Fragmentation and Plinian Eruption of Crystallizing Basaltic Magma ∗ Pranabendu Moitra A,B, , Helge M

Earth and Planetary Science Letters 500 (2018) 97–104

Contents lists available at ScienceDirect

Earth and Planetary Science Letters

www.elsevier.com/locate/epsl

Fragmentation and Plinian eruption of crystallizing basaltic magma ∗ Pranabendu Moitra a,b, , Helge M. Gonnermann b, Bruce F. Houghton c, Chandra S. Tiwary d,e a Department of Geology and Center for Geohazards Studies, University at Buffalo, Buffalo, NY, USA b Department of Earth, Environmental and Planetary Sciences, Rice University, TX, USA c Department of Geology and Geophysics, University of Hawai‘i at Manoa, Honolulu, USA d Department of Material Science and NanoEngineering, Rice University, Houston, TX, USA e Materials Science and Engineering, Indian Institute of Technology Gandhinagar, Gujarat, India a r t i c l e i n f o a b s t r a c t

Article history: Basalt is the most ubiquitous magma on Earth, erupting typically at intensities ranging from quiescently Received 14 December 2017 effusive to mildly explosive. The discovery of highly explosive Plinian eruptions of basaltic magma Received in revised form 30 July 2018 has therefore spurred debate about their cause. Silicic eruptions of similar style are a consequence of Accepted 2 August 2018 brittle fragmentation, as magma deformation becomes progressively more viscoelastic. Magma eventually Available online xxxx crosses the glass transition and fragments due to a positive feedback between water exsolution, viscosity Editor: T.A. Mather and decompression rate. In contrast to silicic eruptions, the viscosity of basaltic magmas is thought to be Keywords: too low to reach conditions for brittle fragmentation. Pyroclasts from several basaltic Plinian eruptions, basaltic Plinian eruption however, contain abundant micron-size crystals that can increase magma viscosity substantially. We explosive magma fragmentation therefore hypothesize that magma crystallization led to brittle fragmentation during these eruptions. oscillatory shear rheometry Using combined oscillatory and extensional rheometry of concentrated particle-liquid suspensions that extensional rheometry are dynamically similar to microcrystalline basaltic magma, we show that high volume fractions of − bubble growth and crystallization particles and extension rates of about 1s 1 or greater result in viscoelastic deformation and brittle conduit model fracture. We further show that for experimentally observed crystallization rate, basaltic magma can reach the empirical failure conditions when erupting at high discharge rates. © 2018 Elsevier B.V. All rights reserved.

1. Introduction of hydrodynamic nature (e.g., Parfitt, 2004; Namiki and Manga, 2008; Valentine and Gregg, 2008; Gonnermann, 2015). The fragmentation and dynamics of Plinian eruptions of basaltic Pyroclastic eruptive deposits from several explosive basaltic magma have been a subject of debate (e.g., Giordano and Ding- eruptions, for example the 1886 eruption of Mt. Tarawera (Sable et well, 2003; Houghton and Gonnermann, 2008; Costantini et al., al., 2009), New Zealand and the 122 BCE eruption of Mt. Etna, Italy 2010; Goepfert and Gardner, 2010). The critical deformation rate (Coltelli et al., 1998; Sable et al., 2006), indicate sustained erup- at which silicate melt dynamically crosses the glass transition is tion columns of approximately 25–30 km height, over durations of ∼ −2 − 10 /τrelax (e.g., Dingwell and Webb, 1989). Here τrelax is the hours, and at mass discharge rates of about 108 kg s 1. They are = relaxation time scale of silicate melts, defined as τrelax η0/G∞, therefore classified as Plinian eruptions. Recent quantitative stud- ∼ where η0 is the shear viscosity at zero frequency, and G∞ 10 ies on these eruptions used existing stress- or strain rate-driven GPa is the shear modulus at infinite frequency. For crystal-poor fragmentation criteria (e.g., Moitra et al., 2013; Campagnola et al., 2 3 basaltic melt η0 is of the order of 10 –10 Pa s (Hui and Zhang, 2016). However due to very small τ , as discussed above, the −8 −7 relax 2007) and τrelax is 10 –10 s. Deformation rates of greater than question remains whether in analogy to their silicic counterparts 5 −1 10 s would be required for basaltic melt to reach the glass tran- these eruptions were associated with brittle fragmentation. The py- sition, which is difficult to reconcile with the dynamics of magma roclasts from these eruptions contain a high abundance of micron- ascent (e.g., Wilson and Head, 1981; Papale, 1999). Consequently, size crystals (up to 90% in volume of the matrix; Fig. 1) that fragmentation during explosive basaltic eruptions is thought to be likely formed as a consequence of water exsolution during eruptive magma ascent (e.g., Hammer and Rutherford, 2002; Goepfert and Gardner, 2010; Arzilli et al., 2015). * Corresponding author at: Department of Geology and Center for Geohazards Studies, University at Buffalo, Buffalo, NY, USA. To assess whether microlites affect the mechanics of magma E-mail address: [email protected] (P. Moitra). fragmentation, this study uses a novel approach by examining https://doi.org/10.1016/j.epsl.2018.08.003 0012-821X/© 2018 Elsevier B.V. All rights reserved. 98 P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104

Fig. 2. Schematic diagrams of experimental methods. (a) The oscillatory shear rheology experiments were performed using a parallel-plate geometry, where Sr is the radius of the disk of sample in between two plates, and h is the sample thickness. (b) The extensional rheometry (tensile test) was performed by placing a cylindrical sample in between two parallel plates, where the upper plate moved upward while the lower plate stayed stationary. L p is the plate separation length and dmid is the diameter at the middle of the cylindrical sample.

