Dynamics and Deposits of Pyroclastic Density Currents in Magmatic and Phreatomagmatic Eruptions Revealed by a Two‐Layer Depth‐Averaged Model

A pyroclastic density current (PDC) is characterized by its strong stratification of particle concentration; it consists of upper dilute and lower dense currents, which generally control the dynamics and deposits of PDCs, respectively. To explain the relationship between the dynamics and deposits for magmatic and phreatomagmatic eruptions in a unified way, we developed a two‐layer PDC model considering thermal energy conservation for mixing of pyroclasts, external water, and air. The results show that the run‐out distance of dilute currents increases with the mass fraction of external water at the source (wmw) owing to the suppression of thermal expansion of entrained air. For wmw ∼ 0.07–0.38, the particle concentration in the dilute current becomes too low to generate the dense current so that the deposits directly form at the bottom of the dilute current in the entire area. These results capture the diverse features of natural PDCs in magmatic and phreatomagmatic eruptions.


Introduction
During explosive volcanic eruptions, a mixture of hot fragmented magma (pyroclasts) and gas ejected as an eruption column from the volcanic vent can collapse and propagate along the ground surface as a pyroclastic density current (PDC). The dynamical features of PDCs are highly variable because they are controlled by the eruption conditions, physical processes of PDCs (e.g., particle sedimentation, ambient air entrainment, and thermal expansion of entrained air), and topography (e.g., Dufek, 2016;Lube et al., 2020). These factors cause PDCs to form extremely diverse deposits (e.g., Branney & Kokelaar, 2002;Cas & Wright, 1987;Fisher & Schmincke, 1984;Sulpizio et al., 2014).
The effect of external water (e.g., groundwater, lakes, and oceans) on eruption styles (magmatic vs. phreatomagmatic eruptions) is a key factor leading to the diverse distribution and sedimentary structures of PDC deposits. Magmatic eruptions produce PDC deposits with high temperatures of ∼700-1200 K, whereas phreatomagmatic eruptions produce deposits with low temperatures of ∼300-700 K (e.g., Koyaguchi & Woods, 1996;Trolese et al., 2017Trolese et al., , 2019. Experimental and numerical simulations of PDCs (Andrews, 2014;Esposti Ongaro et al., 2016;Ishimine, 2005) suggest that the run-out distance of PDCs (i.e., the length of PDC deposits) increases as the source temperature decreases (i.e., the water:magma mass ratio increases). For phreatomagmatic eruptions, PDC deposits show spatial variation in sedimentary structures from poorly sorted massive facies to well sorted (cross-)stratified facies, as seen in base surge deposits (e.g., Wohletz & Sheridan, 1979). Understanding these Abstract A pyroclastic density current (PDC) is characterized by its strong stratification of particle concentration; it consists of upper dilute and lower dense currents, which generally control the dynamics and deposits of PDCs, respectively. To explain the relationship between the dynamics and deposits for magmatic and phreatomagmatic eruptions in a unified way, we developed a two-layer PDC model considering thermal energy conservation for mixing of pyroclasts, external water, and air. The results show that the run-out distance of dilute currents increases with the mass fraction of external water at the source (w mw ) owing to the suppression of thermal expansion of entrained air. For w mw ∼ 0.07-0.38, the particle concentration in the dilute current becomes too low to generate the dense current so that the deposits directly form at the bottom of the dilute current in the entire area. These results capture the diverse features of natural PDCs in magmatic and phreatomagmatic eruptions.
