**Abbreviations**

The following abbreviations are used in this manuscript: RePLaT Real Particle Lagrangian Trajectory model

#### **Appendix A. RePLaT-Chaos Manual**

#### *Appendix A.1. First Steps and Data Format Requirements*

Appendix A.1.1. Launching RePLaT-Chaos

RePLaT-Chaos can be downloaded from the website of the Institute for Theoretical Physics, Eötvös Loránd University [48]. Both a Windows executable installer file and a compressed file

including a Java Archive application (usable also on Linux platforms) are accessible. In the former case, the installer installs the application in the folder RePLaT-Chaos in a user selected location. RePLaT-Chaos can be launched by clicking RePLaT-Chaos.exe in the folder. In the latter case the downloaded zip file should be unpacked, and the application can be launched, e.g., from the command prompt by typing the java -jar RePLaT-Chaos.jar command from folder RePLaT-Chaos. In both cases the folder RePLaT-Chaos has a sub-folder named default which contains the default values for the text fields of the user interface (default/default\_\*.txt files) and the continents.txt file for displaying the map for the simulations. Therefore, this folder and its contents should not be renamed or removed, however, the content of the default\_\*.txt files may be changed preserving their formats. Sample meteorological data in the required format are also available on the website for a 16-day time period overlapping with the Eyjafjallajökull volcano's eruptions in 2010.

On the user interface the menu items and buttons with underlined letter/number can be reached not only by mouse clicks but also by keyboard shortcut Alt + <letter>/<number>.

#### Appendix A.1.2. Input Meteorological Data

For the simulation of the spreading of pollutant clouds meteorological data in Network Common Data Form (NetCDF) format files are needed. Data should be downloaded for the entire globe and at least six hours of time resolution. Such data are accessible in different temporal and spatial resolutions, e.g., from the European Center for Medium-Range Weather Forecasts (ECMWF) datasets [66] (ERA-40, ERA-Interim, etc.). The properties of the files required by RePLaT-Chaos are:


#### Appendix A.1.3. Output Data

RePLaT-Chaos writes the data of the particles of the pollutant clouds to comma-separated values text files with CSV file extension, therefore, these files can be easily read or analyzed with other tools, too. An output file represents again a single time instant and contains one line for each particle. The file name convention is <file name pattern><yyyyMMddhhmmss>.csv. The comma separated data in a line are: *λ*, *ϕ*, *z*, *r*, *ρ*p, *ι*, where


The application computes the length of the pollutant cloud [km] and/or the ratio of the non-escaped particles if the user chooses this/these option(s). It writes the natural logarithms of these quantities to file at the time instants given by the user. The file of the length and the ratio of the non-escaped particles contain lines of format of <yyyyMMddhhmmss><tab>ln(value of the quantity).

#### *Appendix A.2. Running a Simulation*

In RePLaT-Chaos two setup options are available for a new simulation. In the first one parameters both for the simulation and for the pollutant cloud should be given. This screen is accessible via File menu clicking menu item New simulation—set parameters. The other options is New simulation—read parameters. In the latter case, pollutant clouds are not initialized according to user-given parameters rather its particles are read from a file.

#### Appendix A.2.1. Setting the Simulation Parameters

The user chooses either the New simulation—set parameters or the New simulation—read parameters menu item, in both cases the simulation parameters should be given at first (left panel of the screen in Figures A1 and A2). These parameters are the following:


• Max. number of particles: The number of the particles in the simulation, including the inserted ones, does not exceed the given number (format: integer).

By default, the fields are filled with the default setting parameters loaded from default/ default\_simulation\_setup.txt. After overwriting any field, the user can reload the default values by clicking the Default button, new data can be loaded in the fields from a chosen file by clicking the Load button, or the values of the fields can be saved in a new file by clicking the Save button. If there is a wrong or empty text field, data could not be saved: the problem is indicated by an alert window.

At first, for starting a simulation the generation of the simulation setup is required: the user can generate it by clicking the Generate simulation setup button. If there is a wrong or empty text field, similarly to saving, the problem is indicated by an alert window. If there is no wrong or empty text field, a pop-up window indicates that the Simulation setup is generated. Then the disabled right panel (the parameter settings of the pollutant cloud (Figure A1) or the data for reading particles of a pollutant cloud (Figure A2)) becomes enabled.


**Figure A1.** New simulation—set parameters. Starting a new simulation by setting the parameters of the simulation and the pollutant cloud.


**Figure A2.** New simulation - read particles. Starting a new simulation by setting the parameters of the simulation and reading the particles of the pollutant clouds from file.

Appendix A.2.2. Setting the Parameters of the Pollutant Cloud

In case of choosing the New simulation—set parameters option (Figure A1) the user should set the following parameters of the particles which will fill a rectangular cuboid:


By default, the fields are filled with the default setting parameters loaded from default/ default\_particle\_parameter\_setup.txt. The usage of the Default, Save and Load buttons are analogous to the ones in the simulation setup panel.

For starting a simulation the generation of the pollutant cloud is required: the user can generate it by clicking the Generate particles button. If there is a wrong or empty text field, the problem is indicated by an alert window. Otherwise a pop-up window indicates that the pollutant cloud is generated (Number of particles: <particle number>.). Then the disabled bottom right panel (for setting the display properties of the simulation and starting the simulation calculation (Figure A1)) becomes enabled.

