Multiphase flow behaviour and hazard prediction of pyroclastic density currents

Abstract

Pyroclastic density currents (PDCs) are dangerous multiphase flows originating from volcanic eruptions. PDCs cause more than a third of volcanic fatalities globally and, therefore, development of robust PDC hazard models is a priority in volcanology and natural hazard science. However, the complexity of gas–particle interactions inside PDCs, as well as their hostile nature, makes quantitative measurements of internal flow properties, and the validation of hazard models, challenging. Within the last decade, major advances from large-scale experiments, field observations and computational and theoretical models have provided new insights into the enigmatic internal structure of PDCs and identified key processes behind their fluid-like motion. Recent developments have also revealed important links between newly recognized processes of mesoscale turbulence and PDC behaviour. In this Review, we consider how recent advances in PDC research close the gaps towards more robust hazard modelling, outline the need to measure the internal properties of natural flows using geophysical methods and identify critical future research challenges. Greater understanding of PDCs will also provide insights into the dynamics of other natural gravity currents and high-energy turbulent multiphase flows, such as debris avalanches and turbidity currents.

Key points

• We are not yet learning quickly enough about pyroclastic density currents (PDCs) to save lives.

• Recent advances in experimental and computational studies delineate the concentration boundaries that separate dilute, intermediate and concentrated regimes of PDC transport.

• Mass and momentum transfer between dilute and concentrated flow regions, and, thus, the evolving transport behaviour, is controlled by the recently identified intermediate regime.

• Identification of pore-pressure feedbacks in experimental PDCs, combined with multiphase modelling, provide insights into the origin of the high mobility and extremely low effective friction of PDCs.

• New geophysical methods to probe the internal flow structure are becoming available and will provide data to test existing PDC flow models and drive future research.

• The advanced understanding of PDCs gained from combining experimental, computational and field approaches must be used to benchmark, validate and advance PDC hazard models.

Introduction

Pyroclastic density currents (PDCs) represent one of the most dangerous and least understood phenomena of explosive volcanism. PDCs form when hot mixtures of volcanic ash, rock and gas fail to become positively buoyant (with respect to air) and develop ground-hugging gravity currents that entrain, and laterally intrude into, the colder and less dense atmosphere (Fig. 1a). Many volcanic systems worldwide, which display a range of eruptive styles and magnitudes, are associated with frequent PDCs¹, demonstrating that these flows are associated with a wide spectrum of eruption processes, such as collapse of eruptive plumes^2,3 and lava domes^4,5, directed magmatic^6,7 and hydrothermal blasts^8,9, phreatic and phreatomagmatic explosions^10,11, and caldera-collapse events^12,13.

Fig. 1: Conceptual models and multiscale spectra of PDCs.

a | Pyroclastic density current (PDC) from Sinabung volcano in 2014, as an example of a PDC that is dominated by concentrated gas–particle transport (top), and a pyroclastic surge from Stromboli volcano in 2019, exemplifying PDCs with dominantly dilute gas–particle transport (bottom). The left-hand panels display images of the two PDCs, with coloured areas outlining transects that are used to show the conceptual structure of dilute and concentrated PDCs (middle-left panels). The conceptual models of the internal flow structure of concentrated and dilute PDCs reveal their respective height-variant-dominant gas–particle-coupling mechanisms and PDC flow regions. Close-ups of the lower regions of the PDCs (middle-right panels) highlight the presence of gas pore-pressure-modified granular-fluid-based underflow in concentrated PDCs and a bedload region in dilute PDCs. The right-hand panels display generalized vertical profiles of time-averaged velocity (U) and volumetric particle concentration (C) for dominantly concentrated and dilute PDCs (the locations of the velocity profiles are marked by the vertical lines in the middle-left panels). b | The characteristic length scales (blue bars) and timescales (red bars) that govern PDC transport behaviour and their fluid-mechanical meaning (based on refs^199,200). The wide, and partially overlapping, length scales and timescales of PDC transport behaviour result in a broad spectrum of concurrently occurring gas–particle, particle–particle and particle–eddy feedback mechanisms inside PDCs. Part a images courtesy of Photovolcanica (top) and Alberto Lunardi (bottom).

The impacts of PDCs are determined by their high velocities and mobility, resulting in large and uncertain hazard footprints^14,15,16,17; high dynamic pressure, causing partial to complete destruction to the built environment around volcanoes^18,19,20; and their abundance of very fine and hot ash particles, causing asphyxiation and burn risks^21,22. In the last decade, the number of confirmed fatalities from PDC-forming eruptions of Merapi (Indonesia) in 2010 (ref.²³), Ontake (Japan) in 2014 (ref.¹⁰), Sinabung (Indonesia) in 2014 (ref.²⁴) and 2016 (refs^25,26), Fuego (Guatemala) in 2018 (ref.²⁷), Stromboli (Italy) in 2019 and White Island (New Zealand) in 2019 totals at least 650 people. The number of fatalities would have been much higher if, for example, geophysical monitoring in 2010 had not called for a timely mass evacuation at Merapi²⁸ or if the mobile-blast-like PDCs of the 2012 Te Maari eruption had engulfed New Zealand’s most popular hiking track during daytime⁸. Nevertheless, PDCs account for one-third of volcanic fatalities and injuries globally²⁹, and multiple disasters in the last decade raise concerns about the ability to mitigate PDC hazards.

Overall, nearly 100 million people now live at risk from PDCs^30,31, demonstrating why the development of robust hazard modelling and PDC risk-mitigation strategies are essential. Hazard modelling, however, depends on understanding the physical processes that control PDCs, which constitutes a primary research objective in physical volcanology and volcanic hazard sciences.

Investigating the physical processes that operate within PDCs is impeded by three main challenges. First, PDCs are one of the most complex types of natural multiphase flow¹. The fluid and thermodynamic behaviour of PDCs includes concurrent particle–particle and fluid–particle interactions, fluid shear, turbulent fluctuations, and entrainment and mixing of ambient air. The importance of each process is governed by broad and overlapping characteristic length scales and timescales (Fig. 1b), which leads to a wide spectrum of possible, concurrent flow regimes, including compressible turbulent flow, gas pore-pressure-modified granular flow and dry granular flow. Based on the characteristics of PDC deposits, volcanologists have traditionally distinguished two types of PDCs: pyroclastic flows and pyroclastic surges. Flows and surges are often viewed as endmembers of PDC behaviour, which are dominated by gas–particle transport at high or low particle volume concentration, respectively (Fig. 1a). However, understanding of the multiscale and multiphase processes that develop across the spectrum of dynamic flow regimes in PDCs remains incomplete³².

Second, direct observations and measurements of the internal structure of PDCs, which could be used to clarify and characterize the complex array of processes operating within them, remain rare and are limited by the hostile nature of these flows. In addition, the paucity of direct geophysical observations means that estimates of the source parameters that govern PDC initiation and transport (such as volume, velocity, density, temperature and grain-size distribution) are plagued by high uncertainties.

