**3. Methods**

RePLaT-Chaos is a simpler version of the previously developed Real Particle Lagrangian Trajectory (RePLaT) model [29,54,55]. It computes the trajectories of individual spherical particles of realistic size and density, taking into account advection and the role of gravity through the terminal velocity of individual particles. In this sense, RePLaT-Chaos (and RePLaT) differ from the dispersion models which track so-called computational particles, like FLEXPART [44] and HYSPLIT [40,41], i.e., when each particle carries a certain amount of mass assigned to them upon the release, and this mass can be changed, e.g., due to deposition processes. In contrast to this, in RePLaT-Chaos, each particle has its own radius and density (and thus, its own realistic mass), and the effect of gravitational settling is calculated individually for each particle based on its own properties. Consequently, if a particle deposits on the surface, the entire particle remains there, not only a certain ratio of its mass. This individual particle approach is essential in order for the chaotic features of spreading to be studied appropriately. A pollutant cloud in the simulations consists of such kind of particles.

The computational background and the validity of RePLaT-Chaos, the RePLaT model, was tested in a number of cases. By simulating the spreading of volcanic ash injected in the atmosphere during the eruption of the Eyjafjallajökull and Mount Merapi [29,56] a reasonable agreemen<sup>t</sup> was found between the distribution of volcanic ash in the simulations and in the satellite observations at different time instances over days. Furthermore, the simulation of the spreading and deposition of radioactive materials continuously released during the Fukushima Daiichi nuclear power plant disaster showed that the arrival times of the pollution at different remote locations (e.g., Chapel Hill, Richland (USA), Stockhom (Sweden)) coincided with the measurements, and the RePLaT simulations were able to reproduce even the measured concentrations with acceptable accuracy [57].

#### *3.1. Calculation of Particle Trajectories*

For small and heavy aerosol particles it can be shown that a particle is advected by the wind components in the horizontal direction and their vertical motion is influenced by its terminal velocity and the vertical velocity component of air (see, e.g., [29]). RePLaT-Chaos utilizes meteorological data given on a regular longitudinal–latitudinal grid horizontally and at different pressure levels vertically. Therefore, the equations of motion of the particles are written in spherical coordinates in the horizontal direction and in pressure coordinates in the vertical direction in agreemen<sup>t</sup> with the structure of the meteorological data:

$$\frac{d\lambda\_{\rm P}}{dt} = \frac{u(\lambda\_{\rm P}(t), \,\varphi\_{\rm P}(t), \, p\_{\rm P}(t), t)}{R\_{E} \cos \varphi\_{\rm P}} = u\_{\rm rad}(\lambda\_{\rm P}(t), \, \varphi\_{\rm P}(t), \, p\_{\rm P}(t), t),\tag{3}$$

$$\frac{\mathrm{d}\varrho\_{\mathrm{P}}}{\mathrm{d}t} = \frac{v(\lambda\_{\mathrm{P}}(t), \varrho\_{\mathrm{P}}(t), p\_{\mathrm{P}}(t), t)}{R\_{E}} = v\_{\mathrm{rad}}(\lambda\_{\mathrm{P}}(t), \varrho\_{\mathrm{P}}(t), p\_{\mathrm{P}}(t), t), \tag{4}$$

$$\frac{\mathrm{d}p\_{\mathrm{P}}}{\mathrm{d}t} = \omega\left(\lambda\_{\mathrm{P}}(t), \varphi\_{\mathrm{P}}(t), p\_{\mathrm{P}}(t), t\right) + \omega\_{\mathrm{term}}\left(\lambda\_{\mathrm{P}}(t), \varphi\_{\mathrm{P}}(t), p\_{\mathrm{P}}(t)\right) \tag{5}$$

where *λ*p and *ϕ*p are the longitude and latitude coordinates, *<sup>p</sup>*p(*t*) ≡ *<sup>p</sup>*(*<sup>λ</sup>*p(*t*), *<sup>ϕ</sup>*p(*t*), *<sup>p</sup>*p(*t*), *t*) is the pressure coordinate of a particle, *RE* = 6370 km is the Earth's radius, *u* and *v* are the zonal and meridional velocity component of the air in the units of m s<sup>−</sup>1, *u*rad and *v*rad are the same but in units of s −1 fitted to the longitude–latitude coordinates, *ω* is the vertical air velocity component in the pressure system, and *ω*term is the corresponding terminal velocity of the particle in motionless air of the form of

$$
\omega\_{\text{term}} = \begin{cases}
\frac{2}{9} r^2 \frac{\rho\_\text{P}}{\nu} \text{g}^2, & \text{if } \text{Re} \ll 1 \\
\sqrt{\frac{8}{3} \frac{\rho\_\text{P} \rho r}{\text{C}\_\text{D}}} \text{g}^3, & \text{if } \text{Re} \gg 1.
\end{cases} \tag{6}$$

