 
THE DEVELOPMENT OF EXTREMELY HIGH-GRADE IGNIMBRITES: 
THE EFFECTS OF STRAIN HEATING AND IMPLICATIONS OF ASH 
AGGLUTINATION 
 
 
 
 
 
 
by 
 
SAGE KEMMERLIN 
 
 
 
 
 
 
 
 
A THESIS 
 
 
 
Presented to the Department of Earth Sciences 
and the Division of Graduate Studies of the University of Oregon 
 in partial fulfillment of the requirements 
for the degree of 
Master of Science 
 
 
June 2021 
 
THESIS APPROVAL PAGE 
 
 
Title: The Development of Extremely High-Grade Ignimbrites: The Effects of 
Strain Heating and Implications of Ash Agglutination 
This thesis has been accepted and approved in partial fulfillment of the 
requirements for the Master of Earth Sciences degree in the Department of 
Earth Sciences by: 
Josef Dufek Chairperson 
 
Leif Karlstrom Member 
 
Thomas Giachetti Member 
and 
Andrew Karduna Interim Vice Provost for Graduate Studies 
 
Original approval signatures are on file with the University of Oregon Division of 
Graduate Studies. 
Degree awarded June 2021 
 ii 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
©2021 Sage Kemmerlin 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 iii 
THESIS ABSTRACT
Sage Kemmerlin
Master of Science
Department of Earth Sciences
June 2021
Title: The Development of Extremely High-Grade Ignimbrites: The E↵ects of
Strain Heating and Implications of Ash Agglutination
The Miocene Grey’s Landing ignimbrite is a large (  23,000 km3), intensely
welded, highly rheomorphic extremely high-grade ignimbrite. The cause of rheo-
morphism can be attributed to syn-depositional strain heating and post-depositional
gravitational flow of the deposit, but the relative contributions of both are not
fully constrained. We model the Grey’s Landing pyroclastic density current as
a dilute current under the assumption that the particles do agglutinate. The
model informs a one-dimension thermal di↵usion model that allows for progres-
sive aggradation of the pyroclastic density current with strain heating. Three
di↵erent methods to calculate viscosity in the shear zone were used: two from
measurements of the Grey’s Landing ignimbrite and inferences of magmatic vis-
cosity, and one from the granular viscosity calculated from the multiphase model.
The results from the granular viscosity model were the only physically realistic
iv
results but generated virtually no strain heating, implying that post-depositional
flow is primarily responsible for rheomorphism.
v
 
CURRICULUM VITAE 
 
 
NAME OF AUTHOR: Sage Kemmerlin 
 
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED: 
University of Oregon, Eugene  
Georgia Institute of Technology, Atlanta 
 
DEGREES AWARDED: 
Master of Earth Sciences, 2021, University of Oregon 
Bachelor of Science, Earth and Atmospheric Science, 2017, Georgia Institute of 
Technology 
 
 
AREAS OF SPECIAL INTEREST: 
Geophysics 
 Geology 
 
 
PROFESSIONAL EXPERIENCE: 
Graduate Employee, University of Oregon, 2018-2021 Graduate Employee, 
Georgia Institute of Technology, 2017-2018 
Undergraduate Teaching Assistant, Georgia Institute of Technology, 2016-2017 
 vi 
ACKNOWLEDGMENTS
I would like to thank my advisor, Josef Dufek, for his support throughout
the years. I would also like to thank my other committee members, Thomas
Giachetti and Leif Karlstrom, for helping mentor me in graduate school. Addi-
tionally I would like to acknowledge my research group for their constant support
and guidance: Eric Breard, Ryan Cahalan, Gabe Eggers, Amelia Winner, Chris
Harper, Paul Regensburger, Allison Kubo, PJ Zrelak, and Annika Dechert.
I want to thank my dear friends and family (Mom, Amma, Aspen, Zach,
Trouvé, Bowie, Kathy, Amelia, PJ, Aaron, Brittany, Genesee, Benny) for their
ever-persistent support and love. Without you, I would not be here. I love you
all immensely and am forever grateful to have you in my life.
I also want to acknowledge that I am an occupier of the traditional home-
lands and political territories of the Kalapuya People, the First Peoples of the
Willamette Valley. Following treaties between 1851 and 1855, Kalapuya People
were dispossessed of their homeland by the United States government and forcibly
removed to the Coast Reservation in Western Oregon. Descendants of the Kala-
puyan tribes are now citizens of the Confederated Tribes of Grand Ronde and the
Confederated Tribes of Siletz Indians.
Finally, I acknowledge that my site of study, called Grey’s Landing, is located
on the traditional homelands of the Shoshone-Bannock Tribes, the Newe (West-
vii
ern Shoshone) whose descendants are members of the Te-Moak Tribe of Western
Shoshone Indians, and the Goshute whose descendants are members of the Con-
federated Tribes of Goshute Reservation.
viii
This thesis is dedicated to Kathy Trafton, my best friend and, as you say,
plutonic love of my life. I am forever grateful of you and your friendship.
Kathy, without you I would have never been able shift how I carry the weight
of my compounding traumas; never been able to grow despite the weight; never
been able to see outside of it to be proud of my growth. Your fierce kindness
has shown me how to follow suit and to treat myself with dignity and to demand
that others treat me likewise. I had never considered myself anything but a meek
person before I met you. Over the course of these past three years, you have
inspired me so immensely that I am altered forever. I now have strength and
courage I never thought possible before. I now am more outspoken that ever. I
am kinder than ever. I am constantly growing because of how much you inspire
me and encourage me.
Your friendship has shown me what true, unconditional love is. Our friendship
is one that has us constantly bouncing back and forth, encouraging each other to
be the best person we can be. It’s one where we will go to the mat for each other
(and we have so many times).
This thesis is dedicated to you, Kathy. Without you, I don’t think I’d have
made it through any of this. And I know I wouldn’t be the person I am today. I
love you so much, my chaotic, passionate, brave, brilliant friend.
ix
 
 
 
 
TABLE OF CONTENTS 
Chapter Page 
 
 
I. INTRODUCTION ......................................................................................................  1 
II. GREY’S LANDING IGNIMBRITE ..........................................................................  4 
III. SCALING ................................................................................................................  15 
 Scaling of liquid film-coated particle collisions ....................................................  15 
 Scaling of droplet-substrate collisions ...................................................................  20 
IV. AXISYMMETRIC, CONSTANT-VOLUME MODEL OF GREY’S LANDING 
PYROCLASTIC DENSITY CURRENT .........................................................................     25 
V. MULTIPHASE MODEL OF A PYROCLASTIC DENSITY CURRENT THAT PRODUCES 
EXTREMELY HIGH-GRADE IGNIMBRITES ............................................................     34 
VI. THERMAL MODEL OF THE EVOLUTION OF AN EXTREMELY HIGH-GRADE 
IGNIMBRITE DURING AND AFTER DEPOSITION .................................................     38 
 Construction of the thermal model ........................................................................  38 
General effects of velocity at the flow-substrate boundary and emplacement                                                      
temperature on the thermal evolution of a rheomorphic ignimbrite  ..................  47 
VII. APPLICATION OF MODELS TO GREY’S LANDING PYROCLASTIC DENSITY 
CURRENT AND IGNIMBRITE ...................................................................................     54 
VIII. DISCUSSION .......................................................................................................     62 
IX. CONCLUSION ........................................................................................................     64 
APPENDIX: VALIDATION OF AXISYMMETRIC, CONSTANT-VOLUME PDC  
MODEL .........................................................................................................................  75 
 x 
REFERENCES CITED ..................................................................................................  6
 xi 
 
 
LIST OF FIGURES 
 
 
Figure Page 
 
 
1. Stratigraphic sections of Grey’s Landing Ignimbrite and regional map ...............  5 
2. Tectonic evolution of the Rogerson Graben ..........................................................  7 
3. Diagram from Andrews and Branney (2011) depicting the deposition of hot pyro- 
 clasts in a pyroclastic density current  ...................................................................       10 
4. Diagram from Ellis et al. (2015) illustrating the results of their crystal model for 
 three locations with different thicknesses of Grey’s Landing ...............................     13 
5. Illustration of the assumed collisional geometry in the Gollwitzer et al. (2012)  
 scaling .....................................................................................................................     16 
6. The restitution coefficient plotted as a function of the particle Reynolds and  
 Stokes numbers ......................................................................................................     19 
7. Illustration of the assumed collisional geometry in the timescale ratio scaling ...  21 
8. The ratio of the viscous timescale relative to the collisional timescale as a funct- 
 ion of the particle Reynolds and Stokes numbers .................................................     22 
9. The effects of initial volume fraction and initial ash cloud volume on the runout 
 distance of a pyroclastic density current on deposit volume ................................     29 
10. The effect of pyroclastic density current volumetric particle concentration on the 
 runout and deposit volume of an ignimbrite from the constant volume model ..     30 
11. The relative effects of temperature, initial volumetric concentration, and ash 
 particle diameter on the runout distance of the pyroclastic density current in the 
 axisymmetric, constant volume numerical gravity current model .......................     32 
12. Resulting effects of ash particle diameter and initial volumetric concentration on 
 xii 
 (a) the runout of the pyroclastic density current in the axisymmetric, constant 
 volume numerical model, and (b) the velocity of the flow ....................................     33 
13. Delineating the dense, intermediate, and dilute transport regimes in pyroclastic 
 density currents ......................................................................................................     37 
14. Image depicting the discretization of the one-dimensional heat diffusion model  40 
15. A visualization of the numerical model of the thermal evolution of the Grey’s 
 Landing ignimbrite through deposition ................................................................     43 
16. The ranges of possible deposition rates for the Grey’s Landing pyroclastic  
 density current .......................................................................................................     48 
17. Effects of the velocity at the PDC-substrate boundary and the emplacement  
 temperature on the heating and cooling of a Grey’s Landing type deposit ..........     51 
18. Effects of a narrower window of velocity at the PDC-substrate boundary and 
 emplacement temperature on the heating and cooling of a Grey’s Landing type 
 deposit ....................................................................................................................     53 
19. A snapshot of the Grey’s Landing pyroclastic density current multiphase model 
 during transport .....................................................................................................     54 
20. Grey’s Landing deposit thickness as determined by the multiphase model .........  55 
21. The vertical temperature profiles at the cessation of emplacement for six 
 locations at different distances from source, with normalized deposit thicknesses 
 at each location ......................................................................................................     57 
22. The increases in temperature in Grey’s Landing ignimbrite as a product of strain 
 heating ....................................................................................................................     58 
23. The diffusion timescale for Grey’s Landing at varying distances from source .....  61 
24. Schematic from Bonnecaze et al. (1995) depicting a large rectangular  
 experimental water-filled tank with a semi-circular lock used in experiments 
 xiii 
 on constant-volume currents .................................................................................  75 
25. The runout of the axisymmetrically spreading gravity current following the  
 instantaneous release of dense suspensions with constant volume  ....................     77 
 xiv 
LIST OF TABLES 
 
