**1. Introduction**

Air pollution is an important environmental issue, especially, in cases when pollutants travel thousands of kilometers, affecting air quality far away from their initial source. In the last decade several events drew even non-specialists' attention to the potential continental and global impacts of pollutant emissions from natural sources or anthropogenic industrial accidents. For example, in April and May 2010 the eruptions of the Icelandic Eyjafjallajökull volcano resulted in airspace closures across Europe. As a consequence, e.g., between 15–21 April 15 to 90% of the flight routes were cancelled implying also significant economic impacts (see, e.g., [1,2]). According to radar and satellite measurements the plumes from Eyjafjallajökull often reached the height of 5–10 km between 14–18 April and 3–20 May [3,4]. The ash particles and gases injected high into the atmosphere were transported mostly by westerly and northwesterly winds towards Europe, and small particles often travelled thousands of kilometers before being removed from the atmosphere. Ash plumes could be detected from several parts of Europe, including Great Britain, Germany, Poland, the Netherlands and Norway, at 1 to 7 km altitude in plumes of 100 m to 3 km depth and 100 to 300 km width [5]. At the beginning of May, due to the northerly flows in the Atlantic region, the ash plumes reached even the Iberian Peninsula within three to five days at an altitude as high as 11–12 km [6,7], and volcanic plumes from the Eyjafjallajökull eruptions were detected as far as Western Siberia, Russia, about 5000 km away from Iceland, on 20–26 April [8,9]. One year later, in May 2011, the volcanic ash clouds of the Grímsvötn volcano (Iceland) quickly rose to 20–25 km in altitude [10] and reached some part of Greenland and Scandinavia within a few days [11,12], impacting the air traffic in Northern Europe. Furthermore, traces of the volcanic clouds could also be detected in the stratosphere over Western Siberia [9]. In March–April 2011, due to the Fukushima Daiichi nuclear disaster (Japan), radioactive materials were transported in the atmosphere over the Pacific Ocean [13–15] causing measurable concentration even in Europe [16–18] within a few weeks in several countries like Greece [19], Germany [20], and Serbia [21]. In April 2015 the plumes from the Calbuco volcano in Chile rising to 15–23 km height [22,23] reached Argentina and Uruguay [24] and influenced even the development of the Antarctic ozone hole in 2015 [25]. In the same year, in Europe, the intense eruption of Mount Etna [26,27] attracted people's attention in December, and its SO2 plumes circumnavigated the whole Northern Hemisphere in an increasingly stretched filament shape [28].

Massive eruptions with plumes injected into the stratosphere can even have an impact on the global climate. It is the consequence of the ability of small aerosol particles or gases to travel in the atmosphere for a long time—even months or years—before they are removed (see, e.g., [29,30]). Additionally, within this time-frame they become substantially mixed over the hemispheres [31]. For example, the global surface temperature dropped by 0.5–0.7 ◦C for about two years due to a significant reduction of irradiation as a result of the Mount Pinatubo volcano's eruption in 1991 [32–34].

The rapid spread of pollutants in the atmosphere is due to the fact that in 3-D flows, as is the case for the atmosphere, individual particles carry out a so-called chaotic motion [35,36]. Its typical characteristics are (i) the sensitivity to the initial conditions, which implies that initially nearby particle trajectories diverge rapidly, namely, exponentially within a short time, (ii) the particle's motion is irregular, and (iii) the development of complicated but well-organized fractal structures. The chaotic nature implies that initially small and compact pollutant clouds stretch rapidly in time and evolve into a more and more complicated filamentary and tortuous structure (as can also be seen, e.g., on satellite observations and in model simulations in [12,28]). The intensity of the chaoticity of the pollutant spreading can be studied by means of different quantities. One of them is topological entropy [35,37,38], which, in the atmospheric context, characterizes the rate of the stretching of the length of pollutant clouds distorted into filament-like shapes. Topological entropy is also closely related to the unpredictability of the spreading and the complexity of the structure of a pollutant cloud [30,31,39].

Due to the impact of gravity, aerosol particles move downwards on average, hence they can travel in the atmosphere exhibiting the above-mentioned chaotic behavior only for a finite time interval before they are deposited on the ground. This kind of chaos is called transient chaos [37,38]. It can be shown that the time dependence of the number of non-deposited particles starts to decay approximately exponentially after a while. The rate of this exponential decrease is called the escape rate [29,37,38].

