Computing Lagrangian means in multi-timescale fluid flows

时间平均是分析多时间尺度流体流动的重要工具之一，在自然界和工业中普遍存在。时间平均的一个突出用途是将流量分解为快、慢两个部分，以了解与每个时间尺度相关的不同现象。例如，在地球物理流动中，波动动力学与快的部分有关，而慢的动力学可以归结为几种力之间的平衡（地转平衡和静水平衡）。时间平均的另一个应用是过滤掉在数值模拟或观测中没有完全捕捉到的快速变化，从而从剩余的缓慢动态中做出有意义的推断。同样，测量或模拟中固有的噪声和误差也可以通过平均去除。对于流体，时间平均可以通过两种不同的方式进行。最直接的方法是对固定空间点的流量变量的时间序列求平均值，以获得所谓的欧拉平均值。另一种方法是沿粒子轨迹平均流动变量，而不是固定位置，这就得到了拉格朗日平均值。越来越多的研究表明，拉格朗日平均法比欧拉平均法有几个关键的优势。例如，它消除了多普勒频移，这种频移掩盖了背景流和波之间时间尺度的分离。然而，拉格朗日平均的广泛采用受到计算复杂性的阻碍。为了计算数值模型中的拉格朗日平均值，通常在每个时间步长的粒子位置上使用插值速度来跟踪粒子。这是一个计算挑战，需要额外的内存空间（考虑到变量的时间序列存储在所有网格点），并且不适合有效的计算并行化。在这个项目中，我们开发了一种数值方法来规避这些困难。这种新方法是基于部分均值的进化而不是粒子跟踪。部分均值可以看作是比总平均周期更短的间隔内的均值。在这种方法中，我们计算拉格朗日均值作为一组偏微分方程（PDEs）的解，这些偏微分方程描述了这些偏均值的演变。这种范式可能是计算拉格朗日均值的一个突破，因为这些偏微分方程可以以各种方式离散并实时求解（即与动态控制方程同时求解）。因此，与粒子跟踪相比，它们不需要存储任何时间序列，并且大大减少了内存占用。PI提出了该数值技术的基本思想。本提案的目标是1)扩展这种新方法的理论基础，2)开发和实施一套数值方案来解决相关的偏微分方程，3)将它们应用于三维流体模型和实验室数据。

Time averaging is one of the most essential tools in analysing fluid flows with multiple time scales, which are ubiquitous in nature and industries. A prominent use of time averaging is the flow decomposition into fast and slow parts to understand different phenomena associated with each time scale. For instance, in geophysical flows the wave dynamics is associated with the fast part, and the slow dynamics can be reduced to a balance between a few forces (geostrophic and hydrostatic balance). Another application of time averaging is to filter out the fast variations that are not fully captured in numerical simulations or observations to make meaningful inferences from the remaining slow dynamics. Similarly, the noise and error inherent in measurements or simulations are removed by averaging. For fluids, time-averaging can be performed in two different ways. The most straightforward approach is to average time series of flow variables at fixed spatial points, to obtain the so-called Eulerian mean. Another approach is to average flow variables along particle trajectories instead of fixed positions, which gives the Lagrangian mean. Lagrangian averaging has several pivotal advantages over its Eulerian counterpart as illustrated by a growing number of studies. For instance, it removes the Doppler shift that eclipses the separation of time scales between the background flow and waves. However, the widespread adoption of Lagrangian averaging has been hindered by computational complications. To compute the Lagrangian mean in numerical models usually particles are tracked using interpolated velocities at particle positions at every time steps. This is a computational challenge that requires extra memory space (considering time series of variables are stored at all grid points) and is ill-suited for efficient computational parallelisation. In this project, we develop a numerical approach that circumvents these difficulties. This new approach is based on the evolution of partial means instead of particle tracking. Partial means can be viewed as means over shorter intervals than the total averaging period. In this approach, we compute the Lagrangian means as solutions to a set of Partial Differential Equations (PDEs) that describe the evolution of these partial means. This paradigm could be a breakthrough in computing Lagrangian means as these PDEs can be discretised in a variety of ways and solved on-the-fly (i.e. simultaneously with the dynamical governing equations). Hence, they do not require storing any time series and substantially reduce the memory footprint compared to particle tracking. The basic idea of this numerical technique is put forward by the PI. The goals of this proposal are to 1) expand the theoretical underpinnings of this novel method, 2) develop and implement a set of numerical schemes to solve the associated PDEs, and 3) apply them to 3D fluid models and laboratory data.