Table Page 
 
 
1. The values, recorded at 10 km from PDC source, of the maximum increases in 
 temperature due to strain heating and the maximum temperature recorded in the 
 deposit for Grey’s Landing .....................................................................................     59 
2. Comparison of Bonnecaze et al. (1995) constant volume gravity current  
 experiments and axisymmetric, constant volume numerical model results based 
 on Dade and Huppert (1995) .................................................................................     76 
 
 
 
 
 
CHAPTER I: INTRODUCTION
Extremely high-grade ignimbrites are the enigmatic deposits of a subset of pyro-
clastic density currents (PDCs), often generated by large explosive eruptions with
high temperatures and low water contents (Branney, 2002; Branney et al., 2008),
can reach volumes up to hundreds of cubic kilometers. They are commonly thick,
extremely-welded, and can display rheomorphic features (e.g. ductile folds, flow
banding, elongation lineations, and eutaxitic fabric) caused by rapid ductile de-
formation in the deposit. A high degree of welding and rheomorphic features are
not found in the more common lower-grade ignimbrites (Branney, 2002; Branney
and Godchaux, 2004; Robert et al., 2013; Andrews, 2011; Chapin, 1979). Erup-
tions that produce extremely high-grade rheomorphic ignimbrites have not been
witnessed historically (Andrews, 2011).
Rheomorphic ignimbrites occur in a range of diverse settings including intra-
continential (e.g.Snake River Plain ignimbrites, Idaho, USA; B and GP (1982)),
intra-oceanic (e.g. Mogán and Fataga Formations, Gran Canaria; Schmincke
(1974)), rifted continental margin (e.g. Paraná volcanic province, Brazil; Kirstein
et al. (2001), and continental arcs (e.g. Bad Step Tu↵; Branney and Kokelaar
(1992). Rheomorphism is shown by the presence of ductile folds and elongation
lineations (Andrews, 2011). The majority of welded ignimbrites are not rheo-
1
morphic, and many of those that are rheomorphic are only so locally (e.g. Long
Top Tu↵s; Branney and Kokelaar (1994), Rattlesnake Tu↵; Streck and Grunder
(1995)). Rheomorphic ignimbrites are particularly abundant in provinces that
record the continental rhyolitic volcanism, ”Snake River-type” volcanism. This
volcanism is characterized by: (1) extensive layers of stratified ash rather than
typical Plinian pumice-fall layers; (2) unusually long and large-volume rhyolite
lavas rather than typical small rhyolite domes and coulees; (3) ignimbrites that
are better sorted and contain less pumice lapilli (or fiamme) than is typical of ig-
nimbrites elsewhere; and (4) ignimbrites that are more intensely rheomorphic and
predominantly lava-like than is typical elsewhere (Branney et al., 2008; Andrews,
2011).
For rheomorphism to occur, whether it be during deposition of the pyroclastic
density current or in post-depositional remobilization, substantially high temper-
atures must be reached (900-1000 oC) or the current must be rapidly deposited
(Andrews, 2011; Ekren et al., 1984; Bachmann and Lipman, 2000; Lavallée et al.,
2015). Rheomorphism is favored by ”soft” pyroclasts of low-viscosity (Andrews,
2011). The mechanisms which lends to the development of extremely high-grade
ignimbrites are not fully constrained. The current model is in which progres-
sive aggradation of a pyroclastic density current, syn-depositional shearing in the
growing deposit, and gravitational compaction and slumping, lends to the condi-
tions necessary to create widespread rheomorphism (Branney and Kokelaar, 1992;
2
Branney, 2002; Andrews, 2006, 2011; Robert et al., 2013; Lavallée et al., 2015; El-
lis et al., 2015; Knott et al., 2016). Previous studies have aimed to quantify the
history and formation of widespread rheomorphism in extremely high-grade ign-
imbrites, such as the Grey’s Landing ignimbrite (Robert et al., 2013; Andrews,
2011; Lavallée et al., 2015; Manley and Andrews, 2004). However, the significance
of agglutination of partially molten ash particles on the dynamics of large, hot
pyroclastic density currents during both transport and progressive aggradation
deposition is relatively little understood.
3
CHAPTER II: GREY’S LANDING IGNIMBRITE
I want to take this space to acknowledge that the ignimbrite of study, called
Grey’s Landing, is located on the traditional homelands of the Shoshone-Bannock
Tribes, the Newe (Western Shoshone) whose descendants are members of the Te-
Moak Tribe of Western Shoshone Indians, and the Goshute whose descendants
are members of the Confederated Tribes of Goshute Reservation.
The Miocene Grey’s Landing ignimbrite is an intensely welded, extremely high-
grade a super-eruption deposit (  23, 000 km2) in the central portion of the Snake
River Plain in the 15-km-wide Rogerson Graben, Idaho, located in the Yellowstone
hot-spot track as shown in Figure 1 (Andrews et al., 2008; Knott et al., 2016,
2020). Knott et al. (2020) estimated eruption volume dense-rock equivalent to
be   2, 800 km3 with an eruption magnitude of 8.8, making Grey’s Landing the
largest documented eruption in the Snake River Plain province. There is no
evidence of any significant break during the emplacement of the ignimbrite and is
inferred to record a single eruption(Andrews, 2011). The deposit, which ranges in
thickness from 5 to greater than 100 meters, exhibits rheomorphic and lava-like
features (Andrews, 2011; Knott et al., 2020). The ignimbrite rapidly pinches out
southwards from the Snake River Plain and west of the Rogerson Graben. Within
the graben, which is bound by two normal faults, the Grey’s Landing ignimbrite
is entirely welded and intensely rheomorphic even as it thins eastward from being
> 65 m thick to 4 m thick (Knott et al., 2016; Andrews et al., 2008).
4
5
Figure 1: Stratigraphic sections Grey’s Landing Ignimbrite and regional map (left) and isopach map (right), modified from
Knott et al. (2020), depict the extent of the Grey’s Landing ignimbrite. Stratigraphic sections, drawn by Knott et al.(2020)
at their field sites (black squares). Site abbreviations: 3C—Three Creek; RG—Rogerson graben; RC—Rock Creek, Cassia
Hills; OH—Oakley Hills; LFC—Little Fish Creek, Lake Hills; MBH—Mount Bennett Hills. The regional map (middle insert)
illustrates the Yellowstone- Snake River volcanic province (Y-SRP) in the northwesten United States. The image shows
rhyolitic eruptive centers: M—McDermitt; OH—Owyhee-Humboldt; BJ—Bruneau-Jarbidge; TF—Twin Falls; P—Picabo;
H—Heise; Y—Yellow- stone. Other: wSRr—western Snake River Plain. State abbreviations: WA—Washington; ID—Idaho;
MT—Montana; OR—Oregon; CA—California; NV—Nevada; UT—Utah; WY—Wyoming. The black square highlights were
Knott et al. (2020) conducted field work. Grey’s Landing is located in the Twin Falls (TF) eruptive center. The isopach map
shows the thickness distribution of the Grey’s Landing ignimbrite in meters.
The paleotopography, on-top of which Grey’s Landing was emplaced, is not a
uniform, flat surface. The Rogerson Graben, in which much of the ignimbrite sits,
developed between 12 and 8 Ma, during which Grey’s Landing, and other members
of the Rogerson Formation were emplaced (Nash et al., 2006; Bonnichsen et al.,
2008). The Rogerson Graben is bound by two formal faults: (1) the 30-km-long
Brown’s Bench Fault in the west that produced the 400-m-high Brown’s Bench
escarpment and (2) the smaller Shoshone Hills Fault in the east (Figure 2). These
faults were both active at that time. At the time the Grey’s Landing ignimbrite
was emplaced, the graben floor was west-sloping due to the Brown’s Bench Fault
(Andrews et al., 2008; Knott et al., 2016). This lead to thickening of the deposit
to the west and thinning to the east (Figure 2). Andrews et al (2011) and Knott et
al. (2016) argue that the west-sloping graben floor is constistent with W-trending
extensional rheomorphic lineations in the Grey’s Landing ignimbrite and suggests
top-to-the-west downslope rheomorphic shear.
Andrews and Branney (2011) determined four units in the deposit: (1) basal
vitrophyre, (2) lithoidal lava-like rhyolite, (3) thin upper vitrophyre, and (4) a
sometimes-present non-welded ash top (Figure 1). The ignimbrite is pervasively
rheomorphic throughout the entire deposit, including in the lower and upper vitro-
phyres, with recorded strains of 10-1000 based on estimates from stretched vesicles
(Robert et al., 2013; Andrews et al., 2008; Andrews, 2011). The ignimbrite records
progressive aggradation of the pyroclastic density current (Branney and Kokelaar,
6
Figure 2: Diagram from Knott et al. (2016). Tectonic evolution of the Rogerson
Graben as a series of progressive stages (from 10.59 to 8.0 Ma; (a–b) that account for
lateral thickness variations of ignimbrites of the Rogerson Formation. (c) Eastward
thinning of the Grey’s Landing Member at Backwaters.
7
1992; Knott et al., 2016; Andrews, 2006, 2011; Robert et al., 2013; Lavallée et al.,
2015; Ellis et al., 2015), in which a vertically migrating shear zone produced per-
vasive extensional lineation, mylonite-like fabrics, folds, and subparallel intrafolial
sheath folds (Knott et al., 2016; Andrews, 2011). Gravity-driven deformation, due
to maintained hot temperatures (above the glass transition temperature) contin-
ued post-deposition, folding the fabric and causing local autobrecciation in the
upper vitrophyre which cooled more rapidly (Knott et al., 2016; Andrews, 2011).
Snake River-type ignimbrites (Branney et al., 2008) are inferred to have low
magmatic water contents (< 1.5 wt.%) (Almeev et al., 2012). The bulk compo-
sitions of the Grey’s Landing ignimbrite are rhyolitic, ranging from 73.4 to 75.8
wt. % SiO2, and are broadly compositionally homogeneous in both the horizon-
tal and vertical directions (Ellis et al., 2013; Bonnichsen et al., 2008; Ellis et al.,
2015). The trace elemental composition of the rhyoltic glass in the dense basal
and upper vitrophyre are identical, indicating little compositional zonation in the
magma prior to eruption (Ellis et al., 2013, 2015). The trace element abundances
in the glasses are 45-87 ppm Sr, 1050-1230 ppm Ba, and Li of 21-26 ppm in the
basal vitrophyre glass and 17-26 ppm in the upper vitrophyre glass (Ellis et al.,
2015). Ellis et al. (2015) also analyzed the plagioclase from the basal and upper
vitrophyres and from microcrystalline interior of the ignimbrite. All the trace el-
ements, with the exception of lithium remained consistent throughout the section
of deposit. In the plagioclase from the vitrophyres and glassy fallout, Li ranges in
8
concentrations from 2 to 10 ppm. In the crystallized interior of the ignimbrite, the
lithium content reached 68 ppm with an average of 33 ppm. The variation showed
no correlation with any other trace element. Ellis et al. (2015) suggest that the
variation in Li could be due to diachronous cooling in which the vitrophyres cooled
quickly, capturing conditions immediately at emplacement of deposit, before the
interior in which Li content may capture a more complicated suite of processes.
The Grey’s Landing ignimbrite is one of the most thoroughly studied extremely
high-grade ignimbrites. Previous studies have aimed to constrain the duration of
post-depositional rheomorphic, lava-like deformation by utilizing one-dimension
thermal conduction models. These studies calculated durations of less than 2
years to 16 years (Robert et al., 2013; Andrews, 2011; Manley and Andrews, 2004;
Andrews, 2006; Lavallée et al., 2015; Ellis et al., 2015). Variability in the calcu-
lated values are due, in part, to the input of di↵erent emplacement temperatures
as well as due to the incorporation of strain heating. Eruption temperature is the
prime control on the emplacement temperature of a deposit (Freundt, 1999). Min-
eral thermometry suggests magmatic temperatures of   850  C (Andrews, 2011;