Third, although analogue experiments provide insights into the complex processes that govern granular-fluid and turbulent flow^33,34,35, as well as the multiscale and multiphase processes within density currents, they cannot capture the full complexity of natural PDCs. The material properties and the spectrum of gas–particle interactions inside natural PDCs cannot be completely downscaled to the typical scale of laboratory experiments (that is, several metres)^36,37,38, which limits the ability of experimental approaches to inform theoretical and computational flow models.

To overcome the challenges associated with PDC research, and to advance understanding of the enigmatic inner working of PDCs, volcanologists have traditionally applied a combination of field-based observational and sedimentological techniques^32,39, analogue experiments^40,41,42, and theoretical and computational modelling^43,44,45. Over time, the approach to understanding PDCs has diversified (see the Supplementary Information) and research has been stimulated by major PDC-forming eruptions, such as Mount St. Helens in 1980 (refs^46,47,48), Pinatubo in 1991 (ref.⁴⁹), Soufrière Hills in 1995–1999 (refs^50,51,52) and Merapi in 2006 and 2010 (refs^7,53,54); fundamental and technological advances in computational fluid mechanics⁵⁵; community-based efforts to synthesize analogue PDCs in large-scale experiments^35,56,57; and the increased instrumentation of high-risk volcanoes with geophysical sensors^58,59,60. To address uncertainty, community-wide efforts have now started to develop global databases of PDC source conditions (FlowDat⁶¹, DomeHaz⁶² and GLADIS⁶³), which capture frequency versus magnitude relationships of PDC occurrence, mass partitioning during column-collapse-derived PDCs and the volumes of PDCs formed by lava-dome collapse. Nevertheless, examining sedimentary records of past eruptions remains the dominant method in PDC research and continues to guide, apply and test model developments.

In this Review, we discuss recent advances in field-based, theoretical, experimental and numerical studies, which provide insights into the critical complexity of PDC internal flow dynamics. We then consider how geophysical monitoring of natural PDCs can provide insight into their internal flow structure and, thus, constrain experimental and numerical models of PDC dynamics. Finally, through identifying areas of continued uncertainty, we outline future research directions that are required to develop robust hazard models that can reliably assist risk mitigation.

Probing PDC deposits

Recent PDC-forming eruptions provide volcanologists with opportunities to reconstruct the chronology of events^8,10 and to obtain high-resolution data of hazard footprints^20,64, flow paths^17,65 and the distribution of dynamic pressure^20,66 and temperature^67,68,69 before fragile deposits are degraded or removed. In some cases, observations of recent PDCs have enabled estimates of local bulk and frontal velocities^7,53,70, time-integrated bulk-flow densities^20,66,70 and flow temperatures⁶⁷ to be made, allowing broad characterization of PDC behaviour. Data sets from recent eruptions are critical for defining the input and boundary conditions of computational PDC flow models. Furthermore, they provide real-world scenarios for model validation, calibration and advancement. However, to quantitatively understand the internal flow dynamics within PDCs, and the generation of associated hazards, we need to characterize the evolution and vertical distribution of the velocity, density and temperature within PDCs.

Qualitatively, properties such as the velocity, density and temperature of PDCs are, at least partially, preserved in their deposits. Indeed, lithofacies characterizations of PDC deposits propounded the original interpretation that there are two main types of PDCs^71,72,73 (Figs 1a,2): fully dilute, fully turbulent PDCs with thin tractional bedload regions (pyroclastic surges)^{66,74,75,76,77}, and granular-fluid-based PDCs, comprising gas pore-pressure-modified granular flow in a lower underflow region and dilute turbulent transport above (pyroclastic flows)^65,78,79,80.

Fig. 2: Approaches to estimate flow velocity, density and temperature from deposit characteristics.

a | Tripartite and upward-fining deposit of dilute pyroclastic density current (PDC) transport from the 2012 hydrothermal blast of Te Maari (New Zealand). b | The grain-size distribution (GSD) of stratified PDC deposits can be inverted to estimate time-averaged velocity, density and, therefore, dynamic pressure profiles using turbulent-boundary-layer theory⁸⁷. c | Flame structures in the 1980 pumice flow deposit of Mount St. Helens (concentrated PDC deposit). Flame structures are interpreted as ‘frozen’ shear instabilities. d | Analyses of the wavelength of flame structures allows the local slip velocity and sedimentation rate of concentrated PDC transport to be estimated⁹⁷. e | Entrained blocks in the lower metre of the Peach Spring Tuff: large blocks can be entrained by concentrated PDCs, owing to a lift force generated during flow slips across the substrate^98,100. f | Clast trains can be used to calculate the flow-front velocity of concentrated PDCs. g | Charred wood at the base of a block-and-ash flow deposit from Soufrière Hills. h | Charcoal reflectance^54,201 is commonly used to estimate the local flow temperatures at the base and in the lower few metres of PDCs.

Over the past 50 years, however, conceptual models of the internal structure of PDCs have evolved, guided by advances in the fields of gas fluidization, granular flow, aqueous gravity currents and disperse multiphase flows^{1,32,81,82,83}. Although early concepts envisaged a longitudinally evolving flow structure dominated by distinct (dilute or concentrated) transport behaviours^13,71,84, physical modelling and interpretation of PDC deposits point to a strong vertical density stratification in all PDCs^85,86. Whether the density stratification is associated with abrupt changes or with a gradual continuum in particle concentration remains uncertain³⁴. As a result, several strategies have been adopted to quantitatively determine the transport parameters of PDCs from their deposits, building upon the original simple distinction between dilute and concentrated PDCs.

For dilute currents, turbulent-boundary-layer theory and sediment-transport hydraulics can be used to infer flow properties from deposit characteristics. For example, particle size, shape and density information of stratified PDC deposits (Fig. 2a) is regularly used to reconstruct profiles of the local time-averaged velocity and density flow structure^87,88,89 (Fig. 2b). As a result, volcanologists can capitalize on the deposits of dilute PDCs to estimate vertical profiles of destruction-causing dynamic pressure, which is essential for future hazard assessments. However, it remains unclear how gas–particle turbulence, heat, polydispersity and wall shear on natural rough surfaces modify energy dissipation, stratification and particle sedimentation inside dilute PDCs.

In addition, the median diameter of stratified PDC deposits can be related to the eddy rotation speed ΔU (refs^90,91), and to a local mean flow velocity⁹², by approximating the conditions of gas–particle decoupling and turbulent sedimentation from large eddies. Analysis of PDC deposits can, therefore, reconstruct spatial variations in flow velocity from past PDC-forming eruptions. However, there are critical assumptions behind the calculation of mean flow velocity from analysis of stratified PDC deposits, such as the decaying isotropic turbulence spectrum, which require future validation.

Dilute PDCs, in their most hazardous flow-outrun regions, tend to be temporally non-depositional and erosive, which limits the use of the above two methods for hazard assessments. In such cases, however, the local bulk-flow density of dilute PDCs can be estimated through modelling the reattachment distance of temporally airborne PDCs⁹³. Iterative modelling of PDC reattachment has recently been applied to the 1980 blast of Mount St. Helens, confirming independent density estimates.

