briefly outline the general results that
can be obtained from an equation system such as system
25.1. First, the choice of functions f(X, t) must be restricted
to cases likely to have a physical meaning. Instead of starting
from a single initial condition X 0 , we start with a set of
proximate initial conditions, which fill an initial volume V 0
in the phase space. For the ‘physical’ cases, the second
principle of thermodynamics implies that, at time t, the
corresponding states X(t) fill a volume V(t) which decreases
with time (in the case of dissipative systems) or remains
constant (in the case of conservative systems). While the
conservative systems retain the memory of the initial condition, since the volume V(t) remains constant, this information is gradually lost in dissipative systems. Indeed, in
general, this volume tends towards zero as time t approaches
infinity. Climate (like many other physical systems) is a
dissipative system. Figure 25.1 gives examples of typical
behaviors of a system for two different initial conditions.
As dissipative systems gradually forget their initial condition, this may turn out to be positive: because this initial
information is in any case lost after a certain time, this
information is not relevant to the long-term behavior of the
25 Modeling and Paleoclimatology
321