Passive scalars in complex fluid flows: variability and extreme events

流体流动中成分的输送和混合对许多科学和工程领域都具有中心重要性。例如，许多工业过程都涉及溶解在液体中的化学物质的混合，在许多情况下还涉及化学物质的反应。运输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，这些成分不影响流体流动：它们被视为被动标量，它们被给定的流动输送（平流），被分子扩散混合，并可能发生化学反应。它们的浓度演变受平流-扩散-反应方程的支配。如果流动是已知的，这个方程预测标量浓度在时间和空间上的变化。然而，在许多应用程序中，流是混乱的，过于复杂，无法准确地了解。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流体流动的统计数据联系起来，由随机过程建模。本项目将发展这样一种方法。它的主要目的是设计数学工具，使描述标量演化的范围成为可能，这些演化可以从可能的流的集合中得到，而不是对单个流的响应。它的新颖之处在于超越了总体平均值的标准描述，以便充分表征不同流实现之间浓度的可变性。该项目的结果将是：(i)将这种可变性与流动特性（如拉伸特性）联系起来的新的数学结果，以及（ii）基于集成模拟的新的数值方法，以最小的计算成本对可变性进行采样。将特别注意导致浓度极值的罕见事件。例如，当一个标量在随机流中释放时，有很小的概率它只是弱分散，因此它的浓度在很长一段时间内保持高。这类概率具有明显的实际重要性，例如对于评估污染源所造成的危险；它们的可靠估计是项目要解决的挑战之一。三个具有环境意义的应用被选择作为新开发的试验台。这包括：(i)水蒸气在低温地区凝结，（ii）臭氧与活性氯反应消耗殆尽，以及（ii）浮游植物在平流过程中经历逻辑演化。这些应用代表了更广泛的一类问题，其特点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，结果将与许多其他由无限维随机动力系统建模的系统相关。

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.