For concentrated PDCs, identifying relationships between the deposit characteristics and the dynamics of gas–particle transport remains challenging. Recently, an empirically and experimentally derived model linked the formation of granular shear instabilities, such as flame structures, to internal slip velocities^94,95,96. Subsequent identification of vertical variations in calculated slip velocities has been used to provide local estimates of sedimentation rates⁹⁷ during aggradation of the 1980 Mount St. Helens pyroclastic flow⁷⁸ (Fig. 2c,d). Sedimentation rates of 4–32 cm s⁻¹, in combination with estimates of flow duration, demonstrate that PDC deposition is characterized by minute-long periods of non-depositional PDC bypass alternating with seconds-long intervals of strong deposition.

Additionally, experiments have related the dynamic lifting of particles, owing to gradients in gas pore pressure inside flows, to frontal velocity^98,99 (Fig. 2e,f). Analysis of dynamic particle uplift has since been used to estimate the flow-front velocity of enigmatic long-runout PDCs generated in explosive volcanic super-eruptions (5–20 ms⁻¹ for PDCs forming the 18.8-Ma Peach Spring Tuff), which were never directly witnessed¹⁰⁰ (Fig. 2e,f). Furthermore, application to the May 18 1980 PDC of Mount St. Helens suggests surprisingly low fontal velocities of c. 10–13 m s⁻¹ at c. 5–6 km from the vent⁹⁸, in agreement with observed flow-front velocities (7–10 m s⁻¹) at the same distance during the 7 August 1980 eruption¹⁰¹.

The high temperature of PDCs (around 60 °C for the coldest PDCs and up to magmatic temperatures for the hottest PDCs), and, hence, the buoyancy of the interstitial fluid, sets them apart from other natural-gravity currents¹⁰². However, the influence of temperature on the dynamics of PDCs remains poorly constrained. New advances in thermal proxies, including palaeomagnetic and charcoal-reflectance techniques (Fig. 2g,h), allow volcanologists to map temperature variations across PDC footprints^69,103,104 and provide new avenues to characterize PDC stratification conditions⁶⁸.

Despite the large number of insights provided by studying PDC deposits, it is important to consider some shortcomings of the methods described above. For example, deposit-based estimates of PDC velocity, density and temperature cannot capture temporal changes in the bypassing PDC, which may be critical to the evolution of PDCs and the intensity of hazard impacts⁵⁶. Furthermore, internal feedbacks between polydispersity, temperature and pore pressure are expected to occur, but are not currently captured in empirical models of concentrated PDC transport.

Indeed, recognition of the effects of fluid drag in the interplay of polydisperse particle settling, gas pore-pressure generation and upward gas migration has added new momentum to the debate surrounding the origin of tens-of-metres-thick sheets of massive PDC deposits extending tens to hundreds of kilometres from their source. Several distinct hypotheses on the underlying flow-transport mechanism of PDCs^{105,106,107,108}, invoking both dilute and concentrated transport regimes, have been proposed to explain these massive PDC deposits. However, analysis of the effects of fluid drag on the evolving density stratification within PDCs, which cannot be captured in their depositional records, helps to discriminate between deposits formed by dilute and concentrated flow.

An empirical PDC-transport parameter T_de-di discriminates between deposits of dominantly dilute PDCs, which are unable to generate an underflow with elevated gas pore pressure, and concentrated transport regimes, which can generate mobile underflows. The non-dimensional ratio quantifies the flow mobility from a PDC deposit by assessing its runout length, inundation area and the role of the hydraulic permeability, which controls the gas pore pressure and, thus, friction on the PDC in the lower flow boundary:

$${T}_{de-di}=\frac{{A}^{3}{d}_{s,1/2}}{{V}^{5/3}{L}^{2}}$$

(1)

where A and V are the deposit footprint area and volume, respectively, and d_s,1/2 is the effective aerodynamic particle diameter at half of the PDC runout distance L (ref.¹⁰⁹). Deposits resulting from dominantly dilute PDC transport are characterized by T_de-di > 5 × 10⁻³, while deposits emplaced by dominantly concentrated PDCs have T_de-di < 2.5 × 10⁻³.

The enigmatic internal flow structure

In the last decade, volcanologists have pushed the development of large-scale experiments^33,35,110 and 3D multiphase flow models^111,112 to circumvent the limitations associated with PDC deposit interpretations. Numerical multiphase models can generate real-scale flows that interact with topography and capture a large range of gas–particle coupling regimes encompassing the concentrated to dilute spectrum¹. The models demonstrate the temporal and spatial evolution of the internal structure of PDCs, but are limited in spatial resolution by the large computational cost associated with high-resolution simulations. Subgrid models are required to capture microscale and mesoscale processes (turbulent dissipation, aggradation, particle breakage), but inadequate constraints exist on such processes for volcanic mixtures.

Large-scale experiments can bridge the scaling issues encountered with benchtop experiments and the low resolution of numerical multiphase models. Direct measurements of large-scale experiments provide insights into the vertical stratification of PDCs^113,114. In addition, they can be used to focus on specific concentration regimes (dilute^35,114 or concentrated regimes of PDCs¹¹⁵) and study evolving flow behaviours as currents develop particle-concentration gradients¹¹⁰ (Fig. 3a).

Fig. 3: Delineating concentrated, intermediate and dilute transport regimes in PDCs.

a | Discriminating drag-dominated and collision-dominated transport regimes through the timescale ratio D_D defined by refs^91,119. Existing models of the particle–particle collision timescales cannot yet delineate dilute and concentrated transport regimes quantitatively. b,c | A schematic representation of the traditional multiphase flow fields of dilute and dense suspensions (part b) and their associated gas–particle-coupling regimes (part c) relative to the flows’ particle volume concentration. d | A schematic illustration, and approximate range in particle concentration, of the dilute, intermediate and concentrated transport regimes of pyroclastic density currents (PDCs). Details of gas–particle feedback mechanisms in each regime are highlighted. Blue arrows represent fluid streamlines, blue dashed lines highlight large eddies. Yellow stars highlight particle collisions and their relative frequency. e | The threshold concentration (Φ_i/d) at the intermediate-concentrated boundary, where frictional contact stresses equal hydrodynamic stresses. f | Formation of concurrent concentrated, intermediate and dilute transport regimes in large-scale PDC experiments. A contour plot of the vorticity highlights the strong changes in the degree of turbulence. Large variations in the vorticity of the dilute region are associated with large coherent turbulence structures and negative values in the concentrated region are associated with strong downslope-directed shear. A broad range of moderate vorticity values are present in the intermediate regime, highlighting intermediate turbulence intensity. The white line scale in the bottom-right corner is 0.3 m long. Panel f is adapted with permission from ref.⁵⁶, Elsevier.

Delineating dilute, intermediate and concentrated regimes

There is extensive evidence that a single PDC spans a range of concentrations, which can result in abrupt changes in flow behaviour^32,116. The particle volume fraction (Φ_V) threshold between dilute and concentrated suspensions is often placed at Φ_V ~ 10⁻³, delineating two-way from four-way coupling (Fig. 3b,c). In two-way coupling, the motion of particles is affected by the flow of gas around them, and the gas flow, in turn, is affected by the presence of particles¹¹⁷. In four-way coupling, two-way coupling is further modified by particle–gas–particle interactions (gas that is compressed between particles, generating lubrication forces)¹¹⁸ and particle–particle interactions (including collisional and prolonged frictional particle–particle contacts)¹¹⁷.

