Ensembles, Forecasts and Predictability
133
Because the effort ofthe unobserved variables is so important becomes crucial to
investigate and understand the dynamics oftheir evolution. There is a general feeling that this deviations must be small and, though we cannot have any hard proof,
the empirical and operational experience indicate that this is indeed the case. The
growth of the unobserved variables becomes then equivalent to the growth of small
errors. How small errors grow, interact and equilibrate nonlinearly has been much
the history of synoptic meteorology in the last 30 years. The following sections are
devoted to the major conceptual steps along this line.
8.2 Sensitivity to initial conditions
The previous discussion has indicated that the initial evolution of a system is
dominated by the dynamics of the instabilities. It is the number of instabilities triggered, their capability to grow rapidly and the physical character of the instability,
that lead to the development of a system rather than another. However, this initial
phase, the "linear era", is not bound to last. The rapidly growing disturbances outgrow the assumptions of linear theory and when they reach full amplitude they start
to interact nonlinearly with each other and with the "basic flow". At this stage they
can modify the basic state. The description in terms of "basic state" and "perturbations" is not useful anymore, the flow has evolved into a fully turbulent state, dominated by the statistics of the fluctuations. The nonlinear interactions in general
tend to equilibrate and attenuate the explosive growth of the perturbations, but the
boost caused by the initial selection of the set of instabilities will have already
hurled the system towards a particular evolution.
The difference in evolution can be highly sensitive to small errors in the initial
condition. Fig. 8.1 shows some examples obtained using the Lorenz (1963) system.
The Lorenz system is a simple system of three equations that can be obtained from
a drastic simplification of the equation for the motion of the atmosphere. They are
in fact a very basic model for the atmospheric circulation, assuming that there is
only the circumpolar vortex and two longitudinal waves. It was studying the properties of this system that Lorenz made the famous discovery of the sensitivity to
initial condition. The legend says that having to suspend a numerical integration for
lunch, he was too lazy after the break to reintroduce all the digits of the initial condition and to his amazement the evolution tumed out to be completely different.
The system is made up ofthree simple equations that interact nonlinearly with quadratic interactions with each other.
dx
- = - ~x+ yz
dt
dy
dt = - cry + cr z
dz
dt = - xy + py - z
(2)
133
Because the effort ofthe unobserved variables is so important becomes crucial to
investigate and understand the dynamics oftheir evolution. There is a general feeling that this deviations must be small and, though we cannot have any hard proof,
the empirical and operational experience indicate that this is indeed the case. The
growth of the unobserved variables becomes then equivalent to the growth of small
errors. How small errors grow, interact and equilibrate nonlinearly has been much
the history of synoptic meteorology in the last 30 years. The following sections are
devoted to the major conceptual steps along this line.
8.2 Sensitivity to initial conditions
The previous discussion has indicated that the initial evolution of a system is
dominated by the dynamics of the instabilities. It is the number of instabilities triggered, their capability to grow rapidly and the physical character of the instability,
that lead to the development of a system rather than another. However, this initial
phase, the "linear era", is not bound to last. The rapidly growing disturbances outgrow the assumptions of linear theory and when they reach full amplitude they start
to interact nonlinearly with each other and with the "basic flow". At this stage they
can modify the basic state. The description in terms of "basic state" and "perturbations" is not useful anymore, the flow has evolved into a fully turbulent state, dominated by the statistics of the fluctuations. The nonlinear interactions in general
tend to equilibrate and attenuate the explosive growth of the perturbations, but the
boost caused by the initial selection of the set of instabilities will have already
hurled the system towards a particular evolution.
The difference in evolution can be highly sensitive to small errors in the initial
condition. Fig. 8.1 shows some examples obtained using the Lorenz (1963) system.
The Lorenz system is a simple system of three equations that can be obtained from
a drastic simplification of the equation for the motion of the atmosphere. They are
in fact a very basic model for the atmospheric circulation, assuming that there is
only the circumpolar vortex and two longitudinal waves. It was studying the properties of this system that Lorenz made the famous discovery of the sensitivity to
initial condition. The legend says that having to suspend a numerical integration for
lunch, he was too lazy after the break to reintroduce all the digits of the initial condition and to his amazement the evolution tumed out to be completely different.
The system is made up ofthree simple equations that interact nonlinearly with quadratic interactions with each other.
dx
- = - ~x+ yz
dt
dy
dt = - cry + cr z
dz
dt = - xy + py - z
(2)