Abstract
Explosive supereruptions can erupt up to thousands of km^{3} of magma with extremely high mass flow rates (MFR). The plume dynamics of these supereruptions are still poorly understood. To understand the processes operating in these plumes we used a fluiddynamical model to simulate what happens at a range of MFR, from values generating intense Plinian columns, as did the 1991 Pinatubo eruption, to upper endmembers resulting in coignimbrite plumes like Toba supereruption. Here, we show that simple extrapolations of integral models for Plinian columns to those of supereruption plumes are not valid and their dynamics diverge from current ideas of how volcanic plumes operate. The different regimes of air entrainment lead to different shaped plumes. For the upper endmembers can generate local uplifts above the main plume (overplumes). These overplumes can extend up to the mesosphere. Injecting volatiles into such heights would amplify their impact on Earth climate and ecosystems.
Introduction
Explosive supereruptions eject from several hundreds to thousands of km^{3} of magma at extremely high flow rates^{1}. Many of these eruptions have had significant impacts to the climate and ecosystems^{2,3,4,5}. Explosive supereruptions cover areas within hundreds km from the vent with thick pyroclastic flows, blanket continentsize regions with ash, and inject large quantities of aerosols into the atmosphere^{4}. Volatiles injected into the stratosphere can alter the Earth climate on a global scale even causing a volcanic winter that can persist for years to decades^{2,3}. On the other hand, tephra layers associated with these catastrophic events are invaluable chronological markers across the affected regions^{6,7}. The mass erupted during supereruptions is orders of magnitude larger than the biggest eruptions experienced in historic times^{1,4,8,9,10}. Estimates of mass flow rates (MFRs) during these supereruptions, obtained from different independent approaches, suggest that they are extremely high, ranging from 10^{9} to 10^{11} kg/s^{4,8,10,11,12}. Such large MFRs require multiple vents or continuous emission along dykes^{13,14}.
Plume dynamics of explosive supereruptions are not well understood as such large events have not been witnessed. In order to understand how such volumes of material are ejected and dispersed we rely on field evidence and models that can produce the observed deposits. Our current understanding on how plumes of explosive supereruptions behave is from extrapolations of simple integral models developed for describing columns generated from small MFR. These simple models^{15,16,17} are based on the Buoyant Plume Theory (BPT) but the similarity assumption behind has been shown not to be valid for large MFRs^{18}.
Large explosive eruptions produce Plinian columns when the erupted mixtures of fragmented hot magma and gas entrain air, which heats up and expands making the plume buoyant. Above a critical MFR the eruption column becomes unstable^{19} and collapses, producing pyroclastic flows that spreads laterally on the ground. At high MFR, the dilute parts of the hot pyroclastic flows can also become buoyant as they also entrain air, forming a coignimbrite eruption plume that can rise up to the stratosphere carrying massive quantities of elutriated fine ash and volatiles.
Results
Fluiddynamical regimes of eruptive plumes for large MFRs
Here, in order to avoid making unrealistic assumptions, we investigate the plume dynamics using a threedimensional computational fluiddynamical code (see Methods) designed to describe the evolution of volcanic plumes and umbrella clouds^{19}. The code simulates the injection of a wellcoupled mixture of solid pyroclasts (ash) and volcanic gas (assumed to be water vapour) from vents of different shapes above a flat surface into a stratified atmosphere. The model does not consider particle sedimentation and particle decoupling^{20} but captures the plume dynamics (see Methods). In this study we will not consider the effects of the rotation of the Earth on the plume dynamics, which can be very significant for very large eruptions, affecting, among other things, their spreading and shape of the plume^{21,22}. For this reason and the for the sake of simplicity, here we focus on eruptions occurring in the equatorial belt, where these effects are negligible^{4,23} and consider tropical windless atmospheric conditions only (see Methods).
Considering the input parameters reported in Table 1 and atmospheric properties described in Methods, we explored the effects of variable MFRs for different vent geometries, such as a single circular vent, fissure, and vents at different distances. For the sake of simplicity, we focus on the results from a circular vent but these are rather general.
The fluid dynamics of large Plinian columns fed by a MFR~10^{9} kg/s have been described in several studies^{19,23,24}. In these columns, the fountainlike structure (radially suspended flow^{24,25}) generated in the lower part of the column is characterised by a highconcentration of erupted mixture and it is denser than the ambient air. In this region, the erupted mixture mixes with the air in largescale vortexes and this mixture becomes rapidly buoyant (see Fig. 1a and Supplementary Movie 1). Approaching the critical MFR at ~10^{9.5} kg/s (see Methods), the radially suspended flow^{24,25} becomes unstable producing partial collapses, but the main plume still survives. The lower central part of the plume is a Negatively Buoyant Region (NBR), while the area around it is fed with relatively pure air that maintains its buoyancy and efficiently transports the mixture up to the stratosphere (see Fig. 1b and Supplementary Movie 2). The highest velocity remains in the central region generating a mushroom shape plume, with very high mass fractions in the central part up to the top of the plume (see Fig. 1b and Supplementary Movie 2).
