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John A. Knox, Donald W. McCann, and Paul D. Williams

regions of zero or negative potential vorticity. In summary, CAT forecasting remains an unsolved problem ( McCann 2001 ) for spontaneous imbalance theory to address. Williams et al. (2005) employed laboratory experiments and a quasigeostrophic model’s results to explore generation mechanisms of gravity waves in a rotating, two-layer vertically sheared flow. To diagnose wave activity, the authors calculated five dynamical indicators, several of which were originally devised as CAT forecasting indices

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J. Vanneste

generally small. A large time-scale separation leads to a weak coupling between the slow and fast components of the dynamics. For the atmosphere and oceans, which are forced mostly at low frequency, this has the consequence that the level of IGW activity often remains low; as a result, the dynamics can be represented accurately by balanced models, that is, models that filter out IGWs completely. This has been formalized by introducing the notion of slow manifolds, which are submanifolds of the state

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Gregory J. Hakim

velocity field, and replacing ensemble propagation altogether by using fixed covariance matrices, such as time averages. The latter case may reduce probabilistic dynamics to classical balance models (e.g., quasigeostrophy) for appropriately chosen fixed covariance matrices. Evaluating the merit of these approximations, and the relationship of the probabilistic approach to existing balance models, are challenges for future research. 4. Implications In general, the inversion operator (13) is not square

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