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...
Main Author: | |
---|---|
Other Authors: | |
Language: | en_US |
Published: |
2019
|
Subjects: | |
Online Access: | https://hdl.handle.net/2144/38793 |
id |
ndltd-bu.edu-oai-open.bu.edu-2144-38793 |
---|---|
record_format |
oai_dc |
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 |
_version_ |
1719303428109238272 |