Increasing MFR up to 10^{10} kg/s produces a total collapse of the radially suspended flow, which generates continuous fountaining to the ground, feeding pyroclastic density currents and increasing the radius of the hotter NBR, resulting in a basal region (~60 km diameter) from where the large coignimbrite plume will develop (see Fig. 1c). In this case, because of the vigorous rising velocities in the periphery owing to the more effective local air entrainment, the upper central portion of such a large plume has a relatively low mass fraction compared with the outer region (Fig. 1c and Supplementary Movie 3). The resulting plume still maintains a mushroom shape but the plume top has flat, rather than umbonate, cap (Fig. 1c and Supplementary Movie 3).
A further increase of MFR up to 10^{11} kg/s will produce a larger coignimbrite plume (>150 km in diameter). In this case, the vortices entrain air mainly at the periphery of the coignimbrite plume, without affecting the area around the hotter NBR (Fig. 1d and Supplementary Movies 4 and 5). This allows the periphery region of the coignimbrite plume to become much more buoyant and increase its velocity. Because of mass conservation, the vertical velocity in the inner part of the plume decreases. This regime results in the formation of a sort of toroid umbrella (donutlike shape), giving to the plume a depressedcap mushroom shape (i.e., two separate lobes in a 2D crosssection).
Our simulations show that vent geometry has a strong control on the dynamics and stability of the Plinian columns but once coignimbrite plumes are generated the processes are predominantly controlled by the diameter of the coignimbrite plume, which is typically larger that the vent area, i.e., longer than the fissures or the distance between multiple vents (Supplementary Movies 1–6). For these reasons, the described features apply to all vent geometries despite the fact that a circular vent was used for the simulations (see Supplementary Movie 6).
Implications for the assessment of eruption parameters
MFRs are typically estimated from the total plume heights assuming BPT is valid. A similar approach has been extended to coignimbrite plume^{15} and is still largely used in the volcanological community^{16,17}. However, the dynamics of large coignimbrite plumes are markedly different from BPT as their horizontal extension is typically much larger than their height. The simulations indicate that, above a critical MFR, the coignimbrite plume rises from a source with radius \({R_{{{\rm CI}}}}\), increasing with MFR as by \({R_{{{\rm CI}}}} \approx 2.8 \cdot 10^{  4}\sqrt {{{\rm MFR}}}\) (with the coignimbrite plume radius, \(R_{{{C\rm I}}}\), expressed in km and MFR in kg/s, see Fig. 2); this powerlaw dependence of the MFR with runout distance was predicted in simple models of pyroclastic flows^{26}. The mechanisms of air entrainment from such broad sources are profoundly different from, and invalidate the similarity assumption used in, BPT. The difference in the scale of the horizontal extension of plume also affects the behaviour of the upper part of the plume, including the column height and the dynamics in the umbrella cloud.
Here, we focus on the dependence of maximum plume height with MFR (Fig. 3 and Fig. 4 for the dynamics of umbrella cloud). The result shows that from 10^{9} to 10^{10} kg/s the maximum plume heights remain similar and are between 40 and 60 km. Coignimbrite plumes appear steadier than Plinian columns, which show a more oscillatory behaviour (see Supplementary Movies 1–5). The maximum column height and highest mass fraction in the umbrella region is reached for MFR around the critical value, i.e., 10^{9.5} kg/s. Remarkably, the Neutral Buoyancy Level (NBL) remains at ~20 km for all the simulations with MFRs above the critical MFR (see Table 2).
For MFR of ~10^{11} kg/s, maximum plume height is in the peripheral region rather than in the centre, owing to the more efficient entrainment of air from the border of the plume. The maximum height for the bulk mass is at ~50 km but local uplifts, having a diameter of ~30–40 km, develop above the umbrella region (see Supplementary Figure 1) and keep rising up to the mesosphere (60–70 km). In this case, two different effective heights should be considered, one for the bulk mass spreading around the umbrella region and one for the maximum height reached by the local uplifts (we call them “local overplume” hereafter). These local overplumes develop from the base of the periphery of the coignimbrite plumes, because of local heterogeneities in the efficiency of air entrainment, and are characterised by higher velocities and larger mass fractions (see Supplementary Figure 1).
