Heat Transfer in Pyroclastic Density Current‐Ice Interactions: Insights From Experimental and Numerical Simulations

Abstract

Stratovolcanoes are common globally, with high‐altitude summit regions that are often glacier‐clad and intersect the seasonal and perennial snow line. During an eruption, interaction between snow/ice and hot, pyroclastic deposits will potentially lead to extensive melt and steam production. This is particularly pertinent when pyroclastic density currents (PDCs) are emplaced onto and propagate over glacierised substrates. Generated melt and steam are incorporated into the flow, which can cause a transformation from a hot, dry granular flow, to a water‐saturated, sediment‐laden flow, termed a lahar. Both PDCs and ice‐melt lahars are highly hazardous due to their high energy during flow and long runout distances. Knowledge of the physics that underpin these interactions and the transformation to ice‐melt lahar is extremely limited, preventing accurate descriptions within hazard models. To physically constrain the thermal interactions we conduct static melting experiments, where a hot granular layer was emplaced onto an ice substrate. The rate of heat transfer through the particle layer, melt and steam generation were quantified. Experiments revealed systematic increases in melt and steam with increasing particle layer thicknesses and temperatures. We also present a one‐dimensional numerical model for heat transfer, calibrated against experimental data, capable of accurately predicting temperature and associated melting. Furthermore, similarity solutions are presented for early‐time melting which are used to benchmark our numerical scheme, and to provide rapid estimates for meltwater flux hydrographs. These data are vital for predicting melt volume and incorporation into PDCs required to facilitate the transformation to and evolution of ice‐melt lahars.

Full text

Heat Transfer in Pyroclastic Density Current‐Ice
Interactions: Insights From Experimental and Numerical
Simulations
Amelia B. Vale
1
, Luke T. Jenkins
1
, Jeremy C. Phillips
1
, Alison C. Rust
1
, Andrew J. Hogg
2
,
Geoff Kilgour
3
, and Anya Seward
3
1
School of Earth Sciences, University of Bristol, Bristol, UK,
2
School of Mathematics, University of Bristol, Bristol, UK,
3
GNS Science, Taupo, New Zealand
Abstract Stratovolcanoes are common globally, with high‐altitude summit regions that are often glacier‐
clad and intersect the seasonal and perennial snow line. During an eruption, interaction between snow/ice and
hot, pyroclastic deposits will potentially lead to extensive melt and steam production. This is particularly
pertinent when pyroclastic density currents (PDCs) are emplaced onto and propagate over glacierised substrates.
Generated melt and steam are incorporated into the flow, which can cause a transformation from a hot, dry
granular flow, to a water‐saturated, sediment‐laden flow, termed a lahar. Both PDCs and ice‐melt lahars are
highly hazardous due to their high energy during flow and long runout distances. Knowledge of the physics that
underpin these interactions and the transformation to ice‐melt lahar is extremely limited, preventing accurate
descriptions within hazard models. To physically constrain the thermal interactions we conduct static melting
experiments, where a hot granular layer was emplaced onto an ice substrate. The rate of heat transfer through the
particle layer, melt and steam generation were quantified. Experiments revealed systematic increases in melt
and steam with increasing particle layer thicknesses and temperatures. We also present a one‐dimensional
numerical model for heat transfer, calibrated against experimental data, capable of accurately predicting
temperature and associated melting. Furthermore, similarity solutions are presented for early‐time melting
which are used to benchmark our numerical scheme, and to provide rapid estimates for meltwater flux
hydrographs. These data are vital for predicting melt volume and incorporation into PDCs required to facilitate
the transformation to and evolution of ice‐melt lahars.
Plain Language Summary When volcanoes explosively erupt they may produce avalanches of hot,
dry volcanic ash. When these volcanic avalanches occur on snow and glacier‐covered volcanoes, they produce
steam and melt, that can mix with the volcanic avalanche, transforming it to a cool, wet volcanic mudflow. Both
volcanic avalanches and mudflows are extremely destructive and dangerous due to their high speeds and long
flow paths. Historically, these flows have resulted in many fatalities and extensive building and infrastructure
damage. We investigate the conditions under which transformation from volcanic avalanches to mudflows can
occur. We use small‐scale laboratory experiments to measure the transfer of heat, steam, and melt generation
when a hot ash layer is emplaced onto an ice layer. A numerical model is presented to describe this heat transfer
at large scales, like in natural volcanic settings. This can be used to estimate the amount of melt required to cause
this transformation from volcanic avalanche to mudflow. This can help us predict the destructiveness of these
interactive events, and help us convey the hazard to stakeholders, and populations living in regions affected by
volcano‐ice interactions.
1. Introduction
Pyroclastic density currents (PDCs) are multiphase gravity currents composed of hot particles and gas that are
generated by the gravitational collapse of an eruption column or lava dome (e.g., Dellino et al., 2021; Druitt, 1998;
Lube et al., 2015; Sulpizio et al., 2014). They are highly destructive due to their high speeds, ranging from around
10 to >100 m s
1
(e.g., Belousov et al., 2002; Cole et al., 1998; Scharff et al., 2019; Sparks et al., 2002;
Yamamoto et al., 1993), and temperatures, typically ranging between 100 and 700°C (e.g., Banks &
Hoblitt, 1996; Belousov et al., 2002; Cole et al., 2002; Druitt et al., 2002). The intermingling of the solid particles
and fluid (gas) phase to varying extents produces a continuum ranging from dilute (gas‐dominated) to concen-
trated (particle‐dominated) PDCs.
RESEARCH ARTICLE
10.1029/2024JB029321
Special Collection:
Advances in understanding
volcanic processes
Key Points:
•Pyroclast‐ice interactions are investi-
gated using laboratory experiments and
numerical modeling
•The parameter space that governs
melting and steam generation is
estimated
•Meltwater source flux hydrographs
show similarities with rainfall‐driven
lahar source hydrographs
Supporting Information:
Supporting Information may be found in
the online version of this article.
Correspondence to:
J. C. Phillips,
j.c.phillips@bristol.ac.uk
Citation:
Vale, A. B., Jenkins, L. T., Phillips, J. C.,
Rust, A. C., Hogg, A. J., Kilgour, G., &
Seward, A. (2024). Heat transfer in
pyroclastic density current‐ice
interactions: Insights from experimental
and numerical simulations. Journal of
Geophysical Research: Solid Earth,129,
e2024JB029321. https://doi.org/10.1029/
2024JB029321
Received 15 APR 2024
Accepted 18 APR 2024
Author Contributions:
Conceptualization: Jeremy C. Phillips,
Alison C. Rust, Geoff Kilgour
Data curation: Amelia B. Vale
Formal analysis: Amelia B. Vale, Luke
T. Jenkins, Andrew J. Hogg
Funding acquisition: Amelia B. Vale,
Luke T. Jenkins, Jeremy C. Phillips,
Andrew J. Hogg, Geoff Kilgour,
Anya Seward
Investigation: Amelia B. Vale, Luke
T. Jenkins
Methodology: Amelia B. Vale, Luke
T. Jenkins, Anya Seward
© 2024. The Author(s).
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.
VALE ET AL. 1 of 22