Fig. 1. Back-scattered electron (BSE) images of pyroclast sections from Mt. Tarawera, 1886 basaltic Plinian eruption. The inset shows two distinct microlite populations avoid normal forces due to sample placement, the samples were with larger high aspect ratio plagioclase crystals and smaller equigranular to high ∗ pre-sheared. The resultant complex shear modulus, G , can be ex- aspect ratio plagioclase and pyroxene crystals in the glass matrix (gray) surrounding vesicles (black). pressed as

∗ the extensional deformation and fracture behavior of concentrated G = G + iG , (1) liquid-particle suspensions that are dynamically similar to microcrystalline basaltic magma. Using scaling analysis and conduit where G and G are the storage or elastic shear modulus and the model of magma ascent in volcanic conduit, it is demonstrated loss or viscous shear modulus, respectively. that under what conditions crystallizing basaltic magma can reach brittle fragmentation during explosive Plinian style eruptive activ- 2.3. Extensional rheometry ity. The extensional rheometry (tensile tests) was performed us- 2. Experimental methods ing an Instron ElectropulsTM E3000. The suspension was placed between two parallel plates that moved apart in the vertical di- 2.1. Overview rection, resulting in an increase in the gap between the two plates We performed a series of experiments in viscous fluids with (Fig. 2b). In order to obtain a purely uniaxial elongation the gap suspended micron-size solid particles to assess the potential ef- was increased at an exponentially increasing rate (e.g., McKinley fect of high microlite concentrations on the rheological behavior and Sridhar, 2002; White et al., 2010). The position of the upper of basaltic magmas. These suspensions have rheological proper- moving plate is thus given as ties that are dynamically similar to basaltic melt with high volume ˙ fraction of microlites (see Appendix A). The suspensions consisted L p(t) = L0 exp(Et), (2) of silicone oil with a Newtonian shear viscosity of 100 Pa s and 3–10 μm glass spheres at a volume fraction of φx = 0.55, which where L0 is the initial gap length (i.e., the initial length of the ˙ = ˙ is close to the maximum packing limit of φm = 0.56. To character- sample), t is time, and E L p/L p is the axial stretch rate. Equiva- ize the viscoelastic properties of the concentrated suspension and lent to pre-shearing in shear rheology experiments, a small initial − its potential for fracture development, we performed oscillatory stretch rate of ≈0.05 s 1 was applied for few seconds prior to each shear and extensional deformation experiments (Fig. 2). Particle experiment. settling was negligible during our experiments because the char- The elongating fluid in the center of the gap underwent a acteristic settling time was much smaller than the experimental nearly shear-free flow whose effective extensional rate of defor- time scales. mation can be defined as − 2.2. Oscillatory shear rheology ˙ 2 d(dmid) γeff(t) = , (3) dmid dt Oscillatory shear experiments, consisting of amplitude and fre- where d is the diameter at the middle of the suspension fila- quency sweeps (e.g., Heymann et al., 2002; Sumita and Manga, mid ment. The effective Hencky strain is defined as 2008; Namiki and Tanaka, 2017), were performed using an Anton Paar Physica MCR 301TM rotational rheometer with parallel plate d0 geometry and a 1mmgap (Fig. 2a). The amplitude sweep exper- γeff = 2ln , (4) d iments were performed at a fixed angular frequency, ω, in the mid −1 2 −1 −4 −1 range of 10 to 10 rad s and 10 to 10 of total strain, where d0 is the initial diameter of the sample at its midpoint γs. The frequency sweep experiments were performed at angu- and is much greater than the particle diameter. The change in − − lar frequency of 10 1 to 102 rad s 1 and a fixed strain ampli- mid-filament radius with respect to time was calculated from con- − tude of about 10 4, which corresponds to the linear viscoelastic servation of fluid volume and with the aid of photographs of the range obtained from the amplitude sweep experiments. In order to extending cylindrical sample. P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104 99

3.2. Potential implications for basaltic Plinian eruptions

We hypothesize that a viscoelastic rheological response, due to magma expansion, may also be possible during volcanic eruption of micro-crystalline basaltic magmas. We propose that this requires crystal volume fractions close to maximum packing, as well as magma elongation at rates above some threshold value, γ˙cr . In the absence of elongation experiments on highly microcrystalline basaltic melt, we assume dynamic similarity of our experiments −1 and a failure condition of γ˙cr >∼ 1s . Conceptually, we envisage that the basaltic magma crystallizes during eruptive ascend (e.g., Cashman, 1993; Hammer and Ruther- ford, 2002; Goepfert and Gardner, 2010; Applegarth et al., 2013; Shea and Hammer, 2013; Arzilli et al., 2015). As pressure decreases with height within the volcanic conduit, the ascending magma vesiculates due to decrease in the solubility of dissolved volatiles. The resultant volume expansion and acceleration are a

consequence of bubble growth, which stretches the surrounding Fig. 3. Results of oscillatory shear rheology. The storage and loss shear moduli, G and G , respectively, as functions of the angular frequency, ω. The inset shows the crystallizing melt. If the vesiculating magma can reach suﬃciently amplitude sweep tests during which the linear viscoelastic region (G ≥ G ) was high crystallinity and expansion rates, we hypothesize that it may −1 2 −1 observed at lower strain, γs , for 10 < ω < 10 rad s . Multiple crossovers be- fracture in a similar fashion as the suspension in our elongation tween G and G in frequency sweep test are often observed in viscoelastic dense experiments. Assuming that the critical elongation rate for brittle particulate suspensions (e.g., Sumita and Manga, 2008). failure in our experiments provides a reasonable failure criterion for microcrystalline basaltic magma, the question is whether and 3. Results and discussion under which conditions a crystal volume fraction of φx/φm close to 1, together with the failure criterion γ˙cr can be reached during 3.1. Experimental results eruptions. We explore these considerations subsequently.