Robert et al., 2013; Cathey and Nash, 2004).The high temperatures reached in the
ignimbrite have been attributed to either high emplacement and eruption temper-
atures of approximately 900 to 1,000  C (Andrews, 2011; Manley and Andrews,
2004) or to lower end member emplacement temperatures of 850  C where ad-
ditional heat is generated from shearing in deposit by the overriding pyroclastic
9
density current (Robert et al., 2013), shown conceptually in Figure 3.
Figure 3: Diagram from Andrews and Branney (2011) depicting the deposition of hot
pyroclasts in a pyroclastic density current. Particles deposit, agglutinate, and deform
from shearing induced by the overriding current; thus, a vertically migrating shear zone
occurs and produces rheomorphic features.
Andrews and Branney (2011) concluded that welding and early deformation
in the Grey’s Landing ignimbrite occurred rapidly during the deposition of a  
1000  C high-mass-flux PDC. They assume deposition through progressive aggra-
dation of the PDC, creating a  2 m thick vertically-migrating shear zone which
produced and deformed the rheomorphic fabric. They argue that deformation con-
tinued post-depositionally from viscous spreading and downslope, gravity-induced
flow. They argue for cooling of the entire deposit to below the glass transition
10
temperature ( 525-750  C) in much less than two years (Manley and Andrews,
2004; Andrews, 2006), using a one-dimensional conductive cooling model (Manley,
1992). However, Andrews (2006) found the maximum duration of ductile behav-
ior (time above the glass transition temperature) to be 4 years for the ignimbrite
at 30 m thickness, and 16 years for the ignimbrite at 60 meters thickness, with
ductile behavior in the lower and upper vitrophyres of  1 year.
Robert et al. (2013) utilized a one-dimensional instantaneous emplacement
thermal conduction model to investigate the e↵ects of strain heating on the de-
posit, while assuming that the deposit in actuality was formed through progressive
aggradation. Their viscosity measurements showed a high deposition temperature
of > 900  C or 850  C if melt contained come dissolved water ( 0.1 wt. % H2O).
They found possible shear zone residence times of 2-20 hours. They found strain
heating of at least a few tens of degrees to be possible and resulted in peak tem-
peratures of 1030-1100  C.
The Lavallée et al. (2015) study also assumed instantaneous emplacement
of Grey’s Landing with no deformation during emplacement. They note that
these assumptions are simplifications as they believe deposition of pyroclastic
density currents results from progressive aggradation and deposits can experience
deformation during this process (Branney and Kokelaar, 1992). Their model,
a thermal conduction model, assumes an initial homogeneous temperature and
porosity, which they allow to evolve over time. Their welding timescale of the
11
basal vitrophyre agrees with the 12-20 minutes proposed by Robert et al. (2013).
They propose that little cooling took place during eruption and transport unless
strain heating accommodated for the temperature lost from cooling.
Ellis et al. (2015) utilized a one-dimensional finite di↵erence cooling model
with crystal growth to model the thermal evolution of the Grey’s Landing ign-
imbrite. They found a strong contribution of latent heat during the crystalliza-
tion process, however latent heat only a↵ected profiles where the temperature
was maintained long enough for crystals to grow, suggesting a cooling rate be-
tween 10 2.8 Ks 1 and 10 4.5 Ks 1 for the crystalline lithology (Ellis et al., 2015;
Lavallée et al., 2015). Their modeling results were in agreement with Lavallée et
al. (2015) in that the glassy upper and lower vitrophyres passed below the glass
transition temperature and out of the ”rheomorphism window” quickly (Figure
4. Their modeling results showed that it is possible that the vitrophyres cooled
rapidly but the interior of the ignimbrite remained hot and above the glass tran-
sition temperature for far longer (Romine et al., 2012; Ellis et al., 2015). In one
calculation they found in slightly less than two years, the vitrophyres cooled to
less than 500  C, while the interior of the deposit remained above 900  C The
suggest, similar to Andrews (2006), that the hot interior of the ignimbrite could
flow laterally and could develop rheomorphic folding where the deposit overlaid
paleoslopes (Andrews, 2011; Knott et al., 2016).
12
Figure 4: Diagram from Ellis et al. (2015) illustrating the results of their crystal
model for three locations with di↵erent thicknesses of Grey’s Landing. Black indicates
amorphous material, and white indicates fully crystalline material while temperature
contours illustrate the temperature of the relative positions in the ignimbrite as a func-
tion of time.
13
In regard to the Grey’s Landing ignimbrite, a rheomorphic ignimbrite with
great areal extent (  23,000 km2), there is the question of how the pyroclastic
density current was able to travel to such extents if the eruption temperature was
indeed 900 - 1,000  C and ash was partially-to-fully molten (Andrews, 2011; Man-
ley and Andrews, 2004; Freundt, 1999). If instead eruption temperatures were
lower (850  C), would strain heating be able to produce high enough tempera-
tures to reach those recorded in the deposit (Robert et al., 2013)? Following gas
fluidization experiments using polyethyleneglycole (PEG) powders heated above
minimum sintering and melting temperatures, Freundt (1998) concluded that ex-
tended particulate transport is only possible if the pyroclastic density current is
in a dilute state where aggregation is avoided by a reduced number of particle
collisions. However, a dilute and turbulent current would cool rapidly due to
entrainment and mixing with cool ambient air (Andrews, 2014; Branney, 2002;
Benage et al., 2016). In turn this would reduce the flow’s ability to retain high
eruption temperatures and reduce the ability to generate rheomorphism without
significant heat generated from shear (Freundt, 1998). This study will illumi-
nate the dominant controls on the high temperatures recorded in Grey’s Landing
and will give insight into the role ash agglutination plays on the development of
extremely high-grade ignimbrites.
14
CHAPTER III: SCALING
We investigate if partially molten ash particles will agglutinate upon deposition
by studying the collisional interaction of partially-to-fully molten particles with a
flat substrate using scaling relationships. For particle-substrate collisions, where
the radius of the substrate (R) can be considered infinite relative to the radius
of a particle (R), it scales equivalently to that of two colliding particles each
with a radius of 2R (Chesters, 1991). In earth science and other fields, such as
chemical and process engineering, there are limited studies on the collisions of
particles or droplets composed of highly viscous fluids. We therefore utilize two
di↵erent scaling approaches to determine if results were consistent (Kavehpour,
2015; Freundt, 1998; Janssen, 2011; Telling and Dufek, 2012).
1 Scaling of liquid film-coated particle collisions
We first follow the analysis of the head-on collision of liquid-film-coated particles
as laid out in an experimental study of particles impacting a wet surface by Goll-
witzer et al. (2012). This approach has previously been applied to water-coated
ash to determine a likelihood of aggregation (Telling and Dufek, 2012). The scal-
ing explores the relationship between the wet and dry coe cients of restitutions
by characterizing the energy dissipation during an impact. Figure 5 shows the
assumed collisional geometries and the potential outcomes of the scaling.
The restitution coe cient can be used to describe if agglutination will occur.
15
Agglutination
Approach
Liquid film Liquid bridge e = 0
forms n
Solid 
particle en = 1
Repulsion
Figure 5: Illustration of the assumed collisional geometry in the Gollwitzer et al.
(2012) scaling. Particles are assumed to collide head-on and form a liquid bridge, and
then either agglutinate or repel based on the energy of dissipation, the collisional kinetic
energy, and the dry restitution coe cient.
At a value of 0 agglutination occurs; whereas at a value of 1, the particles behave
perfectly elastically and rebound (Telling and Dufek, 2012; Gollwitzer et al., 2012).
The dry ‘intrinsic’ restitution coe cient (eo) is determined experimentally by
dropping solid particles onto a substrate of the same material. The wet, film-
coated, restitution coe cient, relative to the dry restitution coe cient is described
by:
r
E
e = e2   dissn , (1)o Ei
16
where Ei as the collisional kinetic energy and Ediss is the energy of dissipa-
tion. The dissipation energy is the sum of 1) the kinetic energy required to move
interstitial fluid out of the way of the colliding particle, 2) the viscous dampening
experienced by the particle due to the fluid, and 3) the energy required to rup-
ture the capillary fluid bridge during particle rebound (Telling and Dufek, 2012;
Gollwitzer et al., 2012).
We apply this scaling to the Grey’s Landing ignimbrite-forming pyroclastic
density current and additionally use a wide parameter space of di↵erential ve-
locities and liquid film viscosities to compare partially molten ash to a range of
collisional interactions. To examine the aggregation potential in the context of
viscous and inertial interactions of particles in the flow we use the non-dimensional
Stokes and Reynolds number. The particle Stokes number (Stp), the ratio between
the inertia of a particle and the viscosity of the liquid, is described by:
⇢pduStp = , (2)
9µ
where ⇢p is the density of the solid particle, d is the diameter of the solid
particle, u is the di↵erential velocity of the particles, and µ is the viscosity of
the liquid film (Gollwitzer et al., 2012). The particle Reynolds number (Rep), is
described by:
17
⇢
Re = l
du
p , (3)
µ
where ⇢l is the density of the liquid film. We utilize a particle diameter of 1
millimeter, a roughness of 1 x 10 8 m, and a solid ash particle density of 2580
kg/m3 and molten ash density of 2390 kg/m3 calculated using the composition
of the Grey’s Landing ignimbrite in MELTS (Gualda et al., 2012; Ghiorso and
Gualda, 2015; Ellis et al., 2013, 2015). We calculate the viscosity, particle Stokes
number, particle Reynolds number, associated energies, and restitution coe cient
over a range of velocities from 0.01 to 10 m/s. The particle Reynolds number
and particle Stokes number for Grey’s Landing were on the order of 10-6 and 10-7,
respectively. The results of the scaling imply that the melt-coated particles will
agglutinate with the substrate, as shown below in Figure 6. The value for the wet
restitution coe cient for partially molten ash particles is 0.
18
Figure 6: The restitution coe cient plotted as a function of the particle Reynolds
and Stokes numbers. Solid, fully molten, and partially-molten ash from Grey’s Landing
scales in the agglutination space (colored as white here) where the restitution coe cient
is 0.
19
2 Scaling of droplet-substrate collisions
To represent the fully molten case, we apply scaling of the contact time to the
viscous timescale of droplet-substrate collisions. We calculate the Ohnesorge num-
ber (Oh) to verify that the molten ash interactions will be dominated by viscous
timescales (Oh >> 1) as opposed to inertial timescales (Oh < 1) (Kamp et al.,
2017). The Ohnesorge number is described by:
µ
Oh = p , (4)
g R
where g is the gravitational acceleration, R is the radius of the drop,   is
the surface tension, and µ is the viscosity of the melt (Kavehpour, 2015; Kamp
et al., 2017). When applied to the Grey’s Landing pyroclastic density current, the
Ohnesorge number scales far greater than unity and viscous timescales dominate
over inertial timescales (Kamp et al., 2017). Viscous timescales are calculated as
(Kavehpour, 2015):
µR
tv = , (5)
 