Here *r* and *ρ*p are the radius and the density of the particle, respectively, *ν* and *ρ* = *p*/*R*d/*<sup>T</sup>* are the kinematic viscosity and density of air, respectively, *R*d = 287 J kg−<sup>1</sup> K−<sup>1</sup> is the specific gas constant of dry air, *g* denotes the gravitational acceleration, *C*D is the drag coefficient (for particles assumed to be spheres *C*D = 0.4), and Re = 2*rV*/*ν* is the Reynolds number (where *V* is the instantaneous particle velocity). The limit of *r* = 0 μm can be considered as gas "particles", the terminal velocity of which is *ω*term = 0.

The dependence of kinematic viscosity *ν* on temperature *T* and pressure *p* is calculated according to Sutherland's law [58]

$$\nu = \beta\_0 \frac{T^{3/2}}{T + T\_\mathcal{S}} \frac{R\_\mathcal{d} T}{p}. \tag{7}$$

Here *β*0 = 1.458 × 10−<sup>6</sup> kg m<sup>−</sup><sup>1</sup> s<sup>−</sup><sup>1</sup> K−1/2 is Sutherland's constant and *T*S = 110.4 K is a reference temperature.

RePLaT-Chaos solves the differential Equations (3)–(5) by an explicit second-order Runge–Kutta method, i.e., by the second-order Petterssen scheme (applied often also in other Lagrangian dispersion models [40,41,44]). Hence the position **r**(*<sup>t</sup>* + Δ*t*)=[*λ*(*t* + <sup>Δ</sup>*t*), *ϕ*(*<sup>t</sup>* + <sup>Δ</sup>*t*), *p*(*<sup>t</sup>* + Δ*t*)] of a particle at time instant *t* + Δ*t*, using the velocity **<sup>v</sup>**(**r**(*t*), *<sup>t</sup>*)=[*<sup>u</sup>*rad(**r**(*t*), *t*), *<sup>v</sup>*rad(**r**(*t*), *t*), *<sup>ω</sup>*(**r**(*t*), *t*) + *<sup>ω</sup>*term(**r**(*t*), *t*)] at time *t*, reads as:

$$\mathbf{r}\_0(t + \Delta t) = \mathbf{r}(t) + \mathbf{v}(\mathbf{r}(t), t)\Delta t,\tag{8}$$

$$\mathbf{r}(t+\Delta t) = \mathbf{r}(t) + \frac{1}{2} \left( \mathbf{v}(\mathbf{r}(t), t) + \mathbf{v}(\ \mathbf{r}\_0(t+\Delta t), t+\Delta t) \right) \Delta t. \tag{9}$$

The utilized meteorological data should be available on a regular latitude–longitude grid on different pressure levels with a given (e.g., 3 or 6 h) time resolution. Therefore, in order to solve the equations of motion of the particles and to calculate the particle trajectories, the quantities *u*, *v*, *ω*, *T* are interpolated to the location of the particles in each time step. RePLaT-Chaos applies linear interpolation in each of the three directions and in time.

Users have the option to choose between variable time step and constant time step for the trajectory calculation. In the former option, the maximum time step Δ*tC* for each particle is determined based on the grid size and the current atmospheric velocity components as

$$\Delta t\_{\mathbb{C}} = \mathbb{C} \min \left\{ \frac{\Delta \lambda\_{\mathbb{E}}}{|u\_{\text{rad}}(\mathbf{r}(t), t)|} ; \frac{\Delta \varphi\_{\mathbb{E}}}{|v\_{\text{rad}}(\mathbf{r}(t), t)|} ; \frac{\Delta p\_{\mathbb{E}}}{|\omega(\mathbf{r}(t), t) + \omega\_{\text{perm}}(\mathbf{r}(t), t)|} \right\} \tag{10}$$

with *C* = 0.2 where <sup>Δ</sup>*λ*g [rad], <sup>Δ</sup>*ϕ*g [rad] and <sup>Δ</sup>*p*g [Pa] denote the grid size in longitudinal, meridional and vertical direction, respectively. By means of such a choice the smallest features resolved by the meteorological fields are taken into account as pointed out in [59]. The minimal time step Δ*t*min is determined by the user, therefore, Δ*t* = max{<sup>Δ</sup>*tC*; <sup>Δ</sup>*t*min}. If the obtained time step Δ*t* would be larger than the time interval (*<sup>t</sup>*next − *t*) up to the next writing of particle data to file or up to the next reading of new meteorological fields, then the time step is modified as

$$
\Delta t = \min \{ t\_{\text{next}} - t\_\prime \max(\Delta t\_\mathbb{C}; \Delta t\_{\text{min}}) \}. \tag{11}
$$