Appendix A.2.3. Reading the Particles of the Pollutant Cloud From File

In the case of reading the particles of the pollutant cloud (Figure A2), the user chooses the file containing the initial conditions of its particles by clicking the Choose input file button. The default file path is in the file default/default\_particle\_file\_setup.txt. The formats and the values of data in the file should meet the requirements which are listed in Appendix A.1.3. The particle data are read from the file by clicking the Read particles button. In the case of wrong values an alert window indicates the problem: There were invalid data while reading from file <file> <wrong lines>. If every line is correct, a pop-up window indicates that the pollutant cloud is generated (Number of particles: <particle number>.). Then the disabled bottom right panel (for setting the display properties of the simulation and starting the simulation calculation (Figure A2)) becomes enabled.

This way of generating a pollutant cloud is worth applying when the user does not want to initialize the particles filling a rectangular cuboid (mentioned in the previous section) rather than the user needs particles with arbitrary positions. For example, in this way several different particle groups (i.e., different pollutant clouds) initialized at different locations can be tracked simultaneously.

#### Appendix A.2.4. Starting a New Simulation

In the bottom right panel (Figure A1 and Figure A2) the user should check whether if she/he wants to watch the spreading of the pollutant cloud during the simulation calculation (Display during calculation?) and if yes, how many particles of the pollutant cloud should be drawn (Number of particles to display, format: integer). By clicking the Start simulation button the simulation calculation starts and the progress (and the particle cloud if chosen) can be tracked in a new window (Figure A3).

**Figure A3.** Example for tracking the calculation of the simulation with displaying the particle positions. The simulation and the pollutant cloud are initialized with the parameters in Figure A1. The colorbar indicates the altitude of the particles.

#### Appendix A.2.5. Setting the Display Properties of the Simulation

If the user has chosen the display option, the position of the particles of the pollutant cloud is displayed continuously colored according to the vertical coordinate of the particles on a map (Figure A3) during the simulation calculation. Otherwise only a progressbar is visible to show the percentage of the progress of the calculation and the corresponding date and time in the simulation. The user can stop and continue the calculation by clicking the Stop or Continue buttons, respectively, and she/he can load the particle data saved in files for replay by clicking the Load button. The longitudinal and latitudinal boundaries of the map, the vertical boundaries of the coloring and the marker size of the particles can be changed on the left side (formats: real numbers in the given intervals). The settings are applied by clicking the Apply button. If there is a wrong or empty text field, the problem is indicated by an alert window.

#### *Appendix A.3. Replaying a Saved Simulation*

A saved simulation can be loaded by clicking the Load simulation menu item in File menu on the new simulation screen or by clicking the Load button in the case of an ongoing simulation. Then the parameters defining the saved simulation appear in a new window (Figure A4). For ongoing simulation text fields are filled with the parameters of the simulation, otherwise they are filled with the values loaded from the default/default\_load\_setup.txt. The following parameters should be given:



**Figure A4.** Setting simulation data for loading saved simulation.

The selected simulation is loaded by clicking the Load button. If the Length file/Escape file is checked, the time dependence of the natural logarithm of the length of the pollutant cloud/the time dependence of the natural logarithm of the ratio of the non-escaped particles also appears on the display panel (Figures A5 and A6). By default, the forward loop of the spreading of pollutant cloud is displayed according to the given Frame rate. The user can stop (Stop) and continue (Continue) the replay, and can move frame by frame the replay forward/backward by clicking the Previous/Next buttons. The instantaneous position of the pollutant cloud can be saved as an image by the Save image button. The properties of the display can be changed similarly as described in Appendix A.2.5 by clicking the Apply button. Beyond those options the speed of the animation (Frame rate) can be modified, too.

If the user has loaded length data/data for the ratio of the non-escaped particles from file, she/he can select a start and end date and time from two lists in the bottom of the panel in Figure A5 or Figure A6. By clicking the Calculate topological entropy/Calculate escape rate button a line is fitted to the graph between the given time instants using the least squares approach (Sections 3.2 and 3.3) and its slope (i.e., the topological entropy *h* (Figure A5))/(−<sup>1</sup>)<sup>×</sup> slope (i.e., the escape rate *κ* (Figure A6)) appears. The value of the obtained topological entropy *h* in Figure A5 means that in every *h*−<sup>1</sup> = 1.242 days the length of the pollutant cloud stretches by a factor of e ≈ 2.718. Analogously, the value of the escape rate *κ* in Figure A6 implies that within *κ*<sup>−</sup><sup>1</sup> = 3.745 days after the start time of the

fit only e<sup>−</sup><sup>1</sup> ≈ 0.368 of the particles (which are non-escaped at the start time of the fit) are still in the air. The graphs can be saved by clicking the Save image button.

**Figure A5.** Displaying saved simulation and the time dependence of the length of a pollutant cloud. The simulation is 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 is greater than 100 km. The pollutant cloud is initialized as a meridional line segmen<sup>t</sup> of 400 km at 47◦ N, 19◦ E and at the altitude of 5500 m. It consists of 1000 particles at the beginning. The particle radius is 0 μm.

**Figure A6.** Displaying saved simulation and the time dependence of the ratio of non-escaped particles. The simulation is initialized with the simulation parameters on the left of Figure A1 but with an end date of "2010.04.30.06:00:00" and with calculating the ratio of non-escaped particles. The pollutant cloud is initialized as 250 × 250 × 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). The particle radius and density is 7 μm and 2000 kg m<sup>−</sup>3, respectively.