Fig. 3 indicates that under oscillatory shear deformation the 3.2.1. Crystallization dense particulate suspension behaves like a viscoelastic material, Crystallization involves both crystal nucleation and growth. The with a crossover between the storage (elastic) modulus, G , and Avrami equation provides a well-known quantiﬁcation of crystal- the loss (viscous) modulus, G (Fig. 3). The viscoelastic rheologi- lization kinetics, and is given by (e.g., Cashman, 1993) cal response is attributable to particle interactions and jamming at 3 4 volume fractions close to maximum packing. The uniaxial exten- φx = 1 − exp −kv IF t . (5) sional experiments show that the sample undergoes brittle frac- −1 ture at elongation rates greater than about 1s . In contrast, the Here φx is the crystal volume fraction, kv is the shape factor, I is suspending liquid without suspended particles does not fracture the crystal nucleation rate, F is the crystal growth rate, and t is (Fig. 4). Consistent with the oscillatory experiments, we interpret time. Given I and F , the Avrami equation facilitates the calcula- fracturing to be a consequence of viscoelastic behavior at high de- tion of crystal volume fraction, φx, as a function of time. Fig. 5a formation rates. shows existing experimental values for I and F in silicate melts

Fig. 4. Results of uniaxial extension experiments. (a) With increasing elongation rates, the suspension progresses from (I.) simple elongation to (II.) ductile necking and break-up to (III.) brittle fracturing, as shown by the experimental images (white arrow indicates irregular and angular fracturing at higher strain rate). In comparison, the existence of a thin liquid film of the suspending liquid is observed (also indicated by white arrow in the top, right image) at higher strain rates. The error bars indicate the reproducibility of experiments and the error associated with the measurement of sample mid-radius using the experimental images. (b) The experimental image sequences show the progressive extension of the particulate suspension filament with time at various extensional strain rates, as well as the extension of the particle-free suspending liquid (bottom row) at a higher strain rate for comparison. The initial mid-radii at the onset of experiments were 4 mm, 6 mm and 5 mm for cases I, II and III, respectively. The corresponding final mid-radii were 1 mm, 0.5 mm and 2.0 mm. The initial and final mid-radii of the thin filament of particle-free liquid were about 1.5 mm and 0.15 mm, respectively. 100 P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104

Fig. 5. (a) Values of the crystal growth rate, F , and the crystal nucleation rate, I, from decompression–crystallization experiments. The data shown are from the recent compilation by Arzilli et al. (2015), where A2015a and A2015b are from Arzilli et al. (2015), HR2002 is from Hammer and Rutherford (2002), BH2010 is from Brugger and Hammer (2010)and SH2013 is from Shea and Hammer (2013). (b) Calculation of the time required for basaltic melts to reach high crystallinity (i.e., φx → φm ), using the Avrami equation (equation (5)). The size of the symbols span time scales corresponding to 0.60 ≤ φm ≤ 0.90, the observed range of microlites in the pyroclasts from basaltic Plinian eruptions at Mt. Etna and Mt. Tarawera. (c) The change in magma (melt+crystal) viscosity, ηx, to the melt viscosity, ηm , as a function of normalized crystal volume fraction, φx/φm . ηx increases in a highly nonlinear fashion as φx → φm . from the compilation of Arzilli et al. (2015). When used in con- portional increase in decompression rate, and that this dependency junction with the Avrami equation these data provide constraints or feedback can potentially drive conditions toward brittle failure. on the time required for basaltic melt to crystallize to values of φx close to φm, where the latter may depend on crystal shape and 3.2.3. Decompression rate size distribution (e.g., Moitra and Gonnermann, 2015). Here we as- To assess under which conditions brittle failure can occur dur- = = sume that kv 4π /3 for spherical crystals, and φm 0.60–0.90, ing basaltic eruptions, we first consider the relation between de- noting that the specific value does not affect our estimates signifi- compression rate and magma crystallization. In general the decom- cantly. Based on the experimentally determined values of I and F , pression rate of erupting magma encompasses magma-static and Fig. 5b shows that basaltic magma can reach crystal volume frac- viscous stresses. If the magma is to undergo brittle fragmentation, 3 tions close to maximum packing in about 10 s. For storage depths as the volume fraction of crystals approaches the maximum pack- corresponding to a few kilometers this implies that basaltic magma ing fraction, then the value of γ˙cr provides a conditional constraint can extensively crystallize, if it ascends at velocities of several me- on decompression rate. ters per second. In the case of the 1886 eruption of Mt. Tarawera, The momentum balance for a growing bubble, neglecting sur- for example, such velocities are consistent with discharge rate es- − face tension and inertial terms, provides an expression for the timates of about 105 m3 s 1 (Sable et al., 2009), given that the elongation rate, γ˙E (e.g., Proussevitch et al., 1993) eruption appears to have been fed from an approximately 17 km- long dike and assuming a dike-width of up to a few meters. 1 dR 1 γ˙E = = P g − Pm . (7) R dt ηx 3.2.2. Viscosity In a general sense the viscosity of the liquid-particle suspen- Here R is the bubble radius, ηx is the viscosity of the crystalline sion, ηx, increases with increasing φx. This relation can be defined melt, P g is pressure of the gas inside bubbles, Pm is the pressure as (e.g., Mader et al., 2013; Moitra and Gonnermann, 2015) of the surrounding crystalline melt, and t is time. Equation (7)indicates that for γ˙E to attain its critical value, for a given value of − 2 η at φ /φ → 1, requires that (P − P ) also reaches some crit- = − φx x x m g m ηx ηm 1 , (6) ical value. This in turn implies that the rate of decrease in melt φm pressure is larger than the rate of decrease in gas pressure within → where ηm is the viscosity of the pure melt. As φx φm the value bubbles, such that of ηx increases in a highly nonlinear fashion (Fig. 5c). We will show that this increase in viscosity, assuming that the eruption is sus- dPm dPg > . (8) tained and has adjusted to magma crystallization, leads to a pro- dt dt P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104 101

