Chapter 4

Dynamics of magma ascent in the volcanic conduit Helge M. Gonnermann and Michael Manga

Overview ascent rates, even in low viscosity magmas, melt and exsolved gas remain coupled allowing for This chapter presents the various mechanisms rapid acceleration and hydrodynamic fragmen- and processes that come into play within the tation in hawaiian eruptions. volcanic conduit for a broad range of effu- sive and dry explosive volcanic eruptions. Decompression during magma ascent causes 4.1 Introduction volatiles to exsolve and form bubbles contain- ing a supercritical fluid phase. Viscous magmas, In the broadest sense, volcanic eruptions are such as rhyolite or crystal-rich magmas, do not either effusive or explosive. During explosive allow bubbles to ascend buoyantly and may also eruptions magma fragments and eruption inten- hinder bubble growth. This can lead to signifi- sity is ultimately related to the fragmentation cant gas overpressure and brittle magma frag- mechanism and associated energy expenditure mentation. During fragmentation in vulcanian, (Zimanowski et al. 2003). If the cause of frag- subplinian, and plinian eruptions, gas is released mentation is the interaction of hot magma with explosively into the atmosphere, carrying with external water, the ensuing eruption is called it magma fragments. Alternatively, high vis- phreatomagmatic. Eruptions that do not involve cosity may slow ascent to where permeable external water are called “dry,” in which case outgassing through the vesicular and perhaps the abundance and fate of magmatic volatiles,

fractured magma results in lava effusion to prod- predominantly H2O and CO2, as well as magma uce domes and flows. In low viscosity magmas, rheology and eruption rate are the dominant typically basalts, bubbles may ascend buoyantly, controls on eruption style. Eruption styles are allowing efficient magma outgassing and rela- often correlated with magma composition and tively quiescent magma effusion. Alternatively, to some extent this relationship reflects differ- bubbles may coalesce and accumulate to form ences in tectonic setting, which also influences meter-size gas slugs that rupture at the surface magmatic volatile content and magma supply during strombolian eruptions. At fast magma rate.

Modeling Volcanic Processes: The Physics and Mathematics of Volcanism, eds. Sarah A. Fagents, Tracy K. P. Gregg, and Rosaly M. C. Lopes. Published by Cambridge University Press. © Cambridge University Press 2012.

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4.1.1 Getting magma to the surface 4.2 Volatiles Volcanic eruptions represent the episodic or continuous surface discharge of magma from Volatiles play the central role in governing the a storage region. We shall refer to this stor- ascent and eruption of magma, as summarized age region as the magma chamber and to the in Figure 4.1. Dissolved volatiles, in particu- pathway of magma ascent as the conduit. For lar water, have large effects on melt viscosity. magma to erupt, the chamber pressure has to Exsolved volatiles form bubbles of supercritical exceed the sum of frictional and magma-static fluids, which we will also refer to as gas or vapor pressure losses, as well as losses associated bubbles. These bubbles allow for significant with opening of the conduit. The latter may magma compressibility and buoyancy, ultim- be due to tectonic forces and/or excess magma ately making eruption possible (Pyle and Pyle pressure, which in turn are a consequence of 1995, Woods and Cardoso 1997). The decrease in magma replenishment and/or volatile exsolu- pressure associated with magma ascent reduces tion in a chamber within an elastically deform- volatile solubility, leading to bubble nucleation ing wall rock. Magma-static pressure loss is and growth, the latter a consequence of both strongly dependent on magma density, which volatile exsolution and expansion (Sparks 1978). is dependent on volatile exsolution and volume Volatile exsolution also promotes microlite crys- expansion as pressure decreases during ascent. tallization, which in turn affects bubble growth Feedbacks between chamber and ­conduit and rheology – there is a complex feedback can produce complex and unsteady eruptive between decompression, exsolution, and crys- behavior. tallization (Tait et al., 1989). The occurrence and dynamics of explosive eruptions are mediated 4.1.2 The volcanic conduit by the initial volatile content of the magma, and Eruptions at basaltic volcanoes may begin as the ability of gases to escape from the ascending fissure eruptions, with typical dike widths of magma. the order of one meter (Chapter 3; Wilson and Head 1981). However, magma often erupts at the surface through structures that more closely 4.2.1 Solubility resemble cylindrical conduits. The processes by Volatile solubility is primarily pressure depend- which flow paths become focused into discrete ent, with secondary dependence on tempera- cylindrical vents are not well understood. One ture, melt composition, and volatile speciation possibility is that the temperature-dependent (McMillan 1994, Blank and Brooker 1994, viscosity of magma results in a feedback Zhang et al., 2007). Formulations for volatile between a cooling, viscosity increase and reduc- solubility can be thermodynamic or empirical. tion in flow velocity, so that the flow becomes In either case, calibration is achieved using more focused over time (Wylie-Lister 1995). In measurements of dissolved volatile concentra- addition, thermal erosion may also contribute tions in quenched melts, equilibrated with a to flow focusing. (Bruce Huppert 1989). volatile phase of known composition at fixed The conduit during silicic explosive erup- pressures and temperatures. The principal tions is generally thought to be more or less species we consider here are H2O and CO2. cylindrical, at least at shallow depths. However, Solubility of CO2 in silicate melts is propor- the conduit cross-sectional area can change tional to partial pressure, whereas for H2O dis- with depth and time because of wall-rock ero- sociation into molecular H2O and OH results sion, caused by abrasion as pyroclasts collide in a square root dependence on partial pres- with conduit walls, or shear stress from the sure. An empirical formulation for combined erupting magma, or wall collapse (Wilson et al. solubility of H2O and CO2 in rhyolitic melts at 1980, Macedonio et al. 1994, Kennedy et al. 2005, volcanologically relevant conditions is given Mitchell 2005). by (Liu et al., 2005)

