Passive scalars in complex fluid flows: variability and extreme events

流体流动中组分的输送和混合对许多科学和工程领域具有重要意义。例如，许多工业过程涉及溶解在液体中的化学品的混合，在许多情况下还包括反应。运输和混合对几个环境问题也至关重要，例如污染物的扩散和大气温室气体的分布。通常，这些成分不会影响流体流动：然后它们被视为被动标量，由给定的流动输送(平流)，通过分子扩散混合，并可能发生化学反应。它们的浓度变化符合对流-扩散-反应方程。如果流量是已知的，这个方程就可以预测标量浓度在时间和空间上的变化。然而，在许多应用中，流动是混沌的，太复杂了，以至于不能准确地知道。在这种情况下，需要一种概率方法，它将标量浓度的统计与由随机过程建模的流体流动的统计相关联。该项目将开发这样一种方法。它的主要目的是设计数学工具，使之有可能描述从可能的流动的集合中预期的标量演变的范围，而不是对单一流动的反应。它的新奇之处在于超越了集合平均的标准描述，以便充分表征不同流动实现之间浓度的可变性。该项目的结果将是(I)新的数学结果，将这种可变性与流动特征，如拉伸性质，和(Ii)新的数值方法，基于集合模拟，采样的可变性，以最小的计算成本。将特别注意导致集中度极端值的罕见事件。例如，当一个标量在随机流中释放时，它很可能只是微弱地分散，因此它的浓度在很长一段时间内保持较高的浓度。这类概率具有明显的实际重要性，例如对污染源造成的风险进行评估；其可靠估计是该项目解决的挑战之一。三个应用程序都具有环境意义，已被选为新开发的试验台。这些包括：(I)水蒸气，它在低温地区凝结；(Ii)臭氧，它因与活性氯的反应而被消耗；以及(Ii)浮游植物，它在被平流时经历逻辑演化。这些应用代表了一类更广泛的问题，其特征是弱扩散和良好混合的初始条件，所设计的方法可以应用于这些问题。除此之外，这些结果将与许多由无限维随机动力系统建模的其他系统相关。

The transport and mixing of constituents in fluid flows is of central importance to many areas of sciences and engineering. Numerous industrial processes, for instance, involve the mixing and in many cases reactions of chemicals dissolved in fluids. Transport and mixing are also crucial to several environmental issues, such as the dispersion of pollutants and the distribution of atmospheric greenhouse gases. Often, the constituents do not affect the fluid flow: they are then regarded as passive scalars, which are transported (advected) by a given flow, mixed by molecular diffusion, and possibly react chemically. The evolution of their concentration is governed by the advection-diffusion-reaction equation. If the flow is known, this equation predicts how the scalar concentration varies in time and space. However, in many applications, the flows are chaotic and too complex to be known exactly. In this case, a probabilistic approach is needed which relates the statistics of the scalar concentration to the statistics of the fluid flows, modelled by random processes. This project will develop such an approach. Its main aim is to devise mathematical tools that make it possible to describe the range of scalar evolutions that can be expected from an ensemble of possible flows rather than the response to a single flow. Its novelty is to go beyond the standard description in terms of ensemble averages in order to fully characterise the variability of the concentration between different flow realisations. The outcomes of the project will be (i) new mathematical results that relate this variability to flow characteristics such as stretching properties, and (ii) new numerical methods, based on ensemble simulations, that sample the variability at minimal computational cost. Particular attention will be paid to rare events which lead to extreme values of the concentration. For example, when a scalar is released in a random flow, there is a small probability that it disperses only weakly and hence that its concentration remains high for a long time. Probabilities of this type have a clear practical importance, for instance for the assessment of the risk posed by pollution sources; their reliable estimation is one of the challenges addressed by the project. Three applications, all of them with environmental significance, have been chosen to serve as testbeds for the new developments. These involve: (i) water vapour, which condenses in low-temperature regions, (ii) ozone, which is depleted by its reaction with active chlorine, and (ii) phytoplankton, which experiences a logistic evolution while being advected. These applications are representative of a much broader class of problems, characterised by weak diffusion and well-mixed initial conditions, to which the methods devised can be applied. Beyond this, the results will be relevant to a number of other systems modelled by infinite-dimensional random dynamical systems.