Rotating Rayleigh-Benard convection: Attractors, bifurcations and heat transport via a Galerkin hierarchy
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912403951714304 |
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| author | Welter, Roland |
| author_facet | Welter, Roland |
| contents | Motivated by the need for energetically consistent climate models, the Boussinessq-Coriolis (BC) equations are studied with a focus on the averaged vertical heat transport, ie the Nusselt number. A set of formulae are derived by which arbitrary Fourier truncations of the BC model can be explicitly generated, and Criteria are given which precisely guarantee that such truncated models obey energy relations consistent with the PDE. The Howard-Krishnamurti-Coriolis (HKC) hierarchy of such energetically consistent ODE models is then implemented in MATLAB, with code available on GitHub. Several theoretical results are proven to support a numerical analysis. Well-posedness and convergence of the HKC hierarchy toward the BC model are proven, as well as the existence of an attractor for the BC model. Since the rate of convergence is unknown, explicit upper and lower bounds on the attractor dimension are proven so as to provide guidance for the required spatial resolution for an accurate approximation of the Nusselt number. Finally, a series of numerical studies are performed using MATLAB, which investigate the required spatial resolution and indicate the presence of multiple stable values of the Nusselt number, setting the stage for an energetically consistent analysis of convective heat transport. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rotating Rayleigh-Benard convection: Attractors, bifurcations and heat transport via a Galerkin hierarchy Welter, Roland Analysis of PDEs Fluid Dynamics Motivated by the need for energetically consistent climate models, the Boussinessq-Coriolis (BC) equations are studied with a focus on the averaged vertical heat transport, ie the Nusselt number. A set of formulae are derived by which arbitrary Fourier truncations of the BC model can be explicitly generated, and Criteria are given which precisely guarantee that such truncated models obey energy relations consistent with the PDE. The Howard-Krishnamurti-Coriolis (HKC) hierarchy of such energetically consistent ODE models is then implemented in MATLAB, with code available on GitHub. Several theoretical results are proven to support a numerical analysis. Well-posedness and convergence of the HKC hierarchy toward the BC model are proven, as well as the existence of an attractor for the BC model. Since the rate of convergence is unknown, explicit upper and lower bounds on the attractor dimension are proven so as to provide guidance for the required spatial resolution for an accurate approximation of the Nusselt number. Finally, a series of numerical studies are performed using MATLAB, which investigate the required spatial resolution and indicate the presence of multiple stable values of the Nusselt number, setting the stage for an energetically consistent analysis of convective heat transport. |
| title | Rotating Rayleigh-Benard convection: Attractors, bifurcations and heat transport via a Galerkin hierarchy |
| topic | Analysis of PDEs Fluid Dynamics |
| url | https://arxiv.org/abs/2402.14724 |