and are on the order of 104 seconds for Grey’s Landing.
We compare the time of contact to the viscous timescale with the assumption
that if the contact time is larger than the viscous timescale, then the droplets will
agglutinate (Figure 7).
20
Coalesced 
droplet Agglutination
Approach tvisc < 1
tcoll
Liquid tvisc
droplet = 1
tcoll Aggregation
tvisc > 1
tcoll
Repulsion
Figure 7: Illustration of the assumed collisional geometry in the timescale ratio scaling.
Liquid droplets are assumed to collide head-on, and then either agglutinate if the viscous
timescale is smaller than the collisional timescale, aggregate if the time scales are equal,
or repel if the viscous.
We calculate the contact time of particles by:
✓ ◆
k ⌘2  1/2
tcontact = ⇡   , (6)
m 4m2
where k is the elastic coe cient , ⌘ is the normal dashpot damping coe cient,
and m is the mass of the droplet (Silbert et al., 2001). We incorporate the influence
of velocity and the particle diameter by:
2R
tcoll = tcontact + , (7)
u
21
As with the Gollwitzer et al. (2012) approach, we calculate a wide range of
particle Reynolds and Stokes numbers for this timescale approach. The results,
however, are in opposition with the Gollwitzer et al. (2012) scaling as the ra-
tio of the viscous timescale to the collisional timescale is on the order of 105 to
106, meaning that the particles will rebound before they are able to deform and
agglutinate (Figure 8).
Figure 8: The ratio of the viscous timescale relative to the collisional timescale as
a function of the particle Reynolds and Stokes numbers. Solid, molten, and partially-
molten ash from Grey’s Landing scales in the rebound space (represented by the color
bar) where the timescale ratio is greater than 1.
22
Freundt (1998) conducted an experimental fluidization study in which they flu-
idized polyethyleneglycol (PEG) powder in perspex tanks with pumped air on the
tank floor. The air was heated to raise powder beds to the melting temperature
(47-64  C) of the PEG powder. For the higher temperature experiments, at the
melting temperature of PEG powder, they found the final bed consisted of dense
solidified PEG melt at base overlain by welded crust that became increasingly less
welded to fully unwelded with height from the base. They found rapid aggregation
upon exceeding a critical melting temperature occurred for PEG powder particles.
In that same study, Freundt (1998) also conducted turbulent flow experiments for
PEG-spray suspensions. They found sedimentation to be influenced by droplet
coalescence which generated large aggregates. The size of the PEG aggregates de-
creased with distance. The distal aggregates were formed by cooler PEG particles
and were highly fragile. Freundt (1998) argues that aggregation and agglutination
at the base of a pyroclastic density current would separate the flow into particulate
transporting overflow and a viscous depositional underflow which grows through
progressive aggradation (Branney and Kokelaar, 1992). They concluding that
welding of high-grade ignimbrites includes pre-depositional aggregation, syndepo-
sitional aggradation and viscous deformation, through post-depositional shear and
compaction. Freundt (1998) concludes that extensive high-grade ignimbrites can-
not be formed by high-concentration pyroclastic density currents, as the particles
would be sticky (supporting our Gollwitzer et al. (2012)-based scaling) and would
23
”catastrophically agglomerate” even at small coalescence e ciencies. Therefore,
they argue these pyroclastic density currents are dilute with hot sticky particles
(Freundt, 1998).
It would be beneficial for more studies on the interactions of partially-to-fully
molten particle collisions to be conducted. There are limited studies thus far
and they have involved typically extremely fine PEG powder that is finer that
ignimbritic ash (Freundt, 1998). Isolating particles to study interactions and
potential development of liquid bridges at di↵erent percent melt and collisional
velocities would be fruitful to validate methods of scaling and to later be incor-
porated more accurately into multiphase models of pyroclastic density currents.
24
CHAPTER IV: AXISYMMETRIC, CONSTANT-VOLUME MODEL OF
GREY’S LANDING PYROCLASTIC DENSITY CURRENT
We first approach modeling the pyroclastic density current with a simple numerical
model, based o↵ of the constant volume turbidity current model by Dade and
Huppert (1995). We do this to get a sense of the parameter space before utilizing
more complex models (see Chapter 5).The model considers an instantaneously
well-mixed, fine-grained, axisymmetric gravity current released from a lock, which
spreads radially over a non-erodible planar surface. To preserve a constant volume,
entrainment is neglected in the model. The results of this box model inform a one-
dimensional thermal model of an ignimbrite during deposition and throughout its
post-depositional cooling by providing values for the initial volume of the PDC
(Vo), duration of the flow, the runout of the flow (xcritical), the velocity of the
current front (u), and the deposition rate (db/dt) of particles sedimenting out of
the current (Dade and Huppert, 1995; Dade, 2003).
The volume of the pyroclastic density current remains constant; thus at any
given time, the geometry relationship of the radially-spreading flow requires that:
V 2o = ⇡x h, (8)
where Vo is the initial and constant volume, x is the radial distance from the
origin to the flow front, and h is the radially averaged thickness of the spreading
25
PDC. The volumetric particle concentration ( ) decreases with time as particles
sediment out of the flow (Dade and Huppert, 1995). The runout of the PDC is
marked by its buoyancy reversal. The runout distance (xcritical) is described by:
xcritical = ((1   1/2 1/2critical)   ( critical) arctan([(1   1/2 0.3critical)/ critical] )) , (9)
where the critical concentration ( critical) is (Dade and Huppert, 1995):
⇢a   ⇢  = icritical   , (10)⇢p ⇢i
The flow rate of the radially spreading pyroclastic density current, u, is calcu-
lated by:
dx
= u = Fr[g0 (     1/2critical)] , (11)
dt p
where
g0
g(⇢p   ⇢= i) , (12)
p ⇢a
and Fr is the Froude number, which is assumed to be a constant equal to the
square root of 2 (Benjamin, 1968). The density of the ambient air, interstitial
gas, and particles are given by ⇢a , ⇢i , and ⇢p , respectively. Throughout the flow
of the PDC, particles sediment out, gradually creating the ignimbrite (Branney,
26
2002). Particles settle out of the current at a speed calculated by the Stokes
settling velocity (ws):
✓ ◆
⇢   ⇢
ws = (2/9)
p i g(r1/2), (13)
µ
where r is the radius of the particles (Robinson, 1926). The deposition rate of
particles is:
db
=   wsb , (14)
dt h(4/3)⇡(dp/2)3
where b is the number of particles in the flow and dp is the diameter of the
particles in the pyroclastic density current. The initial number of particles in the
flow, bo, is calculated using the volume of the deposit and the diameter of the
particles (Dade and Huppert, 1995):
V
b = so , (15)
(4/3)⇡((dp/2)3)
where Vs is the volume of the solids, which for the Grey’s Landing ignimbrite
is calculated by:
Vs = Vdep   (Vdep ⇤ 0.075), (16)
where Vdep is the volume of the Grey’s Landing ignimbrite deposit. The 0.075
accounts for the porosity in the Grey’s Landing ignimbrite (Lavallée et al., 2015).
27
As the constant volume pyroclastic density current travels radially outward
from the caldera, the numerical model tracks the evolution of current height (h),
volumetric particle concentration ( ), velocity of the current (u), number of par-
ticles in the flow (b), and the deposition rate (db/dt).
We utilize this model to illustrate the e↵ects that initial particle volume con-
centration (volume fraction) and initial ash cloud volume (km3) have on the
runout distance of a pyroclastic density current (km) and its deposit volume (km).
These results are not specific to an ignimbrite deposit and are meant to highlight
the e↵ect that particularly the initial particle concentration has on the pyroclas-
tic density current and deposit. Figure 9 shows that increasing the initial volume
fraction of particles and the volume of the initial ash cloud leads into increases
in both the PDC runout and the deposit volume. However the runout distance is
controlled more strongly by the initial ash cloud volume than by the initial volume
fraction of particles
The general e↵ect of pyroclastic density current volumetric particle concen-
tration on the runout and deposit volume of an ignimbrite is shown in Figure
10.
