Theoretical and Computational Fluid Dynamics

, Volume 32, Issue 6, pp 713–732 | Cite as

The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy

  • Jared P. WhiteheadEmail author
  • Terry Haut
  • Beth A. Wingate
Original Article


Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.


Wave-mean flow interaction Multi-scale asymptotics Slow manifold Reduced models 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Jared P. Whitehead
    • 1
    Email author
  • Terry Haut
    • 2
  • Beth A. Wingate
    • 3
  1. 1.Department of MathematicsBrigham Young UniversityProvoUSA
  2. 2.Center for Applied Scientific ComputingLawrence Livermore National LaboratoryLivermoreUSA
  3. 3.Department of MathematicsUniversity of ExeterExeterUK

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