When PDCs propagate over and are emplaced onto snow or ice they mechanically and thermally scour the
substrate (Pierson et al., 1990; Thouret et al., 2007; Walder, 2000b). These processes can generate steam and
meltwater that are incorporated into the flow, which can fundamentally affect its dynamics and cause trans-
formations in both flow mobility and character (Figure 1). Generation and escape of steam can increase pore fluid
pressures within the granular layer and cause fluidization of the flow, enhancing its overall mobility (Roche
et al., 2002; Rowley et al., 2014). Incorporation of meltwater can affect the friction and cohesion properties of the
bulk particle layer and can transform the flow into an ice‐melt lahar if sufficient melt is generated (Ahmed
et al., 2012; Branney & Gilbert, 1995; Huggel et al., 2007; Walding et al., 2023).
PDC interactions with frozen substrates are difficult to study in situ due to their unpredictable and hazardous
nature and poor preservation potential due to the susceptibility of snow and ice to melt out of deposits, reworking
them in the process (Breard, Calder, & Ruth, 2020). Few evidence‐based field studies of PDC‐ice interactions and
subsequent lahar generation exist. For example, highly detailed investigations were conducted following the
catastrophic 1985 eruption of Nevado del Ruiz. These studies provided constraints on (a) total ice loss and melt
volume (Thouret, 1990), and (b) PDC, tephra fall and lahar events and deposits (Naranjo et al., 1986; Pierson
et al., 1990). Kilgour et al. (2010) also provides a detailed study following the 25 September 2007 eruption of
Ruapehu, which generated small‐volume lahars. These studies provide context for the geophysical scale modeling
described in Section 4.2. Experiments and theoretical modeling complement detailed field studies and can offer
additional insights into the microphysical interactions between PDCs and frozen substrates.
In order to comprehensively investigate the physics underpinning PDC‐ice interactions, the thermal and me-
chanical effects must be disentangled. Our study builds on previous work where the thermal effects are studied in
isolation by considering the scenario where particles are rapidly placed on a horizontal frozen surface, so that
there is no mechanical scour. Few theoretical and experimental studies of hot particle‐ice interactions exist of-
fering insights into the heat transfer from particle to ice layers. Walder (2000a, 2000b) developed a theory for
pyroclast‐snow interactions and thermally‐driven slurry formation based on vertical thermal transfer between a
porous, hot, monodisperse, and polydisperse particle layer and a snow substrate. Experiments where hot particles
were released onto shaved ice revealed a continuum of behaviors. Where no convective bubbling occurred within
the granular layer, the particles melted into the snow as a wetting front rose upwards through the particle layer. In
other cases rising vapor bubbles caused complete convective overturning of the particle layer by fluidization,
thermally scouring, and incorporating the snow, facilitating the transformation from a dry non‐cohesive mass of
particles into a slurry. The latter regimes were favored by higher initial particle temperatures and smaller grain
diameters. Cowlyn (2016) conducted complementary experiments to determine the amount of melting that could
be generated by an individual pyroclast. These experiments provided numerical constraints for the rate of melt and
steam production to inform predictive models for PDC‐ice interaction.
Project administration: Amelia B. Vale,
Jeremy C. Phillips, Alison C. Rust
Software: Luke T. Jenkins, Andrew
J. Hogg
Supervision: Jeremy C. Phillips, Alison
C. Rust, Andrew J. Hogg, Geoff Kilgour
Validation: Luke T. Jenkins
Visualization: Amelia B. Vale, Luke
T. Jenkins
Writing – original draft: Amelia B. Vale,
Luke T. Jenkins
Writing – review & editing: Amelia
B. Vale, Luke T. Jenkins, Jeremy
C. Phillips, Alison C. Rust, Andrew
J. Hogg, Geoff Kilgour, Anya Seward
Figure 1. Redoubt volcano viewed from the northwest following the 4 April 2009 eruption. Annotations show volcanic
processes and deposits. Incisions in the glaciated surface indicate thermal and mechanical scour by pyroclastic density
currents (PDCs). Lahar deposits are observed downstream of the PDC deposits. Photo source: USGS (2009).
Journal of Geophysical Research: Solid Earth
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In this paper we focus on the thermal interactions between a layer of hot particles and ice. The previous
experimental work of Walder (2000b) and Cowlyn (2016) are extended to include the interactions of volcanic and
non‐volcanic particles with ice across the expected thermal range for PDC‐ice interactions. We report on the time
evolution of temperature through the particle layer and the products generated by the interactions between hot
particles and ice. Numerical simulations of this heat transfer are also presented, along with mathematical analysis
at geophysical scales, highlighting the implications and importance of these simulations for natural PDC‐ice
interactions, and ice‐melt lahar generation.
2. Materials and Methods
2.1. Experiments
A series of static melting experiments were conducted, where hot particles were poured onto a horizontal ice
substrate to (a) investigate heat transfer between the particle and ice layers, and (b) quantify melt and steam
generation. These experiments were designed as an analog to thermally‐driven pyroclast‐ice interactions and do
not investigate the role of mechanical scouring in incorporating water and steam into hot density currents. Our
experiment data are freely available in an online repository (Vale et al., 2023).
2.1.1. Materials
Three particle types were used in the experiments: glass ballotini, crushed pumice (acquired from: Specialist
Aggregates, product code: 7803), and an andesitic ash sample from Ruapehu Volcano. Glass ballotini have
frequently been used in granular flow experiments because of their highly regular shape and packing structure,
making them a good particle type for comparison purposes (Roche et al., 2004; Rowley et al., 2014). The natural
samples used in experiments were selected to encompass a range of compositions, from felsic, vesicular volcanic
glass (pumice) to more mafic and heterogeneous volcanic samples (Ruapehu ash). Grain characteristics were
constrained for the three particle types through image acquisition and analysis techniques, described below
(Figure 2, Table 1).
The natural experiment samples were sieved to be within the 500–2,000 μm grain size fractions, while Ballotini
particles were pre‐sorted into 1,000–1,400 μm sieve fractions. Particle grain size and shape distributions were
determined for all three particle types using Dynamic image analysis using a CAMSIZER X2 (Figure 2, Table 1)
(Buckland et al., 2021; Microtrac MRB, 2024). Particle sphericity is a measure of shape, computed for an imaged
particle as the ratio of the area to the perimeter squared, multiplied by a factor of 4π (Retsch Technology, 2023),
which has a maximum value of 1 for a perfect sphere (perfect circle in image, Figure 2). Ballotini particles are
spherical, and regular in shape, with sphericities consistently close to 1. Ruapehu ash samples have sphericities
ranging between 0.8 and 0.9. Pumice are the most variable in shape, with sphericities between 0.6 and 0.9.
A Hitachi S3500‐N scanning electron microscope (SEM) operating in backscattered electron (BSE) mode at the
University of Bristol was used to image and characterize Ruapehu ash and pumice samples particle componentry
and vesicularity. The samples were mounted in epoxy resin, manually ground to grades PSI 240 to 1200, polished
using a Buehler AutoMet 250 autopolisher to grades 9, 3 and 1 μm, and then carbon coated. Pumice samples were
composed almost entirely of glass (>95%), with few crystals present. Ruapehu ash samples were more varied in
composition and texture, with the presence of microlites, phenocrysts and glass (Figure 2). SEM BSE images
were analyzed using ImageJ, an open‐source, Java‐based image processing software (Schneider et al., 2012).
Images were manually edited (i.e., colored black) to remove particle fragments trapped in epoxy in vesicle in-
teriors. Image thresholiding was applied to distinguish between the groundmass and vesicles. The percentage of
vesiculated area in the image was calculated using the “analyze particles” function within ImageJ, similar to Liu
et al. (2017).
A glass pycnometer was filled with water, and subsequently one type of particles were added to the vessel. The
volume of particles was determined by measuring the volume of displaced water (Flint & Flint, 2002). Ballotini
particles were the densest, and pumice particles were the least dense. This is consistent with particle vesicularity
measurements, where ballotini particles had no vesicles, meanwhile pumice particles had a vesiculated area up to
78%. However, for porous samples these densities are maxima because water may enter some of the vesicles.
The packing fraction of the granular layer in a cylindrical vessel was calculated by dividing the bulk density of the
particle layer by the grain density of each particle type. The bulk density was determined by dividing the mass of
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Table 1
Grain Characteristics of the Particle Types Used in Static Melting Experiments (a)
Grain characteristic Glass ballotini Crushed pumice Ruapehu ash
(a)
Grain size range (μm) 1,000–1,400 500–2,000 500–2,000
Sphericity 1 0.6–0.9 0.8–0.9
Density ρ(kg m
3
) 2,500 1,080 2,200
Vesicularity (%) 0 75–78 24–56
Packing fraction (%) 65 65 71
Specific heat capacity c
p
(J kg
1
K
1
) 1,245 3,684 1,201
Thermal conductivity k(W m
1
K
1
) 1.1–1.13 0.75 1.08–1.56
(b)
Diffusivity ratio R
α
3.26–3.35 6.27 2.00–2.89
Conductivity ratio R
k
1.96–2.02 2.96 1.42–2.06
Nusselt number Nu 0.0177–0.0182 0.0267 0.0128–0.0185
Note. We also present the corresponding dimensionless parameters for each material using Equations 7a–7d and taking
d=20 mm (b).
Figure 2. Grain size distributions of experiment particle types, glass ballotini (left column), crushed pumice (central column),
and Ruapehu ash/lapilli (right column), shown with Camsizer X2 imagery for shape analysis and Microscope/scanning
electron microscope (SEM) imagery used for componentry and vesicularity analysis. SEM Voltage 20 kV, and working
distance 22.4 mm.
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particles by the volume of the bed. For these packing calculations the grain density was obtained by multiplying
the solid density (from He pycnometry on powdered samples) by the average solid fraction of the particles based
on vesicle area versus solid fraction analysis of SEM images of polished grains. Analysis of packing fraction
showed that packing densities for pumice and ballotini particles were equivalent, while packing density for
Ruapehu ash was slightly higher.
Bulk particle thermal properties in a water‐saturated state were measured at room temperature using a Portable
Electronic Divided Bar (PEDB, product code: Hot Dry Rocks HDR01), at GNS Science, Taupo, New Zealand.
The PEDB determines a ratio between the thermal gradient across the sample and a known material. With this
method, thermal conductivity measurements are accurate to within ±3.5% (Antriasian, 2009). Specific heat
capacities were also determined by introducing a temperature perturbation and comparing the net thermal energy
absorbed by the sample during thermal re‐equilibration from one steady‐state temperature to another (Antriasian
& Beardsmore, 2014). Thermal conductivity measurements of water‐saturated experiment particle samples
ranged from 0.75 to 1.56 W m
1
K
1
, with pumice particles characterized by the lowest thermal conductivities,
and Ruapehu ash particles the highest.
2.1.2. Experiment Configuration and Procedure
Experiments were initiated by releasing hot particles onto a horizontal layer of ice contained within a cylindrical
alumina beaker whose initial temperature was approximately 20°C (Figure 3). The particle release lasted ∼2 s,
Figure 3. Experimental configuration and model domains. (a) Particles are poured through a thermocouple array onto a horizontal layer of ice. Temperature is recorded
using a ring‐mounted thermocouple array supporting eight type‐k thermocouples set at different heights through the particle layer. Thermocouple heights are denoted by
star markers. (b) Schematic diagrams of our numerical domains. For numerical convenience, we non‐dimensionalize and rescale our dimensionless coordinate system
(left) into separate subdomains (right) for each phase (see Appendix A).
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and over that time a layer of particles of uniform horizontal thickness was formed. The particle mass, and hence
the layer thickness (5–45 mm), was varied. The nominal initial temperature of particles was varied over the range
200–700°C, though the thermocouples recorded the actual initial temperature following particle layer
emplacement. The maximum temperature recorded by each internal thermocouple was used to calibrate our
model. The experiment temperature range was informed by PDC temperatures estimated by direct and proxy
evidence (Banks & Hoblitt, 1996; Cole et al., 2002; Druitt et al., 2002; Lerner et al., 2019; Scott &
Glasspool, 2005).
The evolution of temperature through the particle layer was recorded every second using eight ring‐mounted type‐
K thermocouples. These thermocouples have a temperature recording range from 40 to 1,100 ±1.5°C, and an
experimentally‐determined response time of <3 s (RS product code: 397‐1236). For all experiments, the ther-
mocouples were fixed at several heights from the ice‐particle interface (0 mm) up to 45 mm (surface of the
thickest particle layer) (Figure 3). After 10 min of particle‐ice contact, the particles were separated from the ice,
weighed with the melt, dried, and then reweighed. Melt was calculated as the mass difference between the wet
particles (particles plus meltwater) and dry particles. The mass of steam generated was inferred from the dif-
ference between the mass loss of ice and total meltwater generated. From each experiment, a single measurement
of the total amount of melt and steam produced in the 10 min after the particles first come into contact with the ice
was obtained.
A limited set of experiments were also conducted at lower temperatures (20–200°C) and smaller particle depths
(10–30 mm) using a larger, rectangular cross‐section (330 ×230 mm) apparatus to confirm that the smaller‐scale
(75 mm diameter) static experiments were scalable to larger systems. In the larger scale experiments, particles
were heated in a laboratory oven with a lower thermal range than the furnace used for the small scale experiments.
The oven was used due to capacity constraints. Scaling between experiment configurations is presented in
Supporting Information S1 (Vale et al., 2023).
2.2. Numerical Simulations
We analyze a one‐dimensional domain comprised of hot ash, of thickness d, overlying ice, with initial thickness
H
0
(Figure 3). The coordinate system is upwards positive, with an origin defined such that z=0 marks the initial
position of the ash‐ice interface. Melting of the ice will move the positions of the ash‐ice interface z=s(t), and air‐
ash interface z=d+s(t).
In each solid phase, thermal diffusion is described by
ρjcp,j
∂T
∂t ∂
∂z(kj
∂T
∂z )=0, (1)
where j=[A,I] denotes the solid phase for ash and ice respectively; ρ
j
,c
p,j
, and k
j
are the density, specific heat
capacity, and thermal conductivity of the jth solid phase respectively. For simplicity, we consider thermal
transport in the solid phases only. That is, higher‐order thermal effects associated with the imbibition of meltwater
are neglected. In Section 4.1, we calibrate our model and demonstrate that this is a reasonable assumption.
During melting, conservation of energy requires that the latent heat of melting (L) is balanced by the difference in
heat flux across the ash‐ice interface:
ρILds
dt =kI
∂T
∂z