Plain Language Summary Explosive volcanic eruptions eject mixtures of hot fragmented magma and gas from the volcanic vent and form eruption columns, which can collapse and propagate along the ground surface as pyroclastic density currents (PDCs). The dynamics and deposits of PDCs are extremely diverse depending on the amount of external water (e.g., groundwater, lakes, and oceans) that mixes with magma. To explain the diverse features of the dynamics and deposits of PDCs for various amounts of external water, we developed a two-layer model for PDCs considering thermal energy conservation for mixing of magma and external water. The two-layer model successfully reproduces the dynamics and deposits of PDCs with strong stratification of particle concentrations in a unified way. The results show that the run-out distance of upper dilute currents increases with the increasing amount of external water. For a relatively small proportion of external water, the lower dense current tends to be absent, resulting in the direct formation of the deposits from the dilute current in the entire area. These model predictions are useful to mitigate the diverse hazards caused by natural PDCs under various geological conditions. SHIMIZU ET AL. © 2023. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Dynamics and Deposits of Pyroclastic Density Currents in Magmatic and Phreatomagmatic Eruptions Revealed by a Two-Layer Depth-Averaged Model
• A two-layer pyroclastic density current model with heat conservation for mixing of pyroclasts, external water, and air is developed • In phreatomagmatic eruptions, the upper dilute current flows over longer distances and the lower dense current tends to be absent • Our results explain the diverse features of the dynamics and deposits of natural pyroclastic density currents in a unified way

Supporting Information:
Supporting Information may be found in the online version of this article. diverse features of the dynamics and deposits of PDCs for magmatic and phreatomagmatic eruptions in a unified way is one major volcanological subject.
The relationship between the dynamics of PDCs and their deposits is not straightforward. The major difficulty comes from the fact that PDCs generally have strong stratification in terms of particle concentration (e.g., Branney & Kokelaar, 2002). Such PDCs comprise two main regions; an upper thick region of low particle volume fractions (≲10 −2 ) and a lower thin region of high particle volume fractions (up to ∼0.5). The upper dilute region behaves as a dilute turbulent suspension current that is controlled mainly by settling of particles, entrainment of ambient air, and thermal expansion of entrained air (e.g., Andrews & Manga, 2012). Through these physical processes, the dilute region partially becomes buoyant and lifts off the ground, which can control the run-out distance of the whole PDCs (e.g., Bursik & Woods, 1996;Dade & Huppert, 1996). On the other hand, the lower dense region behaves as a dense gas-pore pressure-modified (i.e., fluidized) granular current that is controlled mainly by particle-particle and gas-particle interactions, frictional interaction between the current and the ground, and deposition at the base (e.g., Lube et al., 2019;Roche et al., 2010). The region directly affects the features of PDC deposits (e.g., sedimentary structure; Branney & Kokelaar, 2002).
To describe the global features of stratified PDCs comprising the upper dilute and lower dense regions and their deposits, numerical two-layer depth-averaged models have been developed (e.g., Doyle et al., 2008;Kelfoun, 2017;Shimizu et al., 2019). In the two-layer models, the continuous stratification of particle concentration and density in PDCs is modeled as upper and lower depth-averaged layers coupled through mass and momentum exchanges on the basis of the idea that the two regions in PDCs are controlled by different physical processes. This paper extends a two-layer model for large-scale PDCs in magmatic eruptions (Shimizu et al., 2019) to both magmatic and phreatomagmatic eruptions. The new model provides a theoretical framework for understanding the relationship between the diverse features of the dynamics and deposits of large-scale PDCs for magmatic and phreatomagmatic eruptions in a unified way.

Methods
We develop a two-layer model for large-scale PDCs in magmatic and phreatomagmatic eruptions by combining the two-layer PDC model of Shimizu et al. (2019) with a thermodynamical model for magmatic and phreatomagmatic eruptions (i.e., the thermal energy conservation for mixing of pyroclasts, volcanic gas (water vapor), external water, and ambient air in the collapsing eruption column; Koyaguchi & Woods, 1996). The model is designed to describe an axisymmetric PDC spreading from a collapsing column on a flat ground surface ( Figure 1). The source column consists of magma (i.e., pyroclasts and volcanic gas), external water, and air entrained into the column. It produces a radial dilute current from the column edge r = r 0 at a constant mass flow rate during time >0, where r is the distance from the center of the column. The formation of the dense basal current is determined by the balance of the particle settling rate at the bottom of the dilute current and the aggradation rate of the deposits at the bottom of the dense current. The deposits progressively aggrade upward from the bottom of the dense current or from the bottom of the dilute current in case a dense current is not formed (see Text S1 in Supporting Information S1 for details).
