Invariant Manifolds, Quasi-Geostrophy and Initialization

R. Vautard Laboratoire de Météorologie Dynamique, Ecole Normale Supérieure, 75231 Paris Cedex 05, France

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Abstract

A critical examination of the foundations of the slow manifold concept is presented. We study the behavior of the simple Lorenz truncated primitive equation model and show that free gravity modes exist for large forcing. For small forcing, the flow is dominated by geostrophic motion and quasi-invariant manifolds are found. However, we present arguments against the existence of a strictly invariant manifold. We show that the bounded derivative conditions are generally not satisfied, even when an invariant manifold exists; smooth functions always possess fast transients. We present a new initialization method based on the analytic expansion of local invariant manifolds and compare it with Machenhauer's method on a simple example.

Abstract

A critical examination of the foundations of the slow manifold concept is presented. We study the behavior of the simple Lorenz truncated primitive equation model and show that free gravity modes exist for large forcing. For small forcing, the flow is dominated by geostrophic motion and quasi-invariant manifolds are found. However, we present arguments against the existence of a strictly invariant manifold. We show that the bounded derivative conditions are generally not satisfied, even when an invariant manifold exists; smooth functions always possess fast transients. We present a new initialization method based on the analytic expansion of local invariant manifolds and compare it with Machenhauer's method on a simple example.

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