z=skA
∂T
∂z



z=s+
, with T(z=s(t)) = Tm,(2)
where T
m
is the melting temperature of ice and q=  k
j
(∂T/∂z)|
z=s
is the heat flux, given by Fourier's law,
evaluated at the ash‐ice interface. This equation—which is often referred to as the Stefan condition—is widely
used in moving‐boundary problems to describe the velocity of a phase‐change interface (e.g., Meirmanov, 2011).
Note that due to our sign convention, melting occurs when ds/dt<0. Therefore, the cumulative melt at time tis
given by (ρ
w
/ρ
I
)s(t), where ρ
w
is the water density.
A Dirichlet condition is required to couple the ash and ice subdomains. The magnitude of the interface tem-
perature must account for occurrence, or absence, of melting. Melting is an isothermal process; meaning that the
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interface temperature is pinned at the melting temperature when Equation 2is negative. When melting terminates,
the heat flux either side of the interface is continuous. Solving Equation 2with ds/dt=0 yields a Dirichlet
condition for the interface temperature that is a weighted arithmetic mean of ash and ice temperatures either side
of the interface.
At the top boundary, heat losses to the air are likely dominated by convection, which are approximated by
equating the surface heat flux to a linear constitutive function that is proportional to the temperature difference at
the surface:
kA
∂T
∂z =γ(TTair), at z=d+s(t),(3)
where T
air
is the ambient air temperature, and γis a dimensional parameter that control the strength of convective
heat losses (e.g., Vollmer, 2009). We assume that the basal boundary is perfectly insulating:
∂T
∂z =0, at z=  H0.(4)
This is likely a reasonable assumption for most geophysical settings, where the heat capacity of ice greatly exceeds
that of ash (ρ
I
c
I
H
0
)/(ρ
A
c
A
d)≫1. This is confirmed in Section 4.1, where we demonstrate that our model cali-
bration improves as d/Hdecreases. In all cases, a uniform initial temperature distribution in the ice (T
I
(t=0, z))
is assumed. When calibrating our model at the laboratory scale, we use an initial temperature distribution
(T
A
(t=0, z)) that is determined experimentally (see Section 4.1). At geophysical‐scales (Section 4.2), we assume
for simplicity that T
A
(t=0, z) is uniform.
2.2.1. Non‐Dimensionalization and Rescaling
To better understand the key controls of volcanically‐induced ice melting, we reduce the number of parameters in
our model by non‐dimensionalizing the governing equations and boundary conditions using rescaled variables
˜z=z/d,˜
t=t/τ,˜
T=T/ϑ, and ˜s=s/d,(5)
in combination with characteristic scales
τ=d2
αA
, and ϑ=ρIL
ρAcp,A
.(6)
Note that the characteristic timescale is diffusive, whereas the characteristic temperature scale is a ratio between
the volumetric latent heat of ice to the volumetric heat capacity of ash.
The remaining dimensionless parameters are
Rα=kIρAcp,A
kAρIcp,I≡αI
αA
,(7a)
Rk=kI
kA
,(7b)
Nu =γd
kA
,(7c)
H=H0
d,(7d)
where α
j
is the thermal diffusivity of the jth phase. To avoid numerical complexities associated with solving
diffusion in a shrinking (ice) domain, we transform our dimensionless coordinate system onto a fixed domain
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using a bilinear mapping (see Figure 3and Appendix A). Dropping the tilde notation used above, the resulting
remapped non‐dimensional governing equations are:
∂T
∂t =ds
dt
∂T
∂η +∂2T
∂η2, f or 0 <η<1, (8a)
∂T
∂t =(1+ν
H+s)ds
dt
∂T
∂ν +Rα
(H+s)2
∂2T
∂ν2, for 1<ν<0, (8b)
ds
dt =Rk
H+s
∂T
∂ν