#### *3.2. Calculation of the Topological Entropy*

Topological entropy is calculated by RePLaT-Chaos as in [30,39]. In order for the length of a pollutant cloud to be appropriately determined, the user should initiate 1-D "pollutant clouds", i.e., line segments or filaments. The length of a filament is the sum of the distances of its neighboring particle pairs:

$$L(t) = \sum\_{i=1}^{n(t)-1} \left| \mathbf{r}\_i(t) - \mathbf{r}\_{i+1}(t) \right| , \tag{12}$$

where **r***i* is the position of the *i*th particle and *n*(*t*) is the number of particles. The distance |**<sup>r</sup>***i*(*t*) − **<sup>r</sup>***i*+<sup>1</sup>(*t*)| in units of km are calculated along grea<sup>t</sup> circles neglecting the vertical stretching which proved to be 10−<sup>2</sup> to 10−<sup>3</sup> times smaller than the horizontal one [31]:

$$|\mathbf{r}\_{i}(t) - \mathbf{r}\_{i+1}(t)| = \arccos\left[\sin\varphi\_{i}\sin\varphi\_{i+1} + \cos\varphi\_{i}\cos\varphi\_{i+1}\cos(\lambda\_{i} - \lambda\_{i+1})\right] \times \frac{180}{\pi} \times 111.1,\tag{13}$$

where *λi* and *ϕi* are the zonal and meridional coordinate of the *i*th particle, respectively. The factor 180 *π* × 111.1 converts the unit from radian to kilometer using the fact that the spherical distance of 1◦ along a grea<sup>t</sup> circle corresponds to a length of 111.1 km along the surface. Note that a filament remains a single filament forever, and cannot split up into two or more branches, because it would require a wind vector that points in more than one direction at some location. Hence the determination of the full (folded) length is unambiguous.

Since subsequent particles may travel far away from each other in time, the length of a pollutant cloud that consists of a finite number of particles may be underestimated compared to a pollutant cloud with the same initial condition consisting of an infinite number of particles (i.e., "continuous" pollutant cloud). This implies that after a certain amount of time, when there are several "cut-off" segments among the particles, the computed length differs from the expected approximately exponential function as its values are lower than the expected ones. Therefore, in order to reduce the underestimation of the length, there is an option for users that if the distance of two neighboring particles becomes larger than a threshold distance defined by the user, a sufficient number of new particles is inserted between them uniformly.

Based on Equation (1) the topological entropy is determined as the slope *h* of a linear least squares fit applied to the natural logarithm of the length *L*(*t*) of the filament for the time interval chosen by the user.

#### *3.3. Calculation of the Escape Rate*

In order to determine the escape rate it is worth tracking the trajectories of a large number of particles until they leave the atmosphere one by one. At each time instant *t* RePLaT-Chaos determines the number *n*(*t*) of the particles still moving in the atmosphere, i.e., the number of the non-escaped particles. Based on Equation (2) the value of the escape rate *κ* is calculated as (−<sup>1</sup>)<sup>×</sup> the slope of a linear least squares fit applied to the natural logarithm of the ratio *n*(*t*)/*n*(0) of non-escaped particles [29,55] for the time interval chosen by the user. It is worth noting that the time *t*0 in Equation (2), after which the exponential decay starts, depends on the initial conditions, the initialization time instant and the properties of the particles as well.