In dilute and concentrated suspensions, momentum exchange can be described by the ratio between the collisional and particle drag response timescales, D_D (refs^91,119):

$${D}_{D}=\frac{{\tau }_{c}}{{\tau }_{p}}$$

(2)

where τ_c is the characteristic timescale between particle collisions and τ_p is the characteristic timescale required for gas–particle dynamic forces to influence the flow behaviour of particles inside the flow. For D_D < 1, particle collisions are too frequent for particles to respond to the gas–particle dynamic forces, while for D_D > 1, gas–particle forces are dominant (Fig. 3a). However, the mechanisms that influence the two timescales, such as local turbulent fluctuations, changes in gas velocity and particle–substrate interaction, are poorly constrained. Thus, Eq. 2 cannot be used to define an exact concentration threshold between concentrated and dilute regimes.

A drastic change in particle-settling behaviour occurs at a particle volume fraction of Φ_V ~ 10⁻² (for monodisperse systems), associated with the onset of mesoscale particle-cluster formation^120,121,122. At particle concentrations above this threshold, heterogeneous drag dominates particle-settling velocities and interparticle collisions are frequent. Thus, a particle volume fraction of Φ_V ~ 10⁻² defines the boundary between the dilute and newly identified intermediate regime (defined by the formation of mesoscale particle clusters) in monodisperse systems¹²¹ (Fig. 3d). In large-scale PDC experiments using natural polydisperse volcanic mixtures, clustering occurs at Φ_V = 1 × 10^−2.6 (G.L., E.C.P.B., T.E.-O. and J.D., unpublished observations), close to the traditional boundary between dilute and dense suspensions¹¹⁹. Therefore, the concentration boundary between the dilute and intermediate regimes in PDCs might range from 10⁻² to 10⁻³, although the role of polydispersity on the onset of mesoscale clustering needs further quantification.

In engineering studies of monodisperse suspensions and large-scale PDC experiments containing polydisperse particles, clustering vanishes at Φ_V > 3–4 × 10⁻¹. At greater particle concentrations, the gas–particle dynamics become dominated by prolonged frictional particle contacts^34,121,123, defining the concentrated regime. The observed intermediate-concentrated boundary (Φ_V ~ 3–4 × 10⁻¹) can be quantified by the condition where the contact (that is, frictional) contributions overcome the hydrodynamic contributions to the granular shear stress. Therefore, by equating frictional and hydrodynamic stress, the intermediate-concentrated boundary can be located at a critical solid fraction of Φ_V = Φ_i/d ~ 3 × 10⁻¹, consistent with experimental findings (Fig. 3e). However, to assess the variability of the intermediate-concentrated boundary for PDCs with different grain-size distributions, we need to derive the material constants and friction laws for pyroclastic mixtures. Concurrent dilute, intermediate and concentrated flow regimes have recently been observed in large-scale PDC experiments³⁴, which demonstrate that all three regimes are present in a wide range of flows and, therefore, the properties of all regimes must be considered (Fig. 3f).

The concentrated regime

The dynamics of gas–particle transport and deposition in concentrated PDCs (${\Phi }_{V}\gtrsim 3\times {10}^{-1}$) have traditionally been investigated through analogue experiments on high-density turbidity currents^38,124, dry granular flows^125,126,127 and gas-fluidized granular flows^40,127,128. Alongside principles of aqueous and aeolian sediment transport, such studies guide current interpretations of PDC deposits^32,39,129. However, the insights gained from the analogue experiments and the application of aqueous and aeolian sediment-transport analogies are limited by conditions that are unique to PDCs, including: high fluid-to-particle density contrasts; large particle-size distributions; and low viscosities of the fluid phase. Conditions specific to PDCs can be demarcated from those of other mass flows through three non-dimensional numbers: the granular Stokes number St_granular, the particle Reynolds number Re_p and the drag-normalized particle-to-fluid density ratio r (ref.¹³⁰):

$$S{t}_{granular}=\frac{\alpha {d}_{p}\sqrt{{\rho }_{s}{P}_{s}}}{{\eta }_{f}}\sqrt{\frac{2}{3}}$$

(3)

$$R{e}_{p}=\frac{\alpha {d}_{p}\sqrt{\frac{2}{3}{P}_{s}{\rho }_{f}{C}_{d}}}{{\eta }_{f}}$$

(4)

$$r=\sqrt{\frac{{\rho }_{s}}{{\rho }_{f}{C}_{d}}}=\frac{S{t}_{granular}}{R{e}_{p}}$$

(5)

where ρ_f and η_f are the fluid density and dynamic viscosity, respectively; ρ_s is the solid density; C_d is the drag coefficient; P_s is the normal stress; d_p is the product of the effective aerodynamic particle diameter (Sauter mean diameter) and the average particle sphericity, chosen to account for the mean hydrodynamic drag of a collection of polydisperse particles^131,132; and $\alpha$ is related to the permeability k of the mixture as follows¹³³:

$$\alpha =\frac{k}{{{d}_{p}}^{2}}$$

(6)

The conditions St_granular = 1, r = 1 and Re_p = 1 delineate three regimes where particle settling is dominated by viscous drag, inertial drag or gravity¹³⁰ (Fig. 4a). Natural PDCs fall exclusively into the gravitational regime, owing to their unique particle-to-gas density ratio, high fine-ash content and low viscosity of the dusty gas phase. Large-scale PDC experiments¹¹⁰ cover a large proportion of the PDC field and, similarly to gas-fluidized granular-flow experiments, are well scaled to PDCs. Conversely, experiments using aqueous fluid phases fall within the viscous drag-dominated and inertial-drag-dominated regimes that characterize natural debris flows and turbidity currents (Fig. 4a). Thus, water–particle analogues should not be used to study concentrated PDC transport behaviour.

Fig. 4: Concentrated transport regimes of PDCs.

a | Different behaviours of dense granular-fluid flows can be delineated using the granular Stokes number (St_granular), the drag-normalized particle-to-fluid density ratio (r) and the particle Reynolds number (Re_p). The fields where viscous drag, inertial drag and gravitational forces dominate are highlighted (based on ref.¹³⁰). Natural pyroclastic density currents (PDCs; with and without elevated pore pressures), and large-scale PDC experiments using natural volcanic particles and air, fall into the gravitational field. Debris flows and high-density particle-laden aqueous gravity currents occupy the viscous drag-dominated and inertial-drag-dominated fields. b | The spectrum of known pore-pressure feedback mechanisms through flow compaction, dilation and shear conditions. The colour bar represents the relative pore pressure in the domain and the blue arrows display the direction for gas advection. Black arrows depict the velocity vector for the solid phase. Schematic velocity profiles for shear-induced dilation and resulting air lubrication reveal that shear gradients disrupt unidirectional and upward-directed gas advection and force strong downwards gas advection into a thin gaseous boundary layer.