Fig. 6. Illustrative example of how variables evolve during basaltic magma ascent in a dike-like conduit (see Appendix B for details). The calculation corresponds to a dike 3 −13.5 −4 14 −3 width and length of 2a = 1.2 m and b = 5 km, respectively. Other conditions are: crystallization rate of IF = 10 s , bubble number density of Nb = 10 m , ◦ −6 maximum crystal packing fraction of φm = 0.60, initial dissolved H2O of co = 3wt%, magma temperature of T = 1100 C, and the initial bubble radius of Ro = 10 m. It is −1 assumed that the magma fragments at the empirically obtained critical strain rate of γ˙cr = 1.0s . Furthermore, it is assumed that there are no bubble and crystal growth after fragmentation, therefore their values are only shown up to fragmentation, along with the value of melt pressure. The model illustrates the interdependencies between crystallization, bubble and crystal bearing multiphase magma viscosity, pressure decrease, and elongation rate. (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

− From the mass balance of a growing bubble (e.g., Proussevitch et where u2/4a2 ≈ 1s 2. This value is within range of the conditional al., 1993), the right hand term in Equation (8)is given by constraint obtained from momentum balance of growing bubble (Equation (11)). We therefore suggest that it may be feasible to dPg 3P g dR attain conditions for brittle failure for basaltic Plinian eruptions. =− =−3P g γ˙E . (9) dt R dt 3.3. Illustrative model Thus, for a critical failure criterion of γ˙E = γ˙cr , dPm ˙ Within the framework of our hypothesis, requirements for brit- > P g γcr. (10) ˙ = dt tle failure of basaltic magma are (1) elongation rates of γE −1 γ˙cr ∼ 1s ; (2) microcrystallinity at volume fractions, φx, near Basaltic Plinian pyroclasts, for example from Mt. Etna and Mt. the maximum packing fraction; and (3) decompression rates of − Tarawera eruptions have vesicularities of ≈50–80% (Sable et al., about 1MPas 1 or greater. In the previous sections we provided a 2006, 2009). If they represent magmatic conditions near fragmen- scaling analysis to demonstrate that these failure conditions might tation, solubility and equation of state indicate gas pressures of be reached during the Plinian eruption of basaltic magma. A key −1 about 1–10 MPa. Such values of P g and γ˙cr = 1s give a condi- consideration is that these conditions can only be achieved as tional failure constraint for decompression rates of the volume fraction of microlites reaches values that are close to their maximum packing fraction. This would be the case after dPm −1 the magma has risen over a significant distance from the cham- > 1MPas . (11) dt ber, assuming that crystallization is occurring during ascent. The strongly nonlinear effect of crystals on viscosity is also expected An order of magnitude estimate of corresponding eruption rates to drive the erupting magma towards decompression rates that are can be obtained from the momentum balance of the erupting increasing proportionally to the viscosity. Once the estimated de- 2 − 3 magma. Assuming basaltic melt viscosity of 10 10 Pa s, as compression rate required for fragmentation is reached, we expect → ∼ φx φm the viscosity of the crystallizing magma will be ηx the aforementioned nonlinearities to drive the system toward frag- 6 − 7 10 10 Pa s (Fig. 5c). At velocities of several meters per second, mentation within seconds. and to first order neglecting any shear-rate dependence of viscosity This is illustrated in Fig. 6, which provides an example calcu- during flow within the conduit, decompression rates are predom- lation of basaltic magma ascent through a dike, and at discharge inantly due to viscous stresses associated with laminar magma rates similar to the 1886 basaltic Plinian eruption of Mt. Tarawera. flow. Assuming that magma is ascending within a dike-like con- The calculations that produced this illustrative example are based duit of width 2a, as appears to have been the case for Mt. Tarawera on mass and momentum balance equations for magma and mag- in 1886, an estimate for decompression rate can be obtained from matic volatiles, together with crystallization based on the Avrami the force balance of steady flow between parallel plates (Bird et equation and shear-rate dependent magma viscosity that accounts al., 2002) for the presence of crystals and bubbles (see Appendix B for details). The model assumes that the eruption is sustained at approx- 2 dPm 3ηxu −1 imately steady discharge rates for a duration that is longer than the = 1MPas , (12) dt 4a2 magma ascent and crystallization time. In other words, the erup- 102 P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104

