Selection of quasi-stationary states in the Navier-Stokes equation on the torus

The two dimensional incompressible Navier-Stokes equation on D-delta := [0, 2 pi delta] x [0, 2 pi] with delta approximate to 1, periodic boundary conditions, and viscosity 0 < v << 1 is considered. Bars and dipoles, two explicitly given quasi-stationary stales of the system, evolve on the time scale O(e(-vt)) and have been shown to play a key role in its long-time evolution. Of particular interest is the role that delta plays in selecting which of these two states is observed. Recent numerical studies suggest that, after a transient period of rapid decay of the high Fourier modes, the bar state will he selected if delta not equal 1, while the dipole will be selected if delta = 1. Our results support this claim and seek to mathematically formalize it. We consider the system in Fourier space, project it onto a center manifold consisting of the lowest eight Fourier modes, and use this as a model to study the selection of bars and dipoles. It is shown for this ODE model that the value of delta controls the behavior of the asymptotic ratio of the low modes, thus determining the likelihood of observing a bar state or dipole after an initial transient period. Moreover, in our model, for all delta approximate to 1, there is an initial time period in which the high modes decay at the rapid rate O(e(-t/v)), while the low modes evolve at the slower O(e(-vt)) rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.