I∂T
∂η



A
<0, with T(η=ν=0) = Tm,(8c)
∂T
∂η =  Nu(TTair), at η=1.(8d)
Note that rescaling Equation 1introduces advective terms to account for the motion of the ash‐ice interface, such
that thermal transport is now described by two coupled advection‐diffusion equations with four principle pa-
rameters: R
α
, which compares the strength of thermal diffusion (α
j
) in the ice and ash; R
k
compares the strength of
thermal conduction in the solid phases; Hcompares the initial thicknesses of the solid phases; and the Nusselt
number Nu is the ratio of convective to conductive heat transfer at the surface of the ash (η=1). Note that the
range of dimensionless parameters for each material are presented in Table 1.
The coupled system of ordinary and partial differential equations is solved using the method of lines (e.g.,
Schiesser, 2012). We use a standard first‐order finite‐volume scheme to discretize our remapped spatial domains;
allowing for the resulting system of equations to be expressed as a series of coupled ordinary differential
equations (ODEs), which are integrated in time using MATLAB’s stiff ODE solver ODE15s (Shampine &
Reichelt, 1997). The ODE solver is provided with the pattern of the Jacobian matrix. This significantly reduces
the computation time by allowing the solver to only evaluate the Jacobian's non‐sparse elements (e.g., Goudarzi
et al., 2016). Our model is validated in Section 4.2, where we demonstrate that our numerical scheme agrees with
similarity solutions that describe early‐time diffusive melting. The convergence of our numerical scheme is
verified by exploring the effect of grid resolution on the melting end state s
∞
≡s(t→∞) in the reference case used
in Section 4.2. For our simulations, we use 2,500 grid cells in each solid phase. At this resolution, simulations take
∼9 s on a single i7-6500U processor, compared with a run time of ∼65 s for simulations that do not utilize the
Jacobian pattern. Further reducing the grid spacing results in variations to |s
∞
| by less than 0.8%. Our numerical
solver and associated plotting scripts are freely available in an online repository (Vale et al., 2023).
3. Experimental Results
3.1. Heat Transfer From Particle to Ice
The time evolution of temperature at set heights through the particle layer was obtained using eight type‐K
thermocouples. For all experiments, the thermocouples captured the initial spike in temperature following
emplacement and the subsequent cooling of the particles as heat was transferred from the particle layer to the
underlying ice and air above. Figure 4provides examples from 200°C Ruapehu ash experiments as our numerical
model is calibrated against this data, however all experiments exhibited the same broad pattern of thermal decay,
but over different timescales. The temperature data for all other experiments can be found in Supporting Infor-
mation S1 (Vale et al., 2023).
The actual maximum particle temperatures recorded by the thermocouples were lower than the furnace tem-
perature because the particles cooled over a period of seconds as they were transported 1.5 m from the furnace,
and then poured ∼10 cm into the ice container. Thicker particle layers and particles characterized by lower
thermal diffusivities retained more heat in transit and release and so attained higher maximum temperatures. The
time at which the maximum temperature was observed varied according to particle type and initial temperature.
At higher maximum temperatures the thermocouples took longer to equilibrate with the particles. Ballotini
particles reached maximum temperatures fastest, meanwhile pumice particles reached maximum temperatures
slowest. This results from the particle's thermal diffusivity. In line with this, ballotini particles also reached
thermal equilibrium fastest, and pumice the slowest. Within individual experiments the maximum temperatures
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obtained by thermocouples varied with position through the particle layer. Typically, the highest temperatures
were recorded in the mid‐particle region, with cooler maximum temperatures recorded closer to the ice and the
particle surface.
The temperature data recorded steam generation signals in two forms, (a) as noise in the thermocouple data, and
(b) as a period of stability around 100°C (see Supporting Information S1, Vale et al., 2023). These signals were
used to determine the duration and intensity of steam generation in experiments. Of the three particle types
examined, pumice particles produced the least steam. This is supported by a general lack of noise or thermocouple
stability around 100°C in the temperature data. Ballotini and Ruapehu ash particles produced significant amounts
of steam in some experiments. The experiment that produced the most steam, B_M250_T700 (34.81 g), generated
steam for over 300 s, or half the total experiment duration, based on duration of the noise signal.
Stepped reductions in the particle temperature followed by stabilizing of the temperature curves were observed in
some experiments (e.g., Figure 4c). These steps were initially recorded close to the particle‐ice interface, but were
subsequently recorded by sequentially higher thermocouples in the particle layer. In the experiments where this
behavior was recorded, these stepped reductions in temperature occurred at different rates between successive
thermocouples, dependent on the initial experiment temperature conditions. We observed these stepped features
across all particle types, most notably in experiments with greater particle layer thicknesses.
3.2. Melt and Steam
Systematic increases in melt with increasing particle layer thickness and temperature were observed across all
particle types (Figure 5). The amount of melt generated was more sensitive to increasing particle temperature for
pumice and Ruapehu ash than for ballotini. The melt data also show a reduction in the rate of increase in melt with
increasing particle layer thickness for all particle types. Steam generation also systematically increased with
particle mass and temperature for ballotini and Ruapehu ash particles, but was negligible for pumice (Figure 5d).
Ballotini particles produced the most steam for any given layer thickness and were more sensitive to particle
temperature than the other particle types.
The generation and escape of steam through the upper particle layer also offered insights into the thermody-
namics of particle‐ice interactions. As expected, steam generation was most productive in the first seconds to
minutes following emplacement, where the duration of steam production was dependent on the initial exper-
iment conditions, with the hottest temperatures and thickest particle layers producing steam for the longest
durations. In some experiments where steam was produced the particles could be fluidized for up to 10 s. In
fluidized experiments, steam escape occurred across the whole surface area of the particle layer. Fluidization of
Ruapehu ash particles also resulted in the elutriation of fines from the particle layer in experiments. In some
other experiments steam escape via the surface was localized and temporally sporadic. The escape of steam via
the upper particle surface also brought melt to the surface with it (see Figure S1 in Supporting Information S1,
Vale et al., 2023).
Figure 4. Thermal evolution for 200°C Ruapehu ash experiments and simulations for d=20 (a), 34 (b), and 45 (c) mm
respectively. Both simulated (solid) and experimental (dashed) curves correspond to thermocouples located at z=[10, 15,
20, and 25] mm above the ash‐ice interface (dark‐light blue).
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4. Modeling Results
4.1. Model Calibration
We calibrate our model using experiments with Ruapehu ash particles heated to 200°C. These experiments
provide two sets of measurements that must be approximated by a well‐calibrated model: (a) thermal evolution of
the ash measured by internal thermocouples; (b) a single measurement of ice melting, recorded after 10 min. In
addition to these experimental constraints, our model calibration is further aided by experimental measurements
of the specific heat capacity and thermal conductivity of the ash. Moreover, the physical properties of ice are well
constrained in the literature. The parameters used in our model calibration are summarized in Table 2. Note that γ
is the only free parameter that is undetermined by experimental measurement or literature values. However, we
fix γ=1 W m
2
K
1
as our results are weakly sensitive to typical variations in γ. We explain the physical
mechanisms related to this weak sensitivity further in Section 4.2.
Analysis of the experimental thermocouple data demonstrates that a significant amount of heat is lost during
transfer of the ash from the oven to the experimental apparatus. This heat loss, which increases for thinner ash
layers (see Figure 6), imparts an initial thermal profile in the ash that must be accounted for in an accurate
calibration. We implement this in our model using a quadratic thermal initial condition in the ash based on the
Figure 5. Panels (a–c) Ice Mass Loss as a function of the mass of heated particles of different initial temperatures. The ice and
particle masses are normalized by their horizontal surface area in contact. Panel (d) Total Meltwater against Ice Mass Loss,
where deviation from a 1:1 slope equates to steam. Particle types are denoted by symbol and correspond to the same symbol
shapes as panels (a–c).
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maximum temperature measured by each internal thermocouple. Specifically,
we use an unconstrained multidimensional nonlinear minimization algorithm
(Nelder & Mead, 1965) to find the quadratic coefficients that correspond to a
minimized total residual between the initial condition and the maximum
thermocouple temperatures. This quadratic profile accounts for both pre‐
experiment heat loss and the initial thermal gradients that redistribute heat
throughout the system.
We compare our calibrated model with experimental data in Figure 4, which
shows the thermal evolution within the ash, and in Table 3, which lists the
proportion of melting after 10 min. Note that we present percentage melting
values to avoid introducing arbitrary length scales into our 1‐D model results.
We find that for all ash thicknesses, our model can accurately simulate the
measured thermal behavior of the system. Our simulations accurately recover
the magnitude and timescale of heat loss within the ash. Moreover, our
calibrated model predicts melting values consistent with those observed in the
laboratory. Our calibration performs best for progressively thinner ash layers.
The growth of these small errors with ash thickness can be attributed to
several effects: (a) maximum experimental temperatures are measured at
progressively later times for thicker ash layers. Therefore, our assumed initial
condition in the ash (a quadratic fit of maximum thermocouple temperatures)
is more appropriate as ddecreases. (b) The basal insulating boundary con-
dition (Equation 4) is typically valid provided that (ρ
I
c
I
H
0
)/(ρ
A
c
A
d)≫1. For thicker ash layers the total thermal
capacity of each phase becomes comparable, meaning that boundary effects begin to impact the dynamics of the
system. (c) The reduced thermal capacity of thinner ash layers induces less melting. Therefore, we expect that our
model, which does not incorporate dynamic effects related to meltwater, to be more valid for thinner layers. (d)
Due to the increased total thermal capacity of ash, steam generation increases with d. For ash temperatures below
400°C, steam generation is negligible in Ruapehu ash samples. However at higher temperatures, and when
ρIcIH0)/ρAcAd)=O(1)or smaller, the latent heat of vapourization becomes non‐negligible to the total thermal
balance of the system.
Note that the overall melting results in our experiments represent a “snapshot” of the of the melting dynamics. In
the model, the end state (i.e. when ds/dt=0) is defined as s∞=lim
t→∞
s(t). For melting to terminate, the ash must
lose sufficient heat to balance the flux terms in the Stefan condition. This is expected at the geophysical scale
where typically d≪H
0
. In this regime, where melting is expected to be negligible relative to the initial thickness
of ice (i.e. s
∞
/H≪1), our model calibration performs best. Naturally, this motivates the use of our calibrated
Table 2
Calibrated Model Parameters
Symbol Value
Ice parameters
Density ρ
I
916 kg m
3
Specific heat capacity c
p,I
2,050 J kg
1
K
1
Thermal conductivity k
I
2.22 W m
1
K
1
Latent heat of melting L333.55 J kg
1
Melting temperature T
m
273 K
Initial temperature T
0,I
253 K
Ash parameters
Density ρ
A
2,200 kg m
3
Specific heat capacity c
p,A
1201 J kg
1
K
1
Thermal conductivity k
A
1.37 W m
1
K
1
Atmospheric heat loss coefficient γ1.0 W m
2
K
1
Note. c
p,A
and k
A
are mean values from five laboratory measurements.
Figure 6. (a) Evolution of the ash‐ice interface plotted for various values of R
α
. The interface initially obeys the early‐time
similarity solution (dashed blue curves) s=  λt
√before transitioning to an end state where melting terminates. (b) Contours
of λ(Equation 11) are plotted to demonstrate its dependence on R
α
and R
k
. In both panels T
A
T
m
≈1.73 and T
I
T
m
≈0.17.
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model to evaluate the evolution of potentially hazardous melt generation
following the emplacement of hot ash onto ice for natural‐scale flows.
4.2. Predictions for Melting During Geophysical Scale Processes
We explore a reference scenario using depths of ash deposited on ice as re-
ported by Pierson et al. (1990) and Kilgour et al. (2010). Our reference
scenario uses length scales of d=0.1 m and H
0
=1 m in conjunction with the
calibrated parameters in Table 2and an initial uniform ash temperature of
200°C. The corresponding dimensionless parameters are listed in Table 4.
Melting occurs in two distinct regimes (Figure 6): early‐time diffusive motion of the ash‐interface, before a
transition to a late‐time regime when melting terminates. Both of these regimes are relevant to volcanic hazards:
the former governs the rate of meltwater supply; while the latter describes the magnitude of melt generation. In
our reference case melting occurs over 0.65–1.56 characteristic (diffusive) time scales (dimensional equivalent of
3.48–8.36 hr). Note that changing the thickness of ice and/or ash will modify the magnitude and duration of
melting, but not the qualitative behavior or existence of these regimes. Before considering the late time regime,
we describe the early‐time transient behavior. In this regime, melting obeys the classical Stefan problem (e.g.,
Meirmanov, 2011) and permits the derivation of a similarity solution for the motion of the ash‐ice interface:
s(t) =  λt
√,(9)
where λis a constant that determines the early‐time melting rate. At early‐times, the motion of the ash‐ice
interface is invariant to the length scales dand H
0
, which allows for the derivation of analytic expressions for
temperature in the ash and ice region respectively:
T=Tm+(TATm)(erf( λ/2) +erf(ξ/2))
erf( λ/2) + 1, for z>s(t),(10a)
T=Tm+(TITm)erfλ/2
Rα
√))+erfξ/2
Rα
√)))
erfλ/2
Rα
√))1, for z<s(t),(10b)
where T
A
and T
I
are the initial ash and ice temperatures respectively, ξ=z/t
√is a diffusive coordinate transform,
and erf(⋅) is the error function. Differentiating these analytical expressions before substituting into the Stefan
condition (Equation 8c) yields an expression for λ, which is given by the solution to
λ
2=Rk(TmTI)eλ2
4Rα