We apply the Dade and Huppert (1995) axisymmetric, constant-volume grav-
ity current model to the pyroclastic density current which produced the Grey’s
Landing ignimbrite (Andrews, 2011; Knott et al., 2020). We utilize minimum de-
posit area (23, 000 km2) and volume (2, 800 km3) of the Grey’s Landing ignimbrite
28
Figure 9: The e↵ects of initial volume fraction and initial ash cloud volume on the
runout distance of a pyroclastic density current and on the deposit volume.
estimated by Knott et al. (2020) to inform our calculations. There is no identifi-
able caldera at Grey’s Landing (Andrews, 2011; Knott et al., 2020) therefore we
follow the assumption made by Knott et al. (2020) of modest caldera dimensions
of one tenth of the areal extent of Yellowstone, leading us to the assumption of
initial PDC radius (x1) of 5,000 meters.
Initial particle density (⇢p) is calculated using the composition of the Grey’s
Landing ignimbrite (Andrews, 2011) in the program Rhyolite MELTS (Gualda
et al., 2012; Ghiorso and Gualda, 2015), for temperatures ranging from 1400  C
to 650  C. The Grey’s Landing ignimbrite is broadly homogeneous in both bulk
and glass compositions (Ellis et al., 2013, 2015). We tested a range of particle
29
Figure 10: The e↵ect of pyroclastic density current volumetric particle concentration
on the runout and deposit volume of an ignimbrite from the constant-volume axisym-
metric model.
diameters, from 1 x 10 4 to 1 x 10 2 meters. In the sieving of the rare, non-welded
parts of large high-grade ignimbites in the Snake River Plain, including Grey’s
Landing, Branney et al. (2008) showed them to be fine-grained, free of pumice
lapilli and fiamme, and better sorted than is typical for massive ignimbites. The
high temperatures reached within the ignimbrite lead to melting of ash particles
and resulted in the lava-like highly rheomorphic texture seen today (Andrews
et al., 2008; Andrews, 2011; Lavallée et al., 2015; Robert et al., 2013).
Initial volumetric concentration and PDC temperature are the other unknown
parameters within the model. We tested the initial volumetric concentration at
three di↵erent values (0.01, 0.001, 0.0005) (Branney, 2002; Choux and Druitt,
30
2002) and tested temperatures over a range spanning 700  C to 1,100  C (Andrews,
2011; Manley and Andrews, 2004; Robert et al., 2013; Lavallée et al., 2015). The
e↵ects of temperature were found to be of negligible significance on flow dynamics
relative to the inital volumetric concentration and particle diameter (Figure 11).
A PDC temperature of 900  C was therefore used for the model as it is close to
the estimated emplacement temperature of the Grey’s Landing vitrophyre (966
 C) (Lavallée et al., 2015).
The e↵ects of particle diameter and initial volumetric concentration on the
Grey’s Landing pyroclastic density current are shown below in Figure 12, where
the longest runout of the pyroclastic density current (1.34x107 meters) occurs
with the smallest ash particle diameter (1x10 4 meters), and the intermediate
initial volumetric concentration (0.001). For all initial volumetric concentrations,
the flows with the smallest particle diameters (1x10 4 meters) had the longest
runouts as well as had the largest amount of particles in the flow. We note that
these results are substantially too large. The maximum runout at Grey’s Landing
is about 80 km from the center (Knott et al., 2020). The velocities recorded are
physically unachievable in a pyroclastic density current (Figure 12).
The utility of the results is in a relative comparison of the e↵ects of particle
diameter, emplacement temperature, and initial volumetric concentration on an
idealized ”box model” flow. The current modeled is lacking integral parts of
pyroclastic density current dynamics such as entrainment, cooling, or interactions
31
Figure 11: The relative e↵ects of temperature, initial volumetric concentration, and
ash particle diameter on the runout distance of the pyroclastic density current in the
axisymmetric, constant volume numerical gravity current model. Blue markers represent
runs with particle diameters of 1 x 10 4 meters, green represents runs with particle
diameters of 1 x 10 3 meters, and magenta represents runs with particle diameters of
1 x 10 2 meters. Circular markers represent runs with initial volumetric concentration
values of 0.005, squares are runs with initial volumetric concentration values of 0.001,
and stars are runs with initial volumetric concentration values of 0.01.
32
Figure 12: Resulting e↵ects of ash particle diameter (1e-4, 1e-3, and 1e-3 meters)
and initial volumetric concentration (0.0005, 0.001, and 0.01) on (a) the runout of the
pyroclastic density current in the axisymmetric, constant volume numerical model and
(b) the velocity of the flow. Temperature in all runs is 900 C
at the base of the flow (Roche et al., 2016; Andrews, 2014; Fauria et al., 2016;
Sher and Woods, 2017).
33
CHAPTER V: MULTIPHASE MODEL OF A PYROCLASTIC DENSITY
CURRENT THAT PRODUCES AN EXTREMELY HIGH-GRADE
IGNIMBRITE
We model the pyroclastic density current which deposits the Grey’s Landing ign-
imbrite with continuum multiphase models. Multiphase models are more realistic
and capture flow dynamics more accurately than their box model counterparts,
such as the axisymmetric, constant-volume model from Chapter 4 (Dufek and
Bergantz, 2007). We utilize a Eulerian-Eulerian (EE) multiphase model in which
the particle phase, treated as a fluid, and the carrier fluid phase, in this case
it is air, have separate conservation equations for mass, momentum, and energy
(Dufek and Bergantz, 2007).
The conservation of mass, or continuity, equation is written as:
@ @
(m↵m⇢) + (m↵m⇢mUi) = 0 (17)
@t @xi
where m↵ is the volume fraction of the mth phase (where if m=1 it is the gas
phase and if m=2 it is the particle phase), m⇢ is the density [kg/m3] of the mth
phase, mUi is the average velocity [m/s] of the mth phase. The subscripts i and j
are indices for spatial direction.
The conservation of momentum is calculated as:
34
@ @ @mP @m⌧
(m↵m⇢mUi) + (
m↵m⇢mUmUj) =   +
ij +mij I +
m
i ↵
m⇢gi (18)
@t @x ij @xi @xj
where mP is the pressure [Pa] of the mth phase, m⌧ij is the stress tensor [Pa]
of the mth phase, mIi is the interphase momentum transfer [kg/m3s], and gi the
gravitational acceleration [m/s2].
The conservation of thermal energy is calculated as:
✓ ◆
@mT @mT @mq
m↵m⇢mc +mp Uj =   +Hgp (19)
@t @xj @xi
where mT is the thermal temperature [K]of the mth phase, mq is thermal heat
flux [J/m2s] of the mth phase, and Hgp is the interphase heat transfer [W/m3].
In multiphase flows, separate boundary conditions must be specified for the
continuous and particle phases. Here, we utilize the ”leaky” boundary conditions
(Dufek and Bergantz, 2007), in which all particles that reach the boundary are
removed from the flow. This assumes the particles are perfectly sticky (no saltation
at boundary) and become part of the substrate upon contact. This model reflect
the Gollwitzer (2012) scaling of Grey’s Landing ash particles shown in Chapter
3. We recognize this as an end-member case and further studies will need to be
conducted either determine which scaling is more accurate or to model the e↵ects
of particles in which the viscous timescale is greater than the collisional timescale
on the dynamics of the pyroclastic density current and subsequent deposit.
35
In order for the pyroclastic density current, with perfectly sticky ash particles,
to reach the great extents seen in the Grey’ Landing deposit, we assume the
current is dilute (  < 10 2) (Weit et al., 2018; Lube et al., 2020). In a dilute
current, particle-particle interactions are rare and the transportation of particles
in the current is dominated by fluid drag and gravity (Figure 13) (Lube et al.,
2020). A dilute pyroclastic density current favors the propagation of the current if
the particles are perfectly sticky as shown in the scaling from Chapter 3. Particles
would be allowed to travel further because of the lack of coalescence and clustering
that occurs during particle collisions. We note again that we utilize an end-
member case here. Future studies may reveal that the scaling relationship between
the viscous timescale and the collisional timescale is more accurate. In this case,
the pyroclastic density current may not be dilute by necessity, as particles may
”bounce” o↵ one another and the substrate during collisions, allowing the current
to potentially exist in an intermediate or dense regime (Breard et al., 2016; Lube
et al., 2020).
36
37