The concentrated regime of PDCs is characterized by a wide range in the granular Stokes number (10⁰–10⁴; Fig. 4a), which can be related to a spectrum of flow behaviours spanning dry granular flow (high St_granular) to pore-pressure-modified granular flow (low St_granular). In dry granular flow, fluid hydrodynamic drag is low and gravity, friction and contact forces dominate. In pore-pressure-modified flow, gravitational forces remain dominant. However, the gas phase modifies particle motion through drag and lift forces, which can partially or fully balance the effect of gravity and, in association with elevated gas pore pressure, reduce the interparticle friction¹³⁴.

Although gas pore pressure cannot yet be measured in natural currents, evidence from field studies^101,108,135, numerical simulations^109,111,136 and experiments^131,137,138 indicate that the low permeability of PDCs enables self-generation and retention of pore pressure for long durations (minutes to hours)^131,139. Specifically, elevated gas pressures can be retained in the concentrated regime of PDCs, as their high polydispersity and abundance of fine ash endows them with a low effective aerodynamic particle size and, subsequently, a low permeability (down to 10⁻¹²–10⁻¹³ m²)^131,139.

A variety of feedback mechanisms between the solid fraction and pore pressure in PDCs¹⁴⁰ are key to the formation and modulation of pore pressure, and occur during local flow compaction and dilation. During compaction, elevated pore pressure can be self-generated by several mechanisms, including: rapid particle settling from intermediate concentrations (10⁻² < Φ_V < 3 × 10⁻¹) by column collapse^141,142; rapid settling of mesoscale clusters¹²¹; hindered settling, where particles fall into interstices in the substrate^143,144; or through local changes in solid fraction within a dense bed¹⁴⁰ (Fig. 4b). Conversely, local dilution associated with the formation of strong vertical shear gradients at the base of a PDC leads to gas moving towards this ‘low’-pressure region¹³⁴ and subsequent air lubrication of the basal portion of the flow (Fig. 4b). The gas flow and drag associated with pore-pressure advection alleviate the high frictional properties of volcanic material, which limits the influence of basal friction^109,134 and enables the dense mixture to propagate for long distances of tens of kilometres and on shallow slopes.

The relative effectiveness and interdependence of the different pore-pressure feedback mechanisms in real-world flows remain to be shown. A major challenge for future studies lies in the development of a granular-fluid rheology that can capture the effects of transient pore-pressure feedbacks (and implementation into large-scale flow models). However, the rheology of hot and polydisperse volcanic mixtures in the concentrated regime is still poorly understood¹⁴⁵. Furthermore, complex transient processes, such as particle breakage⁴⁴, attrition¹³⁷ and segregation¹⁴⁶, propagation over rough substrate^97,100 and changes in shear-rate gradients¹³⁴ owing to slope variations, are still unaccounted for in the concentrated regime.

The dilute regime

The upper flow region of dense PDCs and the major proportion of dilute PDCs are traditionally considered to behave as a fully dilute and fully turbulent suspension. The dynamics of the dilute turbulent regime (Φ_V < 10⁻³ to 10⁻²) is dominated by eddies with diameters comparable to the flow height (excluding the wake) down to the Kolmogorov scale⁸⁵ (Fig. 5a). In this regime, particle–particle interactions are rare and particle transport is dominated by fluid drag and gravity. Eddies modify PDC dynamics in several ways, including: dissipation of energy from large to smaller scales⁹²; delaying particle settling by actively mixing the flow¹⁴⁷; and contributing to ambient-air entrainment^114,148 (Fig. 5a,b).

Fig. 5: The dilute turbulent transport regime.

a | Structure of an experimental pyroclastic surge. Turbulent eddies and Kelvin–Helmholtz instabilities promote entrainment of ambient air. b | Comparison of the velocity structures in ambient-temperature (top) and heated (bottom) dilute pyroclastic density current analogues. While the gravity current structure in the cold flow is well defined, complex velocity fields, buoyancy reversal and lift-off occurs in the hot flow. c | Particle-transport regimes in turbulent eddies. Data taken from a large-scale dilute pyroclastic density current experiment shows the wide range of particle-transport regimes⁵⁶. The five transport regimes of particles in turbulent eddies, as described in refs^91,119, are illustrated as sketches. In turbulent homogeneous transport, particles follow gas streamlines. In transient and turbulent sedimentation regimes, particles detach from streamlines and their settling is delayed by the rotating motions of eddies. In the unrolling and margins regime, particles segregate towards eddy peripheries and mesoscale clusters form. In the fall regime, particles are negligibly affected by the eddies. Part b adapted from ref.¹¹⁴, Springer Nature Limited. Part c adapted with permission from ref.⁵⁶, Elsevier.

PDCs composed of rock and hot gas can only propagate as a gravity current as long as the mixture density exceeds that of the ambient atmosphere¹⁴⁹. The density of PDCs can be modulated through simultaneous entrainment of material from the substrate (which increases its density)¹⁵⁰ and, importantly, dilution and a decrease in density through particle settling and entrainment of ambient air. The resulting changes in density strongly influence the flow velocity.

Turbulence enhances entrainment of ambient air through Kelvin–Helmholtz instabilities at the free surface^114,151,152 (Fig. 5b). Entrainment is largely controlled by the flow stratification^111,152 and modifies the thermal stratification and density of the current. Large-scale experiments have shown that the extent of air entrainment can be parameterized by a value known as the entrainment coefficient, which typically varies between 0.1 and 0.2 and can locally reach values of 0.4–0.5 in the head of the current^56,114,153. The entrainment coefficient is locally variable within a certain flow and values >0.2 are associated with vigorous entrainment of ambient air. Because turbulent entrainment reduces the density contrast between the current and the ambient air, reducing its final runout and lift-off distances, parameterizations of the entrainment coefficient are important in models where air entrainment is not captured by the governing equations.

In addition, settling of particles from turbulent eddies, which further dilutes the current and modifies its density stratification, is critical for the evolution of PDCs in the dilute regime. The Stokes number describes the ratio of a characteristic particle response time (such as the settling timescale) to a characteristic fluid timescale (such as the eddy overturn time):

$$St=\frac{1}{f}\frac{\varDelta \rho {d}^{2}\varDelta U}{18{{\mu }}_{g}\delta }$$

(7)

where ΔU is the eddy rotation speed, d is the particle diameter, $\delta$ is the eddy diameter, μ_g is the dynamic viscosity of the gas and Δρ is the difference in density between the solid and the gas. f is a drag factor related to the particle Reynolds number. For St ≪ 1, particles couple with the gas and closely follow fluid streamlines, for St ~ 1, particles segregate towards eddy peripheries and for St ≫ 1, particles decouple from the turbulent mixture. Another important parameter is the stability number (Σ_T), which assesses the gravitational force acting on particles within dilute PDCs and evaluates whether particles will separate from eddies.

$${\varSigma }_{T}=\frac{{U}_{T}}{\varDelta U}$$

(8)

where U_T is the settling velocity of particles in the fluid. Particles only separate from eddies when Σ_T ≫ 1.