πRα
√(erf(λ
2
Rα
√)+1)+(TATm)eλ2
4

π
√erfλ
2)1).(11)
We solve Equation 11 using MATLAB’s nonlinear root finding algorithm fzero. We demonstrate the accuracy
of our numerical scheme by overlaying these similarity solutions for different values of R
α
in Figure 6a. The
similarity solution's independence of length scales means that the early‐time
melting rate is determined by the interplay between R
α
,R
k
,T
m
,T
I
, and T
A
only. Solving Equation 11 allows for efficient exploration of this parameter
space. Given that the principal temperatures trivially modulate the initial
melting rate (e.g., λmonotonically increases with T
A
and T
I
), we consider the
impact of R
k
and R
α
in Figure 6b. Here we show that λmonotonically in-
creases and decreases with R
α
and R
k
respectively. Increasing R
k
linearly
increases the first term in Equation 8c, which reduces the strength of melting
by decreasing the heat flux differential across the ash‐ice interface. The
melting rate increases with R
α
due to the increased relative strength of thermal
diffusion in the ice. Furthermore, by reducing the relative strength of thermal
diffusion in the ash, convective heat losses to the atmosphere are transmitted
Table 3
Measured and Simulated Melt After 10 Min
d (mm) H
0
(mm) Melt (%) experiment Melt (%) modeled
(a) 20 24.4 34.5 35.5
(b) 34 26.1 43.7 45.4
(c) 45 28.2 48.8 42.6
Note. Labels (a), (b) and (c) correspond to the subplots in Figure 4.
Table 4
Dimensionless Reference Parameters Used for Analysis of Geophysical
Scale Melting
Dimensionless parameter Symbol Value
Diffusivity ratio R
α
2.28
Conductivity ratio R
k
1.62
Nusselt number Nu 0.073
Lengthscale ratio H10
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to the ash‐interface at a slower rate, thus delaying the transition from early‐time self similar melt propagation to
late‐time termination of melting (see below).
At intermediate‐times, the motion of the ash‐ice interface deviates from the early‐time s∝t
√scaling toward a
late‐time state (s
∞
) when melting stops. This transition develops as convective heat losses at the upper boundary
(η=1) begin to impact thermal diffusion across the ash‐ice interface. Eventually, the ash has lost sufficient heat
that the heat fluxes across the ash‐ice interface balance and melting terminates. Accordingly the ash‐ice interface
will remain motionless unless provided with latent heat to resume melting. At intermediate‐ and late‐times,
thermal transport and associated motion of the ash‐ice interface, is coupled to the inherent length scales of the
system, and also to the parameters that govern the rate of heat loss within the system. The impacts of select
parameters (R
α
,H,T
A
T
m
, and Nu) on the total (i.e., end state) melt production are illustrated in Figure 7. As in
the early‐time regime, increasing R
α
, which strengthens thermal diffusion in the ice, results in increased total
melting (Figure 7a). For relevant geophysical settings where H≫1, |s
∞
| is invariant to the ice thickness
(Figure 7a). This is because, when melting is small relative to the initial ice thickness, heating of the ice remains
confined away from the basal boundary which therefore does not impose any scale effects on the total melt
production. In Figure 7b we highlight the impact of the ash temperature and Nusselt number on s
∞
. Intuitively,
|s
∞
| monotonically increases with the ash temperature above a threshold temperature which is required to supply
latent heat across the ash‐ice interface. This threshold is derived by setting λ=0 in Equation 11, which yields
TATm>Rk(TmTI)