Figure 13: Delineating the dense, intermediate, and dilute transport regimes in pyroclastic density currents (figure originally
from (Lube et al., 2020)).a) discriminating drag-dominated and collision-dominated transport regimes through a time-scale
ratio not used in this study. b-c) a schematic presentation of the traditional multiphase flow fields of dilute and dense
suspensions (b) and their associated gas-particle coupling regimes (c) relative to the flows’ particle volume concentration.
d) a schematic presentation and approximate range in particle concentration of the dilute, intermediate and dense transport
regimes of PDCs. Details of gas-particle feedback mechanisms in each regime are highlighted in encircled ‘zooms’. Blue
arrows show fluid streamlines, blue dashed lines highlight large eddies. Yellow stars highlight particle collisions and their
relative frequency. e) the threshold concentration at the intermediate-dense boundary where (frictional) contact stresses
equals hydrodynamic stresses.
CHAPTER VI: THERMAL MODEL OF THE EVOLUTION OF AN
EXTREMELY HIGH-GRADE IGNIMBRITE DURING AND AFTER
DEPOSITION
We investigate the e↵ects of strain heating on the temperatures in the Grey’s
Landing ignimbrite at various distances from the source. We also model the post-
depositional cooling to determine if the deposit stays above the glass transition
long enough for gravitational-induced flow, post-deposition, could significantly
contribute to rheomorphism in the ignimbrite.
3 Construction of the thermal model
The general unsteady one-dimensional heat di↵usion equation is described as:
✓ ◆
 T    T
⇢c = k + S, (20)
 t  y  y
where ⇢ is density, c is the heat capacity, and k is the conductivity of the
material, T is temperature, t is time, y is the position, and S is a heat source
term. Here, ⇢ and c are dependent on temperature. Thermal conductivity in rhy-
olites was shown by Romine et al. (2012) to vary little over a range of magmatic
temperatures and water contents and is assumed here to be constant. The dis-
cretization of the one-dimensional unsteady heat di↵usion equation in this study
utilizes a fully implicit scheme. It is solved by the Tridiagonal Matrix Algorithm
38
(TDMA) for the temperature at each position in time, encompassing the unsteady
heat di↵usion as well as the thermal input from strain heating (Patankar, 1980).
Following Patankar’s (1980) derivations, the fully implicit discretization equation
can be written as:
aPTP = aSTS + aNTN + b, (21)
where
k
a sS = , (22)
(@y)s
k
aN =
n , (23)
(@y)n
⇢c  y
ao = p , (24)
P  t
b = Sc y + a
o T o , (25)
P P
aP = aS + aN + a
o   SP y, (26)P
This series of equations represents the relationship, shown in Figure 14, be-
tween the temperature at point P (TP ) and its neighbors to the north (TN) and
39
to the south (TS) within the current time step, as well as their respective coe -
cients, and a linearized source term, b, which incorporates the coe cient ao and
P
the temperature of point P at the previous time step (T o). The variables ks andP
kn refer to the conductivities at the interfaces. The locations of the parameters
(@y)s and (partialy)n are shown in Figure 14. Sc, a constant external heat source,
and SP , an external heat force are set to zero in this study.
Figure 14: The one-dimensional heat di↵usion discretization depicting node points
P,S, and N; interfaces s and n, distances between nodes: (@y)n and (@y)s, and the
control volume  y.
We calculate the influence of heating generated by viscous dissipation within
the shear zone to investigate if the deposit undergoes significant strain to induce
rheomorphism. Volumetric strain heating (Qstrain) inW/m3, or the heat generated
from viscous dissipation, is given by:
40
Q 2strain = ⌘ ⇥ " , (27)
where ⌘ is the apparent viscosity [Pa · s] and " is the strain rate (Robert et al.,
2013):
 u
" = , (28)
 y
where u is the velocity in the pyroclastic density current and y the height of
the current. We set the strain rate to exponentially decrease with depth from the
top (the flow-substrate boundary) in the shear zone. The shear zone thickness
(hsz), which is vertically migrating during deposition) is set to be 2 meters in
thickness (Andrews, 2006). The maximum shear zone thickness is 2 meters based
on the wavelength ( 1 m) of F1 folds observed in the deposit and the maximum
thickness ( 1 m) of the folded layers in the ignimbrite (Andrews, 2011).
We calculate the increase in temperature (SP) at each node in the shear zone
by:
Q
SP = strain
 t
, (29)
⇢cp
using Q 3strain [W/m ] at that node based on the temperatures and viscosities
of the previous time step, the time step ( t), and the density (⇢) [kg/m3] and
heat capacity (cp) [J/kgK] at that node which depend on the temperature of
41
the node at the previous time step. The values of density and heat capacity
are determined from MELTS modeling of Grey’s Landing and uses groupings of
temperature as determined from the model to give temperature-dependent density
and heat capacity. If the node is the newly emplaced node, representing the
growing deposit through progressive aggradation, the viscosity, density, and heat
capacity are based on the emplacement temperature.
We add this increase in temperature to the temperature of that node at the
previous time step. The time steps are determined from an input file which gives
information from the pyroclastic density current such as deposition rate [m/s],time
[s], distance [m], shear rate [s 1], and emplacement temperature [C].
The deposition of the Grey’s Landing member is treated as a series of small,
instantaneous emplacement events. The e↵ects of compaction are neglected in the
model for simplicity, but we note its importance on the welding of an ignimbrite
(Quane and Russell, 2005).
The depositing ignimbrite, shown schematically in Figure 15, overlies a basal
unit with a far-field temperature boundary held at 25  C. The basal unit is defined
by thermal properties di↵erent from those in the Grey’s Landing ignimbrite. The
density (⇢ ), heat capacity (c), thermal conductivity (k), and viscosity (µ) of
the Grey’s Landing deposit are assumed to be dependent on temperature. The
liquidus was found to be at 1000.20  C.
42
Figure 15: A visualization of the numerical model of the thermal evolution of the
Grey’s Landing ignimbrite through deposition. The basal unit is of a constant thick-
ness (h1) with a far-field temperature value (T0) and its temperature varies in space
and time (T1). The Grey’s Landing unit deposits in increments, as informed from the
pyroclastic density current model. The thickness (h) of the deposit grows in time, and
the temperature of the deposit (T ) evolves in time and space. The vertically-migrating
shear zone has a constant thickness (hsz) and an evolving temperature (Tsz).
43
We utilize three di↵erent measurements of viscosity to calculate strain heating
within the shear zone of Grey’s Landing. In dilatometric measurement exper-
iments performed in a study by Lavallée et al. (2015), the viscosity of Grey’s
Landing ignimbrite vitrophyre melt when temperatures are above the glass tran-
sition temperature (870  C) and below 960  C was found to be:
9601
log10(⌘) =     3.545, (30)T 195.7
Viscosity [Pa · s] is only included in the calculation of strain heating. We cap
the upper limit of viscosity at the value for 960   C (Lavallée et al., 2015). When
the temperature of a node is less than the glass transition temperature, we assume
no strain heating occurs there.
We additionally calculate the thermal evolution of the Grey’s Landing ign-
imbrite with the equation for viscosity as determined from parallel-plate viscom-
etry experiments conducted by Robert et al. (2013) to measure the apparent
viscosity of the basal Plinian ash-fall tu↵ at Grey’s Landing. They found the
apparent viscosity to be described by:
13441
log10(⌘) =   4.5, (31)
T   304.5
We test another calculation of viscosity in the thermal model (Dufek and
Bergantz, 2007), one in which the viscosity within the shear zone is assumed to
44
be the granular viscosity as outputted by the multiphase model:
pµ =p µf +p µk/c (32)
where pµf is the e↵ective viscosity in the frictional regime of the pyroclastic
density current:
P fr sin
2p 
pµf = ⇣ ⌘ , (33)2
p
4sin2p pI @ Ui2D + @xi
where P f is the frictional pressure, p  is the specularity, pI2D is the second
invariant of rate of strain tensor [s 2], pUi is the average velocity of the particle
phase [m/s] (Dufek and Bergantz, 2007). However, if the flow is dilute enough,
this term contributes very little.
pµk/c is the total kinetic-collisional viscosity in the PDC and is described by:
   
