arXiv subjects

Roland Welter

Publications and source records attributed to Roland Welter.

4 recordsLinked to original sources

arXiv2025

Energetic consistency and heat transport in Fourier-Galerkin truncations of free slip 3D rotating convection

This paper examines the effects of energetic consistency in Fourier truncated models of the 3D Boussinesq-Coriolis (BC) equations as a case-study towards improving the realism of convective processes in climate models. As a benchmark we consider the Nusselt number, defined as the average vertical heat transport of a convective flow. A set of formulae are derived which give the ODE projection of the BC model onto any finite selection of modes. It is proven that projected ODE models obey energy relations consistent with the PDE if and only if a mode selection Criterion regarding the vertical resolution is satisfied. It is also proven that the energy relations imply the existence of a compact attractor for these ODE's, which then implies bounds on the Nusselt number. By contrast, it is proven that a broad class of energetically inconsistent models admit solutions with unbounded, exponential growth, precluding the existence of a compact attractor and giving an infinite Nusselt number. On the other hand, certain energetically inconsistent models can admit compact attractors as shown via a simple model. The above formulas are implemented in MATLAB, enabling a user to study any desired Fourier truncated model by selecting a desired finite set of Fourier modes. All code is made available on GitHub. Several numerical studies of the Nusselt number are conducted to assess the convergence of the Nusselt number with respect to increasing spatial resolution for consistent models and measure the distorting effects of inconsistency for more general solutions.

physics.flu-dyn↗

arXiv2025

Rotating Rayleigh-Benard convection: Attractors, bifurcations and heat transport via a Galerkin hierarchy

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.

math.AP↗

arXiv2023

Time averages and periodic attractors at high Rayleigh number for Lorenz-like models

Revisiting the Lorenz '63 equations in the regime of large of Rayleigh number, we study the occurrence of periodic solutions and quantify corresponding time averages of selected quantities. Perturbing from the integrable limit of infinite $ρ$, we provide a full proof of existence and stability of symmetric periodic orbits, which confirms previous partial results. Based on this, we expand time averages in terms of elliptic integrals with focus on the much studied average `transport', which is the mode reduced excess heat transport of the convection problem that gave rise to the Lorenz equations. We find a hysteresis loop between the periodic attractors and the non-zero equilibria of the Lorenz equations. These have been proven to maximize transport, and we show that the transport takes arbitrarily small values in the family of periodic attractors. In particular, when the non-zero equilibria are unstable, we quantify the difference between maximal and typically realized values of transport. We illustrate these results by numerical simulations and show how they transfer to various extended Lorenz models.

math.DS↗

arXiv2020

Asymptotic approximation of a modified compressible Navier-Stokes system

We study the long time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of the heat equation in terms of Hermite functions in which $n^{th}$ order approximations can be computed for solutions with $n^{th}$ order moments. We then obtain existence of solutions to the mcNS system and show that the approximation in terms of Hermite functions gives the leading order terms in the long-time asymptotics, and under certain assumptions can be evaluated explicitly.

math.AP↗