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1 Introduction
This paper is concerned with the predictability of the atmosphere and
oceans on timescales of days to decades. A variety of phenomena will be
discussed, from individual weather events through weather regimes and
El Nino, to decadal ocean-atmosphere fluctuations and climate change.
However, no matter what timescale or phenomenon is being considered,
we shall be studying processes which are believed to be fundamentally
chaotic.
A chaotic system can be defined as one whose evolution is sensitive
to initial conditions. However, this is not to say that the unpredictability associated with this sensitivity is only of importance for initial-value
problems. For example, determining the impact on climate of doubling
CO2 is not primarily an initial value problem. Nevertheless, the fact that
the climate is chaotic has fundamental implications for the predictability
of this type of question. Just as initial conditions for a weather forecast
are not perfectly accurate, and hence can only be specified completely in
terms of some probability distribution, so also the formulation of a climate
model (associated with the physical parametrisations in particular) is only
approximate, and again can only be specifed completely in terms of some
stochastic distribution. The instabilities that amplify uncertainties in the
initial state, may also amplify uncertainties in model formulation.
We start in section 2 with a discussion on predictability of initial value
problems. As suggested above, a fundamental quantity in this discussion
is the forecast probability density function (PDF). The evolution of this
PDF can be described in the first phase of the forecast by linearised dynamics. The semi-major axes of the PDF are given by the dominant singular
vectors of the linear evolution operator (using the so- called Mahalanobis
inner product). We relate these singular vectors to more familiar quantities associated with eigenmode growth on the one hand, and to Lyapunov
exponent growth on the other. We also discuss the relationship of singular
vectors and so-called breeding vectors.
In section 3, we apply the methodology developed in section 2 to study
predictability associated with a variety of phenomena on timescales ranging
from days to seasons. In particular, the singular vector instability of individual extratropical weather systems, and of the coupled ocean-atmosphere
El Nino/Southern Oscillation, is studied. For example, we demonstate the
endemic upscale energy cascade associated with extratropical predictabil-
1 Introduction
This paper is concerned with the predictability of the atmosphere and
oceans on timescales of days to decades. A variety of phenomena will be
discussed, from individual weather events through weather regimes and
El Nino, to decadal ocean-atmosphere fluctuations and climate change.
However, no matter what timescale or phenomenon is being considered,
we shall be studying processes which are believed to be fundamentally
chaotic.
A chaotic system can be defined as one whose evolution is sensitive
to initial conditions. However, this is not to say that the unpredictability associated with this sensitivity is only of importance for initial-value
problems. For example, determining the impact on climate of doubling
CO2 is not primarily an initial value problem. Nevertheless, the fact that
the climate is chaotic has fundamental implications for the predictability
of this type of question. Just as initial conditions for a weather forecast
are not perfectly accurate, and hence can only be specified completely in
terms of some probability distribution, so also the formulation of a climate
model (associated with the physical parametrisations in particular) is only
approximate, and again can only be specifed completely in terms of some
stochastic distribution. The instabilities that amplify uncertainties in the
initial state, may also amplify uncertainties in model formulation.
We start in section 2 with a discussion on predictability of initial value
problems. As suggested above, a fundamental quantity in this discussion
is the forecast probability density function (PDF). The evolution of this
PDF can be described in the first phase of the forecast by linearised dynamics. The semi-major axes of the PDF are given by the dominant singular
vectors of the linear evolution operator (using the so- called Mahalanobis
inner product). We relate these singular vectors to more familiar quantities associated with eigenmode growth on the one hand, and to Lyapunov
exponent growth on the other. We also discuss the relationship of singular
vectors and so-called breeding vectors.
In section 3, we apply the methodology developed in section 2 to study
predictability associated with a variety of phenomena on timescales ranging
from days to seasons. In particular, the singular vector instability of individual extratropical weather systems, and of the coupled ocean-atmosphere
El Nino/Southern Oscillation, is studied. For example, we demonstate the
endemic upscale energy cascade associated with extratropical predictabil-