The conditions of the source dilute mixture are given as follows. Hot magma with temperature T m = 1000 K and water mass fraction w m = 0.03 mixes with cold external water with temperature T w = 273 K and is ejected from the vent; this study considers a wide range of mass fractions of external water in the mixture of magma and external water (w mw = 0-0.6). The ejected material mixes with ambient air with temperature T a = 273 K to form a collapsing column (air mass fraction at the column edge n a0 = 0.07), which in turn forms a dilute current. We assume that the collapsing column generates a homogeneous dilute mixture for simplicity (Shimizu et al., 2019), although the collapsing column can generate a dense basal current at the bottom (e.g., Valentine, 2020). The conservation of thermal energy between magma, external water, and air at atmospheric pressure gives the mass fractions of water vapor n v0 and liquid water n w0 and the temperature T 0 as a function of w mw at r = r 0 (Figure 1c), where the vaporization of liquid water, the condensation of water vapor, and the resulting latent heat are taken into account. As w mw increases from 0 to ∼0.2, T 0 decreases from ∼950 to ∼373 K (∼100°C), n v0 increases, and n w0 remains zero, because all the external water mixed with the magma vaporizes. When w mw is larger than ∼0.2, T 0 hardly changes because T 0 is less than 100°C and water vapor and liquid water coexist; n v0 decreases and n w0 increases as w mw increases from ∼0.2. The other conditions of the source dilute mixture (i.e., the thickness h 0 , velocity u 0 , and column radius r 0 ) are obtained from the magma discharge rate (̇m = 10 9 kg s −1 ), the Richardson number of the dilute current at r = r 0 (Ri 0 = 1), and the aspect ratio of h 0 to r 0 (a 0 = 0.2).
The basic equations of the two-layer PDC model developed by Shimizu et al. (2019) are extended by considering the condensation and vaporization of water in the dilute mixture (i.e., in the source and the propagating current) with low temperatures below 100°C. To obtain the spatiotemporal variation of the mass fraction of liquid water in the dilute current, the condensation and vaporization rates of water are considered in the equation system on the basis of a moist eruption column model (Koyaguchi & Woods, 1996). Both the dynamics and deposits of two-layer PDCs are strongly controlled by the sedimentation process characterized by the settling speed of particles at the bottom of the dilute current (W s ) and the aggradation speed of the deposits at the bottom of the dense current (D); our representative simulations assume W s = 0.5 m s −1 (i.e., the terminal velocity for the particle diameter of 0.1 mm and the particle density of 2,600 kg m −3 ) and D/W s = 1.22 × 10 −3 . To investigate the effects of external water, we perform parametric study for w mw = 0-0.6. We also assess the effects of the uncertainties of other parameters (i.e., w m , T m , n a0 , ̇m , Ri 0 , a 0 , W s , and D/W s ) on our conclusion. For details see Texts S1-S3 in Supporting Information S1.

Results
Representative results for phreatomagmatic eruptions (w mw = 0.3) show that as a dilute current spreads radially from the column edge r = r 0 (Figures 2a and 2b; Movie S1), the density of the dilute current decreases through particle settling, air entrainment, and thermal expansion of entrained air. When the frontal region of the dilute current becomes lighter than the ambient air to lift off the ground (i.e., a co-ignimbrite ash plume forms), the front of the dilute current stops spreading and the dilute current converges to a steady state (Figure 2c), where the sum of the radial mass flux of particles from the front of the dilute current to the co-ignimbrite ash plume and the total particle settling rate at the bottom of the dilute current is balanced by the radial mass flux of particles from r = r 0 . The particles settling from the bottom of the dilute current form the deposit.