−3 tion has adjusted to crystallization. It can be seen that viscosity Nb Bubble number density per unit volume of melt (m ) increases by more than an order of magnitude over a distance of a Oh Ohnesorge number few 100 meters, even though φx only increases by a small fraction p Pressure of gas–pyroclast flow (Pa) as φx → φm. As a consequence, pressure decreases by tens of MPa P g Gas pressure inside bubble (Pa) over the same distance. This leads to a rapid increase in the rate Pm Melt pressure outside bubble (Pa) of bubble growth, the volume fraction of exsolved volatiles, ascent Pe Peclet number − rate, and most importantly to a critical elongation rate. The key Q Magma discharge rate (kg s 1) point of this illustrative example is that the evolution to brittle rp Particle radius (m) failure occurs over a narrow depth interval, due to the nonlin- R Bubble radius (m) ear dependence of viscosity on crystal content and the feedback Re Reynolds number for magma ascent in conduit between viscosity, decompression rate and magma expansion. The Rep Particle Reynolds number system thus evolves invariably toward failure conditions, assuming S Half distance between two adjacent bubbles (m) that discharge rates are sufficiently high and magma crystallizes Sr Radius of the sample-disk in shear experiments (m) sufficiently fast. St Stokes number t Time (s) ◦ 4. Conclusion T Magma temperature ( C) ◦ T K Absolute temperature in Peclet number ( C) − Explosive and dangerous Plinian style eruptions of basaltic u Ascent rate (m s 1) −1 magma have been a subject of debate. Using a novel approach of u f Ascent rate at fragmentation (m s ) −1 combined oscillatory and extensional rheometry, we have investi- u p Velocity of gas-pyroclast flow (m s ) 3 gated the viscoelastic properties of dense particulate suspensions Vmo Initial melt volume surrounding bubble (m ) 3 that are dynamically similar to microcrystalline basaltic magma, Vm Melt volume surrounding bubble (m ) as preserved in basaltic Plinian pyroclasts from Mt Etna, 122 BCE 3 V x Crystal volume in melt surrounding bubble (m ) and Mt Tarawera, 1886. We find that such a suspension fractures − z Depth (m) brittly at elongation rates of about 1s1 or greater, while the γeff Effective extensional strain in tensile experiments particle-free suspending liquid does not fracture at such rates. We ˙ −1 γeff Effective extensional strain rate (s ) in tensile experiments propose that basaltic magma with high volume fractions of crys- −1 γ˙cr Critical rate of deformation for fragmentation (s ) tals may also undergo brittle failure at such elongation rates, which γs Shear strain are otherwise not sufficiently high to fracture crystal-free basalts. −1 γ˙s Shear rate (s ) This study provides a basis for future investigations of viscoelas- −1 γ˙E Extensional rate (s )inmelt+crystal matrix surrounding tic rheology and extensional deformation of expanding multiphase growing bubble − magma during explosive volcanic eruptions. Using scaling analy- σ Surface tension (N m 1) sis and numerical model of magma ascent in volcanic conduit, we φ Volume fraction of vesicles or gas show that high crystal volume fractions and critical elongation rate φ f Gas volume fraction at fragmentation can potentially be reached during Plinian eruptions of crystallizing φm Maximum packing fraction of crystals basaltic magma. φmb Maximum packing fraction of bubbles φp Gas volume fraction in gas-pyroclast flow Notation φx Volume fraction of crystals in the groundmass a Fissure half width (m) η Viscosity of magma (Pa s) b Fissure length (m) ηm Viscosity of basalt melt (Pa s) − − B Gas constant (J mol 1 K 1) ηl Viscosity of suspending liquid (Pa s) + + + c Concentration of dissolved volatiles inside melt (wt%) ηr Ratio of melt crystal bubble viscosity to melt crystal c Concentration of volatiles at the vapor–melt interface (wt%) viscosity R + Ca Capillary number ηx Viscosity of melt crystal matrix (Pa s) −3 d Initial diameter of suspension filament under extension (m) ρ Density of magma (kg m ) 0 − Density of gas (kg m 3) dmid Mid-diameter of suspension filament (m) ρg − −3 D Diffusivity of H O(m2 s 1) ρl Density of suspending liquid (kg m ) H 2 − ˙ −1 3 E Axial stretch rate (s ) ρm Density of basalt melt (kg m ) fm Friction factor τrelax Relaxation time scale (s) − g Gravitational acceleration (m s 2) − F Crystal growth rate (m s 1) G∞ Infinite shear modulus (Pa) Acknowledgements G Storage modulus (Pa) G Loss modulus (Pa) We thank T. Mather for editorial handling of the manuscript, as ∗ G Complex shear modulus (Pa) well as L. Wilson and an anonymous reviewer for their thoughtful h Thickness of the sample in shear experiments (m) comments. This research was not supported by any specific grant − − I Crystal nucleation rate (m 3 s 1) from funding agencies in the public, commercial, or not-for-profit k Boltzmann constant sectors. kv Volumetric factor in Avrami equation Appendix A. Dynamic similarity L0 Initial gap between plates (m) L p Position of upper moving plate (m) ˙ −1 L p Speed of upper moving plate (m s ) To assess the dynamic similarity between our experimental M Mach number conditions and the conditions during volcanic eruptions of crys- −1 M w Molecular weight of water (kg mol ) talline basaltic magma, we define the rheophysical properties of n Flow index the particulate suspension under shear using the dimensionless P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104 103