#### *3.4. RePLaT-Chaos in a Nutshell*

RePLaT-Chaos is a desktop application with user-friendly graphical user interface and simulates the atmospheric spreading of pollutant clouds in the time interval and with simulation setups given by the user. Pollutant clouds consist of individual particles, the number of which is determined by the user. The initial position, size, and other properties of the pollutant cloud (and its particles) can be set up on the user interface, and pre-generated pollutant clouds can also be read for the simulations. For the spreading calculations, meteorological files containing the appropriate meteorological data

that overlap the defined time interval are required. RePLaT-Chaos determines the new position of each particle of the pollutant cloud from Equations (3)–(5) in each time step based on the meteorological data and writes the particle data to file. Furthermore, there are options for computing the length of the pollutant cloud or the ratio of the particles not deposited from the atmosphere. These data are needed for the calculation of the two quantities characterizing the chaoticity of the spreading: the topological entropy and the escape rate. RePLaT-Chaos provides an opportunity to replay simulations saved in files and to determine the above-mentioned two chaotic measures. The detailed manual for the application can be found in Appendix A. In the next section, the applicability of RePLaT-Chaos are presented, drawing attention to the main features of the large-scale atmospheric spreading of pollutants and its chaotic characteristics.

#### **4. Results from RePLaT-Chaos Simulations**

#### *4.1. Spreading of a Volcanic Ash Cloud Emitted during the Eyjafjalljökull Volcano's Eruption*

The Eyjafjalljökull volcano in Iceland showed an increased seismic activity in the spring of 2010. After the first eruption on 20 March, one of the most intense eruptions happened on 14 April 2010 [60]. For about four days, the vertical extent of the emitted ash columns often exceeded the height of 4 to 5 km, with the top of the column occasionally reaching even the altitude of 10 km according to weather radar, LIDAR, and satellite measurements (see, e.g., [3,4]). The mean size and density of the particles which travelled across Europe were found to be between *r* ≈ 0.1 to 10 μm and p ≈ 2000 kg m<sup>−</sup>3, respectively (see, e.g., [5,61–63]).

Based on these data, to ge<sup>t</sup> a first impression about the main characteristics of atmospheric pollutant spreading by means of RePLaT-Chaos, Figure 1 shows the simulation of the spreading of a single, initially compact ash cloud of height of 4 km injected into the atmosphere due to the eruption of the Eyjafjallajökull on 14 April 2010 at 06:00 UTC. The ash cloud in the simulation consisted of 2.7 × 10<sup>4</sup> particles with *r* = 5 μm and p = 2000 kg m<sup>−</sup>3.

Figure 1 illustrates that within a few days the ash cloud travels over Scandinavia and reaches Eastern Europe due to being transported by the northwesterly winds of a high pressure system located south of Iceland at the beginning and then moving towards Scandinavia. Figure 1 demonstrates well that the spreading of volcanic ash clouds (and any atmospheric pollutants) differs from the dispersion of dye droplets on clothes. The latter is of a slowly growing circular shape, while Figure 1 shows that an important feature of atmospheric pollutant spreading is the rapid distortion of an initially small and compact cloud into an increasingly stretched, filament-like shape, extending to a region of some thousands of kilometers within a few days. As mentioned in the Introduction, the observed rapid stretching of pollutant clouds is a consequence of the chaotic nature of the spreading. Therefore, the rate of the stretching is a possible measure of the strength of chaos which will be illustrated through some examples in Section 4.2.

At the beginning (see Figure 1a), the top of the ash cloud reaches the altitude of 9 km (cyan color). However, due to the impact of gravity, the particles descend in the atmosphere more or less continuously (but not uniformly), and after two days they reach the altitude of about 4–6 km (green color, Figure 1c). Within three days, the altitude of the ash cloud in an extended region decreases even below 2–3 km (yellow color, Figure 1d). After 10 days a large number of particles are found to be deposited on the ground (black color, Figure 1f) across Siberia. The deposition distribution shows another important characteristic, typical of chaotic phenomena, namely that it is inhomogeneous with filamentary structure, with denser and sparser regions. Additionally, it can be also seen that particles do not fall out from the atmosphere at almost the same time as a coherent patch but rather some parts of the ash cloud are deposited by the 7th day after the eruption already (Figure 1e), while several particles are still in the middle of the troposphere, at an altitude of about 5 km (green) even after 10 days (Figure 1f). As it is introduced in Section 1, this kind of deposition dynamics is characteristic to

transiently chaotic phenomena. The measure of the rapidity of deposition processes will be discussed in Section 4.3 in detail.

**Figure 1.** Simulation of the spreading of volcanic ash particles from the Eyjafjallajökull volcano's eruption at different time instants in the form of "year.month.day.hour:minute" indicated in the panels' label (**<sup>a</sup>**–**f**). 30 × 30 × 30 particles of *r* = 5 μm and p = 2000 kg m<sup>−</sup><sup>3</sup> are initiated in a rectangular cuboid of size of 100 km × 100 km × 4 km at 63.63◦ N, 19.6◦ W, at the altitude of 7 km on 14 April 2010 at 06 UTC. Simulation is initialized with the parameters on the left of Figure A1. Colorbar indicates the altitude of the particles, black color marks deposited particles.

