https://doi.org/10.1130/G47045.1

Manuscript received 25 September 2019 Revised manuscript received 17 December 2019 Manuscript accepted 26 December 2019

© 2020 Geological Society of America. For permission to copy, contact [email protected].

A critical magma chamber size for volcanic eruptions Meredith Townsend1 and Christian Huber2 1Department of Earth Sciences, University of Oregon, 100 Cascade Hall, Eugene, Oregon 97403, USA 2Department of Earth, Environmental, and Planetary Sciences, Brown University, 324 Brook Street, Providence, Rhode Island 02906, USA

ABSTRACT Flegrei, Laguna del Maule, Aso, and Santorini We present a model for a coupled magma chamber–dike system to investigate the condi- (Townsend et al., 2019). tions required to initiate volcanic eruptions and to determine what controls the size of erup- One aspect missing from existing magma tions. The model combines the mechanics of dike propagation with internal chamber dynam- chamber models is dike propagation. For many ics including crystallization, volatile exsolution, and the elastic response of the magma and volcanoes, eruptions initiate through fssures fed surrounding crust to pressure changes within the chamber. We fnd three regimes for dike by dikes, so eruptions depend on the mechanics growth and eruptions: (1) below a critical magma chamber size, eruptions are suppressed of the dike and whether it can reach the surface. because chamber pressure drops to lithostatic before a dike reaches the surface; (2) at an Coupled dike-chamber models such as those of intermediate chamber size, the erupted volume is less than the dike volume (“dike-limited” Segall et al. (2001), Rivalta and Segall (2008), eruption regime); and (3) above a certain chamber size, dikes can easily reach the surface Rivalta (2010), and Buck et al. (2006) were de- and the erupted volume follows a classic scaling law, which depends on the attributes of the veloped for diking events at basaltic rift zones magma chamber (“chamber-limited” eruption regime). The critical chamber volume for an like Kilauea (Hawaii, USA) and Iceland. While eruption ranges from ∼0.01 km3 to 10 km3 depending on the water content in the magma, some of these models address the infuence of depth of the chamber, and initial overpressure. This implies that the frst eruptions at a vol- volatiles on magma compressibility, the full cou- cano likely are preceded by a protracted history of magma chamber growth at depth, and pling between pressure change during growth of that the crust above the magma chamber may have trapped several intrusions or “failed an intrusion and phase changes in the magma are eruptions.” Model results can be combined with feld observations of erupted volume, pres- typically neglected, and none of these models sure, and crystal and volatile content to provide tighter constraints on parameters such as have been adapted to understand more silicic, the eruptible chamber size. water-rich arc volcanoes. In this paper, we pres- ent a new dike-chamber model that considers

INTRODUCTION ­signifcantly infuence Ver by increasing β by both internal magma chamber dynamics and Determining what allows magma to reach more than an order of magnitude (Huppert and propagation of dikes, with the goal of under- the surface and what controls the amount and Woods, 2002). This model points to the impor- standing how these processes affect the occur- rate of volcanic output has profound implica- tance of internal magma chamber processes rence and size of eruptions at silicic volcanoes. tions for the assessment of volcanic hazards. on the erupted volume, like volatile exsolution Eruption rates, styles, and sizes are infuenced concurrent with the eruption. However, it ne- MODEL by overpressure in crustal magma chambers as glects some aspects of magma dynamics (e.g., The model we present combines the magma well as the dynamics of ascent through dikes or crystallization caused by exsolution during de- chamber lumped model of Degruyter and Huber conduits that connect magma chambers to the compression) as well as the processes involved (2014) with the dike-chamber model of Segall surface (e.g., Cashman, 2004; Gonnermann and in the propagation of a dike and transport of et al. (2001). The chamber is a spherical volume Manga, 2007; Moran et al., 2011; Wong et al., magma to the surface. of eruptible magma containing silicate melt, 2017). Huppert and Woods (2002) presented an The scaling law of Huppert and Woods crystals, and dissolved and exsolved water. The idealized model for effusive eruptions fed by (2002) has been supported by subsequent mod- crystal volume fraction in the chamber evolves chamber overpressure; assuming a fxed conduit els that take into account conduit physics (e.g. according to a melting curve parameterized for geometry, they found a scaling law for the total Anderson and Segall, 2011; Melnik and Sparks, dacite after Huber et al. (2009), and the water

