The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy
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.
KeywordsWave-mean flow interaction Multi-scale asymptotics Slow manifold Reduced models
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- 1.Ariel, G., Engquist, B., Kim, S.J., Tsai, R.: Iterated averaging of three-scale oscillatory systems. Technical Report ICES REPORT 13-01, The Institute for Computational Engineering and Sciences, The University of Texas (2013)Google Scholar
- 3.Bogoliubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Non-linear Oscillations. Gordon and Breach, New York (1961)Google Scholar
- 11.Chen, Lin-Yuan, Goldenfeld, Nigel, Oono, Y.: The Renormalization group and singular perturbations: multiple scales, boundary layers and reductive perturbation theory. Phys. Rev. E54, 376–394 (1996)Google Scholar
- 17.Gander, M.J.: 50 years of time parallel time integration. In: Carraro, T., Geiger, M., Korkel, S., Rannacher, R. (eds.) Multiple Shooting and Time Domain Decomposition. Springer, Berlin (2015)Google Scholar