Peclet number, Pe, Stokes number, St, and particle Reynolds num- magma ascent time scale (Slezin, 2003). Because exposures of ber, Rep , defined as (Stickel and Powell, 2005, and references basaltic Plinian eruption vent, such as at Tarawera, typically in- therein) dicate a few-meter wide and partially back-filled feeder dike (Sable et al., 2009), we assume magma ascent within an ideal- 6πηr3 γ˙ ρ r2 γ˙ ρ r2 γ˙ l p l p p p ized dike. Volatile exsolution and bubble growth cause a small Pe = , Rep = , St = . (A.1) kTK ηl ηl degree of cooling (Proussevitch and Sahagian, 1998). This is ap- − − Here k (= 1.38 × 10 23 JK 1) is the Boltzmann constant, a is the proximately offset by the latent heat of crystallization (Mastin and Ghiorso, 2001), and therefore isothermal conditions are assumed. suspended particle radius, T K is the absolute temperature, ηl is The model comprises the following set of equations: the suspending liquid viscosity, ρl is the density of the suspending liquid, and ρp is the particle density. Pe is the ratio between d(ρu) hydrodynamic forces and forces of Brownian motion, Rep is the ra- = 0(B.1) dz tio between inertial and viscous forces, and St is the ratio between −1 the characteristic time scales of particle motion and shear defor- dPm fm = − ρ g − ρu2 1 − M2 (B.2) mation of the suspension. For the oscillatory shear experiments, dz 4a the shear rate can be calculated as ωγ (Mezger, 2006), where ω s dR R 2σ is the angular frequency and γs is the shear strain or strain ampli- = P − P − (B.3) − − g m tude (Fig. 3). Thus for a range of shear rates (10 4 − 101 s 1), with dz 4ηxu R − − particle density of 2500 kg m 3, liquid density of ≈1000 kg m 3, − dc =−4π RDH (c cR ) + dφx Vmo average particle radius of ≈ 6 μm and liquid viscosity of 100 Pa s, c (B.4) dz uVm dz Vm we find that Pe 1, St 1 and Rep 1. This indicates that the dPg 3BTρm D H (c − cR ) 3P g dR experimental conditions were non-Brownian, with a strong liquid- = − (B.5) 2 particle coupling, whereas the inertial forces were negligible. For dz R M w u R dz basaltic magma with 10–100 μm microlites, erupting at ascent ve- dφx − = 3 3 − 3 4 locities of about 1ms 1 in a volcanic conduit of about 10–100 m (1/u)4kv IF t exp kv IF t (B.6) dz in radius, or in an idealized dike of 1–10 m width, an order of −2 − − φ magnitude estimate of shear rates is 10 3–100 s 1. This corre- = − x ˙ n−1 ηx ηm 1 γs (B.7) sponds to Pe 1, St 1 and Re 1, which are similar to φm p the experimental conditions. −4/5 − 1 − 12 η2Ca2 φ φmb For deformation under extension the ratio of the viscous to cap- η 5 r = 1 − (B.8) r − 12 2 φ illary and inertial forces is defined by the Ohnesorge number, Oh, 1 5 Ca mb − as (Villermaux, 2012) φ 2 = − x ˙ n−1 η η ηmηr 1 γs (B.9) Oh = , (A.2) φm ρld0σ ρg φ f u f = ρgpφpu p (B.10) where η is the viscosity of particulate suspension, d is the initial 0 (1 − φ )u = (1 − φ )u (B.11) diameter of the suspension filament, and σ is the surface tension. f f p p ≈ − + + − 2 + 2 For an initial diameter of d0 0.005 m for the filament in the pm(1 φ f ) p g φ f ρm(1 φ f )u f ρg φ f u f ∼ 4 − 5 elongation experiments, a suspension viscosity of 10 10 Pa s, 2 2 − − − = p + ρ (1 − φ )u + ρ φ u . (B.12) density of ≈ 103 kg m 3, and surface tension of ≈ 10 2 Nm 1, we m p p gp p p estimate that Oh 1. This suggests that the capillary and inertial Equation (B.1)is the mass balance for magma in the dike. Equa- forces were negligible in the experiments. This is similar to mag- tions (B.2) and (B.3)are the momentum balance equations for matic conditions where Oh is also 1, assuming a melt and crys- magma flow in the dike and for bubble inside the magma, respec- tal bearing bubble wall thickness of about 1–10 μm, surface ten- tively. Equation (B.4)represents the diffusion of volatiles inside −1 −1 5 6 sion of about 10 Nm , and viscosity of about 10 to 10 Pa s. bubbles, where the second term on the right hand side represents Micro-crystals in basaltic Plinian pyroclasts (e.g., Fig. 1) are of- the rate of change in dissolved H2O concentration due to micro- ten bi- or poly-modal in size and shape, as opposed to the spher- lite formation and subsequent reduction in melt volume, Vmo − V x, ical particles in our analog experiments. This effects the relation- where V x (= Vmoφx) is the total volume of crystals, and φx is the ship between φx and viscosity. It can, however, be accounted for by volume fraction of crystals. The diffusion of H2O into the bubble normalizing φx relative to the maximum packing fraction φm. The is calculated using the mean field approximation (e.g., Toramaru, maximum packing fraction, φm, may be 0.6or greater (Moitra and 1995). Equation (B.5)represents the change in gas pressure inside Gonnermann, 2015) for a distribution of large, high aspect ratio bubbles (Proussevitch et al., 1993). Equation (B.6)is the Avrami crystals and very small high aspect ratio or equigranular crystals, equation (Cashman, 1993) and represents the rate of change in as can be observed in Fig. 1. Therefore, the microlite content of crystal volume fraction of the melt. The shear-rate dependent ≤ ≤ 0.6 φx 0.9in the basaltic Plinian pyroclasts from Mt. Etna 122 change in magma viscosity, due to the presence of crystals and BCE and Mt. Tarawera 1886 (Sable et al., 2006, 2009) are likely bubbles, is included through Equation (B.7)(Moitra and Gonner- near the maximum packing. To this end, our experiments were mann, 2015) and Equation (B.8)(Pal, 2003), respectively. ηx is the