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

TL;DR

This paper investigates how energetic consistency in Fourier-Galerkin truncated models of 3D rotating convection affects heat transport, demonstrating that consistent models have bounded heat transfer while inconsistent ones can exhibit unbounded growth.

Contribution

It provides a criterion for mode selection ensuring energetic consistency and demonstrates the impact on heat transport bounds in Fourier-truncated models.

Findings

Consistent models obey energy relations and have bounded Nusselt number.

Inconsistent models can exhibit unbounded exponential growth in heat transport.

Numerical results show convergence of Nusselt number with increasing resolution for consistent 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.