The time step Dt cannot be freely chosen. Obviously, the
smaller the Dt, the longer it will take to get to the result.
However, there is a maximum time step, equal to c ÂDx,
where Dx is the chosen spatial resolution and c is the characteristic speed of propagation of the information from one
point to another. This is called the Courant, Friedrichs and
Lewy criterion (or CFL criterion), named after the mathematicians who formulated it. Thus, obtaining a simulation
with a fine spatial resolution takes a long time because it
requires not only calculations to be made on more points, but
also a smaller time step. A compromise must therefore be
made between spatial resolution and time taken to produce
the simulation. Paleoclimate studies which require long
simulations over several hundreds or even thousands of
years often used models of coarser resolution rather than
those used for climate forecasts into the next century.
However, nowadays, there are specific projects in which the
same models are used to compare the mechanisms of past
and future climate changes.
Within the atmospheric general circulation models, two
types of processes are often differentiated: dynamic processes and physical processes. The first type deals with the
evolution of the circulation and can only be calculated from
the three-dimensional spatial distribution of other variables,
such as temperature. It is through the use of the dynamic laws
[Eqs. (25.2)–(25.5)] that we can run the simulation forward,
time step by time step. The second type are calculated for
each vertical column separately for a given time step. These
are mainly radiation, clouds, precipitation and surface
exchanges. The distribution of the three-dimensional variables used at the dynamic stage of integration is obviously
closely dependent on the evaluation of the physical processes
for each vertical column. The dynamic and physical calculations are therefore carried out alternately, sometimes using
different time steps. Taking the example of the atmospheric
model included in the IPSL_CM6 model used in the
Sixth IPCC Assessment Report (publication planned for
2021) the time steps are from 430 s (high-resolution version
with 50 km and 79 vertical levels) to 2 min (low-resolution
version with 300 km and 39 vertical levels) for the dynamic
processes and 15 min for the physical ones.
In a general circulation model, we try to achieve the best
representation of both types of processes. Circulation is
calculated based on the basic laws of fluid mechanics,
expressed for the particular case of a thin atmospheric layer
surrounding a rotating planet. We have seen how these
equations can be simplified for this specific context, based
on the characteristic scales of global atmospheric circulation.
However, there is a second type of simplification inherent to
the construction of a model, and this is related to the physical
processes defined above. The fine details of these processes
are not always well understood. Moreover, their characteristic spatial scale is often much too small for them to be
explicitly represented in current models, whose spatial resolution is of the order of a hundred kilometers. Therefore,
the modeler will not attempt to represent the process in
detail, but rather to represent its impact on the atmospheric
characteristics at the resolution of the model. Thus, for
example, each cloud is not represented individually; rather
the impact of clouds on the radiative balance and on precipitations is formulated. This is called parametrization
of a subgrid process. These parameterizations represent
simplifications of reality in the sense that we have an
incomplete knowledge of it and the process itself is not
represented but rather its impact at the relevant scale. The
parameterizations, as well as the methods of discretization of
the equations used, along with the spatial and temporal
resolutions, constitute the main characteristics of a model.
Returning to the methods of discretization of the equations governing the evolution of the state of the atmosphere,
two approaches can be identified. The first is a description of
the atmosphere in a finite number of points, generally
organized into a three-dimensional grid. These are called
‘grid-point models’ or ‘grid-box models’, with the ‘box’
referring to the smallest unit volumes of the grid. The resolution of the model is defined by the size of this box, or by
the number of points used to describe the longitudes, latitudes, and the number of vertical levels. There are many
examples of grids, among which grids whose points are
regularly spaced in terms of longitude and latitude, and
grids whose points are regularly spaced in terms of longitude
and the cosine of latitude. In general, the vertical levels are
not evenly distributed. In particular, they need to be closer
together in the boundary layer of the atmosphere, the layer
closest to the surface.
A second type of approach involves using spherical
harmonics to describe the variations in the atmosphere on the
horizontal plane. The grid point method is retained for the
vertical dimension. These ‘spectral’ methods are particularly suited to the atmosphere, which forms a continuum on
the surface of a sphere. The calculations for this method are
faster, in particular due to the fact that the first and second
derivatives on the horizontal can be easily expressed for this
type of decomposition. The spectral models are well suited
for the representation of waves in the atmosphere with a
smaller number of degrees of freedom than in the grid point
models. The advantages of the spectral models are, however,
less significant for fine resolutions, as there are many calculations, especially for physical processes, which still have
to be carried out on a grid model. In general, the number of
points in the grid exceeds the number of degrees of freedom
in the spectral method so as to avoid problems with aliasing.
These grids therefore give the impression of a finer resolution than the real number of degrees of freedom of the
model. This is why the description of the resolution of these
models refers to the number and type of harmonics chosen.