conducted using a suspension with (0.55) close to (0.56). φx φm melt+crystal viscosity, and ηr is the ratio of melt+bubble+crystal viscosity to ηx. Thus, the bubble and crystal bearing magma Appendix B. Modeling of eruptive magma ascent viscosity, η in Equation (B.9), is based on the “effective medi- um” approach (e.g. Truby et al., 2015). For the post-fragmentation The model represents the ascent of crystallizing basaltic magma gas-pyroclast flow, mass balance of the gas phase and suspended during eruptions and includes growth of bubbles, assumed to pyroclasts are solved using equations (B.10) and (B.11), respec- be monodisperse, spherical and homogeneously distributed (e.g., tively. Equation (B.12)represents the momentum balance of the Proussevitch and Sahagian, 1998). The model is assumed time- gas-pyroclast flow as a whole with a modified friction factor (e.g., invariant, because the eruption duration was longer than the Melnik et al., 2005; Koyaguchi et al., 2010). The viscosity of the 104 P. Moitra et al. / Earth and Planetary Science Letters 500 (2018) 97–104 gas-pyroclast flow varies as a function of gas and pyroclast volume Houghton, B.F., Gonnermann, H.M., 2008. Basaltic explosive volcanism: constraints fractions (Mastin and Ghiorso, 2000). from deposits and models. Chem. Erde, Geochem. 68 (2), 117–140. In the above equations u = Q /(ρ2ab) is the magma ascent Hui, H., Zhang, Y., 2007. Toward a general viscosity equation for natural anhydrous and hydrous silicate melts. Geochim. Cosmochim. Acta 71 (2), 403–416. velocity, where 2ab is the cross-sectional area of the dike. Further- Kerrick, D., Jacobs, G., 1981. A modified Redlich–Kwong equation for H O, CO , = − + = 2 2 more, ρ ρm(1 φ) ρg φ is the density of magma with φ and H2O–CO2 mixtures at elevated pressures and temperatures. Am. J. Sci. 281, R3/S3 the gas volume fraction calculated from the bubble radius, 735–767. Koyaguchi, T., Suzuki, Y.J., Kozono, T., 2010. Effects of the crater on eruption column R, and the half distance between two adjacent bubbles, S. fm (= dynamics. J. Geophys. Res., Solid Earth 115 (B7). 24/Re + 0.01) is the friction factor for the idealized dike (Wilson Mader, H.M., Llewellin, E.W., Mueller, S.P., 2013. The rheology of two-phase magmas: = and Head, 1981) and Re ρu2a/η is the Reynolds number. The a review and analysis. J. Volcanol. Geotherm. Res. 257, 135–158. rate of shear, γ˙s, used to calculate the shear-rate dependent vis- Mastin, L.G., Ghiorso, M.S., 2000. A Numerical Program for Steady-State Flow of cosity of multiphase magma is defined as u/a in the model, where Magma–Gas Mixtures Through Vertical Eruptive Conduits. USGS Open-File Re- u is the magma ascent rate and a is the dike half width. M is port 00-209, pp. 1–61. Mastin, L.G., Ghiorso, M.S., 2001. Adiabatic temperature changes of magma–gas mix- the Mach number (e.g., Melnik et al., 2005). The solubility of H2O tures during ascent and eruption. Contrib. Mineral. Petrol. 141 (3), 307–321. is calculated as function of pressure, temperature and melt com- McKinley, G.H., Sridhar, T., 2002. Filament-stretching rheometry of complex fluids. position (Dixon, 1997). D H is the diffusivity of H2O (Zhang et al., Annu. Rev. Fluid Mech. 34 (1), 375–415. Melnik, O., Barmin, A.A., Sparks, R.S.J., 2005. Dynamics of magma flow inside vol- 2007). Vmo = 1/Nb is the initial melt volume surrounding one bub- canic conduits with bubble overpressure buildup and gas loss through perme- ble and N is the bubble number density per unit volume of melt. b able magma. J. Volcanol. Geotherm. Res. 143 (1–3), 53–68. The gas density, ρg , is calculated as a function of P g using a mod- Mezger, T.G., 2006. The Rheology Handbook: For Users of Rotational and Oscillatory ified Redlich–Kwong equation of state (Kerrick and Jacobs, 1981). Rheometers. Vincentz Network GmbH & Co KG. The complete description of symbols is provided under ‘Notation’. Moitra, P., Gonnermann, H., 2015. Effects of crystal shape-and size-modality on magma rheology. Geochem. Geophys. Geosyst., 1–26. Initial conditions for Fig. 6 are dissolved H2O content of 3wt% Moitra, P., Gonnermann, H.M., Houghton, B.F., Giachetti, T., 2013. Relating vesicle (Gamble et al., 1990), corresponding H O saturation pressure 2 shapes in pyroclasts to eruption styles. Bull. Volcanol. 75 (2), 1–14. −6 (Dixon, 1997), initial bubble radius of 10 μm, and zero ini- Namiki, A., Manga, M., 2008. Transition between fragmentation and permeable out- tial volume fraction of crystals. We assumed a straight-sided dike gassing of low viscosity magmas. J. Volcanol. Geotherm. Res. 169 (1–2), 48–60. with the exit condition at the surface as choked, defined by Mach Namiki, A., Tanaka, Y., 2017. Oscillatory rheology measurements of particle- and bubble-bearing fluids: solid-like behavior of a crystal-rich basaltic magma. Geo- number of M = 1 (Woods and Koyaguchi, 1994). It was matched phys. Res. Lett. 44 (17), 8804–8813. by varying the magma discharge rate, Q . The system of Equa- Pal, R., 2003. Rheological behavior of bubble-bearing magmas. Earth Planet. Sci. ® tions (B.1)–(B.6)was coupled and solved using MATLAB solver Lett. 207 (1–4), 165–179. ode15s (Shampine and Reichelt, 1997). Papale, P., 1999. Strain-induced magma fragmentation in explosive eruptions. Na- ture 397 (6718), 425–428. Parfitt, E.A., 2004. A discussion of the mechanisms of explosive basaltic eruptions. J. References Volcanol. Geotherm. Res. 134 (1), 77–107. Proussevitch, A.A., Sahagian, D.L., 1998. Dynamics and energetics of bubble growth Applegarth, L., Tuffen, H., James, M., Pinkerton, H., 2013. Degassing-driven crystalli- in magmas: analytical formulation and numerical modeling. J. Geophys. Res., sation in basalts. Earth-Sci. Rev. 116, 1–16. Solid Earth 103 (B8), 18223–18251. Arzilli, F., Agostini, C., Landi, P., Fortunati, A., Mancini, L., Carroll, M., 2015. Plagio- Proussevitch, A.A., Sahagian, D.L., Anderson, A.T., 1993. Dynamics of diffusive bubble clase nucleation and growth kinetics in a hydrous basaltic melt by decompres- growth in magmas: isothermal case. J. Geophys. Res. 98 (B12), 22283. sion experiments. Contrib. Mineral. Petrol. 