Projectability

Brian Hoskins

University of Reading and The Grantham Institute for Climate Change, Imperial College London, UK

Our scientific interaction over the past 40 years has been stimulating, Tim, and Jackie and I have very much enjoyed our social interaction with Gill and you.

The talk you asked me to give was on the 1988 PhD thesis of my student, Zhang Zuojun. The PhD topic I had suggested to him was stimulated by the 1983 Simmons et al paper on barotropic normal modes of the zonally asymmetric DJF mean flow, and also the Farrell papers on the initial value problem. These led to the idea that we should delve deeper into the nature of the instability and the implications for the initial value problem.

The big insight that we gained was on the initial value problem in the situation in which the operator is not self-adjoint. In this case the eigen-vectors are not orthogonal, and their structures can be quite similar. The adjoint of the first eigen-vector is orthogonal to all the other eigen-vectors, and the optimum projection onto the first eigen-vector is obtained by choosing initial conditions in the direction of its adjoint. We defined the projectability as the inverse cosine of the angle between the first eigen-vector and its adjoint. For cases in which at least one other eigen-vector has a similar structure and the angle between them is small, the projectability becomes large and the structure of the optimum initial condition is very different from that of either eigen-vector.

You had found that the 500 hPa streamfunction anomaly that correlated with ECMF forecast skill in the medium range. When you looked at the linear stability of the high and low skill cases, you were upset to find that the first eigen-vector of the high skill case had a slightly larger growth rate than that for the low skill case. When you heard about our work, you suggested that Zuojun look at your two flows. He found that the projectability was some four times larger for the low skill case. This, rather than the slightly different growth rate, was the crucial factor for the first two months.

Zuojun disappeared to the computer industry and we never wrote up this work, but I am glad that, in your words, this helped you know that you were on the right track!