Particle–eddy interaction in the dilute regime can be categorized into five gas–particle transport regimes, based on theoretical modelling and observations from large-scale experiments, namely: unrolling and margins of eddies (St > 1; Σ_T < 3), turbulent sedimentation (St < 1; Σ_T > 0.3), turbulent transient (0.1 < St < 1; Σ_T < 0.3), turbulent homogeneous (St < 0.1; Σ_T < 0.3) and fall (St > 1; Σ_T > 3)^56,91 (Fig. 5c). Large-scale experiments show that multiple gas–particle transport regimes coexist in a single current and dominate in spatially distinct portions of the flow (Fig. 5c). The results from large-scale experiments have important implications for numerical models of dilute PDCs, which traditionally assume that gas and particle phases are nearly homogeneously suspended inside a PDC and have negligible feedbacks. Future research should be focused on understanding the mechanism and associated constitutive relationships that govern gas–particle transport within and out of large eddies, to characterize particle-settling conditions in lower-order PDC flow models.

The intermediate regime

The recently discovered intermediate regime (10⁻³ to 10⁻² < Φ_V < 0.3) is characterized by an inhomogeneous distribution of particles, resulting in mesoscale particle clustering^154,155. Currently, three mechanisms of cluster formation are recognized (Fig. 6a): collision-induced, drag-induced and wake-induced. In collision-induced clustering, particle–particle frictional and inelastic contacts dissipate the granular temperature of the mixture and, therefore, the energy required to homogenize the gas–particle suspension^156,157.

Fig. 6: The intermediate transport regime.

a | The three mechanisms of mesoscale cluster formation. In collision-induced clustering, the granular temperature strongly declines. During drag-induced clustering, particles in denser regions settle faster than particles in dilute regions and incorporate these particles into larger clusters. During wake-induced clustering, drag reduction in the wake of large particles grows fast-settling clusters. b | Contour plot of the vertical velocity component V in the lower 0.45 m of a large-scale dilute pyroclastic density current experiment. Fast-settling mesoscale clusters are associated with high vertical velocities. c | Vertical velocity against flow height for the location marked by the blue line in part b. The vertical grey bar represents the settling velocity of the mean grain-size diameter of the mixture in this flow region. d | High-resolution multiphase simulation of a settling suspension of spherical glass beads in air (mean particle volumetric concentration of 7 vol.%). The suspension segregates into band-like high-concentration domains of mesoscale clusters (red arrow) separated from low-concentration neighbouring regions (blue arrow). Cluster-induced turbulence is generated through the fast settling of mesoscale clusters displacing air upwards. e | The frequency spectrum of local gas pressure for the simulation shown in part d. The typical thickness of mesoscale clusters associated with different gas-pressure values are displayed on the second y-axis. The frequency of clusters (and average cluster thickness) decline with decreasing gas pressure. The decay of the gas pressure with increasing frequency follows a slope of −5/3 (blue line), which is generally observed in homogeneous turbulent fluid flows with a cascade of eddies that dissipate the excess turbulent energy. Part d adapted with permission from ref.¹⁰⁹, Wiley.

Alternatively, in the presence of a fluid, viscous drag and mean drag effects can result in drag-induced instabilities and clustering. Viscous drag relates to the increased drag force operating on two particles in the vicinity of each other, which decreases their relative mean velocity prior to collision and increases the probability of clustering. The mean drag effect refers to an instability driven by the volumetric particle concentration control on the slip (the presence of relative velocity) between gas and particles¹⁵⁸. As gas tends to bypass regions with higher particle concentrations, resulting in a decrease of drag on these particles, clusters grow as particles in denser regions fall rapidly and capture particles from slower-settling, more dilute regions.

Finally, wake-induced clustering, where reduction of drag on a trailing particle in the wake of another particle leads to clustering, is especially important in fines-rich polydisperse systems, owing to the stronger relative influence of drag on smaller particles compared with larger particles^159,160. Wake-induced clustering will only occur if the particle Reynolds number in a polydisperse suspension exceeds a critical value of 275.

Regardless of the mechanism, mesoscale clustering changes the properties of volcanic mixtures and can enhance settling in intermediate-concentration regimes^34,120,155. In large-scale PDC experiments, dendritic mesoscale clusters, which attain lengths of several decimetres, settle at terminal velocities twofold to threefold that of particles in dispersed suspensions³⁴ (Fig. 6b,c). Mesoscale clustering can, as a result, explain the formation of elevated gas pore pressure inside PDCs¹¹¹ and corresponds to what has been long hypothesized as rapid suspension sedimentation⁵¹. Furthermore, high-resolution multiphase simulations of settling suspensions show that the formation of clusters self-generate gas turbulence through gas-pressure fluctuations¹⁶¹ (Fig. 6d). The turbulence energy resulting from a spectrum of millimetre-thick to centimetre-thick clusters is similar to that known to occur for the inertial range of eddies in fully turbulent flows (Fig. 6e). Therefore, simulations of settling suspensions appear to contradict the common assumption that fluid turbulence in gravity currents is hampered in dense suspensions with particle volume fractions exceeding ~1 vol.% (ref.³²).

The intermediate regime is the least understood of the three concentration regimes presented here. The understudied nature of the intermediate regime is, in part, due to its transient nature, which makes it hard to investigate experimentally. Moreover, in the intermediate regime, gas–particle, particle–gas–particle and particle–particle interactions occur simultaneously, adding intrinsic multiphase complexity. It is, however, a regime that requires attention, as its controls the coupling between the dilute and concentrated regimes, as well as density stratification and particle settling in PDCs. In addition, mesoscale clustering in the intermediate regime influences the rate of mass loss (through sedimentation) and momentum (through viscosity and friction). Consequently, drag law corrections and subgrid models that account for mesoscale clustering must be developed. Implementation of such subgrid models, within both two-layer depth-averaged models and multiphase models, will help to predict the flow kinematics, the capacity to overcome topographic obstacles and the PDC final runout distance.

Insights from geophysical measurements

Three-dimensional multiphase simulations and laboratory experiments provide new insights into the flow structure of PDCs. However, we still lack comprehensive measurements of real-world flows to test and advance theory-based and experiment-based knowledge of PDCs. Although the energy and destructive impact of PDCs hamper direct observations, remote geophysical measurements provide an opportunity to probe the physics and structure of flows during their propagation.

Geophysical observations of PDCs are sparse, primarily because of the unpredictability of PDCs and the uncertainty and logistical challenges in optimally placing sensors before an eruption. Nevertheless, geophysical data are required to constrain the macroscale and microscale processes operating within PDCs, and, thus, validate and improve existing empirical, analytical and numerical flow models before they can be confidently applied to forecast PDC hazards. Furthermore, direct geophysical measurements of the length scales, timescales and energy scales of hot gas–particle transport inside PDCs will inform fluid-dynamic and thermodynamic scaling relationships, understanding of gas–particle dynamics in the dilute, intermediate and concentrated regimes, and guide the focus of future laboratory and computational PDC experiments.