Rα
√.(12)
We also demonstrate that, relative to other dimensionless parameters, the Nusselt number has a limited impact on
the total melt production. By strengthening convective heat losses to the atmosphere (increasing Nu), melt
generation reduces, but only by a small fraction. This relative insensitivity to Nu is because thermal transport in
the ash is dominated by conduction into the ice. This is expected for 0 <Nu <1, and further explains the weak
sensitivity to Nu in our model calibration. Note also that as the experiments have not reached end state melting,
and therefore as Equation 11 is invariant to Nu we expect our calibration to be insensitive to variations in the
Nusselt number. In Figure 7b our reference case (Nu ∼0.073) will lie between the dashed black and uppermost
orange curves. The proximity of our reference case to the convection free (Nu =0) limit demonstrates that
convective heat losses to the atmosphere are essentially negligible over the time scale of melting for typical
geophysical parameters and temperatures considered.
5. Discussion
When PDCs are emplaced onto snow and ice substrates, they rapidly transfer heat from the particle layer into the
substrate due to large temperature gradients between the two media. This heat transfer generates melt and steam
Figure 7. End state position of the ash‐ice interface versus (a) R
α
, with several different thickness ratios plotted to
demonstrate the invariance of total melting to H; (b) Difference in ash and melting temperature T
A
T
m
, also plotted for
various values of Nu which highlights the weak dependence on the Nusselt number. The dashed black and blue curves
represent the Nu =0 limit and melting threshold (see Equation 12) respectively.
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which can be incorporated into the flow transforming its mobility and character. The role of melt and steam in
PDC‐ice interactions differ, as do their production timescales. Melting is a continuous process for as long as (a)
the hot ash can supply latent heat (via a difference in heat flux across the ash‐ice interface), and (b) there is a
supply of ice to melt. The production and incorporation of melt can cause a PDC to transform from a dry granular
flow into a saturated, sediment‐laden flow, termed an ice‐melt lahar (Thouret et al., 2007). Both melt and steam
production depend on the initial temperature gradient and the thermal diffusivity of the particles in contact with
the ice but steam production requires heating of water above its vaporization temperature and therefore terminates
earlier than melting. The production of steam in PDC‐ice interactions can result in elevated pore fluid pressures,
fluidization and convective overturning of the particle layer. Fluidization increases the mobility of the layer, while
convective overturning sustains the heat transfer from the particle layer to the ice as hot particles from the interior
of the particle layer are brought into contact with the ice, essentially “resetting” the thermal boundary condition at
the ice‐particle interface (Walder, 2000a). Fluidization and convective overturning of the particle layer ceases
when the temperature of the particles in contact with the ice drops below the temperature required for steam
production.
In natural volcanic settings PDCs scour the ice thermally and mechanically, but before the physical coupling
between the thermal and mechanical mechanisms can be considered, the thermal interactions must first be iso-
lated. In the previous sections we presented a series of systematic static melting experiments, along with a
calibrated 1‐D numerical model and mathematical analysis to resolve the rate of heat transfer from a static hot
particle layer to an ice substrate, and to quantify melt. The model can be used to derive a time‐series of melt
generation (see Section 5.3). From this, ice‐melt lahar source hydrographs can be generated, which can be used as
an input in surface flow hazard models. From here on, we discuss heat transfer in particle‐ice interactions,
including the generation and role of melt and steam. Experimental insights and how the 1‐D model can be applied
at geophysical scales, including constraining the ice‐melt lahar hazard are also considered. Finally, the limitations
of this investigation are reviewed and recommendations for further work are suggested.
5.1. Heat Transfer in Particle‐Ice Interactions
Static melting experiments, where a layer of hot particles were emplaced onto a horizontal ice layer were con-
ducted to investigate heat transfer from particle to ice, as an analog to the thermal interactions between hot
pyroclasts and ice. Our 1‐D model simulates heat transfer between two solid phases, (a) a hot ash layer, and (b) an
underlying ice layer. The model is calibrated with experiments using Ruapehu ash heated to 200°C. The
experiment data provided two constraints for the model, (a) the time evolution of temperature through the particle
layer, and (b) a mass of melt at the end of the 10 min experiment. In all experiments the thermocouples recorded
an initial spike in the temperature as particles were emplaced onto the ice, followed by subsequent gradual cooling
as heat was transferred from the particle layer into the ice substrate. Our model captures the magnitude and
timescale of the heat loss from the ash. With our geophysical extension this model can be used to determine the
cooling timescales of deposited pyroclastic material.
5.2. Melt and Steam Generation in Particle‐Ice Interactions
The heat transfer from the particle layer to the ice substrate initiates the production of melt and steam. The
quantities of melt and steam produced in experiments were determined by the particle type, initial particle
temperature, and layer thickness. The differing sensitivities of particles to layer thickness and temperature in
generating melt and steam are caused by variations in both bulk particle and individual grain characteristics,
specifically grain shape, vesicularity, and thermal diffusivity. Ballotini and Ruapehu ash experiments produced
comparable quantities of melt, but ballotini particles produced more steam. Pumice experiments produced the
least melt and negligible steam at all temperatures investigated. Steam generation was not present in all exper-
iments, and was negligible for all cases where the particles were initially cooler than 400°C. For natural samples
with T
A
(t=0, z)>400°C the bulk of ice that is melted or evaporated is measured to be in the liquid phase
(Figure 5d); meaning that, melting is the dominant phase transition over the range of temperatures experienced
during PDC emplacement. Therefore for simplicity, higher‐order terms related to vaporization are not included in
our 1‐D model. Our numerical model successfully predicts melt generation to c. 5% where steam production is
negligible (Table 3). With our geophysical extension, we can predict melt generation at geophysical scales and
provide a time‐series of this melt generation. This informs ice‐melt lahar genesis, which is discussed further in
Section 5.3.
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5.2.1. Melt
In experiments, melt systematically increased with initial particle temperature and layer thickness. The melt data
showed a reduction in the rate of increase in melt with increasing particle layer thickness for all particle types
(Figure 5). Two potential explanations for this melt curve flattening are proposed. First, the observed flattening
could result from the limited timescale of the experiment such that heat from particles in the upper region of a
thick particle layer did not have time to transfer heat to the particle‐ice interface. A second possibility is that the
observed flattening relates to changes in the partitioning of energy within the system with increasing particle layer
thickness. With thinner particle layers the thermal energy melted the underlying ice layer and caused a single
phase change from ice to meltwater. With thicker particle layers the increased thermal energy can be expended
through (a) heating of meltwater to higher temperatures, and (b) meltwater vaporization. The ratio of this energy
partitioning will depend on the initial experiment conditions, including particle layer thickness and temperature.
The particle type (and so particle porosity and thermal diffusivity) is also important; for example, all experiments
with pumice particles produced negligible steam.
Temperature data from the experiments also provided insights into the movement of melt through the particle
layer. In several experiments, a stepped reduction in particle temperature followed by temperature stabilization
was recorded by the thermocouples. This was interpreted to be an upward‐moving meltwater front. This melt-
water front was cooler than the dry particles, which caused a step in the temperature time‐series as the ther-
mocouple came into contact with the melt, and a stabilization as the pore space surrounding the thermocouple
became occupied by melt. These stepped reductions in temperature were recorded by sequentially higher ther-
mocouples as the meltwater front rose through the particle layer. These meltwater fronts were observed for all
three particle types, and rates of movement ranged from 0.04 to 0.59 mm/s. An example of these stepped tem-
perature time‐series can be seen in Figure 4c. As our model does not include a fluid phase, it cannot capture these
perturbations. However, our model performs well at reproducing the leading‐order thermal decay measured in the
experiments.
The principal mechanism proposed for this upward‐moving wetting front is particle sinking, displacing ponded
ice‐surface melt which is generated from the downward wasting of the ice layer. Walder (2000b) similarly re-
ported on the presence of an upward‐moving meltwater front as particles passively melted into the underlying
snow. This hypothesis was tested for ballotini based on measured packing fraction, where 65% of any given
volume is spheres and the remaining 35% is external pore space. The ballotini experiment with a 45 mm particle
layer thickness and an initial temperature of 200°C produced 72.1 g melt during the experiment, equivalent to
16.76 mm melt within our experiment apparatus if no particles were present. Taking into account the packing
fraction of ballotini for the first 45 mm and conservation of mass, this calculation produces a meltwater height of
46.0 mm, which exceeds the particle layer thickness by 1.0 mm. Melt ponding was observed above the particle
surface in this experiment. An additional mechanism for the upward transport of meltwater, relating to steam
escape, is discussed in Section 5.2.2.
5.2.2. Steam
For simplicity, higher‐order terms related to steam generation are not included in our numerical model. It remains
important however to understand the role of steam in PDC‐ice interactions, in terms of both flow mobility and
character, as PDC emplacement temperatures can be higher than the temperatures required to generate steam, for
example, Mount St. Helens (Banks & Hoblitt, 1996), and Soufrière Hills Volcano (Cole et al., 2002; Scott &
Glasspool, 2005). Initial particle temperatures in some experiments were sufficiently high to generate steam,
revealing a range of additional behaviors pertaining PDC‐ice interactions. High temperature experiments revealed
the existence of steam‐driven melt incorporation, fluidization, and elutriation of fines. Evidence for the presence
of steam in experiments was recorded by the thermocouple data in the form of (a) data noise and (b) stability
around 100°C. Temperature data noise was interpreted as sporadic and localized steam generation caused by
meltwater contacting particles above the minimum temperature required to produce steam. Stability in the
temperature time‐series around 100°C was interpreted as sustained boiling and generation of steam caused by
convective overturning of the particle layer in contact with the ice, resetting the thermal boundary condition.
Additional evidence for steam generation was provided through experiment footage (see Supporting Informa-
tion S1, Vale et al., 2023).
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Several features observed during the experiments and in the resulting data provided evidence of steam‐driven
melt incorporation into the particle layer. Within the first few seconds of some high temperature experiments,
we observed vapor‐supported melt lenses skittering on the particle surface. After a few seconds the vapor bubbles
burst, leaving a saturated area on the particle surface. Thermocouple data indicates that this boiling is occurring at
or close to the particle‐ice interface as the data show sustained temperature stability around 100°C in the ther-
mocouples closest to the ice. This steam‐driven melt mixing observation is also consistent with observations by
Walder (2000b), who reported that under some conditions steam generation can cause complete convective
overturning of the particles, which drives thermal scouring of the substrate, mixing, and slurry formation.
The thermocouple data provided further evidence for steam‐driven melt mixing. In some experiments where
steam escaped through the particle layer the temperature profile steps were recorded at multiple successive
thermocouples within the same 1 s period, that is, apparent simultaneous temperature reductions recorded by