volume of erupted material (Ver): 2005). Crystallization and volatile exsolution solubility follows the parameterization of Dufek in the conduit affect the mass fux and behavior and Bergantz (2005) and Zhang (1999). The Ver=VP ch β∆ , (1) of an eruption, but the fnal amount of erupted chamber resides in a colder viscoelastic crust, material remains a function of magma chamber but because we focus on single eruptions that

where Vch is chamber volume, β is the effec- conditions and is well approximated by the scal- occur on short time scales, the crust is essen- tive compressibility of the system (host rocks ing law. Comparison of predicted and observed tially elastic and the thermodynamic state of the and magma), and ΔP is the total pressure drop erupted volumes lends further support for the chamber is affected primarily by magma with- in the chamber during the eruption. With- scaling law, e.g. Mount St. Helens (Anderson drawal leading to heat loss and decompression. in this framework, magmatic volatiles may and Segall, 2011; Mastin et al., 2008), Campi Recharge rates are generally much slower than

CITATION: Townsend, M., and Huber, C., 2020, A critical magma chamber size for volcanic eruptions: Geology, v. 48, p. XXX–XXX, https://doi.org/10.1130/ G47045.1

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Downloaded from https://pubs.geoscienceworld.org/gsa/geology/article-pdf/doi/10.1130/G47045.1/4940897/g47045.pdf by Brown University user on 07 February 2020 eruption rates, so recharge is neglected here. The dike is a vertically oriented half-­ eruptive fssure, which we set here to be 0.5a

The change in mass in the chamber with time ellipsoid, with half-length a and half-height b. (we expand on qerupt in the Data Repository, and

(dMc/dt) is balanced by mass fux into the dike, Initially, the dike dimensions are small com- show that changing this fraction does not impact

qdike, which we assume is proportional to the pared to the chamber depth d, with a0 = 0.02d the results). We do not consider erosion of the

pressure difference between the chamber, Pc, and b0 = a0/2. Dike growth is governed by the fssure or the development of more cylindrical

and dike, Pd: pressure gradients available to drive magma shapes, which may impact the evolution of the with viscosity η to fow laterally and vertically eruptive fux (e.g., Wadge, 1981; Aravena et al., dMc through the fracture (Segall et al., 2001): 2018). =−qPdike =−γ ()cdP . (2) dt Initially, the dike (Pd) and chamber (Pc) pres- 2 da δ  ∆Pd  sure Pd(0) = Pc(0) = Plit + Pcrit, where Plit is litho- =   , (5) The constant γ represents the “conductivity” dt 3η  a  static pressure and Pcrit is the magma overpres- between the chamber and dike and depends on sure required to initiate a dike. Pcrit depends on the geometry of their connection. For simplic- 2 factors such as the strength of the crust, the prev- db δ  ∆Pd  ity, the results presented here are calculated =−∆∆ρρgP,,in which d > ∆ gb alence of preexisting fractures, and the shape of dt 3η  b  with γ = 1 kg/Pa·s; we show in the GSA Data the chamber and whether stress is concentrated 1 db Repository that this value only modestly affects otherwise = 0, (6) along the chamber walls (Gregg et al., 2012; the time-integrated results. dt Jellinek and DePaolo, 2003; Rubin, 1995). To

Conservation of water mass, Mw, and en- account for these possible variations, we ex-

thalpy, H, leads to: where δ is the maximum aperture of the dike, plore a range of Pcrit from 10 to 50 MPa. As the