Satellite measurements represent one endmember of geophysical observations and can provide valuable information during ongoing eruptions. For example, a satellite infrared sensor detected the May 18 1980 lateral blast at Mount St. Helens and provided key information on the PDC front velocity¹⁶². However, satellite coverage rarely has sufficient temporal resolution to record the evolution of a phenomenon that, in most cases, lasts only minutes. As a result, ground-based techniques, such as infrasound or Doppler radar, are usually preferred.

PDCs generate acoustic and infrasound waves in the atmosphere and seismic waves in the solid earth^{163,164,165,166,167,168,169}. Infrasound and seismic signals have both been used to detect and locate PDCs and estimate their velocities^59,163. However, the limited spatial resolution of acoustic, seismic and infrasound measurements means that their signals are currently interpreted through conceptual models of the potential PDC structure, rather than validating or refining such conceptions. The main challenge to quantitatively interpreting infrasound and seismic signals is understanding how flow energy is distributed into turbulent fluctuations, by linking the frequency content of the infrasound and seismic signals to dynamic processes of PDC behaviour, its downstream evolution and interactions with natural topography¹⁶⁶.

The thermal energy content of PDCs influences the air-entrainment rate¹⁵² and buoyancy, and, ultimately, controls the runout and hazard. Information on transport temperatures primarily comes from direct temperature measurements of PDC deposits and thermal proxies preserved in the deposits, such as remnant magnetism^170,171, charcoal reflectance^172,173 and rind thickness on clasts^174,175. Direct geophysical measurements using portable thermal infrared cameras have the potential to provide additional, important information about PDC cooling processes associated with the entrainment of ambient air. However, portable thermal cameras need to overcome restrictions associated with the opacity of the gas–particle mixture in infrared wavelengths. As a result, infrared measurements of moving PDCs have, so far, been limited to the detection and tracking of the PDC front¹⁷⁶.

Alternatively, probing PDCs with active-source approaches, such as Doppler radar measurements, might provide more information about the interior structure and velocity of the current^177,178. The potential of Doppler radar has been demonstrated in powder-snow-avalanche research, where microwave radars characterized the internal density stratification of the flow¹⁷⁹. However, a major challenge to radar measurements on active volcanoes is ensuring the equipment has an optimal viewing geometry prior to a PDC event. One study has successfully used Doppler radar to measure the velocity distributions of both dilute and concentrated regimes in a PDC at Colima⁵⁸. Unfortunately, the instrument was destroyed by another sequence of PDCs in 2015, illustrating another major challenge associated with placing valuable geophysical equipment in proximity to active events.

Despite challenges, geophysical measurements offer some of the only means to measure active currents directly, and developing geophysical techniques should be a major focus of future research (Fig. 7a,b). To optimize the insights gained from geophysical measurements of PDCs, their signals must be calibrated experimentally (for example, radar scattering off of different concentrations of volcanic ash). Small-scale and large-scale PDC experiments, as well as numerical calculations, will likely be needed to help interpret geophysical observations.

Fig. 7: Application of geophysical observations and modelling to PDC research.

a | A schematic illustration of the possible geophysical measurements, such as infrared imagery and seismic signals, that should be made on real-world pyroclastic density currents (PDCs). Currently, interpretation of geophysical signals is guided by conceptual flow models. b | Infrasound and seismic signals generated by a propagating PDC in 1992 at Mount Unzen (Japan). c | Numerical solution of a 3D model of continuous Plinian-column collapse (mass eruption rate = 10⁸ kg s⁻¹, time = 600 s)¹⁹⁸. d | A 3D model of a single collapse impulse (dam break; total mass = 1.6 × 10⁹ kg)¹⁹⁸. e | A calibrated integral box model for a single collapse impulse (total mass = 1.6 × 10⁹ kg)¹⁹⁸. The colour scale in parts c and d represents the particle volume fraction in a logarithmic scale. Variations in the area covered by the PDC models, and their interaction with topography, reflect the initial flow concentration and inertia, and the total mass involved in the PDCs. Part b adapted with permission from ref.¹⁶³, Japan Meteorological Agency. Parts c and d adapted with permission from ref.¹⁹⁸, Elsevier.

Video data from large-scale experiments might also help the continued development of optical techniques, such as particle image velocimetry and videogrammetry, to describe turbulent intensity and ash-cloud volume¹⁸⁰. Furthermore, although direct measurements of the internal properties of natural PDCs are very challenging, even a single instrumented measuring site would provide an enormous amount of information to constrain physical and numerical models of PDC transport.

The potential of direct geophysical measurements has been demonstrated in the snow-research community, where powder snow avalanches are artificially triggered and their height-variant and time-variant dynamics probed at a single instrumented site¹⁸¹. As such, direct geophysical measurements of PDCs will provide a driver for future research and a formidable test bench for models, especially when employed for hazard assessment.

PDC hazard modelling

Insights from geophysical measurements, and improved knowledge of the internal dynamics of PDCs, enable increasingly accurate numerical models of PDCs to be created. The main aim of PDC hazard models is to generate forward predictions that can be used, in conjunction with field observations and mapping, to assess and mitigate potential hazards (Fig. 7c–e). The most common approach¹⁸² is to map, in a numerical simulation for a given PDC scenario, the maximum flow dynamic pressure, temperature, in-air particle concentration, the inundation area and the thickness of the deposits¹⁵. Such modelling has multiple challenges, which include uncertainty in the source properties (location, volume, mass flow rate, temperature, duration, grain-size distribution) and boundary conditions (including topography).

There are several key challenges associated with developing a satisfactory model for mapping PDC hazard variables. For example, models must account for a wide range of dynamic scales, flow density stratification, a non-linear granular rheology, broad particle sizes and density distributions, and complex interactions with the substrate and three-dimensional morphology.

Although simplified hazard models with empirically calibrated parameters (such as density stratification and variations in the particle-size distributions) have been developed for other geophysical granular mass flows (such as rock avalanches)^183,184, similar empirically calibrated hazard models are currently not available for PDCs. Even in cases where scientists had a relatively broad statistical ensemble of observations to calibrate empirical parameters (such as during the 1995–2010 eruption of Soufrière Hills volcano)¹⁸⁵, hazard prediction has been hindered by three main aspects of volcanic flows.

First, the diversity of PDC generation mechanisms, and the broad range of possible eruption sizes, results in extremely large uncertainties associated with the initial conditions of the flow. Second, frequent unpredictable transitions from concentrated to intermediate or dilute regimes can generate highly mobile turbulent flows, which might decouple from the main current^186,187. Transitions between regimes and decoupling from the main flow might also occur in response to topographic slope changes, channels and obstacles. Third, the substantial thermal energy contained within PDCs influences internal dynamics and often controls the size of the hazard region. The third point, in particular, is important, since even dilute ash clouds with low dynamic pressure can be lethal at temperatures >100 °C (ref.¹⁸⁸).

