86
This does not imply that I think that the climate necessarily has a lowdimensional attractor. Rather, for some purposes, I believe the use of
relatively simple models can be helpful to illustrate basic processes. On
the other hand, as (singular vector) calculations in the body of the text
suggest, there are other circumstances where comparison with a turbulent
fluid may be more appropriate.
2 Predictability of the first kind
2.1 The forecast probability density function
As mentioned in the introduction, we can distinguish two basic types of
prediction. Following Lorenz (1975), predictions ofthe first kind are initial
value problems (e.g. medium-range weather forecasts with an atmosphere
model, or seasonal forecasts with a coupled ocean-atmosphere model). Predictability of the first kind is therefore concerned with the question of how
uncertainties in the initial state evolve during the forecast and limit its
skill.
The predictability of a system is strongly dependent on its stability
properties. If the system is particularly unstable, then any uncertainty in
the initial state that projects significantly onto one of these instabilities
will severely limit the skill of an initial-value forecast. Let us try to be
more precise. Suppose we represent the uncertainty in the initial condition
for a prediction of the first kind in terms of a PDF in some finite mdimensional phase space. For example, for each direction in phase space,
let us assume this PDF to be normally distributed about our best estimate
of the initial state. The standard deviation of this normal distribution will,
in general, vary with direction. However, we can define a metric on the
phase space (local to the initial condition) so that the standard deviations
are independent of direction (see section 2.7 for a more explicit description
of this). With respect to this metric, the PDF will now be isotropic, and
isopleths of the PDF will bound an m-dimensional ball (see Fig 1a).
A quantitative measure of predictability can be defined in relation to
the properties of the evolution of this initial PDF. In the early part of
the forecast, error growth is governed by linear dynamics. During this
period, initially spherical isopleths of the PDF will evolve to bound an
m-dimensional ellipsoidal volume (see Fig 1b). The major axis of the ellipsoid corresponds to a phase-space direction which defines the dominant
This does not imply that I think that the climate necessarily has a lowdimensional attractor. Rather, for some purposes, I believe the use of
relatively simple models can be helpful to illustrate basic processes. On
the other hand, as (singular vector) calculations in the body of the text
suggest, there are other circumstances where comparison with a turbulent
fluid may be more appropriate.
2 Predictability of the first kind
2.1 The forecast probability density function
As mentioned in the introduction, we can distinguish two basic types of
prediction. Following Lorenz (1975), predictions ofthe first kind are initial
value problems (e.g. medium-range weather forecasts with an atmosphere
model, or seasonal forecasts with a coupled ocean-atmosphere model). Predictability of the first kind is therefore concerned with the question of how
uncertainties in the initial state evolve during the forecast and limit its
skill.
The predictability of a system is strongly dependent on its stability
properties. If the system is particularly unstable, then any uncertainty in
the initial state that projects significantly onto one of these instabilities
will severely limit the skill of an initial-value forecast. Let us try to be
more precise. Suppose we represent the uncertainty in the initial condition
for a prediction of the first kind in terms of a PDF in some finite mdimensional phase space. For example, for each direction in phase space,
let us assume this PDF to be normally distributed about our best estimate
of the initial state. The standard deviation of this normal distribution will,
in general, vary with direction. However, we can define a metric on the
phase space (local to the initial condition) so that the standard deviations
are independent of direction (see section 2.7 for a more explicit description
of this). With respect to this metric, the PDF will now be isotropic, and
isopleths of the PDF will bound an m-dimensional ball (see Fig 1a).
A quantitative measure of predictability can be defined in relation to
the properties of the evolution of this initial PDF. In the early part of
the forecast, error growth is governed by linear dynamics. During this
period, initially spherical isopleths of the PDF will evolve to bound an
m-dimensional ellipsoidal volume (see Fig 1b). The major axis of the ellipsoid corresponds to a phase-space direction which defines the dominant