several thermocouples (e.g., Figure S2b in Supporting Information S1, Vale et al., 2023). This sporadic steam
escape is caused by dry particles, above the vaporization temperature, coming into contact with cool meltwater.
This escape of steam via the upper particle surface also brought melt to the surface with it. This was recorded in
experiment footage as localized “flash” wetting of the particle surface (e.g., Supporting Information S1,
Figure 3b, Vale et al., 2023). The thermocouple data, in combination with these observations highlight the
important role of steam for efficient melt incorporation and mixing within the particle layer.
Granular layer mobility is known to increase when pore fluid pressures are high (e.g., Breard, Dufek, et al., 2020;
Druitt et al., 2007; Roche et al., 2002). Measuring pore fluid pressures was not possible in our small‐scale ex-
periments, however, fluidization, which occurs as a result of high pore pressure was able to be directly observed
for various initial temperatures and layer thicknesses. Fluidization occurs when the upward flux of gas is suf-
ficient to support the weight of the particles above, reducing interparticle contacts, causing the layer to behave in a
fluid‐like manner (Sparks, 1976). Based on the grain size range of experiment particles, and assuming a density
for steam of effectively zero, our experiment particles relate to groups B and D of the Geldart fluidization
classification system, which behave as a continuum (Geldart, 1973). Fluidization of the particle layer will have an
influence on flow mobility in dynamic particle‐ice interaction settings. In experiments fluidization occurred
instantaneously following emplacement, and endured for several seconds. Ballotini particles were most readily
fluidized across the widest range of initial conditions. This is consistent with the inferred measurements of steam
production being greatest for ballotini particles.
Where fluidization occurred in Ruapehu ash experiments, fines were elutriated from the particle layer and
spattered up the beaker sides. Elutriation of fines is a phenomenon frequently reported in pyroclastic literature, for
example, Wilson (1980), Fisher (1995), and Kelfoun and Gueugneau (2022). Evidence for fines elutriation is
recorded in PDC deposits in the form of fines depletion (Brand et al., 2014) and elutriation pipes (Pacheco‐Hoyos
et al., 2020; Stinton et al., 2014). The presence of this phenomenon in the experiments highlights that the initial
particle temperature range in experiments was sufficient to reproduce naturally occurring behaviors.
Steam was not produced in all experiments, nor was it accounted for in our model. Nevertheless, steam plays an
important role in the mobility of and melt incorporation into the particle layer due to vapor‐driven fluidization and
steam escape, which drives the efficient mixing of melt through the layer. Extrapolating these observations to
dynamic settings and geophysical scales, it is anticipated that the presence of steam may increase the mobility and
potential runout distances of PDCs, and accelerate the transformation from PDC to ice‐melt lahar.
5.3. Geophysical Scale Melting and Constraining Ice‐Melt Lahar Hazard
Where melt supply is limited and incorporation predominantly consists of eroded frozen matter, a PDC can
transform into a mixed avalanche (Breard, Calder, & Ruth, 2020; Lube et al., 2009; Pierson & Janda, 1994). If
sufficient melt is generated and mixing occurs, this layer can transform from hot, dry granular matter, to a
saturated, sediment‐laden flow, or ice‐melt lahar (Kilgour et al., 2010; Major & Newhall, 1989; Pierson
et al., 1990; Waythomas, 2014).
Ice‐melt lahars have historically represented some of the most hazardous volcanic flows (Brown et al., 2017). For
example, during the 1985 eruption of Nevado del Ruiz, Colombia, PDCs were emplaced onto the summit area
resulting in thermal and mechanical scour of the snow and ice substrate, removing 16% of the surface area and 9%
of the total volume of the summit ice cap (Thouret, 1990). The incorporation of melt, snow, and ice into these
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PDCs transformed them into ice‐melt lahars that flowed down valleys on the volcano flanks. The propagation of
these lahars through populated areas resulted in ∼25,000 fatalities (Naranjo et al., 1986), highlighting the extreme
hazards posed by PDC‐ice interactions and ice‐melt lahars, and the need for robust modeling of these events.
Our experiments and numerical model provide constraints on the magnitude and timescales of melting in PDC‐ice
interactions. This can be used to inform source conditions for the generation of ice‐melt lahars. Physics‐based
simulations of lahars (and other debris flows) used in hazard assessment typically use volumetric flux [L
3
/T]
hydrographs as source conditions, which provide time‐series fluxes of water and entrained solids (e.g., Jenkins
et al., 2023). This source flux is typically distributed over an area A[L
2
], meaning that the prescribed flux has units
[L/T]. In cases of deposition of hot, static ash, our model can provide a time‐series of melt generation. Given that
s(t) represents the total melt [L] generated in a time t, the meltwater flux q
w
[L/T] is given by
qw=ρw
ρI
ds(t)
dt ,(13)
where ρ
w
is the water density. In Section 4.2, we established that the early‐time motion of the ash‐ice interface
obeys s∼λt
√. Substituting this into Equation 13 and taking ρ
w
/ρ
I
≈1 yields the early‐time form of the meltwater
flux, which obeys qw(t)∼ λ/(2t
√). In Figure 8we plot redimensionalized hydrographs that correspond to the
simulations shown in Figure 6. Here, the meltwater flux decay follows the qw∝1/t
√early‐time scaling, before
sharply dropping off to zero as melting terminates. Our model predicts that melting occurs over nearly 10 hr, and
the magnitude and duration of q
w
exceeds well‐established empirical thresholds for analogous rainfall‐driven
debris flows (e.g., Guzzetti et al., 2008). By combining hydrographs generated by our model with calibrated
empirical thresholds for lahar/debris flow initiation, insights can be provided into the triggering conditions of
melt‐driven lahars. Furthermore, where melt‐driven lahar genesis is expected, our model provides a physical basis
for the general form of melt‐driven source hydrographs. Note that a slight perturbation is required to avoid a
singularity in q
w
at t=0. However, morphodynamic flow solvers typically require a short “ramp‐up” period to
avoid instabilities in the source region. Above, homogeneous ash emplacement, distributed over an area A, is
considered. However, in natural settings, the thicknesses and thermal properties of the ash and ice will vary
laterally. Given that the lateral extent of ash greatly exceeds the ash thickness (L/d≫1), melting dynamics and
overall heat loss is dominated by vertical thermal transport. Therefore, it would be a reasonable assumption to
neglect terms related to lateral thermal transport, and instead divide the system into a series of 1‐D subdomains.
Figure 8. Example hydrographs calculated by dimensionalizing the curves in Figure 6a. Note that early‐time melting
produces a meltwater flux that decays with q
w∼1/
t
√.
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5.4. Limitations and Recommendations for Further Work
In the static melting experiments, the interactions between hot granular media and ice are investigated. In nature,
where PDCs are emplaced onto frozen substrates, the substrate surface will typically be comprised of varying
thicknesses of less‐dense snow, and underlain by ice. Ice is used in our experiments for two reasons: (a) snow is
difficult to reproduce in a laboratory environment, and (b) ice is a less complex substance and efforts were made
to simplify the research problem for modeling purposes. Further research into PDC interactions with frozen
substrates should consider how heat transfer, melt and steam generation would differ if the substrate consisted of
snow, rather than ice. The experimental works of Walder (2000a, 2000b) investigated hot particle‐snow in-
teractions and provided useful insights into thermal scour of snow substrate by convective vapor bubbling, and
passive melting at lower temperatures. However, this research did not quantify the rate of heat transfer, or the
generated melt and steam, limiting comparison with our numerical model.
Similar to Walder (2000b), our experiments isolated the thermal interactions between particle and ice, and did not
account for mechanical scouring, which can occur in natural PDC‐ice interactions. The static experiments pre-
sented here represent an important precursory investigation to dynamic experiments from phenomenological and
numerical perspectives because in nature some emplaced particles will form a static bed, and therefore some
passive melting will occur. Furthermore, prior to these experiments the dynamics and timescales of melt and
steam generation were unknown. In nature where mechanical scour processes also operate, mechanical abrasion
of the substrate can generate ice fragments which can be incorporated into the hot granular layer. This may
enhance the production of meltwater as heat transfer from the hot granular layer to fragmented ice will be more
efficient, compared with unfragmented ice, due to the larger surface area to volume ratio of the ice fragments
(Stroberg et al., 2010). To investigate the effects of coupled thermal and mechanical scouring, hot granular layer
interactions with ice should be studied in dynamic flowing configurations.
Though higher‐order terms related to steam generation were not included in our numerical model, it was
nonetheless able to accurately reproduce the leading order dynamics measured in the experiments. This is
not considered an insurmountable limitation as there are currently no means to quantify steam in natural
PDC‐ice interactions, and steam generation at geophysical scales using the ratio of melt to steam generation
in the small‐scale laboratory experiments under different initial conditions can be estimated. At geophysical
scales, it is most critical that (a) melt volume generated from pyroclast‐ice interactions can be quantified,
and (b) the physical role of steam for efficient melt incorporation, and its effects on flow mobility are
understood. The static melting experiments in combination with the numerical model work to resolve these
requirements.
6. Conclusions
A series of static melting experiments were conducted, where hot particles were emplaced onto an ice
substrate as an analog to PDC‐ice interactions. These experiments isolated the thermal interactions, as in
order to fully understand the thermal and mechanical coupling in PDC‐ice interactions, a detailed physical
knowledge of particle‐ice interactions in the simplest configuration must first be generated. The experiments
revealed that melt and steam systematically increase with increasing particle temperature and layer thick-
ness. The experiments were capable of reproducing phenomena seen in natural‐scale flows, including
fluidization and fines elutriation, indicating that the initial temperature conditions were within a represen-
tative natural range. Experiments also provided insights into melt mixing mechanisms. Based on the
thermocouple data, in combination with visual observations, steam was identified to play critical role in the
rate of melt incorporation through the particle layer. This has implications for the rate of transformation
from PDC to ice‐melt lahar.
A 1‐D numerical model of heat transfer, which was calibrated against Ruapehu ash experiments is also presented.
This model accurately predicts melt generation to within ∼5%. We also provided analytical similarity solutions
for our numerical model at early‐times and at typical geophysical scales, along with an example meltwater flux
hydrograph which can be used to inform source conditions for simulations of melt‐driven lahars. The ability to
predict melt generation at geophysical scales when a PDC is emplaced onto an ice substrate represents a sig-
nificant advancement toward robust modeling of the ice‐melt lahar hazard.
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Appendix A: Bilinear Remapping of the Numerical Model
A common approach in obtaining numerical solutions to moving boundary problems is to rescale the governing
equations onto a fixed domain. Although introducing additional mathematical complexity (in the form of
advective transport terms), solving advection‐diffusion problems on a fixed domain removes numerical com-
plexities associated with solving coupled PDEs on evolving domains (e.g., time‐dependent grids and resolution).
Bilinear mapping is adopted. That is, a separate linear transform is applied to each solid phase. The location of key
interfaces in these rescaled domains (ηfor ash, νfor ice) are summarized in the table below. These rescaled
domains are solved separately and coupled together through a shared Direchlet boundary condition for tem-
perature and the Stefan condition, which scale advective terms that account for the motion of the ash‐ice interface.
z η
Air‐ash 1 +s(t) 1
Ash‐ice s(t) 0
z ν
Ash‐ice s(t) 0
Ice‐rock H
0
1
A1. Remapping the Ash Subdomain z⇔η
The ash subdomain is rescaled using the linear mapping
η=zs.(A1)
Accordingly, the spatial and temporal derivatives from independent variables (z,t) to (η,t) must be transformed.
The transformed first and second spatial derivatives are
∂
∂z