ΔPd is the overpressure at the dike inlet, and dike grows, Pd drops, facilitating fow from the dMw Δρ is the density difference between the bulk chamber to the dike. The model terminates when =−qdike ()χχde+ , (3) dt magma and crust, and g is gravity. In this work, Pc = Pd (there is no longer a pressure gradient), magma density evolves as a function of crystal or when there is no longer enough overpressure dH and volatile fraction, but is always positively at the dike inlet to prevent freezing, i.e., when =−cTqQdike− cond , (4) dt buoyant. The maximum aperture of the dike, δ, Pd < Plit + 2 MPa (Rubin, 1995). This criterion is is governed by elastic deformation of the host a simplifed way to approximate effects of heat

where χd and χe are dissolved and exsolved wa- rock: loss in the dike, because we do not explicitly ter mass fractions in the magma, respectively, model thermodynamic processes in the dike. c is heat capacity, T is temperature, and Q is 21bP()− ν ∆ d Magma density and gas fraction evolve in the cond δ = , (7) the rate of heat fow from the chamber into the µEk() chamber, which sets infow conditions for the surrounding crust. The frst term in Equation 4 dike, but once the magma enters the dike its represents heat advected out of the chamber by where μ and v are the elastic stiffness and Pois- properties are fxed. By neglecting vesiculation fow into the dike, and the second term repre- son’s ratio of the crust, respectively, and E(k) in the dike, we may underestimate the total vol- sents heat conducted from the chamber into the is the elliptic integral of the second kind with ume erupted, so these results should be consid- crust (effectively negligible on the time scale of modulus k (Segall et al., 2001). ered minimum bounds. In addition, the magma most eruptions). If the dike reaches the surface (b = d), we ac- viscosity η is constant, and the thermal effects of

count for the eruptive fux qerupt when conserving viscosity are not considered. Thus in our results

mass in the dike (Md): (see the Data Repository), and in similar models 1GSA Data Repository item 2020120, expanded (Segall et al., 2001; Anderson and Segall, 2011), governing equations, an analysis of the effects of dMd η only affects the time evolution of an eruption =−qq, (8) magma viscosity, dike-chamber conductivity, and fs- dt dike erupt and not the total erupted volume. sure length, and a derivation of the effective magma Equations 2–6 and 8, expanded on in the compressibility, is available online at http://www.geo- society.org/datarepository/2020/, or on request from where qerupt depends on the vertical pressure gra- Data Repository, compose a set of six ordinary [email protected]. dient, the dike aperture, and the length of the differential equations for Pc, Pd, T, exsolved wa-

Figure 1. Erupted magma volumes (left) and erupted mass (right) as a function of chamber volume for different initial volume fractions of exsolved

volatiles (εg) and differ- ent chamber depths. Blue dots represent 4 km depth, red dots represent 10 km depth. Gray shaded areas correspond to saturated cases (initial exsolved volatile fraction is >0; “wet”) or undersaturated cases (exsolved volatile fraction = 0 and no exsolu- tion takes place during the eruption; “dry”). Magma overpressure required to

initiate dike, Pcrit, = 10 MPa.

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Downloaded from https://pubs.geoscienceworld.org/gsa/geology/article-pdf/doi/10.1130/G47045.1/4940897/g47045.pdf by Brown University user on 07 February 2020 Figure 2. Eruption volume,

Ver (circles), and dike volume (diamonds) versus chamber volume,

Vch, scaled by magma com- pressibility, β, and initial chamber overpressure, ΔP. Colors correspond to different depths and water contents. In regime 1 (left-

hand side), Ver = 0, and volume of dikes follows the scaling of Huppert and Woods (2002; black line). In regime 2 (“dike- limited eruption regime”),

Ver = 0 but is signifcantly less than that indicated by the scaling law. In regime 3 (“chamber-limited erup-

tion regime”), Ver follows the scaling law and is signifcantly greater than dike volume.

ter content εg, and dike height b and length a. of β). Because volatiles exsolve more readily the scaled chamber volume VchβΔP confrms that