The complex relationship between plume height and MFR in Fig. 3 suggests that for large eruption intensities (MFR > 10^{9} kg/s) we cannot use column height estimations to assess the value of MFR^{15,27}. To compare our results from the threedimensional simulations with those of the simple BPT integral models^{15,27}, we estimated the main mean variables^{20}, such as mixture density difference, \({\mathrm{\Delta }}\rho\), vertical velocity, U, temperature, T, and mass fraction ξ (see Supplementary Figure 2) and extracted optimal parameter values (see Fig. 3 and Table 3). These BPT models do not adequately describe coignimbrite plumes but if they are used as extrapolations the effective entrainment coefficient, k, should be properly tuned and not assumed as an invariant. Since variations of H_{NBL} with MFR from sustained Plinian column to fully coignimbrite plume are almost negligible (H_{NBL} ~15–20 km, see Table 2), accordingly to BPT, this implies that the effective entrainment coefficient should increase with MFR and have significantly different values for the two regimes.
The total column height, H_{t}, varies with MFR (see Table 2), rising from \(H_{\rm t} \cong 40  45\) km at \({\mathrm{MFR}} = 10^9\) kg/s to \(H_{\rm t} \cong 50  60\) km at \({\mathrm{MFR}} = 10^{9.5}\) kg/s, then decreasing \(H_{\rm t} \cong 40\) km at \({\mathrm{MFR}} = 10^{10}\) kg/s and increasing again to \(H_{\rm t} \cong 50  60\) km at \({\mathrm{MFR}} = 10^{10.5}\) kg/s. To estimate plume height ranges, we considered a fraction mass of \(\xi = 10^{  3}\) for the dilute upper region and \(\xi = 10^{  1}\) for the lower values, which refer to a more concentrated plume region. This implies that the behaviour of the plumes above NBL and the plume height is controlled by an effective entrainment coefficient, k_{U}, which is different from the entrainment coefficient, k_{L}, that governs the air entrainment below NBL (see Fig. 3). The total heights obtained from threedimensional simulations cannot be described by the BPT model (for the optimal input values see Table 3) even if the value of k_{U} is empirically tuned. The maximum height is reached by the oscillating local overplume at MFR > 10^{10.5} kg/s. It is inferred that the total column height (reported in Fig. 3) is generated from a large amplitude gravity wave that is excited by the intensive plume. This again confirms the difficulty of simple integral models to capture such complex plume dynamics.
Discussion
Our results have enormous implications for the assessment of the dynamics of supereruptions. For the most extreme MFRs^{13}, ~10^{11} kg/s, similar to those estimated for the Young Toba Tuff (YTT) eruption^{4}, the total plume height increases beyond the stratosphere (up to ~60–70 km) owing to the development of local overplumes above the umbrella region, even though the NBL remains at ~20 km. The bulk mass spreads in the umbrella region between ~20 and ~50 km, and the local overplumes develop above the main umbrella region (see Supplementary Figure 1). All previous studies relied on the results of the Woods and Wohletz model^{15} and used a relatively low plume height (~30–40 km) but our simulations indicate that volatiles and fine ash can be transported up to 70 km. This has important implications for the effects of eruptions, like YTT, on the Earth’s climate^{2,3,28}. Climate models are sensitive to the injection height because atmosphere stratification and availability of H_{2}O influence the conversion of SO_{2} into sulphate, which governs the climate response. Injections into the stratosphere affect the albedo of the atmosphere on the order of decades^{2} but the longevity of SO_{2} in the mesosphere could be considerably longer.
Our simulations also show that partially collapsing plumes, generated slightly above the critical MFR value, can reach heights of up to 50–60 km and efficiently transport the mixture up to the stratosphere (see Fig. 1b and Supplementary Movie 2). This results in effective upward transport of a large mass fraction of fine ash that is generated from the pyroclastic flows, enriching the fine ash content of the umbrella cloud with respect to the groundhugging pyroclastic flows. This can be the case for eruptions similar to the Campanian Ignimbrite event for which a MFR of ~10^{9} kg/s was empirically estimated for the initial Plinian phase and ~2–5×10^{9} kg/s for the coignimbrite phase^{12,29}.
