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nonlinearity prevent the breeding vector from converging to a Lyapunov
vector.
It can be asked whether the dominant singular vectors and breeding
vectors correlate with one another. In general they do not. At initial
time the singular vectors are dominated by sub-synoptic scales, whilst the
breeding vectors are dominated by synoptic scales (for stationary basic
states they would be given by the synoptic-scale eigenmodes). As such
their spatial correlation is close to zero. At optimisation time, the singular
vectors evolve towards synoptic scales and the correlation with the dominant breeding vectors increases, though only to values of about 0.2 (in the
T21QG model).
Using these results, and those from the last section, I believe that it
is possible to make the following conclusions about the relationship between the breeding vectors and the PDF of analysis error. Firstly, since
the spectrum of breeding vectors is extremely fiat, there is no particular
reason why the analysis error should project more onto the leading breeding vectors. Of course, if one had enough breeding vectors (and if they
were suitably orthogonalised) then any perturbation, including the analysis error, should project into the space spanned by these vectors. However,
it appears from the results above that the number of such vectors would
have to be a significant fraction of the phase space dimension.
In fact, in my opinion, even if the spectrum of breeding vectors was
in fact much steeper, it is questionable whether the analysis error would
ever rotate into the direction of a leading breeding vector. The reason for
this is to do with the role of observations in the analysis cycle. As discussed eg by Hollingsworth (1987) and Daley (1991), operational analyses
blend observations with a first-guess field in a scale-dependent manner. On
large scales, the observations carry more weight than the first guess field,
whilst on small scales the first guess carries more weight than the observations. Therefore, in my opinion, to represent the role of observations,
one would require, within each breeding cycle, the breeding vector to undergo a phase-space rotation (and not just a renormalisation as is actually
done). This phase-space rotation would continually 'frustrate' the breeding
vector's attempt to rotate towards some dominant Lyapunov direction.
The NMC assumption that analysis errors rotate into a selection of preferred directions can be compared with the ECWMF philosophy in which
( cf section 2g) it is assumed that there are, in fact, no preferred phasespace directions for the analysis error (i. e. with respect to a suitable inner
nonlinearity prevent the breeding vector from converging to a Lyapunov
vector.
It can be asked whether the dominant singular vectors and breeding
vectors correlate with one another. In general they do not. At initial
time the singular vectors are dominated by sub-synoptic scales, whilst the
breeding vectors are dominated by synoptic scales (for stationary basic
states they would be given by the synoptic-scale eigenmodes). As such
their spatial correlation is close to zero. At optimisation time, the singular
vectors evolve towards synoptic scales and the correlation with the dominant breeding vectors increases, though only to values of about 0.2 (in the
T21QG model).
Using these results, and those from the last section, I believe that it
is possible to make the following conclusions about the relationship between the breeding vectors and the PDF of analysis error. Firstly, since
the spectrum of breeding vectors is extremely fiat, there is no particular
reason why the analysis error should project more onto the leading breeding vectors. Of course, if one had enough breeding vectors (and if they
were suitably orthogonalised) then any perturbation, including the analysis error, should project into the space spanned by these vectors. However,
it appears from the results above that the number of such vectors would
have to be a significant fraction of the phase space dimension.
In fact, in my opinion, even if the spectrum of breeding vectors was
in fact much steeper, it is questionable whether the analysis error would
ever rotate into the direction of a leading breeding vector. The reason for
this is to do with the role of observations in the analysis cycle. As discussed eg by Hollingsworth (1987) and Daley (1991), operational analyses
blend observations with a first-guess field in a scale-dependent manner. On
large scales, the observations carry more weight than the first guess field,
whilst on small scales the first guess carries more weight than the observations. Therefore, in my opinion, to represent the role of observations,
one would require, within each breeding cycle, the breeding vector to undergo a phase-space rotation (and not just a renormalisation as is actually
done). This phase-space rotation would continually 'frustrate' the breeding
vector's attempt to rotate towards some dominant Lyapunov direction.
The NMC assumption that analysis errors rotate into a selection of preferred directions can be compared with the ECWMF philosophy in which
( cf section 2g) it is assumed that there are, in fact, no preferred phasespace directions for the analysis error (i. e. with respect to a suitable inner