We consider the two-dimensional Navier-Stokes equation on the (possibly) asymmetric torus, D_δ = [0,2𝜋δ] × [0,2𝜋], both with and without stochastic forcing. Absent external force, the vorticity is known to reach a rest state of zero. There exists at least three so called "quasi-stationary states" which attract nearby solutions at rates faster than the global decay rate. The system evolves toward one of these three qualitatively different transient states for long times while the system overall tends toward the final rest state. We develop a finite-dimensional model of the associated deterministic vorticity equation to show how the selection of the dominant quasi-stationary state depends on the aspect ratio of the domain, given by δ. This is followed by formal analysis of the problem as a perturbation from the symmetric domain. Once the selection mechanism for the deterministic model is characterized, stochastic forcing is added to the reduced system. Numerical analysis shows the dominant quasi-stationary state is consistent with what is seen in the deterministic setting. Finally through multiscale averaging methods, the leading order dynamics of the stochastically forced finite-dimensional model for δ close to one is studied. As a result we formally obtain leading order asymptotics of statistics of interest, including the selection mechanism.
| Identifer | oai:union.ndltd.org:bu.edu/oai:open.bu.edu:2144/38793 |
| Date | 12 November 2019 |
| Creators | Cooper, Eric |
| Contributors | Beck, Margaret A. |
| Source Sets | Boston University |
| Language | en_US |
| Detected Language | English |
| Type | Thesis/Dissertation |
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