In recent years, stochastic parametrizations have been ubiquitous in modelling uncertainty in fluid dynamics models. One source of model uncertainty comes from the coarse graining of the fine-scale data and is in common usage in computational simulations at coarser scales. In this paper, we look at two such stochastic parametrizations: the Stochastic Advection by Lie Transport (SALT) parametrization introduced by Holm and the Location Uncertainty (LU) parametrization introduced by M\'emin. Whilst both parametrizations are available for full-scale models, we study their reduced order versions obtained by projecting them on a complex vector Fourier mode triad of eigenfunctions of the curl. Remarkably, these two parametrizations lead to the same reduced order model, which we term the helicity-preserving stochastic triad (HST). This reduced order model is then compared with an alternative model which preserves the energy of the system, and which is termed the energy preserving stochastic triad (EST). These low-dimensional models are ideal benchmark models for testing new Data Assimilation algorithms: they are easy to implement, exhibit diverse behaviours depending on the choice of the coefficients and come with natural physical properties such as the conservation of energy and helicity.

翻译：近年来，随机参数化在流体动力学模型的不确定性建模中已广泛应用。模型不确定性的一个来源来自细尺度数据的粗粒化处理，并且在较粗尺度的计算模拟中普遍使用。本文研究了两种随机参数化方案：Holm提出的Lie传输随机平流（SALT）参数化和Mémin提出的位置不确定性（LU）参数化。尽管这两种参数化方案可用于全尺度模型，但我们通过将其投影到旋度的复向量Fourier模态三体上，研究了它们的降阶版本。值得注意的是，这两种参数化方案导出了相同的降阶模型，我们称之为螺旋度保持随机三体（HST）。随后，将该模型与另一种保持系统能量的替代模型（称为能量保持随机三体（EST））进行了比较。这些低维模型是测试新数据同化算法的理想基准模型：它们易于实现，根据系数选择表现出多样化的行为，并具备能量和螺旋度守恒等自然物理特性。