Asymptotics of a Slow Manifold

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Abstract

Approximately invariant elliptic slow manifolds are constructed for the Lorenz-Krishnamurthy model of fast-slow interactions in the atmosphere. As is the case for many other two-time-scale systems, the various asymptotic procedures that may be used for this construction diverge, and there are no exactly invariant slow manifolds. Valuable information can however be gained by capturing the details of the divergence: this makes it possible to de. ne exponentially accurate slow manifolds, identify one of these as optimal, and predict the amplitude and phase of the fast oscillations that appear for trajectories started on it. We demonstrate this for the Lorenz-Krishnamurthy model by studying the slow manifolds obtained using a power-series expansion procedure. We develop two distinct methods to derive the leading-order asymptotics of the late coefficients in this expansion. Borel summation is then used to de. ne a unique slow manifold, regarded as optimal, which is piecewise analytic in the slow variables. This slow manifold is not analytic on a Stokes surface: when slow solutions cross this surface, they switch on exponentially small fast oscillations through a Stokes phenomenon. We show that the form of these oscillations can be recovered from the Borel summation. The approach that we develop for the Lorenz-Krishnamurthy model has a general applicability; we sketch how it generalizes to a broad class of two-time-scale systems.

Original languageEnglish
Pages (from-to)1163-1190
Number of pages28
JournalSiam Journal on Applied Dynamical Systems
Volume7
Issue number4
DOIs
Publication statusPublished - 2008

Keywords

  • slow manifold
  • exponential asymptotics
  • atmospheric waves
  • ORDINARY DIFFERENTIAL-EQUATIONS
  • PERTURBED HAMILTONIAN SYSTEM
  • LORENZ-KRISHNAMURTHY MODEL
  • GRAVITY WAVE GENERATION
  • EXPONENTIAL SMALLNESS
  • BALANCED MOTION
  • ROSSBY NUMBER
  • EXISTENCE
  • VARIABLES
  • DYNAMICS

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