pµk/c =p ↵p⇢ pvkin +p vcoll (34)
where p↵ is the volume fraction of the particle phase, p⇢ is the density of
the particle phase [kg/m3], pvkin is the kinetic shear viscosity, and pvcoll is the
collisional shear viscosity (Dufek and Bergantz, 2007).
By testing the first two methods, we get a sense of the absolute maximum,
but likely unrealistic strain heating possible with the viscosity from Equation 30
and Equation 31, and the minimum strain heating possible with the much lower
45
viscosity value of the granular viscosity.
Once the deposit has been fully emplaced, we calculate the di↵usion timescale:
L2
⌧ = , (35)
K
where L is a characteristic length scale (here, the deposit thickness), and K
is the thermal di↵usivity which we calculate with the maximum density and heat
capacity recorded in the deposit during emplacement. Once the deposit cools
past the glass transition temperature (870  C), it is assumed to be fully solidified
and therefore rheomorphism will have ceased throughout the extent of the deposit
(Lavallée et al., 2015). This assumed to occur within the di↵usion timescale.
The models allow us to qualitatively investigate the thermal evolution of ex-
tremely high-grade ignimbrites, such as Grey’s Landing, with the e↵ects of strain
heating during emplacement as well as calculate the post-depositional cooling.
We will be able to determine if rheomorphism occurs predominately during em-
placement of the ignimbrite, as a result of shearing, or it occurs primarily during
post-depositional gravity-driven flow of the ignimbrite, as a result of the mainte-
nance of high temperatures in the deposit.
46
4 General e↵ects of velocity at the flow-substrate boundary and em-
placement temperature on thermal evolution of a rheomorphic ign-
imbrite
We quantify the e↵ects that emplacement temperature and velocity at the pyro-
clastic density current-substrate boundary have on the maximum amount of heat
generated from strain at a single node and the time it takes for the ignimbrite to
fully cool to the glass transition temperature (870  C) (Lavallée et al., 2015).
First we test a range of volumetric particle concentrations for the pyroclastic
density currents as well as particle size (given in diameter) on the deposition rate
of the PDC (in m/s):
 ws td = , (36)
 d
where d is deposit thickness (in meters),   is the volume fraction of particles
in the flow, ws is the settling velocity (m/s) which will be a function of particle
diameter,  t is the time interval (s), and  d is the volume fraction of particles in
the deposit. We assume  d is one, meaning there is no pore space in the deposit.
In actuality, the porosity of Grey’s Landing is measured to be 0.77 (Lavallée et al.,
2015). We plot the resultes of the calculations in Figure 16.
47
Figure 16: The ranges of possible deposition rates for the Grey’s Landing pyroclastic
density current as calculated from the volumetric particle concentration ( ) and the
particle diameter (in meters). The blue region represents the selected deposition rate
we used in created a regime diagram
The concentrations of the Grey’s Landing PDC are unknown. We selected
to use the deposition rate of 2.8x10 4 m/s produced by a volumetric particle
concentration of 1 x 10 3 and a particle diameter of 100 microns. We selected a 100
micron particle diameter for our study as estimates of median diameter of grains
at Grey’s Landing is between 2-3   (125-250 micron) (Branney et al., 2008). More
studies would need to be conducted in the future to determine which deposition
rate is the most accurate. We chose a more dilute particle concentration since
we are utilizing a perfectly sticky multiphase model in which particle agglutinate
upon contact. A more dilute flow would allow particles to travel farther instead
48
of clustering and sedimenting out of the flow quickly and subsequently sticking to
the boundary. Future studies should investigate the e↵ects of di↵erent volumetric
concentrations and particle-particle and particle-substrate collisions as describes
in Chapter 3.
We compare unsteady one-dimensional thermal model results, with the depo-
sition rate of 2.8x10 4 m/s, located one quarter (20 km) of the estimated furthest
runout distance for the Grey’s Landing PDC (Knott et al., 2020). We test the
end-member maximum thermal e↵ects of flow velocity at the current-substrate
boundary and the emplacement temperature on the deposit. We utilize the vis-
cosity (Equation 30 from Lavallée et al. (2015) in the calculation of strain heat-
ing. We approximate the strain rate here with the velocity at the flow-substrate
boundary divided by the height of the deposit at that time. This does not include
exponential decay as is described in Chapter 6 and executed in Chapter 7. This
gives us the maximum possible strain heating and thus thermal e↵ects. We test
a velocity parameter space of 1, 0.1, 1 x 10 2, 1 x 10 3, 1 x 10 4, and 1 x 10 5
m/s, and temperatures of 850, 900, 966 (the vitrophyre emplacement temperature
found by Lavallée et al. (2015)), 1000, and 1050  C to cover possible emplacement
temperatures (Andrews, 2006, 2011; Robert et al., 2013; Ellis et al., 2015; Lavallée
et al., 2015).
The e↵ects of emplacement temperature and velocity at the flow-substrate
boundary are shown in Figure 17. For all emplacement temperatures, a velocity
49
of 1 m/s produced substantial and physically unrealistic increases in temperature
from strain heating; subsequently, the maximum increases in recorded temperature
from the emplacement temperature and time for the entire deposit to cool to the
glass transition temperature (870  C) (Lavallée et al., 2015). These increases are
the largest for the lowest emplacement temperature of 850  C; where the maximum
temperature increase from strain heating is 5.18x105  C, the maximum increase
in temperature above the emplacement temperature is 4.88x105  C the cooling
duration is 489 years. A velocity of 0.1 m/s creates an unrealistic value for an
emplacement temperature of 850  C such as generated increases in temperature
from shearing at 1330  C. However, for the other emplacement temperatures, the
results are consistent with previous studies (Andrews, 2006, 2011; Robert et al.,
2013; Ellis et al., 2015; Lavallée et al., 2015). We conclude that a velocity of 1
m/s is physically unrealistic and creates shear rates that are beyond what is likely
possible for this type of flow. For this viscosity and shear rate this implies a 850  C
is also physically unrealistic. The value, being below that of the glass transition
temperature, significantly e↵ect viscosities in the deposit, leading to unrealistically
high strain heating values, which the model assumes occurs regardless of the
temperature. The ability to shear a deposit that it already below the suggested
glass transition temperature Lavallée et al. (2015) would be di cult and unlikely.
We conclude that the minimum possible emplacement temperature is the glass
transition temperature.
50
Figure 17: E↵ects of the velocity at the PDC-substrate boundary and the emplacement
temperature on the heating and cooling of a Grey’s Landing type deposit. Results were
generated through a one-dimensional unsteady conduction model located 20 km from
PDC source. The velocity at the boundary and the emplacement temperature did e↵ect
the thermal evolution on (a) the timescale (duration) for the deposit to fully cool to the
glass transition temperature (870  C; Lavallée et al. (2015)); (b) the maximum increase
in temperature from strain heating; and (c) the maximum in increase of temperature
(in Celsius) generated from strain heating.
51
When the emplacement temperature of 850  C and the velocity of the flow at
the flow-substrate boundary of 1 m/s are removed, the thermal e↵ects of velocity
and emplacement temperature are shown more clearly (Figure 18. The largest
increases in temperature from strain heating and maximum increase of tempera-
ture relative to emplacement temperature occur at the larger velocities (e.g. 0.1
m/s) and lower temperatures (e.g. 900  C). These results (Figure 17) suggest that
there are e↵ects from shearing in the deposit during deposition and those e↵ects
will change depending on the initial emplacement temperature and velocity at the
flow-substrate boundary. However, we note again that these results are maximum
possible strain heating. Previous studies have not investigated these e↵ects before
as they occur during progressive aggradation of the pyroclastic density current
and particularly with thermal properties of the deposit that evolve with temper-
ature (Andrews, 2006, 2011; Robert et al., 2013; Ellis et al., 2015; Lavallée et al.,
2015).
52
Figure 18: E↵ects of a narrower window of velocity at the PDC-substrate boundary
and emplacement temperature on the heating and cooling of a Grey’s Landing type
deposit. Results were generated through a one-dimensional unsteady conduction model
located 20 km from PDC source. The velocity at the boundary and the emplacement
temperature did e↵ect the thermal evolution on (a) the timescale (duration) for the
deposit to fully cool to the glass transition temperature (870  C; Lavallée et al. (2015));
(b) the maximum increase in temperature from strain heating; and (c) the maximum
in increase of temperature (in Celsius) generated from strain heating.
53
CHAPTER VII: APPLICATION OF MODELS TO GREY’S LANDING
PYROCLASTIC DENSITY CURRENT AND IGNIMBRITE
We apply our multiphase model and our thermal model to a Grey’s Landing-type
pyroclastic density current. The multiphase model, notably, does not include to-
pography such as the western dipping floor of the Rogerson Graben or the Brown’s
Bench Escarpment. The modeled PDC flows over a flat, horizontal surface.
Figure 19: A snapshot of the Grey’s Landing pyroclastic density current multiphase
model during transport. The colorbar shows the log of the volume fraction of particles
in the current at the time the snapshot was taken.
The multiphase model of the Grey’s Landing pyroclastic density current (flow
”snapshot” shown in Figure 19 utilizes a single grain size (125 micron), an initial
volume fraction of 0.04, an initial height of 200 meters, an initial velocity of
20 m/s, and an initial temperature of 966  C (the emplacement temperature of
the Grey’s Landing vitrophyre Lavallée et al. (2015)). The model ran for 3,158
54
seconds (approximately 52.6 minutes) and had a PDC runout distance of 59,900
meters. The runout distance is shorter than the longest runout (approximately 80
km) seen at Grey’s Landing (Figure 1), but is consistent with the average runout
(approximately 60 km).
We analyzed the results of the multiphase model at six distal locations (100,
1000, 10000, 20000, 30000, and 40000 meters from source) in order to assess the