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5668 − 55.99 p Figure 4.1 Schematic illustration of conduit processes.  w 3/2 . Cpcc=  ++()0.4133 pww0.002041 p  (a) In effusive eruptions of silicic magma (high viscosity, low  T  ascent rate) gas may be lost by permeable flow through (4.2) porous and/or fractured magma. (b) During (sub)plinian eruptions bubble walls rupture catastrophically at the Here C is total dissolved H O in wt.%, C is dis- fragmentation surface and the released gas expands rapidly w 2 c solved CO in ppm and T is temperature in as the flow changes from a viscous melt with suspended 2 bubbles to gas with suspended pyroclasts. (c) Extensive loss Kelvin. pw and pc are the partial pressures in of buoyantly rising bubbles occurs during effusive eruptions MPa of H2O and CO2, respectively. This formu- of low-volatile content, low-viscosity magma. (d) Coalescence lation is also approximately applicable to other and accumulation of buoyant bubbles, followed by their melt compositions (Zhang et al., 2007), but more rupture at the surface, produces strombolian explosions in accurate models are available (Dixon 1997, slowly ascending low-viscosity magmas. (e) Bubbles remain Newman and Lowenstern 2002, Papale et al., coupled to the melt in low-viscosity, hawaiian eruptions, 2006). Equilibrium concentrations of dissolved which are characterized by relatively high ascent rates, CO and H O, based on this formulation, are hydrodynamic fragmentation, and sustained lava fountaining. 2 2 shown in Figure 4.2. Notice that almost all CO2 exsolves at >100 MPa for rhyolite and at >25

MPa for basalt, whereas most H2O dissolves at 3/2 Cpww=0.0012439 <100 MPa for rhyolite and at <25 MPa for basalt. 354.94 pp+−9.623 1.5223 p3/2 Consequently, processes in the shallow conduit + ww w (4.1) T are predominantly affected by H2O exsolution. +−p 1.08441×−01−−45pp.362 × 10 c ()ww4.2.2 Diffusivity Volatile diffusivities in silicate melts are best and characterized for H2O. A recent formulation for

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2 –1 H2O diffusivity (m s ) in rhyolite with H2O con- tents � 2 wt % is (Zhang and Behrens 2000)

 10661 p  DC= exp −−17.14 − 1.772 m . (4.3) wr w  T T 

A formulation for higher H2O concentrations is provided by Zhang et al. (2007), who also give the

following equation for H2O diffusivity in basalt

 19110 DC= exp −−8.56 . (4.4) wb w  T 

CO2 diffusivity, Dc, is smaller than water diffu- sivity over a range of conditions (Watson 1994).

The most reliable formulation for Dc in silicate melts is based on argon diffusivity, which is

essentially identical to CO2 at volcanologically relevant conditions (Behrens and Zhang 2001, Nowak et al., 2004)

17367 + 1.0964 p D = −−18.239 m c T

()855.20+ .2712 pCmw + . T

4.2.3 Pre-eruptive volatile content of magmas Water Basaltic magmas show a wide range of water contents ranging from >0.5 wt.% to 6–8 wt.% (Johnson et al., 1994, Wallace 2005), with non- arc basalts generally being considerably dryer than arc magmas, which have highly variable water contents. Silicic magmas also show a wide range of water contents up to 6 wt.%, and pos- sibly higher (Carmichael 2002).

