Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus

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 state...

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Main Author: Cooper, Eric
Other Authors: Beck, Margaret A.
Language:en_US
Published: 2019
Subjects:
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Online Access:https://hdl.handle.net/2144/38793
id ndltd-bu.edu-oai-open.bu.edu-2144-38793
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spelling ndltd-bu.edu-oai-open.bu.edu-2144-387932019-12-15T03:05:35Z Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus Cooper, Eric Beck, Margaret A. Mathematics Bar Dipole Navier-Stokes Qusi-stationary Vorticity 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. 2019-12-13T19:08:12Z 2019-12-13T19:08:12Z 2019 2019-11-12T20:01:52Z Thesis/Dissertation https://hdl.handle.net/2144/38793 en_US
collection NDLTD
language en_US
sources NDLTD
topic Mathematics
Bar
Dipole
Navier-Stokes
Qusi-stationary
Vorticity
spellingShingle Mathematics
Bar
Dipole
Navier-Stokes
Qusi-stationary
Vorticity
Cooper, Eric
Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus
description 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.
author2 Beck, Margaret A.
author_facet Beck, Margaret A.
Cooper, Eric
author Cooper, Eric
author_sort Cooper, Eric
title Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus
title_short Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus
title_full Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus
title_fullStr Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus
title_full_unstemmed Selection of quasi-stationary states In the 2D Navier-Stokes equation on the torus
title_sort selection of quasi-stationary states in the 2d navier-stokes equation on the torus
publishDate 2019
url https://hdl.handle.net/2144/38793
work_keys_str_mv AT coopereric selectionofquasistationarystatesinthe2dnavierstokesequationonthetorus
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