vertical thermal evolution of the deposit across extent of the ignimbrite. The sub-
sequent deposit thinned and was emplaced at cooler temperatures with distance
from the source, as is expected, shown in Figure 20.
Figure 20: Grey’s Landing ignimbrite deposit thickness as determined by the multi-
phase model. The colorbar shows emplacement temperature of the deposit .
We calculated the thermal evolution of the Grey’s Landing PDC deposit, as
informed by the multiphase model, three times - each with a di↵erent viscosity:
55
the viscosity of the vitrophyre measured by Lavallée et al. (2015), the viscosity
of the basal ash fall measured by Robert et al. (2013), and the granular viscosity
calculated in the multiphase model Dufek and Bergantz (2007), as described in
Chapter 6. Figure 21 shows the vertical temperature profile following the cessation
of emplacement, for each of the distal locations and for each of the viscosities
(shown as subfigures).
The temperatures recorded in at the end of emplacement are e↵ected by strain
heating during emplacement, as the pyroclastic density current overrides the grow-
ing deposit, shearing the substrate. In Figure 22 we show (a) log of the maximum
temperature increase in the ignimbrite at a node as a product of strain heating
and viscosity calculation, (b) the ratio of the maximum increase in temperature
at a node as a product of strain heating and the emplacement temperature, and
(c) the log of the ratio of the maximum temperature recorded in the deposit at
the end of emplacement and the emplacement temperature.
56
Figure 21: The vertical temperature profiles at the cessation of emplacement for
six locations at di↵erent distances from the source, with normalized deposit thickness
at each location. The temperature profiles from di↵erent calculations of viscosity in
the shear zone are shown as separate subfigures: (a) viscosity of the Grey’s Landing
vitrophyre as calculated by Lavallée et al. (2015), (b) viscosity of the basal ash flow
at Grey’s Landing as calculated by Lavallée et al. (2015), and (c) granular viscosity
calculated from the multiphase model of the Grey’s Landing PDC.
57
Figure 22: The increases in temperature in Grey’s Landing ignimbrite as a product of
strain heating. Each subfigure also shows the di↵erence in these temperature increased
based on the viscosity calculation within the two meter shear zone: where (a) log of
the maximum temperature increase in the ignimbrite at a node as a product of strain
heating and viscosity calculation, (b) the ratio of the maximum increase in temperature
at a node as a product of strain heating and the emplacement temperature, and (c)
the log of the ratio of the maximum temperature recorded in the deposit at the end of
emplacement and the emplacement temperature.
58
The temperatures recorded in Grey’s Landing were approximately 900-1100
 C. The maximum increases in temperature due to strain heating, as well as the
maximum temperature recorded in the deposit is shown in Table 1 for 10 km from
the PDC source.
Lavallée et al. (2015) Robert et al. (2013) granular
viscosity
T strain max (C) 8.73e3 6.93e8 0.82e-6
T max at end
of emplacement (C) 1.09e4 7.14e8 965.73
Table 1: The values, recorded at 10 km from PDC source, of the maximum
increases in temperature due to strain heating and the maximum temperature
recorded in the deposit for Grey’s Landing. Each column describes the results for
a di↵erent method of calculating viscosity in the shear zone: viscosity measured
by Lavallée et al. (2015), viscosity, measured by Robert et al. (2013), and the
granular viscosity calculated from the multiphase model.
The increases in temperature as a product of strain heating are physically un-
realistic for viscosity calculations in the shear zone for those measured by Lavallée
et al. (2015) and Robert et al. (2013). The results from the Robert et al. (2013)
59
viscosity being orders of magnitude above temperatures recorded in Grey’s Land-
ing. The temperature increases from strain heating when granular viscosity is
used are on the order of 10 8 to 10 10 and the changes to thermal evolution of
the deposit relative to standard conduction from instantaneous emplacement are
negligible.
We calculate the di↵usion timescale for each x location from the source and
present them in Figure 23. Di↵usion timescale decreases with deposit thickness.
We assume in this timescale, the temperatures will cool past the glass transition
temperature. The di↵usion timescale for the thicker regions is on the order of
decades to nearly 140 years for the most proximal case. There is potential for the
deposit to maintain heat long enough in the interior for rheomorphism to occur
due to gravitation-induced flow.
60
Figure 23: The di↵usion timescale for Grey’s Landing at varying distances from the
source.
61
CHAPTER VIII: DISCUSSION
In the case of hot (966  C), sticky ash particles which agglutinate upon contact
in a dilute Grey’s Landing pyroclastic density current, we find that the current
is able to travel distances that match well with the average distal extents of
the deposit (60 km from center). The eruption occurs in just under an hour
and emplaces an ignimbrite as thick as 44.6 meters. We note the thickest area
occurs very proximal to the source yet the maximum thickness produced by our
model is less than half of the maximum thickness recorded in the actual Grey’s
Landing ignimbrite (approximately 100 m). However, Grey’s Landing sits on top
of a western sloping graben which may have contributed to deposit thickening
particularly if high temperatures (above the glass transition) were maintained
within the deposit.
The results of our thermal model support that vast majority of the interior
of the ignimbrite could be held above the glass transition temperature for con-
siderable periods of times; years in the thicker portion of the deposit where the
emplacement temperature was above the glass transition temperature. The model,
based on strain heating calculations using the granular viscosity determined by
the multiphase model, generated virtually no temperature increases from strain
heating for a starting temperature of the PDC of 966  C. This, as well as the long
cooling times, supports the hypothesis that the vast majority of rheomorphism
62
occurs post-deposition at Grey’s Landing ignimbrite. More tests of this model at
lower eruption and emplacement temperatures needs to occur to say more con-
clusively if strain heating could play a more substantial role. The general e↵ects
of strain heating model, shown in Chapter 6.2, imply that at cooler temperatures
that are closer to the glass transition temperature, the temperature increases from
strain heating become more significant.
63
CHAPTER IX: CONCLUSION
Our study is limited and does not cover the whole spectrum of studies that need
to occur in order to fully characterize how ash agglutination e↵ects the transport
of the Grey’s Landing pyroclastic density current, how the deposit reached it
recorded temperatures of approximately 900-1000  C, constrain all situations that
could lead to shear heating, or include the topography of the underlying Rogerson
Graben on the PDC flow and on post-depositional flow (Lavallée et al., 2015; Ellis
et al., 2015; Andrews, 2011, 2006; Robert et al., 2013). Future studies would need
to include: (1) experiments of partially-to-fully molten highly viscous particle-
particle and particle-substrate collisions to better constrain under what conditions
partially-to-fully molten ash agglutinates, (2) use the saltation boundary in the
multiphase model to test the other end-member flow-substrate boundary e↵ects,
(3) use the experimental results and multiphase model to determine if the Grey’s
Landing pyroclastic density current needed to be dense or dilute, and (4) test
a larger swathe of eruption and subsequently emplacement temperatures in the
multiphase to fully constrain the e↵ects of strain heating at Grey’s Landing, (5)
model the post-depositional cooling with fine resolution to determine how the
deposit cools past the glass transition temperature.
64
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APPENDIX A. Validation of axisymmetric, constant-volume PDC model 75
*APPENDIX A. Validation of axisymmetric, constant-volume PDC model
A validation of the axisymmetric, constant-volume gravity current numerical
model based on that of Dade and Huppert (1995) was undertaken to ensure the
model captured physically accurate behavior. The model, which considers an
instantaneously well-mixed, fine grained, axisymmetric gravity current released
from a lock which spreads radially over a non-erodible planar surface, is compared
to experiments conducted by Bonnecaze et al. (1995) in which silicon carbide
particles were released from a semi-circular lock with a 10 cm radius into a water-
filled rectangular tank with dimensions of 150 cm by 250 cm and a water depth
of 14 cm. The schematic is shown below in Figure 24
Figure 24: Schematic from Bonnecaze et al. (1995) depicting a large rectangular
experimental water-filled tank with a semi-circular lock used in experiments on constant-
volume currents.
Particle density (⇢) was 3217 kg/m3 and particle diameter (dp) was 37 mi-
APPENDIX A. Validation of axisymmetric, constant-volume PDC model 76
crons. Initial volume (VO) of the current was 0.006874 m3 with an assumed initial
temperature of 25 C. Initial volume fraction ( ) was 0.2. The model was set to
have spatial steps of 0.01 meter. In the experiment and the model, the critical
concentration (phicrit) was assumed to be zero.
Comparisons between the theoretical predictions of the model and the exper-
imental results from Bonnecaze et al. (1995) are presented in Table 2 where the
runout distance (in meters) and the settling velocity of the particles matches well
between the model and experiments.
Numerical model Bonnecaze et al. (1995) experiment
Runout distance (m) 0.77 0.76
Settling velocity (m/s) 1.7e-3 1.5e-3
Table 2: Comparison of Bonnecaze et al. (1995) constant volume gravity current
experiements and axisymmetric, constant volume numerical model results based
on Dade and Huppert (1995)
The non-dimensionalized numerical model results for the Bonnecaze et al.(1995)
experiments are shown in Figure 25.
APPENDIX A. Validation of axisymmetric, constant-volume PDC model 77
Figure 25: The runout of the axisymmetrically spreading gravity current follows the
instantaneous release of dense suspensions with constant volume. The figure shows
the dimensionless distance, R = r/r1, as a function of the dimensionless time, T =
t/t1 of the numerical axisymmetric, constant-volume gravity current model using the
geometries and experiemental values from Bonnecaze et al. (1995)