Title: Energetic consistency and heat transport in Fourier-Galerkin truncations of free slip 3D rotating convection Show Chinese title

Title: 能量一致性和傅里叶-伽辽金截断的自由滑移3D旋转对流中的热传输

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. Show Chinese abstract

Abstract: 本文考察了在三维Boussinesq-Coriolis（BC）方程的Fourier截断模型中能量一致性的影响，作为改进气候模型中对流过程现实性的案例研究。 我们以Nusselt数作为基准，定义为对流流动的平均垂直热传输。 推导出一组公式，给出了BC模型在任意有限模式选择上的常微分方程投影。 证明了如果且仅当满足关于垂直分辨率的模式选择标准时，投影的常微分方程模型服从与偏微分方程一致的能量关系。 还证明了能量关系意味着这些常微分方程存在紧致吸引子，从而推导出Nusselt数的界限。 相比之下，证明了一类广泛的能量不一致模型允许具有无界指数增长的解，排除了紧致吸引子的存在，并导致无穷大的Nusselt数。 另一方面，通过一个简单模型展示了某些能量不一致模型可以存在紧致吸引子。 上述公式已在MATLAB中实现，使用户能够通过选择所需的有限Fourier模式集来研究任何所需的Fourier截断模型。 所有代码均可在GitHub上获得。 进行了若干次关于Nusselt数的数值研究，评估了一致模型中Nusselt数随着空间分辨率增加的收敛性，并测量了更一般解中不一致性引起的失真效应。