Delineating the interaction and separation of concentrated, intermediate and dilute regimes is, therefore, not only a puzzling problem in the theory of granular fluids but also one of the major challenges for modelling PDC hazards^15,189. For example, mesoscale clustering in the intermediate regime can enhance sedimentation and modify concentration and temperature stratification, ultimately controlling the dynamic pressure and heat-transfer capacity of a PDC. Therefore, fully accounting for the broad range of recognized gas–particle feedback mechanisms operating within dilute, intermediate and concentrated transport regimes, and including appropriate rheological models that capture the effects of pore-pressure feedback mechanisms on flow mobility, constitute major challenges for future PDC flow models.

Accounting for gas–particle feedback mechanisms is especially important when models are used to assess hazards and support decision-making for risk mitigation and volcanic-crisis management. In such cases, uncertainties in the initial and boundary conditions, as well as uncertainties associated with the flow dynamics, gas–particle feedbacks and other internal PDC processes, should be quantified and managed in a probabilistic framework. However, owing to their high computational costs, two-dimensional and three-dimensional multiphase flow models^45,152, which can effectively describe PDC generation and coexistence of the three different flow regimes for single-scenario simulations^43,112, are non-optimal for use in a probabilistic framework. Furthermore, there is a developing general consensus that depth-averaged models^{190,191,192,193,194,195,196}, which are computationally simple, cannot provide a comprehensive view on all PDC regimes^64,197.

One possible compromise is application of multilayered models, in which PDC stratification is simplified into a concentrated, basal layer underlying a dilute ash cloud. In a multilayered model, mass, momentum and energy exchange in the intermediate regime between the two layers can be described by empirical models^{190,191,192,193,194}. However, to obtain reliable predictions of flow behaviour from a multilayered model, calibration of empirical models in a multilayer formulation is essential (for example, constraining the fluxes of mass, momentum and energy within and through the boundaries of the dilute, intermediate and concentrated regimes).

As such, an emerging method of modelling PDC dynamics that might be particularly useful for probabilistic hazard assessments involves the calibration of low-dimensional models with two-dimensional and three-dimensional numerical simulations. Calibration of low-dimensional models by higher-dimensional models could incorporate complex features of PDCs, such as the deposition and erosion rates at the lower flow boundary, particle sedimentation and elutriation rates from the dilute ash cloud and concentrated basal layer, and the atmospheric air-entrainment rate. For example, Fig. 7e displays a calibrated integral (0D) model, where the effects of topographic barriers on PDC invasion at Campi Flegrei (Italy) are computed by using 3D models (Fig. 7c,d) with different boundary and initial conditions (steady fountaining and single collapse)¹⁹⁸. Although simplified models can be calibrated to fit complex model results, input and boundary conditions, the universality of the calibrated parameters (the ability of models calibrated for a particular PDC scenario to produce meaningful results in non-calibrated cases) has to be verified.

To address the challenge of developing robust and accurate PDC hazard forecasts, a broad international initiative (Validation and Benchmarking of Pyroclastic Density Currents initiative of the International Association of Volcanology and Chemistry of the Earth’s Interior) has recently commenced to intercompare existing models and to evaluate their inaccuracies in a broad range of PDC flow conditions. The goal is to develop a consensual view on PDC model validation and benchmarking¹⁹⁹, to identify the optimal approach to hazard modelling and to provide guidance on the applicability of models at different levels of approximation.

Summary and future perspectives

Owing to their multifaceted perils, extreme and poorly mitigable hazard intensities, and high risk exposure in growing and expanding society, PDCs remain one of the greatest volcanic hazards. Recently, the combination of data from field studies, large-scale and benchtop-scale experiments, and theoretical and numerical models, has provided critical new insights into the internal dynamics of PDCs. A holistic approach, considering evidence from multiple subdisciplines in PDC research, is required to address the complex, enduring gaps in understanding the flow and hazard behaviour of PDCs. Below, we highlight several challenges for future research.

Current granular-fluid rheologies that are used in numerical hazard models are based on simplified monodisperse systems. To achieve reliable and rapid hazard forecasts of the concentrated and intermediate transport regimes in PDCs, granular-fluid rheologies must be derived for polydisperse gas–particle mixtures and implemented into existing flow models. Polydisperse gas–particle rheologies will need to address microphysical processes, such as particle interactions with small-scale fluid motion (that is, on the size of an individual particle) and the organization of permeability structures inside flows, which are neither resolved nor parameterized in current models.

Fundamental uncertainties exist on how energy fluctuations are generated, transported and dissipated in PDCs, including the modification of coherent turbulence structures owing to the influence of polydisperse particles, clustering and drag changes. As a result, it remains uncertain how, in dilute and intermediate transport regimes, gas–particle turbulence controls the sedimentation rate, associated momentum loss, flow stratification and buoyancy reversal. In addition, it is still unknown how the granular temperature influences the rheology of concentrated flows. Future research into the generation of energy fluctuations would also add important complexity to newly developed multilayer continuum flow models and inform the constitutive equations for mass, momentum and energy transport within and between layers.

Heat is one of the primary killers in PDC disasters. However, the consequence of temperature modifications on the internal energy, viscosity, density and compressibility of flows and, subsequently, on flow resistance, entrainment and large-scale thermal flow evolution, are poorly understood. Therefore, future laboratory and computational experiments considering thermal effects on scales ranging from the largest eddy to the size of individual particles, in combination with careful characterizations of the thermal properties of natural volcanic material, are needed. The design and thermodynamic scaling of such multidisciplinary studies should be guided by, and ultimately validated against, remote or direct geophysical measurements of the development and evolution of thermal stratification in real-world flows.

Analysis of PDC deposits might offer indirect insights into the characteristics and conditions of hazardous PDCs. However, future field studies are required to link time-variant erosion, deposition and bedform development in the lower flow boundary to the unknown characteristics of internal velocity and density stratification and turbulent near-bed boundary stresses in PDCs. Combined large-scale experiments and high-resolution multiphase simulations could add critical complexity and time-dependence to the simplified relationships between deposit properties and the time-averaged vertical structure that are used in turbulent boundary layer and turbulent sedimentation models.

The international community has started to systematically validate computational flow models against measured flow conditions in large-scale experiments¹⁹⁹. Critical analysis of computational flow models constitutes an important coming of age of PDC research and will define the strengths, weaknesses and limits of current modelling strategies, and test existing models so that they can be deployed confidently for public safety. However, large-scale experiments cannot capture the full complexity of natural flows. Therefore, although it is certainly an ambitious endeavour, we feel that the strongest scientific advance in PDC research would result from a multinational and multi-instrument effort to measure the internal properties of real-world PDCs to test and advance experiment-based and theory-based knowledge.

Geophysical measurements of the vertical PDC structure, including velocity, density and temperature fluctuations, and its evolution over natural topography, should be guided by approaches currently utilized in the research of snow avalanches and turbidity currents¹⁸¹. Recent advances in remote and direct geophysical measurement techniques should be used to consider the diversity of PDC behaviour resulting from the natural spectra of flow magnitudes and initiation mechanisms. The resulting data would not only inform realistic model benchmarks but would also provide a more complete picture of the range of multiscale and multiphase processes operating inside PDCs across dilute, intermediate and concentrated flow regime boundaries.