Ensembles, Forecasts and Predictability
135
behaviour of the system. The sensitivity can be more or less pronounced, even disappear, depending on the values of the parameters.
If the nonlinear terms are eliminated from the Lorenz system the dramatic
spread disappears. The bottom panel of Fig. 8.1 shows the case in which the nonlinear interaction in the Lorenz system are artificially tumed off. After the same
period of time the same initial condition as in the top panel are all bunched together
in a small area rather than being dispersed all over the possible range ofvalues. We
have reduced the Lorenz system to a linear system, that does not exhibit a strong
sensitivity to initial condition. Linear systems are the simplest example of equations that do not exhibit this dramatic sensitivity, but there are also nonlinear systems that keep neighboring initial conditions close together also at later times.
8.3 The traditional paradigm
In simple terms, the atmospheric circulation appears to be composed by a circumpolar vortex, characterized by the appearance of transient deviations from the
zonal symmetry that can have time scales of a few days or months. The first physically and mathematically consistent interpretation of the atmospheric variability
was offered by Chamey (1947) in a famous paper in which he sketched the interpretative paradigm that has guided science for the following 40 years.
In Chamey's vis ion the atmospheric motions could be seen as made up ofthe circumpolar current (the basic state) plus deviations that would draw their energy
from the basic state itself. The deviations would be created continuously by the
intrinsic instability of the basic state and they would grow linearly in the initial
low-amplitude phase, but they would then interact with each other and ultimately
with the basic state itself in a complex process of nonlinear equilibration. The fimdamental process was however the initial growth of the perturbation that would set
the stage, fixing the time scales and the energetic ofthe whole process.
The approach is now well described in textbooks (Pedlosky, 1979), but we can
synthesize here in slightly different terms for the sake of our discussion. The equation of motion can be linearized around a time independent basic state,
dx
- +Lx = O
dt
(3)
where the operator L is representing the linearized equation of motion and the vector x is representing the state variable. In finite terms, the equation (2) is a vector
equation for the time evolution of the vector x, representing, for instance the gridpoint values in a finite-difference model or the spectral harmonics in a spectral
model, the finite representation of the operator L is usually a matrix, A, and the
entire equation can be written as
dx
- +Ax = O
dt
(4)
135
behaviour of the system. The sensitivity can be more or less pronounced, even disappear, depending on the values of the parameters.
If the nonlinear terms are eliminated from the Lorenz system the dramatic
spread disappears. The bottom panel of Fig. 8.1 shows the case in which the nonlinear interaction in the Lorenz system are artificially tumed off. After the same
period of time the same initial condition as in the top panel are all bunched together
in a small area rather than being dispersed all over the possible range ofvalues. We
have reduced the Lorenz system to a linear system, that does not exhibit a strong
sensitivity to initial condition. Linear systems are the simplest example of equations that do not exhibit this dramatic sensitivity, but there are also nonlinear systems that keep neighboring initial conditions close together also at later times.
8.3 The traditional paradigm
In simple terms, the atmospheric circulation appears to be composed by a circumpolar vortex, characterized by the appearance of transient deviations from the
zonal symmetry that can have time scales of a few days or months. The first physically and mathematically consistent interpretation of the atmospheric variability
was offered by Chamey (1947) in a famous paper in which he sketched the interpretative paradigm that has guided science for the following 40 years.
In Chamey's vis ion the atmospheric motions could be seen as made up ofthe circumpolar current (the basic state) plus deviations that would draw their energy
from the basic state itself. The deviations would be created continuously by the
intrinsic instability of the basic state and they would grow linearly in the initial
low-amplitude phase, but they would then interact with each other and ultimately
with the basic state itself in a complex process of nonlinear equilibration. The fimdamental process was however the initial growth of the perturbation that would set
the stage, fixing the time scales and the energetic ofthe whole process.
The approach is now well described in textbooks (Pedlosky, 1979), but we can
synthesize here in slightly different terms for the sake of our discussion. The equation of motion can be linearized around a time independent basic state,
dx
- +Lx = O
dt
(3)
where the operator L is representing the linearized equation of motion and the vector x is representing the state variable. In finite terms, the equation (2) is a vector
equation for the time evolution of the vector x, representing, for instance the gridpoint values in a finite-difference model or the spectral harmonics in a spectral
model, the finite representation of the operator L is usually a matrix, A, and the
entire equation can be written as
dx
- +Ax = O
dt
(4)