Even though, as the above examples demonstrate, volcano eruptions and industrial accidents may have an impact on a continental and global scale far away from their initial location, many of the students and non-experts are not familiar with the above-mentioned main properties of large-scale atmospheric pollutant spreading and deposition. For example, a common misconception is that pollutant clouds disperse in the atmosphere like dye blobs on clothes. There are some freely available atmospheric dispersion models with which simulations can be carried out, such as the Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) model [40,41] which has also a web based user interface, or the Lagrangian analysis tool LAGRANTO [42,43]. Nevertheless, the available models are principally designed for researchers, providing several options for dispersion calculations, and they often do not have user-friendly graphical user interface, as it is the case, e.g., for the FLEXible PARTicle (FLEXPART) dispersion model [44] and for the FALL3D [45–47]. To our knowledge, none of these models are designed to investigate the chaotic features of atmospheric spreading. Therefore, in this study we introduce a Lagrangian model called the Real Particle Lagrangian Trajectory model– Chaos version (RePLaT-Chaos), which specifically aims to demonstrate the chaotic behavior of pollutants. It is freely downloadable from [48]. Due to its easy-to-understand graphical user interface, it is also

a suitable tool for students and for other non-experts who are interested in atmospheric spreading phenomena and would like to study this in an interactive way by monitoring the spreading process on maps. Similar educational tools on environmental topics which affect our everyday life have become popular nowadays. For desktop applications which allows students to explore the subject of climate change, see, e.g., Educational Global Climate Model (EdGCM) [49] or Planet Simulator (PlaSim) [50]).

The paper is organized as follows. In Section 2 the two chaotic quantities, the topological entropy and the escape rate, which can be determined by means of the RePLaT-Chaos application, are introduced. Section 3 presents the equations of motions for trajectory calculations, the computation of the topological entropy and escape rate, and a brief overview of the RePLaT-Chaos application. Section 4 demonstrates the applicability and possibilities of RePLaT-Chaos on different examples. It includes a simulation of the spreading of a volcanic ash cloud emanated from the Eyjafjalljökull volcano's eruption. Furthermore, case studies regarding the topological entropy and escape rate are also presented in order to ge<sup>t</sup> an impression about their meaning and their magnitudes in different cases. Section 5 summarizes the chaotic characteristics of atmospheric pollutant spreading observable using RePLaT-Chaos and the main features of the application. Appendix A provides a detailed manual for the RePLaT-Chaos application, an overview of the user interface, including the description of its pages, and presents the options for starting new or loading saved simulations. It also contains instructions on how to obtain the topological entropy and the escape rate by means of RePLaT-Chaos.

#### **2. Chaotic Quantities**

#### *2.1. Topological Entropy*

In dynamical systems theory, topological entropy is a measure of the complexity of the motion [35,36]. Besides its abstract interpretations, its property which is the easiest to capture in measurements is that it also represents the growth rate of the length of line segments. The existence of the topological entropy is a basic property of chaos. A possible definition of chaos is that "a system is chaotic if its topological entropy is positive" [35,36].

As is mentioned in the Introduction, due to the chaotic nature of spreading, pollutant clouds stretch rapidly in time. The growth of the length *L* of a pollutant cloud in time *t* is approximately exponential after some days, i.e.,

$$L(t) \sim \exp(ht). \tag{1}$$

Here *h* is called the topological entropy [35,36,51–53], or the stretching rate in the atmospheric context [30,39]. Topological entropy is the rate of the exponential increase of the filament length. It is a measure of chaoticity, i.e., it quantifies the complexity and irregularity of the advection of a pollutant cloud: the larger the topological entropy, the more quickly the pollutant cloud stretches, the more complicated the shape in which it develops, the more foldings and meanders it contains, and the larger the geographical area the pollutant cloud covers.

## *2.2. Escape Rate*

Transient chaos means that chaotic behavior takes place only for a finite duration. This is the case for the spreading of aerosol particles in the atmosphere [35,37,38]. In this kind of systems, there exists a time-dependent set, the so-called chaotic saddle, which is responsible for the chaotic motion. The trajectories initialized on this saddle would never leave the saddle and carry out chaotic motion for an infinite amount of time. The chaotic saddle is a zero-measure set with fractal structure. As a consequence, in computational simulations using random initial conditions the probability for an initial condition to be located exactly on the chaotic saddle is zero and the trajectories sooner or later leave (i.e., "escape") any arbitrary pre-selected region of the saddle. In the context of the atmospheric pollutant spreading problem, this region can be chosen as the entire atmosphere, therefore, escaping means the deposition of particles. After a sufficiently long time *t*0, the decay in the ratio *n*(*t*)/*n*(0) of survivor particles is approximately exponential in transiently chaotic systems:

$$n(t)/n(0) \sim \exp(-\kappa t) \text{ for } t > t\_0. \tag{2}$$

The coefficient *κ* is called the escape rate [35,37,38] and in the case of pollutant spreading it characterizes the speed of the deposition. Larger escape rate implies faster deposition process, i.e., more particles leaving the atmosphere up to a given time instant. In general, the average lifetime of the particles after *t*0 can be estimated by *κ*<sup>−</sup>1.