170 (5–6), 55. Sable, J.E., Houghton, B.F., Del Carlo, P., Coltelli, M., 2006. Changing conditions of Bird, R., Stewart, W., Lightfoot, E., 2002. Transport Phenomena, second edition. John magma ascent and fragmentation during the Etna 122 BC basaltic Plinian erup- Wiley & Sons, Inc. tion: evidence from clast microtextures. J. Volcanol. Geotherm. Res. 158 (3–4), Brugger, C.R., Hammer, J.E., 2010. Crystallization kinetics in continuous decompres- 333–354. sion experiments: implications for interpreting natural magma ascent processes. Sable, J.E., Houghton, B.F., Wilson, C.J.N., Carey, R.J., 2009. Eruption mechanisms J. Petrol. 51 (9), 1941–1965. during the climax of the Tarawera 1886 basaltic Plinian eruption inferred Campagnola, S., Romano, C., Mastin, L., Vona, A., 2016. Confort 15 model of con- from microtextural characteristics-of the deposits. In: Special Publications of duit dynamics: applications to Pantelleria Green Tuff and Etna 122 BC eruptions. International Association of Volcanology and Chemistry of the Earth’s Interior, Contrib. Mineral. Petrol. 171 (6), 1–25. pp. 129–154. Cashman, K.V., 1993. Relationship between plagioclase crystallization and cooling Shampine, L.F., Reichelt, M.W., 1997. The MATLAB ODE suite. SIAM J. Sci. Comput. 18 rate in basaltic melts. Contrib. Mineral. Petrol. 113 (1), 126–142. (1), 1–22. Coltelli, M., Carlo, P.D., Vezzoli, L., 1998. Discovery of a Plinian basaltic eruption of Shea, T., Hammer, J.E., 2013. Kinetics of cooling- and decompression-induced Roman age at Etna volcano, Italy. Geology 26 (12), 1095–1098. crystallization in hydrous mafic-intermediate magmas. J. Volcanol. Geotherm. Costantini, L., Houghton, B.F., Bonadonna, C., 2010. Constraints on eruption dy- Res. 260, 127–145. namics of basaltic explosive activity derived from chemical and microtextural Slezin, Y.B., 2003. The mechanism of volcanic eruptions (a steady state approach). J. study: the example of the Fontana Lapilli Plinian eruption, Nicaragua. J. Vol- Volcanol. Geotherm. Res. 122 (1–2), 7–50. canol. Geotherm. Res. 189 (3–4), 207–224. Stickel, J.J., Powell, R.L., 2005. Fluid mechanics and rheology of dense suspensions. Dingwell, D.B., Webb, S.L., 1989. Structural relaxation in silicate melts and non- Annu. Rev. Fluid Mech. 37, 129–149. newtonian melt rheology in geologic processes. Phys. Chem. Miner. 16 (5), Sumita, I., Manga, M., 2008. Suspension rheology under oscillatory shear and its 508–516. geophysical implications. Earth Planet. Sci. Lett. 269 (3), 468–477. Dixon, J.E., 1997. Degassing of alkalic basalts. Am. Mineral. 82 (3–4), 368–378. Toramaru, A., 1995. Numerical study of nucleation and growth of bubbles in viscous Gamble, J.A., Smith, I.E., Graham, I.J., Kokelaar, B.P., Cole, J.W., Houghton, B.F., Wilson, magmas. J. Geophys. Res., Solid Earth 100 (B2), 1913–1931. C.J., 1990. The petrology, phase relations and tectonic setting of basalts from the Truby, J., Mueller, S., Llewellin, E., Mader, H., 2015. The rheology of three-phase sus- Taupo Volcanic Zone, New Zealand and the Kermadec Island Arc–Havre Trough, pensions at low bubble capillary number. Proc. R. Soc. Lond. A, Math. Phys. Eng. SW Pacific. J. Volcanol. Geotherm. Res. 43, 253–270. Sci. 471 (2173), 20140557. Giordano, D., Dingwell, D., 2003. Viscosity of hydrous Etna basalt: implications for Valentine, G., Gregg, T., 2008. Continental basaltic volcanoes—processes and prob- Plinian-style basaltic eruptions. Bull. Volcanol. 65, 8–14. lems. J. Volcanol. Geotherm. Res. 177 (4), 857–873. Goepfert, K., Gardner, J.E., 2010. Influence of pre-eruptive storage conditions and Villermaux, E., 2012. The formation of filamentary structures from molten silicates: volatile contents on explosive Plinian style eruptions of basic magma. Bull. Vol- Pele’s hair, angel hair, and blown clinker. C. R., Méc. 340 (8), 555–564. canol. 72 (5), 511–521. White, E.E.B., Chellamuthu, M., Rothstein, J.P., 2010. Extensional rheology of a shear- Gonnermann, H.M., 2015. Magma fragmentation. Annu. Rev. Earth Planet. Sci. 43, thickening cornstarch and water suspension. Rheol. Acta 49 (2), 119–129. 431–458. Wilson, L., Head, W.J., 1981. Ascent and eruption of basaltic magma on the earth Hammer, J.E., Rutherford, M.J., 2002. An experimental study of the kinetics of and moon. J. Geophys. Res. 86, 2971–3001. decompression-induced crystallization in silicic melt. J. Geophys. Res., Solid Woods, A.W., Koyaguchi, T., 1994. Transitions between explosive and effusive erup- Earth 107 (B1). tions of silicic magmas. Nature 370 (6491), 641–644. Heymann, L., Peukert, S., Aksel, N., 2002. Investigation of the solid–liquid transition Zhang, Y., Xu, Z., Zhu, M., Wang, H., 2007. Silicate melt properties and volcanic erup- of highly concentrated suspensions in oscillatory amplitude sweeps. J. Rheol. 46 tions. Rev. Geophys. 45 (4), 1–27. (1), 93–112.