SINGULAR VECTORS, METRICS, AND ADAPTIVE OBSERVATIONS

Singular vectors of the linearized equations of motion have been used to study the instability properties of the atmosphere-ocean system and its related predictability. A third use of these singular vectors is proposed here: as part of a strategy to target adaptive observations t o ''sensitive'' parts of the atmosphere. Such observations could be ma de using unmanned aircraft, though calculations in this paper are moti vated by the upstream component of the Fronts and Atlantic Storm-Track Experiment. Oceanic applications are also discussed. In defining this strategy, it is shown that there is, in principle, no freedom in the choice of inner product or metric for the singular vector calculation. However, the correct metric is dependent on the purpose for making th e targeted observations (to study precursor developments or to improve forecast initial conditions). It is argued that for predictability st udies, where both the dynamical instability properties of the system a nd the specification of the operational observing network and associat ed data assimilation system are important, the appropriate metric will differ from that appropriate to a pure geophysical fluid dynamics (GF D)problem. Based on two different sets of calculations, it is argued t hat for predictability studies (but not for GFD studies), a first-orde r approximation to the appropriate metric can be based on perturbation energy. The role of observations in data assimilation procedures (con straining large scales more than small scales) is fundamental in under standing reasons for the requirement for different metrics for the two classes of problems. An index-based tensor approach is used to make e xplicit the role of the metric. The strategy for using singular vector s to target adaptive observations is discussed in the context of other possible approaches, specifically, based on breeding vectors, potenti al vorticity diagnosis, and sensitivity vectors. The basic premises un derlying the use of breeding and singular vectors are discussed. A com parison of the growth rates of breeding and singular vectors is made u sing a T21 quasigeostrophic model. Singular vectors and subjective pot ential vorticity (PV) diagnosis are compared for a particular case stu dy. The areas of sensitivity indicated by the two methods only partial ly agree. Reasons for disagreement hinge around the fact that subjecti ve PV diagnosis emphasizes Lagrangian advection, whereas singular vect or analysis emphasizes wave propagation. For the latter, areas of sens itivity may be associated with regions of weak PV gradient, for exampl e, mid to lower troposphere. Amplification of singular vectors propaga ting from regions of weak PV gradient to regions of strong PV gradient is discussed in terms of pseudomomentum conservation. Evidence is sho wn that analysis error may be as large in the lower midtroposphere as in the upper troposphere.