We solve the equations in MATLAB software at lower pressures, we also see greater Ver at the primary infuence of volatiles is to change using the ode15s solver. Results were validated shallower depths (Fig. 1). However, this is also β (Fig. 2). by comparing simplifed cases with the original partly due to dikes being initially closer to the The main effect of the dike is to limit the

model results of Degruyter and Huber (2014) surface and thus consuming less of the with- amount of magma that erupts. Only when Vch >

and Segall et al. (2001). Eruption volumes Ver drawn magma to reach it. Vcrit can eruptions occur. Vcrit can vary between are calculated by integrating q over time, and Figure 2 compares V to the scaling relation- ∼0.01 km3 and 10 km3 depending on chamber erupt π er the dike volumes are calculated by Vd = abδ. ship by Huppert and Woods (2002) (Equation depth, water content, and initial overpressure 3 Because we do not model conduit physics like 1). Three regimes are apparent: below a criti- Pcrit (Fig. 3). In general, Vcrit is greater for deeper

bubble growth, Ver is akin to a dense-rock equiv- cal chamber volume (Vch < Vcrit), no eruptions chambers, in part because of the requirement for

alent (DRE). occur, but the volume of dikes Vd follows the dikes to reach the surface and in part because of

scaling law (Equation 1). At intermediate Vch, the increased solubility of volatiles. In Figure 3,

RESULTS eruptions occur, but the majority of magma leav- we see that Vcrit in the driest and wettest systems

The fnal volume of magma erupted, Ver, is ing a chamber is stored in the dike. Beyond this (3 and 7 wt% H2O) is only weakly sensitive to

primarily a function of the chamber volume Vch. regime (Vch >> Vcrit), Ver is closely approximated depth. In the dry case, magma never reaches vola-

Figure 1 shows Ver versus Vch for simulations by the scaling law VchβΔP. In this regime, Vd << tile saturation to form a compressible fuid phase;

run at different depths (4–10 km) and initial Ver and dikes do not pose a signifcant limit on the depth dependence of Vcrit is solely a refection

exsolved water contents (εg = 0% [and under- eruptions. Moreover, the collapse of calculated of the requirement for dikes to reach the surface.

saturated during dike growth], 1%, and 10% Ver for different amounts of water with respect to Similarly, in the wettest case, magma always con-

volume fractions). When Vch is less than a criti-

cal volume, Vcrit, eruptions do not occur because

pressure Pc or Pd drops below a critical threshold

(either Pc = Pd or Pd < Plit + 2 MPa) before dikes

reach the surface. Once Vch > Vcrit, Ver increases

nonlinearly with Vch before increasing linearly at greater Vch (Fig. 1). Figure 3. Critical cham-

Eruption volume Ver is sensitive to the pres- ber volume, Vcrit, to ence of an exsolved volatile phase. While there initiate eruptions at the is little difference in V between magmas with surface as function of er chamber depth (x-axis), εg = 1% and εg = 10%, magma chambers with water content (red = 3 εg = 1% can produce eruptions more than two wt%, green = 5 wt%, orders of magnitude greater than chambers con- blue = 7 wt%), and initial taining no exsolved volatiles (Fig. 1). The jump chamber overpressure, P (circles 10 MPa, in V between dry and wet magmas occurs be- crit = er squares = 50 MPa). cause the exsolution of a low-density volatile phase in wet magmas maintains greater pressure in the chamber, effectively increasing the com- pressibility β by more than an order of magni- tude (see the Data Repository for the derivation

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Downloaded from https://pubs.geoscienceworld.org/gsa/geology/article-pdf/doi/10.1130/G47045.1/4940897/g47045.pdf by Brown University user on 07 February 2020 Figure 4. Chamber volumes (colored values) as function of eruption size (x-axis), depth (y-axis), and volatile saturation state (left, under- saturated; right, saturated). Black boxes are examples of inferred chamber volume from post-caldera Santorini (Greece) (Parks et al., 2012; Degruyter et al., 2016); Monte Nuovo eruption of Campi Flegrei (Italy) (Di Vito et al., 1987; Forni et al., 2018); 2011 eruption of Cordón Caulle (Chile) (Singer et al., 2008; Jay et al., 2014); Laguna del Maule (Chile) throughout the Holocene (Hildreth et al., 2010; Singer et al., 2014; Andersen et al., 2017); and the twin eruptions at Quizapu (Chile) (Hildreth and Drake, 1992; Ruprecht et al., 2012). White space covers

parameters for which no eruptions are predicted to occur (chamber volume [Vch] smaller than critical chamber volume to initiate eruption at surface [Vcrit]).

