## 5 Discussion  
The results of the EC runs at Changbaishan show that the areas of invasion from PDCs are influenced by the different vent position only in scenarios 1 and 2. This feature reflects the relatively complex morphology of the top of the volcano.Therefore, we conclude that the up-to-400 m high caldera rim surrounding the Heaven Lake and the caldera breach to the north control the distribution of the PDCs invasion areas on the upper flanks of the volcano, at least for PDCs originated by the collapse of 3-10 km high eruptive columns. The water outlet feeding the Erdaobaihe River could play a role in transforming the potential PDCs flowing in the Erdaobaihe valley in lahars. This conclusion is supported by the field data related to the Millennium eruption, where the PDC-lahar transition has been recognized in the field [51]. Therefore, this type of hazard should also be taken into account at Changbaishan. In the runs, we have assumed that the water of the Heaven Lake does not play a  
Figure 8: Probability map of invasion by PDCs in scenario 4 (Ht = 30 km) (probability scale at the bottom) superimposed to the China-North Korea infrastructural network. role in the dynamics of the eruptive column, e.g., magma-water interaction, or PDC flow dynamics. This is a limit of our analysis which cannot be quantified. However, with the exception of vent 1, which is located at the center of the lake, the other 16 vents are located outside the lake area (Figure 3). Our results from runs with the larger \(\Delta\)H/L values give us a rough estimate of the dispersion areas of potentially denser flows. According to equation 2, the flow mobility is directly expressed by the angle \(\phi\), a measure of the energy decay rate. This is because larger values of \(\phi\) imply faster energy decay rate and less mobile flows [23]. In this regard, we analyze the relationship between the median, larger and smaller \(\Delta\)H/L values according to the database of Ogburn [25] and Ogburn and Calder [24], and the distance from the center of the Changbaishan caldera taken as common reference point (Figure 9). In scenario 1, the maximum distance is within 9-12 km, while in scenario 2, it ranges between 13 and 22 km. In scenarios 3 and 4, the distances are in the ranges 22-42 km and 38-85 km. As shown in Figure 9, this increasing difference between the maximum distances theoretically travelled by PDCs with different \(\Delta\)H/L values in each scenario does reflect not only the different volumes of PDCs, but also the occurrence of a flat, obstacle-free, topography moving away from the caldera. In addition, the results of the scenario 4 runs (Figure 4) show that some PDCs are potentially able to partly fill the valleys located around the Changbaishan plain in the Gaina basaltic plateau and Sino-Korean Craton (Figure 1). The results of the runs allow us to obtain some empirical relations between the calculated area covered by PDCs and Ht as a function of the different vents and selected \(\Delta\)H/L. These relations are shown in Figure 10, where the values Ht adopted for the different scenarios are reported versus the areas covered by PDCs. In the diagram, the best-fit lines are obtained by interpolating the minimum, median, and maximum areas values in the different scenarios. These lines, even if obtained by interpolating few points, may be used to obtain an order of magnitude estimate of the area potentially affected by PDC invasion if data on the column height are available and the assumption that PDCs originate by column collapse holds. The empirical relations of Figure 10 clearly show that  
Figure 9: \(\Delta\)H/L vs. maximum distance from the center of the caldera travelled by PDCs according to the results of simulations in scenarios 1–4. The volumes V corresponding to the selected \(\Delta\)H/L values are reported and calculated from the equation \(\Delta\)H/L = 0.11 (V\({}^{-0.14}\)) of [24]. the higher the column height, the smaller the difference between the maximum and minimum area potentially affected by PDC.