The onset of zonal modes in two-dimensional Rayleigh–Bénard convection
<p>We study the stability of steady convection rolls in two-dimensional Rayleigh–Bénard convection with free-slip boundaries and horizontal periodicity over 12 orders of magnitude in the Prandtl number <span data-mathjax-type="texmath"><span tab...
| Главные авторы: | , , |
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| Формат: | Journal article |
| Язык: | English |
| Опубликовано: |
Cambridge University Press
2022
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| _version_ | 1826308446537908224 |
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| author | Winchester, P Howell, PD Dallas, V |
| author_facet | Winchester, P Howell, PD Dallas, V |
| author_sort | Winchester, P |
| collection | OXFORD |
| description | <p>We study the stability of steady convection rolls in two-dimensional Rayleigh–Bénard convection with free-slip boundaries and horizontal periodicity over 12 orders of magnitude in the Prandtl number <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>6</mn></mrow></msup><mo>&#x2264;</mo><mi>P</mi><mi>r</mi><mo>&#x2264;</mo><msup><mn>10</mn><mn>6</mn></msup><mo stretchy="false">)</mo></math>">(10−6≤Pr≤106)(10−6≤Pr≤106)</span></span> and 6 orders of magnitude in the Rayleigh number <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>8</mn><msup><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C0;</mi></mrow><mn>4</mn></msup><mo>&lt;</mo><mi>R</mi><mi>a</mi><mo>&#x2264;</mo><msup><mn>10</mn><mn>8</mn></msup><mo stretchy="false">)</mo></math>">(8π4<Ra≤108)(8π4<Ra≤108)</span></span>. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x0393;</mi><mo>=</mo><mn>2</mn></math>">Γ=2Γ=2</span></span>, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>P</mi><mi>r</mi><mo>,</mo><mi>R</mi><mi>a</mi><mo stretchy="false">)</mo></math>">(Pr,Ra)(Pr,Ra)</span></span> plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>r</mi><mo stretchy="false">&#x2192;</mo><mn>0</mn></math>">Pr→0Pr→0</span></span> and find that the steady convection rolls become unstable almost instantaneously, eventually leading to nonlinear relaxation osculations and bursts, which we can explain with a weakly nonlinear analysis. In the complementary large-<span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>r</mi></math>">PrPr</span></span> limit, we observe that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.</p> |
| first_indexed | 2024-03-07T07:19:37Z |
| format | Journal article |
| id | oxford-uuid:4865d4ea-a07c-4161-ad4f-4afbbfb218e9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:19:37Z |
| publishDate | 2022 |
| publisher | Cambridge University Press |
| record_format | dspace |
| spelling | oxford-uuid:4865d4ea-a07c-4161-ad4f-4afbbfb218e92022-09-23T10:29:15ZThe onset of zonal modes in two-dimensional Rayleigh–Bénard convectionJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4865d4ea-a07c-4161-ad4f-4afbbfb218e9EnglishSymplectic ElementsCambridge University Press2022Winchester, PHowell, PDDallas, V<p>We study the stability of steady convection rolls in two-dimensional Rayleigh–Bénard convection with free-slip boundaries and horizontal periodicity over 12 orders of magnitude in the Prandtl number <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>6</mn></mrow></msup><mo>&#x2264;</mo><mi>P</mi><mi>r</mi><mo>&#x2264;</mo><msup><mn>10</mn><mn>6</mn></msup><mo stretchy="false">)</mo></math>">(10−6≤Pr≤106)(10−6≤Pr≤106)</span></span> and 6 orders of magnitude in the Rayleigh number <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>8</mn><msup><mrow class="MJX-TeXAtom-ORD"><mi>&#x03C0;</mi></mrow><mn>4</mn></msup><mo>&lt;</mo><mi>R</mi><mi>a</mi><mo>&#x2264;</mo><msup><mn>10</mn><mn>8</mn></msup><mo stretchy="false">)</mo></math>">(8π4<Ra≤108)(8π4<Ra≤108)</span></span>. The analysis is facilitated by partitioning our modal expansion into so-called even and odd modes. With aspect ratio <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x0393;</mi><mo>=</mo><mn>2</mn></math>">Γ=2Γ=2</span></span>, we observe that zonal modes (with horizontal wavenumber equal to zero) can emerge only once the steady convection roll state consisting of even modes only becomes unstable to odd perturbations. We determine the stability boundary in the <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>P</mi><mi>r</mi><mo>,</mo><mi>R</mi><mi>a</mi><mo stretchy="false">)</mo></math>">(Pr,Ra)(Pr,Ra)</span></span> plane and observe remarkably intricate features corresponding to qualitative changes in the solution, as well as three regions where the steady convection rolls lose and subsequently regain stability as the Rayleigh number is increased. We study the asymptotic limit <span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>r</mi><mo stretchy="false">&#x2192;</mo><mn>0</mn></math>">Pr→0Pr→0</span></span> and find that the steady convection rolls become unstable almost instantaneously, eventually leading to nonlinear relaxation osculations and bursts, which we can explain with a weakly nonlinear analysis. In the complementary large-<span data-mathjax-type="texmath"><span tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>r</mi></math>">PrPr</span></span> limit, we observe that the zonal modes at the instability switch off abruptly at a large, but finite, Prandtl number.</p> |
| spellingShingle | Winchester, P Howell, PD Dallas, V The onset of zonal modes in two-dimensional Rayleigh–Bénard convection |
| title | The onset of zonal modes in two-dimensional Rayleigh–Bénard convection |
| title_full | The onset of zonal modes in two-dimensional Rayleigh–Bénard convection |
| title_fullStr | The onset of zonal modes in two-dimensional Rayleigh–Bénard convection |
| title_full_unstemmed | The onset of zonal modes in two-dimensional Rayleigh–Bénard convection |
| title_short | The onset of zonal modes in two-dimensional Rayleigh–Bénard convection |
| title_sort | onset of zonal modes in two dimensional rayleigh benard convection |
| work_keys_str_mv | AT winchesterp theonsetofzonalmodesintwodimensionalrayleighbenardconvection AT howellpd theonsetofzonalmodesintwodimensionalrayleighbenardconvection AT dallasv theonsetofzonalmodesintwodimensionalrayleighbenardconvection AT winchesterp onsetofzonalmodesintwodimensionalrayleighbenardconvection AT howellpd onsetofzonalmodesintwodimensionalrayleighbenardconvection AT dallasv onsetofzonalmodesintwodimensionalrayleighbenardconvection |