Figure 4.2 Solubility of CO2 and H2O (dashed contours) and equilibrium solubility degassing paths (solid lines) Carbon dioxide

for typical rhyolitic and basaltic magmas undergoing CO2 content of magmas is more difficult to con- closed-system dagassing. (a) Rhyolite magma with no initial strain, because of its low solubility at shallow exsolved volatiles at 850 °C, initial pressure of 200 MPa, and depths (Fig. 4.2). Measurements of H2O and CO2 initial dissolved H2O content of 4 wt.%, using the solubility in melt inclusions, in conjunction with solubil- model of Liu et al. (2005). (b) Basaltic magma with no initial ity relations, suggest that arc magmas contain exsolved volatiles at 1300 °C, initial pressure of 200 MPa, and several wt.% of CO2, much of which exsolved pre- initial H2O content of 0.7 wt.%, using the solubility model of Dixon (1997). eruptively, for example during vapor-saturated fractional crystallization (Papale 2005; Wallace 2005). Thus, arc magmas may already contain,

or have lost, significant amounts of exsolved CO2

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prior to eruption. CO2 of undegassed mid-ocean 4.3.1 Nucleation ridge basalts and ocean island basalts appears to Pressure decreases as magma ascends. If the be about 1 wt.% or less (Hauri 2002). Because CO2 concentration of dissolved volatiles exceeds the exsolves at greater pressures than H2O, relative equilibrium solubility at a given pressure, the proportions of CO2 and H2O in erupted gases, melt is supersaturated, a requirement for bub-

glasses and, melt inclusions can provide con- ble nucleation. Supersaturation, �ps, is defined straints on depths of magma degassing. Because as the difference between actual pressure and typical CO2 contents of silicic magmas are con- the pressure at which the concentration of dis- siderably lower than H2O, the latter dominates solved volatiles would be in equilibrium with eruption dynamics. the co-existing vapor phase. The supersatur- ation required for nucleation corresponds to the Sulfur energy that must be supplied to increase the sur- H2S and SO2 are, after water and carbon diox- face area between two fluids. It is a consequence ide, the most abundant magmatic volatiles. S of surface tension, γ, a measure of the attractive concentrations are, at a minimum, several thou- molecular forces that produce a jump in pressure sand ppm in arc and back-arc magmas (Wallace across a curved interface between two fluids. This 2005). Volcanic sulfur emissions are spectro- so-called Laplace or capillary pressure is 2γ/R for graphically readily detectable and, therefore, a spherical bubble of radius R. Supersaturation play an important role in monitoring of active is typically produced when fast magma ascent volcanoes via remote sensing. Emissions from allows insufficient time for volatile diffusion into

active volcanoes, especially during voluminous existing bubbles, which occurs when τdif/τdec ≫ 1

explosive eruptions, have significant environ- (Toramaru, 1989). Here, τdec = pm/pṁ is the char- 2 mental and atmospheric impacts (Bluth et al., acteristic decompression time, τdif = (S – R) /D is 1993). Measured SO2 fluxes from erupting vol- the characteristic time for volatile diffusion, S canoes often exceed by one to two orders of is the radial distance from bubble center to the magnitude the amounts thought to be dissolved midpoint between adjacent bubbles, D is volatile

in the magma prior to eruption. This “excess diffusivity in the melt, pm is pressure of the melt,

sulfur” problem is sometimes attributed to an and pṁ is the decompression rate. exsolved S-bearing volatile phase accumulating Bubble nucleation can be heterogeneous or within the magmatic system prior to eruption homogeneous. In heterogeneous nucleation, (Wallace et al., 2003). crystals provide substrates that facilitate nucle- ation because of the lower interfacial energy Chlorine and Fluorine between solid and vapor than between melt and

