# Passive scalars in complex fluid flows: variability and extreme events

### Abstract

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.

### Planned Impact

The immediate beneficiaries of the research will be the applied mathematicians, physicists and engineers engaged in work on the transport and mixing of scalars. The tools of analysis developed to assess the variability of scalars will be useful to them in the many contexts where the fluid flows are too complex to be entirely predictable and hence need to be treated probabilistically. Because the work of this community is, in many instances, closely linked to applications, it is anticipated that the impact of the research will extend beyond academic researchers and reach industry and environmental agencies. The main message of the research, namely that fluctuations in a fluid flow lead to a large variability in the chemical or biological activity, a variability which can be diagnosed by a combination of analytical and numerical tools, is indeed an important one for those who need to evaluate the risk posed by the release of pollutants in the environment. Specific outcomes of the research, importance-sampling techniques and particle-mesh numerical methods in particular, are also expected to have a direct impact on the modelling of scalars employed by operational forecasters and climate modellers. The three examples of reacting systems that have been chosen as testbeds for the development of new methods (water vapour, ozone and plankton) are all the subject of an intense research activity motivated by their clear environmental importance. It is anticipated that the project, while achieving its aims in the development of new general methods, will answer some scientific questions specific to each these problems and hence impact each of the relevant disciplines.

Haynes P
(2014)

*Dispersion in the large-deviation regime. Part 1: shear flows and periodic flows*in Journal of Fluid Mechanics
Haynes P
(2014)

*Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number*in Journal of Fluid Mechanics
Tsang YK
(2017)

*The effect of coherent stirring on the advection-condensation of water vapour.*in Proceedings. Mathematical, physical, and engineering sciences
Tzella A
(2014)

*Front propagation in cellular flows for fast reaction and small diffusivity.*in Physical review. E, Statistical, nonlinear, and soft matter physics
Tzella A
(2015)

*FKPP Fronts in Cellular Flows: The Large-Péclet Regime*in SIAM Journal on Applied Mathematics
Tzella A
(2016)

*Dispersion in Rectangular Networks: Effective Diffusivity and Large-Deviation Rate Function.*in Physical review lettersDescription | A theory describing the spreading of chemical constituents in periodic flows has been developed. This goes beyond standard theory in describing the tails of the concentration distribution using the probabilistic theory of large deviations. Another strand of work has considered the propagation of chemical fronts in fluid flows, leading to new predictions for the front speed for the widely used FKPP model. Finally new results, still to be published, concern the spreading of chemicals on networks such as the street networks of cities, again treated using large-deviation methods. |

Exploitation Route | The large deviation theory can be exploited to predict the spreading of pollutant in river, seas and in cities. |

Sectors |
Environment,Other |