#### *4.2. Stretching of the Pollutant Clouds—The Topological Entropy*

Section 4.1 has shown that even an initially cuboid-shaped pollutant cloud soon becomes distorted into a tortuous, filamentary shape due to the chaotic nature of atmospheric spreading, and the extension of the cloud grows rapidly. To quantify this growth, by means of RePLaT-Chaos application, the time-dependence of the length increase of 1-D pollutant clouds (i.e., lines or filaments) can be measured. The stretching rate of the length, the topological entropy *h* in Equation (1), quantifies the intensity of the underlying chaotic dynamics which the pollutant cloud is subjected to during spreading.

To ge<sup>t</sup> an impression of the meaning and consequences of the value of the topological entropy *h*, Figure 2 illustrates the distribution of two meridional line segments (having the same length at the emission) after 10 days and the corresponding curves of their length increase. Both cases show that the length of the filaments indeed grows in an approximately exponential manner in time (Figure 2b,d) after a few days (as a line in the semi-logarithmic plot). In Figure 2b the slope of the linear fit is *h* = 0.808 day−1, while in Figure 2d the slope is found to be about 56% smaller, *h* = 0.357 day−1. These values mean that in every *h*−<sup>1</sup> = 1.238 and 2.801 days the length of the pollutant cloud stretches by a factor of e ≈ 2.718, respectively. With *h* being in the exponent in Equation (1), this approximately double factor between the topological entropies results in the fact that the length of the filaments after 10 days is about 1.242 × 10<sup>6</sup> km for the filament initiated in Europe and 9.660 × 10<sup>3</sup> km for the one emitted in Africa (calculated as exp(14.032) and exp(9.176), respectively, reading the length data on April 24 at 6 UTC from the graphs.). The nearly 100-fold difference in their length (and the corresponding deviation of their topological entropies) obviously implies remarkably different distribution patterns at the end of the simulation. While *h* = 0.357 day−<sup>1</sup> in Figure 2c is associated with a slightly crumpled filament which has not travelled far away from its initial location as it is drifting slowly with the trade winds near the Equatorial region, the filament in Figure 2a has a much more complicated shape with several foldings and meanders that cover a considerable part of the Northern Hemisphere. We note that, in general, larger topological entropy values and more intense spreading characterize the pollutant clouds initiated at the mid- and high latitudes than the ones start in the tropical region [30,31]. This is a consequence of the enhanced cyclonic activity in the extratropics associated with intensified shearing and mixing effects on the pollutant clouds. It is also worth noting that certain atmospheric features can be identified based on the pattern formed by the particles of the pollutant clouds: e.g., in Figure 2a south of Greenland and east of Scandinavia the trace of two cyclones can be noticed drawn by the spiral formations of the pollutant cloud.

**Figure 2.** (**<sup>a</sup>**,**<sup>c</sup>**) The advection pattern of a pollutant cloud after 10 days, and (**b**,**d**) the time dependence of the logarithm of the length of the pollutant clouds, respectively. The simulations are initialized with the simulation parameters on the left of Figure A1 but with top and bottom reflection coefficients of 1 and inserting new particles if the distance of two particles became greater than 100 km. The pollutant clouds are initialized as meridional line segments of 400 km at the altitude of 5500 m at (**<sup>a</sup>**,**b**) 47◦ N, 19◦ E and (**<sup>c</sup>**,**d**) 10◦ S, 19◦ E. They consist of 1000 particles at the beginning. Their initial position is indicated by the thick black lines to which arrows are pointing. The particle radius is 0 μm. Colorbar indicates the altitude of the particles. In panel (**b**,**d**) the black line indicates a linear fit to the logarithm of the length for the time interval from 16 April, 6 UTC to 24 April, 6 UTC. Its slope is (**b**) *h* = 0.808 day−<sup>1</sup> and (**d**) *h* = 0.357 day−1.