The results for phreatomagmatic eruptions have the following two differences from typical results for magmatic eruptions (w mw = 0; Figures 2d-2f; Movie S2). First, the lift-off of dilute currents with large w mw is delayed (i.e., the run-out distance of dilute currents increases with w mw ; the red curve in Figure 3a). Second, the dense current can become absent for phreatomagmatic eruptions (i.e., the run-out distance of dense currents decreases as w mw increases from 0 to ∼0.07, it remains zero for w mw ∼0.07-0.38, and it increases as w mw increases from ∼0.38; the blue curve in Figure 3a).
The increase in the run-out distance of dilute currents with w mw is due to low temperatures in currents (i.e., the result of the reduced thermal expansion of entrained air and the increase in the mass fraction of liquid water). The dilute current density ρ is approximated as p/((n a + n v )RT), where p is the atmospheric pressure, n a and n v are the   , t); red), dense current (h H (r, t); blue), and deposits (z b (r, t); black) are shown. The run-out distance of steady-state dilute current and that of steady-state dense current are expressed by r ∞ and r ∞H , respectively. mass fractions of entrained air and water vapor, R is the gas constant of the mixture of air and water vapor, and T is the dilute current temperature. This approximated equation remains valid under the conditions of the present simulations, where the volume of the solid-liquid mixture is negligibly small compared with that of the gas phase. For large w mw (low T), the dilute current entrains a large amount of air and has large n a before ρ becomes smaller than the ambient air density ρ a ; consequently, ρ/ρ a is maintained above 1 over long distances (Figures 4a-4c). Furthermore, as w mw increases from ∼0.2, the mass fraction of liquid water n w increases (n v decreases; Figure 1c), resulting in the increase in ρ/ρ a and the delay of lift-off of dilute currents (Figures 4a-4c). The delay of lift-off of dilute currents also leads to the decrease in proportions of co-ignimbrite ash-fall deposits with w mw (Figure 3d). The absence of any dense current for w mw ∼ 0.07-0.38 (see the gray region in Figure 3a) is explained by the balance of the particle settling rate at the bottom of the dilute current at r = r 0 (ϕ s0 W s ) and the aggradation rate of the deposits at the bottom of the dense current (ϕ sD D) (i.e., ϕ s0 W s < ϕ sD D; Shimizu et al., 2019). In this balance, ϕ sD and D/W s are independent of w mw (the black dashed line in Figure 4d). Therefore, the condition for the absence of dense currents depends on the solid volume fraction in the dilute current at r = r 0 (ϕ s0 ); ϕ s0 decreases as w mw increases from 0 to ∼0.15 owing to the increase in n v0 , and it increases as w mw increases from ∼0.15 owing to the decrease in T 0 and the increase in n w0 (Figures 1c and 4d). In the range of ϕ s0 < ϕ sD D/W s (w mw = 0.07-0.38), the dense current is absent throughout r > r 0 . Our result is consistent with previous experimental and numerical studies (Breard et al., 2018;Lube et al., 2015;Valentine, 2020). Although the above results are quantitatively affected by the uncertainties of parameters other than w mw , they are not changed qualitatively (see Text S2 in Supporting Information S1).

Geological Implications
We have obtained two fundamental results: the run-out distance of dilute currents increases as w mw increases (T 0 decreases), and the dense current tends to be absent for intermediate values of w mw . These results can provide a unified framework for understanding the diverse features of the dynamics and deposits of PDCs in magmatic and phreatomagmatic eruptions.
