Ensembles, Forecasts and Predictability
139
delicate problem, but choosing a norm that corresponds to the energy of the perturbation usually gives good results.
The loss of the self-adjointness also produces strange effects on the finite time
behaviour. The normal modes dominate the asymptotic behaviour of the evolution
of a system like eq. (l), but they are not able to dominate the evolution uniformly
from the start. This effect is illustrated in Fig. 8.2, which depicts synthetically the
differences between long time and short time behaviour. The top two lines describe
the eigenvalue distribution (middle) and the distribution ofthe singular values (top)
for the propagator and for the matrix A. The examples shown are constructed with
a distribution of singular values, typical of barotropic problems (solid line in the
top row), that generates a peculiar distribution of singular values ofthe propagator
(dotted lines) (see Navarra, 1987). The most amplifying mode correspond to the
largest value and in general amplifying modes have values larger than one, whereas
decaying modes have factor less than one. The top lines show the distribution of
eigenvalues in the complex plane. The problems have been stabilized by adding
suitable dissipation and so all the real parts are confined to the negative semiplane.
By all accounts of the normal mode paradigm this flow is stable (asymptotically
stable), since at long times the perturbations are exponentially damped.
The example shows clearly what happens with non self-adjoint operators. At
finite times the evolution may be amplifying in the general sense we have defined
above, but the long time behaviour is controlled by the eigenvalue spectrurn. In the
case of the atmosphere it is the characteristic at finite times that are more relevant,
since at longer time, non-linear interactions will take over and so it is not at alI
clear that the modes will have time to evolve linearly into a asymptotic state. In fact
the identification of normal modes in models and/or observations has always been
difficult (GalI, 1976). Normal mode theory provided realistic estimation of time
and space scales and a useful interpretative tool, but the identification of more specific prediction ofthe theory has been more controversial.
8.5 Ensembles
The sensitivity to initial conditions and the uncertainties in the development of
initial errors, including both unobserved variables and real errors, lead to the consequence that a single integration may not really be indicative of the evolution of
the atmosphere. It is likely that a single forecast may take a wrong development
simply because it did not contain the right combination of uncertainties. In other
words a single integration is not an adequate sampling of the possible values of the
uncertainties around the nominal initial condition.
It becomes therefore necessary to sample more widely around the initial condition, usually perturbing the initial state in one way or another. The simplest
approach would include just random perturbations of the initial conditions, but
other techniques have been developed that tend to use "optimal" perturbations, in
other words perturbations that excite the errors that grow fast. The finite time insta-
139
delicate problem, but choosing a norm that corresponds to the energy of the perturbation usually gives good results.
The loss of the self-adjointness also produces strange effects on the finite time
behaviour. The normal modes dominate the asymptotic behaviour of the evolution
of a system like eq. (l), but they are not able to dominate the evolution uniformly
from the start. This effect is illustrated in Fig. 8.2, which depicts synthetically the
differences between long time and short time behaviour. The top two lines describe
the eigenvalue distribution (middle) and the distribution ofthe singular values (top)
for the propagator and for the matrix A. The examples shown are constructed with
a distribution of singular values, typical of barotropic problems (solid line in the
top row), that generates a peculiar distribution of singular values ofthe propagator
(dotted lines) (see Navarra, 1987). The most amplifying mode correspond to the
largest value and in general amplifying modes have values larger than one, whereas
decaying modes have factor less than one. The top lines show the distribution of
eigenvalues in the complex plane. The problems have been stabilized by adding
suitable dissipation and so all the real parts are confined to the negative semiplane.
By all accounts of the normal mode paradigm this flow is stable (asymptotically
stable), since at long times the perturbations are exponentially damped.
The example shows clearly what happens with non self-adjoint operators. At
finite times the evolution may be amplifying in the general sense we have defined
above, but the long time behaviour is controlled by the eigenvalue spectrurn. In the
case of the atmosphere it is the characteristic at finite times that are more relevant,
since at longer time, non-linear interactions will take over and so it is not at alI
clear that the modes will have time to evolve linearly into a asymptotic state. In fact
the identification of normal modes in models and/or observations has always been
difficult (GalI, 1976). Normal mode theory provided realistic estimation of time
and space scales and a useful interpretative tool, but the identification of more specific prediction ofthe theory has been more controversial.
8.5 Ensembles
The sensitivity to initial conditions and the uncertainties in the development of
initial errors, including both unobserved variables and real errors, lead to the consequence that a single integration may not really be indicative of the evolution of
the atmosphere. It is likely that a single forecast may take a wrong development
simply because it did not contain the right combination of uncertainties. In other
words a single integration is not an adequate sampling of the possible values of the
uncertainties around the nominal initial condition.
It becomes therefore necessary to sample more widely around the initial condition, usually perturbing the initial state in one way or another. The simplest
approach would include just random perturbations of the initial conditions, but
other techniques have been developed that tend to use "optimal" perturbations, in
other words perturbations that excite the errors that grow fast. The finite time insta-