Methods
Computational fluiddynamical model
To simulate fluid dynamics of volcanic plumes, we used the pseudo gas model by Suzuki et al.^{19}, in which the momentum and heat exchanges between the volcanic ash and gas are assumed to be so rapid that the velocity and temperature are the same for all phases. The fluid dynamics model solves a set of partial differential equations describing the conservation of mass, momentum, and energy, and a set of constitutive equations describing the thermodynamics state of the mixture of volcanic ash, gas, and air. The computational model was designed to reproduce the injection of a mixture of volcanic ash and volcanic gas from a vent into a stratified atmosphere. For this study typical tropical atmosphere conditions were used as initial atmospheric conditions (see Supplementary Figure 3). For given temperature gradients of atmosphere, the atmospheric density and pressure were calculated from the hydrostatic relationship. Initial wind velocity was set to be zero in the whole computational domain. Free slip conditions were applied at the ground boundary, and fixed inflow conditions at the vent. At the other boundaries, mass, momentum, and energy fluxes were assumed to be continuous (i.e., free outflow/inflow conditions). The use of the latter conditions can avoid the wave reflection at the boundaries even when the initial atmospheric conditions, such as hydrostatic pressure and density, are perturbed by the eruption.
The governing equations were solved numerically by the Roe scheme^{30} with MUSCL (Monotone Upstreamcentred Scheme for Conservation Laws) interpolation^{31} for spatial integration and time splitting method for time integration. The present model with these schemes has a thirdorder accuracy in space and secondorder accuracy in time. We used a generalised coordinate system in the computational domain.
Computational settings and simulation strategy
The grid size near the vent was set to be sufficiently smaller than the vent diameter (i.e., D_{0}/20, where D_{0} is the vent diameter) in order to resolve the flow structures and turbulent mixing at a low altitude, whereas it was increased at a constant rate up to 300 m/grid with the distance from the vent. We confirmed that the flow regime and plume height are basically unaffected by the numerical resolutions by changing the grid sizes (see Supplementary Figure 4).
The computations were carried out on the Earth Simulator (NEC SXACE; 64 GFLOPS/core) at the JAMSTEC and also on the Fujitsu PRIMERGY CX400 (22 GFLOPS/core) at the Research Institute, Kyushu University. Each simulation took 52–280 h with 512 cores.
In order to investigate how variation of MFR affects the plume dynamics, we carried out a set of simulations with variable vent size, following the method described by Suzuki et al.^{25}. The vent radius ranged from 600 to 6000 m. The pressure, density, and velocity at the vent were kept fixed among the simulations. The pressure at the vent was assumed to be in equilibrium with the atmospheric pressure. We assumed a magmatic temperature of 1053 K and water content of 6 wt.%, similar to the magmatic properties in the Pinatubo 1991 eruption^{23}. Considering these values of pressure, temperature, and density the equation of state gives a density at the vent of 3.5 kg/m^{3}. The exit velocity was assumed to be 256 m/s, corresponding to a fixed Mach number of 1.5. As a result, the MFR ranges from 10^{9} to 10^{11} kg/s.
It is important to stress that in this study we kept fixed the magmatic properties such as magma temperature and water content. The variation of these properties can lead to different critical conditions between the flow regimes described in the main text. However, the qualitative features of each flow regimes would be same in the possible ranges of magmatic temperature and water content for magmatic eruptions. We also assumed the equilibrium of pressure and the supersonic flow as the conditions at the vent. The disequilibrium and sub/supersonic flow can change the flow structures near the vent, which results in the change of the final distance of PDC and therefore the change of transition between the flow regimes^{32,33}. In addition, we ignore the nonequilibrium effects between the volcanic ash and gas phases. However, the nonequilibrium effects are less relevant in the strong eruptions rather than in the weak eruptions^{20}.
Data availability
The authors declare that all data supporting the findings of this study are available in the article and in Supplementary Information. Additional information is available from the corresponding author upon request.
Change history
22 August 2018
We became aware of a mistake in the data displayed in the original version of Fig. 3. Specifically, the lines showing the relationship between column height and MFR for MFR larger than 10^{10} kg/s, were based on simulations where the exit gas fraction was assumed to be an unrealistic value of 0.76 rather than the correct value of 0.33. This has been corrected in both the PDF and the HTML versions of the Article. The text was written on the basis of the correct plots, and so this error does not affect the original discussion or conclusions of the Article. The authors apologize for the confusion caused by this mistake.
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Acknowledgements
A.C. was partially supported by a grant of the International Research Promotion Office Earthquake Research Institute, the University of Tokyo. Y.J.S. and T.K. were partially supported by KAKENHI (grand nos. 25750142 and 17K01323). We warmly thank V.C. Smith for revising the English and very helpful suggestions.
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A.C. and Y.J.S. designed the simulation set. Y.J.S. performed the runs and processed the results. A.C., Y.J.S., and T.K. analysed and interpreted the results. A.C. wrote the manuscript with input from all the coauthors.
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Costa, A., J. Suzuki, Y. & Koyaguchi, T. Understanding the plume dynamics of explosive supereruptions. Nat Commun 9, 654 (2018). https://doi.org/10.1038/s41467018029010
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