ACKNOWLEDGMENTS tains an exsolved fuid phase regardless of depth; logic and petrologic data (P-T-X-volatiles Funding for this work was provided in part by the U.S. Vcrit in these cases is also controlled by dike propa- and DRE ­volume) to infer the size of the National Science Foundation grant EAR-1760004 to gation. For intermediate water content (5 wt%), ­underlying chamber­ that fed a particular erup- Huber and Townsend. All data from the volcanic sys- tems used to produce Figure 4 come from published we see a greater spread of Vcrit with depth, refect- tion. In Figure 4, we compile examples from ing the combined infuence of dike propagation several well-studied volcanic episodes such literature cited in the references. MATLAB code for the model will be made available upon request. We and pressure-dependent water solubility. Finally, as the Holocene phase at Laguna del Maule thank Paul Segall and two anonymous reviewers for Vcrit also depends on the initial chamber overpres- (Chile) and the post-caldera eruptions at San- constructive reviews that improved the manuscript.

sure Pcrit (Fig. 3). For a given water content and torini. Based on the average size of eruptions 3 depth, increasing Pcrit from 10 to 50 MPa reduces at Laguna del Maule, ∼0.1–1 km (Hildreth REFERENCES CITED Vcrit by almost one order of magnitude (Fig. 3). et al., 2010; Singer et al., 2014), and the likely Andersen, N.L., Singer, B.S., Jicha, B.R., Beard, B.L., This is in agreement with Segall et al. (2001), water-saturated conditions of the erupted prod- Johnson, C.M., and Licciardi, J.M., 2017, Pleisto- cene to Holocene growth of a large upper crust- who found that the ratio of magmastatic head to ucts (Andersen et al., 2017), Vch inferred from initial overpressure controlled the ability of dikes the model is between 5 and 20 km3 (Fig. 4), al rhyolitic magma reservoir beneath the active ∼ Laguna del Maule volcanic feld, central Chile: to reach the surface, with greater overpressures in reasonable agreement with estimates from Journal of Petrology, v. 58, p. 85–114, https://doi​ leading to dikes that erupt more easily. a gravity survey by Miller et al. (2017). Al- .org/10.1093/petrology/egx006. though the eruptions of post-caldera Santorini Anderson, K., and Segall, P., 2011, Physics-based DISCUSSION AND CONCLUSIONS are smaller on average, at ∼0.001–0.02 km3 models of ground deformation and extrusion rate at effusively erupting volcanoes: Journal of The existence of a critical chamber size (Parks et al., 2012), the inferred Vch of ∼5–10 Geophysical Research, v. 116, B07204, https:// needed to produce eruptions raises questions km3 is on a similar scale (Fig. 4), refecting doi.org/10.1029/2010JB007939. about how volcanoes develop. For many silic- the dry post-caldera conditions in the chamber Aravena, A., Cioni, R., de’ Michieli Vitturi, M., ic systems, subvolcanic magma chambers are at Santorini­ compared to Laguna del Maule Pistolesi, M., Ripepe, M., and Neri, A., 2018, thought to grow at depths of 7–10 km or 2 (Degruyter et al., 2016). Evolution of conduit geometry and eruptive pa- ∼ ∼ rameters during effusive events: Geophysical kbar (Huber et al., 2019); prior to signifcant Although we focus on eruption potential and Research Letters, v. 45, p. 7471–7480, https:// in situ crystallization and fractionation, mag- eruption size, the dike-chamber model presented doi​.org/10.1029/2018GL077806.

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