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point instability. In the Lorenz model (4.5), the outset (eg Thompson and
Stewart, 1991) of the saddle is symmetric with respect to the two Lorenz
regimes. In the modified model (3.2) it becomes biased towards one of the
regimes.
A statement of these results in general terms leads to the following
nonlinear paradigm (Palmer, 1993). The influence of a weak forcing fa on
a nonlinear system (such as the climate) is greatest in regions of phasespace where the dominant singular value is large. On the other hand, the
response of the system to fa is greatest in regions where the local PDF is
a maximum, and singular values are small. To first order, the response to
fa will be a change in the value of the PDF at the maxima.
This analysis is broadly consistent with recent studies of systematic errors in weather prediction and climate models. For example, as shown
by Molteni and Tibaldi (1990), medium-range systematic errors tend to
have large-scale equivalent barotropic structures which correspond reasonably well with the regime structures discussed in section 2. In order to
understand how such errors arise, ie what systematic forcing errors these
structures are most sensitive to, we need to estimate either a pseudo-inverse
or a sensitivity gradient of that error pattern over an ensemble of trajectory portions, that, together, cover the climate attractor. Such studies are
underway (see section 3.5 and section 6).
It should be noted that this paradigm is very hard to prove from first
principles; even the notion of existence and uniqueness of a PDF is hard
to establish. Nevertheless below we shall use the paradigm as a possible
means of interpreting climate variability.
4.3 Application to atmospheric forcing by tropical SST anomalies
Up to now I have imagined fa to represent a model uncertainty. However,
we can apply the paradigm more generally. Consider a prediction of the
second kind, mentioned earlier; the response of an atmospheric GCM to
imposed SSTs. Fig 20 a,b shows the 1000hPa temperature and 500hPa
height difference in the wintertime climate of (a recent version of) the
ECMWF model using firstly SSTs from the EI Nino winter 1986/87, and
secondly using SSTs for the winter 1988/89.
Applying the nonlinear paradigm,the forcing associated with the imposed SST changes will have increased certain atmospheric regimes, and
point instability. In the Lorenz model (4.5), the outset (eg Thompson and
Stewart, 1991) of the saddle is symmetric with respect to the two Lorenz
regimes. In the modified model (3.2) it becomes biased towards one of the
regimes.
A statement of these results in general terms leads to the following
nonlinear paradigm (Palmer, 1993). The influence of a weak forcing fa on
a nonlinear system (such as the climate) is greatest in regions of phasespace where the dominant singular value is large. On the other hand, the
response of the system to fa is greatest in regions where the local PDF is
a maximum, and singular values are small. To first order, the response to
fa will be a change in the value of the PDF at the maxima.
This analysis is broadly consistent with recent studies of systematic errors in weather prediction and climate models. For example, as shown
by Molteni and Tibaldi (1990), medium-range systematic errors tend to
have large-scale equivalent barotropic structures which correspond reasonably well with the regime structures discussed in section 2. In order to
understand how such errors arise, ie what systematic forcing errors these
structures are most sensitive to, we need to estimate either a pseudo-inverse
or a sensitivity gradient of that error pattern over an ensemble of trajectory portions, that, together, cover the climate attractor. Such studies are
underway (see section 3.5 and section 6).
It should be noted that this paradigm is very hard to prove from first
principles; even the notion of existence and uniqueness of a PDF is hard
to establish. Nevertheless below we shall use the paradigm as a possible
means of interpreting climate variability.
4.3 Application to atmospheric forcing by tropical SST anomalies
Up to now I have imagined fa to represent a model uncertainty. However,
we can apply the paradigm more generally. Consider a prediction of the
second kind, mentioned earlier; the response of an atmospheric GCM to
imposed SSTs. Fig 20 a,b shows the 1000hPa temperature and 500hPa
height difference in the wintertime climate of (a recent version of) the
ECMWF model using firstly SSTs from the EI Nino winter 1986/87, and
secondly using SSTs for the winter 1988/89.
Applying the nonlinear paradigm,the forcing associated with the imposed SST changes will have increased certain atmospheric regimes, and