The two dimensional incompressible Navier-Stokes equation on D δ := [0, 2πδ] × [0, 2π] with δ ≈ 1... more The two dimensional incompressible Navier-Stokes equation on D δ := [0, 2πδ] × [0, 2π] with δ ≈ 1, periodic boundary conditions, and viscosity 0 < ν 1 is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale O(e −νt) and have been shown to play a key role in its long-time evolution. Of particular interest is the role that δ 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 be selected if δ = 1, while the dipole will be selected if δ = 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 δ 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 δ ≈ 1, there is an initial time period in which the high modes decay at the rapid rate O(e −t/ν), while the low modes evolve at the slower O(e −νt) rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.
The two dimensional incompressible Navier-Stokes equation on D δ := [0, 2πδ] × [0, 2π] with δ ≈ 1... more The two dimensional incompressible Navier-Stokes equation on D δ := [0, 2πδ] × [0, 2π] with δ ≈ 1, periodic boundary conditions, and viscosity 0 < ν 1 is considered. Bars and dipoles, two explicitly given quasi-stationary states of the system, evolve on the time scale O(e −νt) and have been shown to play a key role in its long-time evolution. Of particular interest is the role that δ 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 be selected if δ = 1, while the dipole will be selected if δ = 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 δ 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 δ ≈ 1, there is an initial time period in which the high modes decay at the rapid rate O(e −t/ν), while the low modes evolve at the slower O(e −νt) rate. The results for the ODE model are proven using energy estimates and invariant manifolds and further supported by formal asymptotic expansions and numerics.
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Papers by Eric Cooper