8.4 The Coupled Manifold 141
8.4 The Coupled Manifold
The analysis of the ensemble experiments with multiple realization showed that the
PRO method is very efficient at identifying the influence of one field on the other, but
this is a somewhat easy case. The presence of multiple cases with the same forcing,
the so-called ensemble, makes it possible to apply other techniques for identifying
the portion of variance that is linked to the external forcing itself. Methods like
separation of variance (Rowell 1997) or even the simple usage of the ensemble
mean can give a good indication of the characteristics of the forced response, even
in case they miss the detailed separation of the time series of the field themselves.
All these approaches however fail when we are confronted with the case of a single
realization.
There are several cases in which executing ensembles is impossible or forbidden
by the terms of the physical problem. In general this can happen when the two
fields that we want to examine are part of the same dynamically linked problem
and they cannot be separated in a “forcing” and a “response”. Coupled atmosphere–
ocean climate simulations are in this class with regard to our examples of marine
temperatures (SST) and atmospheric geopotential Z, since in this case the evolution
of the SST is not prescribed externally but is partially determined by the geopotential
itself. We then have to investigate to what extent the geopotential exerts control over
the SST.
Another notable example are observation records that are not reproducible. In
many cases experiments can be repeated and statistical ensembles can be con-
structed but in geophysical application observations cannot be reproduced in a strict
sense. The Earth atmosphere and ocean constantly evolve and our record of observa-
tions in time is a single realization of the Earth climate. The situation is very similar
to a numerical simulation performed with a coupled atmosphere–ocean model, also
in this case no parameter can be considered “external” and traditional separation of
variance methods fail.
However the formulation of the Forced Manifold is sufficiently general that it
can be used also in the case in which we have a single realization. Nothing in the
formulation we have used in (8.3)or(8.8) is linked to the availability of multiple
realizations. We can set up the problem also for single data sets Z and S. The only
victim is probably the name, since in this case we do not have “forcing” field and
“response” field and calling it “Forced Manifold” does not seem very appropriate.
We can still separate the field in sectors, but now we have a mutual effect of one field
on the other, then the name Coupled Manifold rather Forced Manifold seems more
appropriate. The Free Manifold, instead, maintains its meaning of variance that is
free from the influence of the field under examination.
The results are shown in Fig. 8.9. We present here the results obtained both under
the problem Z D AS and S D BZ. The top line shows the Coupled Manifold for
Z and the corresponding Free Manifold. We can see a familiar pattern of locations
where the variance of Z is highly influenced by the variations of the SST in the
region. The non-local nature of the analysis means that we can conclude only that
the various geographical locations in Z are globally influenced by the entire region