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current of the Canaries, the predominance of the Trade Winds, the West African upwelling and
the suspended dust (Schutz, 1979). These are the true elements of the Canary Islands climate.
1. Complex Systems and Mathematics.
A complex system is one where many different spatial and temporal scales are needed
for an accurate description of its behavior. Climate is indeed a complex system: Temporal
scales range from the infinitesimal time neded for a phase change to the thousands of years of
residence time of sea water in the deep Ocean basins. On the other hand, spatial scales vary
from the few !lm of the incoming short wave Solar radiation to the 10 7 m scale of the Rossby
and Kelvin waves involved in the El Nino-Southern Oscillation phenomenon. It is a wise
advice to distinguish three groups of variables according to their temporal scales (Morantine
and Watts, 1994; Watts etal. 1994):
a) The "natural" scales of the phenomenon are those of the macroscopic variables selected by
the observer for a description of the system status. Climate can be described, as was pointed
out above, by the same variables employed in models of the atmosphere, supplemented with
those describing tsea-air interaction. As a result, a climate model is the coupling of an
atmospheric GCM and a marine one, each one with its ordinary natural scales (Figure 3). Ice
caps are included in the marine part of the model.
b) The "slow" variables are those that can be considered as constants when compared with the
state variables. Their evolution can not be observed with the resolution accorded to the space
variables. As a rule, they represent boundary conditions or physical constants and can be
considered as parameters. In climate studies most lithospheric topographic constraints belong
to this type, as well as the geological timescale. The detection of "truly slow" variables is one
of the main concerns of the climate study (Grassberger,1986; Lorenz, 1986; Temam, 1995)).
Other physical barriers, like the tropopause, the thermocline, etc., may belong or not to this
class depending on the problem considered.
c) "Fast" variables are those whose fluctuation with respect to the state variables is detected as
noise by the observer. Variables belonging to this class could be eliminated from the system if
their influence is proved to be irrelevant. This not a simple problem: It is related with the
ergodicity or not of the weather-climate system and is an open field of study (Fraedrich and
Ziehmann-Schlumbohm,1994).
The above classification allows the modeller to concentrate on the selection and
behavior of a number of variables notably smaller than that of all interesting ones, as was
already pointed in the Introduction.
Scale considerations are most relevant when translating a mathematical model into a
computational one. The discretization step introduces a non physical, spurious parameter, i.e.
the mesh size of the computational grid (to be precise, a vector mesh size concept must be
provided) (Adams et aI., 1993). Therefore the output of a computational climate model can
yield physically senseless results because of the unresolved features of subgrid scale. A well
known example are climate change experiments: These can show extra heat sources not
consistent with the Physics when run under the hypothesis that C<=>2 concentration is doubled.
This problem is an interesting one within Bifurcation Theory.
The Canary Islands climate system, which must be studied at "regional" scales, is a
typical case of nearly-subgrid system. The usual climate models employ a rectangular grid with
a linear mesh around 300 km, in such a way that four adjacent rectangles would cover the
entire Archipelago, and any island lies completely within one single rectangle. Vertical
discretizations with 10 to 20 atmospheric levels are not fine enough to resolve many of the
small scale features which determine the Islands climate.
current of the Canaries, the predominance of the Trade Winds, the West African upwelling and
the suspended dust (Schutz, 1979). These are the true elements of the Canary Islands climate.
1. Complex Systems and Mathematics.
A complex system is one where many different spatial and temporal scales are needed
for an accurate description of its behavior. Climate is indeed a complex system: Temporal
scales range from the infinitesimal time neded for a phase change to the thousands of years of
residence time of sea water in the deep Ocean basins. On the other hand, spatial scales vary
from the few !lm of the incoming short wave Solar radiation to the 10 7 m scale of the Rossby
and Kelvin waves involved in the El Nino-Southern Oscillation phenomenon. It is a wise
advice to distinguish three groups of variables according to their temporal scales (Morantine
and Watts, 1994; Watts etal. 1994):
a) The "natural" scales of the phenomenon are those of the macroscopic variables selected by
the observer for a description of the system status. Climate can be described, as was pointed
out above, by the same variables employed in models of the atmosphere, supplemented with
those describing tsea-air interaction. As a result, a climate model is the coupling of an
atmospheric GCM and a marine one, each one with its ordinary natural scales (Figure 3). Ice
caps are included in the marine part of the model.
b) The "slow" variables are those that can be considered as constants when compared with the
state variables. Their evolution can not be observed with the resolution accorded to the space
variables. As a rule, they represent boundary conditions or physical constants and can be
considered as parameters. In climate studies most lithospheric topographic constraints belong
to this type, as well as the geological timescale. The detection of "truly slow" variables is one
of the main concerns of the climate study (Grassberger,1986; Lorenz, 1986; Temam, 1995)).
Other physical barriers, like the tropopause, the thermocline, etc., may belong or not to this
class depending on the problem considered.
c) "Fast" variables are those whose fluctuation with respect to the state variables is detected as
noise by the observer. Variables belonging to this class could be eliminated from the system if
their influence is proved to be irrelevant. This not a simple problem: It is related with the
ergodicity or not of the weather-climate system and is an open field of study (Fraedrich and
Ziehmann-Schlumbohm,1994).
The above classification allows the modeller to concentrate on the selection and
behavior of a number of variables notably smaller than that of all interesting ones, as was
already pointed in the Introduction.
Scale considerations are most relevant when translating a mathematical model into a
computational one. The discretization step introduces a non physical, spurious parameter, i.e.
the mesh size of the computational grid (to be precise, a vector mesh size concept must be
provided) (Adams et aI., 1993). Therefore the output of a computational climate model can
yield physically senseless results because of the unresolved features of subgrid scale. A well
known example are climate change experiments: These can show extra heat sources not
consistent with the Physics when run under the hypothesis that C<=>2 concentration is doubled.
This problem is an interesting one within Bifurcation Theory.
The Canary Islands climate system, which must be studied at "regional" scales, is a
typical case of nearly-subgrid system. The usual climate models employ a rectangular grid with
a linear mesh around 300 km, in such a way that four adjacent rectangles would cover the
entire Archipelago, and any island lies completely within one single rectangle. Vertical
discretizations with 10 to 20 atmospheric levels are not fine enough to resolve many of the
small scale features which determine the Islands climate.