25 Modeling and Paleoclimatology
327
smaller the Dt, the longer it will take to get to the result.
However, there is a maximum time step, equal to c ÂDx,
where Dx is the chosen spatial resolution and c is the characteristic speed of propagation of the information from one
point to another. This is called the Courant, Friedrichs and
Lewy criterion (or CFL criterion), named after the mathematicians who formulated it. Thus, obtaining a simulation
with a fine spatial resolution takes a long time because it
requires not only calculations to be made on more points, but
also a smaller time step. A compromise must therefore be
made between spatial resolution and time taken to produce
the simulation. Paleoclimate studies which require long
simulations over several hundreds or even thousands of
years often used models of coarser resolution rather than
those used for climate forecasts into the next century.
However, nowadays, there are specific projects in which the
same models are used to compare the mechanisms of past
and future climate changes.
Within the atmospheric general circulation models, two
types of processes are often differentiated: dynamic processes and physical processes. The first type deals with the
evolution of the circulation and can only be calculated from
the three-dimensional spatial distribution of other variables,
such as temperature. It is through the use of the dynamic laws
[Eqs. (25.2)–(25.5)] that we can run the simulation forward,
time step by time step. The second type are calculated for
each vertical column separately for a given time step. These
are mainly radiation, clouds, precipitation and surface
exchanges. The distribution of the three-dimensional variables used at the dynamic stage of integration is obviously
closely dependent on the evaluation of the physical processes
for each vertical column. The dynamic and physical calculations are therefore carried out alternately, sometimes using
different time steps. Taking the example of the atmospheric
model included in the IPSL_CM6 model used in the
Sixth IPCC Assessment Report (publication planned for
2021) the time steps are from 430 s (high-resolution version
with 50 km and 79 vertical levels) to 2 min (low-resolution
version with 300 km and 39 vertical levels) for the dynamic
processes and 15 min for the physical ones.
In a general circulation model, we try to achieve the best
representation of both types of processes. Circulation is
calculated based on the basic laws of fluid mechanics,
expressed for the particular case of a thin atmospheric layer
surrounding a rotating planet. We have seen how these
equations can be simplified for this specific context, based
on the characteristic scales of global atmospheric circulation.
However, there is a second type of simplification inherent to
the construction of a model, and this is related to the physical
processes defined above. The fine details of these processes
are not always well understood. Moreover, their characteristic spatial scale is often much too small for them to be
explicitly represented in current models, whose spatial resolution is of the order of a hundred kilometers. Therefore,
the modeler will not attempt to represent the process in
detail, but rather to represent its impact on the atmospheric
characteristics at the resolution of the model. Thus, for
example, each cloud is not represented individually; rather
the impact of clouds on the radiative balance and on precipitations is formulated. This is called parametrization
of a subgrid process. These parameterizations represent
simplifications of reality in the sense that we have an
incomplete knowledge of it and the process itself is not
represented but rather its impact at the relevant scale. The
parameterizations, as well as the methods of discretization of
the equations used, along with the spatial and temporal
resolutions, constitute the main characteristics of a model.
Returning to the methods of discretization of the equations governing the evolution of the state of the atmosphere,
two approaches can be identified. The first is a description of
the atmosphere in a finite number of points, generally
organized into a three-dimensional grid. These are called
‘grid-point models’ or ‘grid-box models’, with the ‘box’
referring to the smallest unit volumes of the grid. The resolution of the model is defined by the size of this box, or by
the number of points used to describe the longitudes, latitudes, and the number of vertical levels. There are many
examples of grids, among which grids whose points are
regularly spaced in terms of longitude and latitude, and
grids whose points are regularly spaced in terms of longitude
and the cosine of latitude. In general, the vertical levels are
not evenly distributed. In particular, they need to be closer
together in the boundary layer of the atmosphere, the layer
closest to the surface.
A second type of approach involves using spherical
harmonics to describe the variations in the atmosphere on the
horizontal plane. The grid point method is retained for the
vertical dimension. These ‘spectral’ methods are particularly suited to the atmosphere, which forms a continuum on
the surface of a sphere. The calculations for this method are
faster, in particular due to the fact that the first and second
derivatives on the horizontal can be easily expressed for this
type of decomposition. The spectral models are well suited
for the representation of waves in the atmosphere with a
smaller number of degrees of freedom than in the grid point
models. The advantages of the spectral models are, however,
less significant for fine resolutions, as there are many calculations, especially for physical processes, which still have
to be carried out on a grid model. In general, the number of
points in the grid exceeds the number of degrees of freedom
in the spectral method so as to avoid problems with aliasing.
These grids therefore give the impression of a finer resolution than the real number of degrees of freedom of the
model. This is why the description of the resolution of these
models refers to the number and type of harmonics chosen.
25 Modeling and Paleoclimatology
327