t=∂η
∂z



t
∂
∂η =∂
∂η ,(A2)
and
∂2
∂z2



t=∂η
∂z



t
∂
∂η
∂
∂η =∂2
∂η2.(A3)
Finally, the transformed temporal derivative is given by
∂
∂t



z
⇒∂
∂t



η+∂η
∂t



z
∂
∂η



t=∂
∂t ∂s
∂t
∂
∂η .(A4)
A2. Remapping the Ice Subdomain z⇔ν
The same procedure is now performed in the ice region, which is rescaled using the linear mapping:
ν=zs
H+s,(A5)
whose first and second spatial derivatives are
∂
∂z



t=∂ν
∂z



t
∂
∂ν =1
H+s
∂
∂ν ,(A6)
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and
∂2
∂z2



t=∂
∂z



t[1
H+s
∂
∂ν]=1
(H+s)2
∂2
∂ν2.(A7)
The transformed temporal derivative is given by
∂
∂t



z
⇒∂
∂t



ν+∂ν
∂t



z
∂
∂ν



t=∂
∂t (1+ν
H+s)∂s
∂t
∂
∂ν .(A8)
Applying these transformed spatial and temporal derivatives to each subdomain yields our system of rescaled,
non‐dimensional governing Equations 8a–8d.
Data Availability Statement
The experiment temperature, melt and steam data, along with executable matlab codes and numerical simulation
data reported within this study are freely available in a Zenodo repository via https://doi.org/10.5281/zenodo.
8278922 (Vale et al., 2023).
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Acknowledgments
ABV acknowledges the support from a
University of Bristol Postgraduate
Scholarship and industrial sponsor GNS
Science. LTJ and JCP acknowledge
funding from the UKRI Global Challenges
Research Fund Grant NE/S009000/1. JCP
acknowledges the support of a University
of Bristol Research Fellowship. LTJ, JCP,
and AJH acknowledge funding from
Research England for a University of
Bristol Research project “From Everyday
to Extreme: Strengthening Resilience to
Frequent Flash Floods in Perú.” GK and
AS are supported by the New Zealand
Ministry of Business, Innovation and
Employment (MBIE) through the Hazards
and Risk Management programme
(Strategic Science Investment Fund,
contract C05X1702). The authors
gratefully acknowledge insightful
conversations with J. Walder and J.
Cowlyn. We thank D. Jessop and three
anonymous reviewers for their thoughtful
reviews, which improved this manuscript.
We are also grateful to Yves Bernabé for
editorial handling of this manuscript.
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