#### *4.3. Deposition of the Particles—The Escape Rate*

Section 4.1 draws attention to the fact that the lifetime of particles even in an initially small pollutant cloud may be quite different. The reason behind the observed differences is that the particles do not fall directly purely vertically from their initial position onto the ground, but travel along complicated trajectories due to the chaotic nature of spreading. In this way their vertical movement is affected by both their terminal velocity and the local instantaneous vertical component of the air. Both the terminal velocity *ω*term (Equation (6)) through kinematic viscosity *ν* and/or air density  and the vertical air velocity *v* ( Equation (5)) depend on the position of the particle and the time instant. For light aerosol particles the value of the upward directional vertical air velocity often exceeds the downward effect of their terminal velocity, thus, besides falling downwards on average, these particles

have more chance to move also upwards in the atmosphere with the flows. The chaotic nature of spreading implies that nearby particles may reach remote locations within short times where they are also subjected to different vertical velocities, therefore, they may be deposited at considerably different time instants and locations.

In order to study the process of deposition, Figure 3a,b shows how the ratio of non-deposited particles initially distributed uniformly over the globe at the altitude of 5.5 km changes in time. At the beginning of the simulation, a short plateau can be seen for both particles of radius *r* = 7 μm (Figure 3a) and *r* = 9 μm (Figure 3b), which indicates that a certain time is needed even for the "fastest falling" particles to reach the surface. After a short transient, the plateau is followed by an approximately exponential decrease in the ratio of non-escaped particles, the rapidity of which is characterized by the escape rate *κ* in Equation (2). The escape rate is smaller (*κ* = 0.278 day−1) for smaller particles (Figure 3a) and larger (*κ* = 0.489 day−1) for larger particles (Figure 3b), as expected naively, but it depends on the atmospheric conditions, too. In fact, the dependence of *κ* on *r* for *r* ≤ 10 μm particles proved to be quadratic in a recent research studying aerosol particles with a realistic density of 2000 kg m<sup>−</sup><sup>3</sup> [55]. This is in harmony with the fact that the updrafts and downdrafts in the atmosphere approximately balance each other's effect on the particles, thus particles in rough average fall with their terminal velocity *ω*term, which depended quadratically on *r* for these small particles with Re 1 (Equation (6)). The obtained *κ*s imply that after the exponential decay takes place, after *κ*<sup>−</sup><sup>1</sup> = 3.597 and 2.045 days, only a proportion of e<sup>−</sup><sup>1</sup> ≈ 0.368 of the particles can still be found in the air, respectively. It is worth noting that the reciprocal of *κ* is often considered to be a rough estimate of the average lifetime of typical particles in the exponentially decreasing stage [37,38].

Figure 3a,b also confirms an interesting observation made in Section 4.1 that even identical particles may often have significantly different lifetimes. For example, the first particles in Figure 3a leave the atmosphere on 15 April, only one day after the emission, while after more than two weeks, on 30 April, there are still 1.4% of the particles (exp(−4.256) from the data of the graph) drifting in the atmosphere. Simulating the atmospheric spreading of a larger number of particles with *r* = 7 μm for a longer time period, it turns out that a ratio of 10−5–10−<sup>6</sup> of the particles is able to survive more than two months in the atmosphere, as well as that the initial location of the long- and short-living particles folds into each other in thin filaments in a fractal structure in extended regions [55].

Figure 3c demonstrates that the inhomogeneity and irregularity in the pattern of the deposited particles in Figure 1f is not the consequence of the initially small extension of the volcanic ash cloud studied in Section 4.1. The filamentary deposition pattern with denser and sparser regions, typical for transient chaos, can also be seen even for particles initially distributed completely uniformly over the whole globe.

**Figure 3.** (**<sup>a</sup>**,**b**) The time-dependence of the logarithm of the ratio of non-escaped particles. The simulations are initialized with the simulation parameters on the left of Figure A1 but with an end date of "2010.04.30.12:00:00" and with calculating the ratio of non-escaped particles. The pollutant clouds are initialized as 300 × 300 × 1 particles at 0◦ N, 0◦ E and at the altitude of 5500 m with an extension of 4 × 10<sup>4</sup> km × 2 × 10<sup>4</sup> km × 0 m (i.e., covering the entire globe uniformly at a single altitude). The particle density is 2000 kg m<sup>−</sup><sup>3</sup> and the particle radius is (**a**) 7 μm and (**b**) 9 μm, respectively. The black lines indicate linear fits to the logarithm of ratio of non-escaped particles for the time interval from 19 April, 12 UTC to 30 April, 12 UTC. The (−<sup>1</sup>)×slopes are (**a**) *κ* = 0.278 day−<sup>1</sup> and (**b**) *κ* = 0.489 day−1. (**c**) The deposition pattern of the particles in panel (**b**) at 30 April 2010 at 12 UTC. Colorbar indicates the altitude of the particles, black color marks deposited particles.