The results for cold dilute currents with large w mw explain the low emplacement temperatures of PDC deposits for phreatomagmatic eruptions (e.g., Trolese et al., 2017Trolese et al., , 2019. They also explain the broad dispersal of PDC deposits (i.e., the long run-out distance of PDCs) for phreatomagmatic eruptions. It is widely known that the run-out distance of PDCs correlates well with the magma discharge rate ̇m (e.g., Giordano & Cas, 2021;Roche et al., 2021;Shimizu et al., 2019). Our results (e.g., Figure 3a) imply that the run-out distance of PDCs in phreatomagmatic eruptions are systematically longer than that in magmatic eruptions for given ̇m . Roche et al. (2021) analyzed data from some large-scale PDCs to determine the relationships between the run-out distance and ̇m and showed that dilute PDCs over water (the 161 ka Kos, 7234 BP Koya (unit C3), AD 1883 Krakatau, 2050 BP Okmok II, and Tosu-Aso 4II-1 PDCs) tend to have about twice longer run-out distances than dilute PDCs on land for given ̇m . This tendency has been interpreted to be due to the delay of particle settling owing to steam produced at the bottom of hot PDCs in contact with water (Dufek et al., 2007;Roche et al., 2021). Considering that some of the dilute PDCs over water (the 161 ka Kos, 7234 BP Koya (unit C3), AD 1883 Krakatau) could be generated by phreatomagmatic eruptions, their longer run-out distances can be explained partly by the presence of external water; our result ( Figure 3a) suggests that, when only the thermal effect of the external water is taken into account, the twice longer run-out distances are accounted for by w mw = 0.15-0.20. More quantitative evaluation of each mechanism awaits further study.
Our results for the presence and absence of dense currents explain diverse sedimentary structures in PDC deposits for magmatic and phreatomagmatic eruptions. The area where only the dilute current is present (i.e., the dense current is absent) becomes wider as w mw increases (Figure 3a). The deposits in the area with the dense current are produced by aggradation from the bottom of the dense current (the proximal area of Figure 3b), whereas the deposits in the area without the dense current are produced directly by particle settling from the bottom of the dilute current (the distal area of Figure 3b). Generally, the differences in the flow-particle interactions within the basal boundary layer between dilute and dense currents are thought to reflect the wide variety of sedimentary structures in PDC deposits (Branney & Kokelaar, 2002). Deposition from the bottom of dilute currents forms (cross-)stratified facies (called "surge facies") through alternating series of tractional bedload transport and shifting sandwaves (Brosch & Lube, 2020) whereas deposition from the dense current forms massive, poorly sorted facies (called "flow facies") due to inhibited traction (Branney & Kokelaar, 2002). Thus, our results for large w mw explain the observation that (cross-)stratified facies are widely produced in phreatomagmatic eruptions regardless of ̇m (De Rita et al., 2002;Valentine & Fisher, 2000;Wohletz, 1998). Furthermore, our results for the absence of dense currents for intermediate values of w mw (Figures 3a and 3c) explain natural PDC deposits with (cross-)stratified facies in the entire area.

Conclusions
Our two-layer PDC model evaluates the effect of strong stratification of particle concentrations in PDCs and captures the diverse features of PDCs in magmatic and phreatomagmatic eruptions. The run-out distance of upper dilute currents for phreatomagmatic eruptions is longer than that for magmatic eruptions for a given magma discharge rate. The sedimentary structures for phreatomagmatic eruptions are controlled by two factors related to the variation of w mw : (a) the liquid water is absent (i.e., n w = 0) for w mw ≲ 0.2, whereas it is present (i.e., n w > 0) for w mw ≳ 0.2 (see Figure 1c for n w0 ), and (b) the dense current is absent for w mw ∼ 0.07-0.38, whereas it is present for w mw ≲ 0.07 or ≳0.38 (see Figure 3a). The combination of these factors, particularly the latter factor, accounts for the diversity of sedimentary structures (e.g., the broad distribution of the (cross-)stratified facies) commonly observed in phreatomagmatic PDC deposits. The model is expected to be a useful tool to analyze the dynamics and deposits for large-scale PDCs in magmatic and phreatomagmatic eruptions in a unified way.

Data Availability Statement
All data and post-processing scripts used to produce the figures of this paper are available in Zenodo (Shimizu, 2023; https://doi.org/10.5281/zenodo.7928713).