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

TL;DR

This work develops and analyzes energetically consistent Fourier-Galerkin truncations of the 3D Boussinesq-Coriolis equations with free-slip boundaries, focusing on how balance laws influence long-time heat transport. By constructing explicit spectral bases and mode-selection criteria, it proves the existence of global attractors and uniform Nu bounds for consistent truncations, while showing energetically inconsistent truncations can exhibit runaway growth or unphysical Nu. The study distinguishes two inconsistency types and demonstrates, via numerical experiments and three hierarchies (\ell^{\infty}, \ell^{1}, HKC), that vertical resolution is a key factor in achieving reliable Nu predictions. The accompanying MATLAB code enables systematic exploration of truncations, balancing physical fidelity with computational feasibility for climate-relevant reduced models.

Abstract

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.

Figures (10)

• Figure 1: For $\mathsf{P} =10$, $\mathsf{k}_1 = \mathsf{k}_2 = 1$, a depiction of a fluid flow from the $\boldsymbol{\tau} = (1,5)$ model with $\mathsf{R} = 2000, \mathsf{S} = 0$, with the color indicating the temperature and the arrows indicating the velocity. Full video available at https://www.youtube.com/watch?v=qCHgRDqIOGM.
• Figure 2: A diagram depicting the dependencies and work flow structure of the code repository. Yellow boxes indicate folders and blue boxes indicate scripts or functions. Black lines indicate folder locations, red lines indicate code dependencies and blue lines indicate output locations.
• Figure 3: Convergence of the finite time Nusselt numbers as a function of time for the $8^{th}$ model in the $\ell^1$ hierarchy at several Rayleigh numbers.
• Figure 4: Nusselt comparison for the $\ell^{\infty}$ hierarchy (a) for $0 \leq \mathsf{R} \leq 5000, \mathsf{S} = 0$ and (b) for $\mathsf{R} = 5000$, $0 \leq \mathsf{S} \leq 400$.
• Figure 5: Nusselt comparison for the $\ell^{1}$ hierarchy, (a) for $0 \leq \mathsf{R} \leq 5000, \mathsf{S} = 0$ and (b) for $\mathsf{R} = 5000$, $0 \leq \mathsf{S} \leq 400$.

Theorems & Definitions (17)

• Proposition 2.1
• Lemma 2.2
• proof
• Proposition 3.2
• proof
• Proposition 3.4
• Lemma 3.5
• proof
• Remark 3.6
• Definition 4.1