Passive scalars in complex fluid flows: variability and extreme events

流体流动中组分的输运和混合对许多科学和工程领域都具有核心重要性。例如，许多工业过程涉及溶解在流体中的化学品的混合和在许多情况下的反应。迁移和混合对若干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。通常情况下，组分不影响流体流动：它们被视为被动标量，通过给定的流动进行输送（平流），通过分子扩散混合，并可能发生化学反应。其浓度的演变受对流-扩散-反应方程控制。如果流量是已知的，这个方程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，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数据与由随机过程建模的流体流的统计数据联系起来。本项目将制定这样一种办法。它的主要目的是设计数学工具，使人们有可能描述的范围内的标量演变，可以预期从一个合奏的可能的流量，而不是一个单一的流量的响应。它的新奇在于超越了集合平均值的标准描述，以充分考虑不同流动实现之间浓度的变化。该项目的成果将是（i）将这种变化与拉伸特性等流动特性联系起来的新数学结果，以及（ii）基于集合模拟的新数值方法，以最小的计算成本对变化进行采样。将特别注意导致浓度极值的罕见事件。例如，当标量在随机流中释放时，它很可能只微弱地分散，因此其浓度在很长一段时间内保持较高。这类概率具有明显的实际重要性，例如对于评估污染源造成的风险;可靠地估计这些概率是该项目要解决的挑战之一。三个应用程序，他们都具有环境意义，已被选为新开发的试验台。这些措施包括：（一）水蒸气，在低温区域凝结;（二）臭氧，由于与活性氯反应而消耗;（二）浮游植物，在平流中经历逻辑演变。这些应用程序是一个更广泛的一类问题的代表，其特点是弱扩散和良好的混合初始条件，设计的方法可以应用。除此之外，结果将是相关的一些其他系统建模的无限维随机动力系统。

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.