Measurable Cl and F are also present in volcanic vapor. Typically, �ps is a few MPa for heteroge- gases, with Cl concentrations in the range of neous nucleation (Hurwitz and Navon, 1994, hundreds to thousands ppm in arc and back-arc Gardner and Denis, 2004, Gardner, 2007) and magmas (Wallace 2005). Because S is less sol- ~10 to 100 MPa for homogeneous nucleation, in uble than Cl, which in turn is less soluble than F which nucleation sites are lacking (Mangan and (Carroll and webster, 1994), these gases are also Sisson, 2000). used to constrain depths of volatile exsolution Classical nucleation theory (Hirth et al., 1970) and magma degassing histories (Allard et al., predicts a very strong dependence of the nucle- 2005). ation rate J (number of bubbles per unit volume

per unit time) on �ps and γ

3 4.3 Bubbles  16πγ ψ  (4.5) J ∝−exp  2 .  3kTBs∆p  The “birth,” “life,” and “death” of bubbles are

key to understanding the dynamics of volcanic Here kB is the Boltzmann constant and γ is typ- eruptions and are the subject of this section. ically 0.05 to 0.3 N m–1, with a dependence on

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both temperature and composition of the melt ∂Ci ∂Cii1 ∂C  2 ∂Ci  + vr = 2  Dri  , (4.9) (Bagdassarov et al., 2000, Mangan and Sisson, ∂t ∂rr∂r  ∂r  2005). ψ is a geometrical factor that ranges between 0 and 1, depends on θ, the contact angle between bubble and crystal, and is given by with vr as the radial velocity of melt at radius r. The boundary condition at r = S is дCi/дr = 0. At 2 r = R the prescribed concentration is obtained (21− coscθθ)( + os ) ψ = . (4.6) from a suitable solubility model with the add- 4 itional assumption that dissolved volatiles at J determines the number density of bubbles per the melt–vapor interface are locally in equilib- rium with the vapor at pg. unit volume of melt, Nd and if bubbles nucleate Under most conditions bubble growth is at large �ps, they do so at a high rate. either limited by (1) the rate of viscous flow of 4.3.2 Growth the melt to accommodate the ensuing volume increase, (2) diffusion rate of volatiles to the As ambient pressure, pm, decreases during magma ascent, bubbles may grow if volatiles melt–vapor interface where they can exsolve, exsolve and the vapor expands. The momen- or (3) the change in solubility caused by decom- tum balance for a growing bubble is obtained pression. For each growth limit it is possible to by neglecting inertial terms in the Rayleigh– simplify the solution of the governing equations Plesset Equation (Scriven, 1959). This is justified or to derive analytical growth laws. because large melt viscosities and short length scales result in Reynolds numbers for bubble Modeling of bubble growth growth ≪1. The resulting equation for a bubble Explicit modeling of diffusive bubble growth surrounded by a shell of melt with constant vis- during magma ascent was first presented cosity is (Proussevitch et al., 1993) by Sparks (1978), based on the equations of Scriven (1959) and Rosner and Epstein (1972). 2 2γ  1 R  Subsequently, Proussevitch et al. (1993) applied pp−+= 4η v − . (4.7) gmR 0 R  R S3  a model that included the diffusion of volatiles to bubble growth in magmas. This model was

Here, pg is the pressure inside the bubble, η0 is enhanced by Sahagian and Proussevitch (1996)

the Newtonian melt viscosity, and vR is the radial and Proussevitch and Sahagian (1998) to include velocity of the melt–vapor interface, i.e., bubble the thermal effects associated with the heat of wall. An idealized uniform packing geometry volatile exsolution and the work done against is assumed, so that each bubble can be repre- the surrounding melt by bubble expansion. sented as a sphere of radius R surrounded by a Figure 4.3 shows the results for the diffusive spherical melt shell of thickness S – R. In this bubble growth in a rhyolite containing both

formulation the time-dependent pm couples bub- H2O and CO2. ble growth with magma ascent. Mass conserva- tion of volatiles requires that Viscous limit

In the viscous limit τvis/τdec ≫ 1 and bubble C 32  ∂ i  growth is retarded, following an exponen- dR()ρρgm=4RD∑ i   dt, (4.8) i  ∂r  rR= tial growth law (Lensky et al., 2004). Here τvis =

η0/(pg – pm) is the characteristic viscous time- where r is radial distance from the center of the scale. The viscous limit to bubble growth is bubble, t is time, ρg is the pressure-dependent the consequence of fast decompression rates vapor density, ρm is melt density, Di is the concen- relative to the viscous deformation of melt by tration dependent diffusivity of dissolved volatile the growing bubbles, and is most significant 9 species i, and Ci is the mass fraction of dissolved at melt viscosities in excess of 10 Pa s (Sparks volatile i. Volatile diffusion is governed by et al., 1994), which in the absence of significant

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amounts of crystals, is achieved for silicic melts 1 at shallow depths and low H2O content. Because

b of retarded bubble growth, pg decreases at a φ 0.8 slower rate than pm, resulting in the build-up of

overpressure with pg – pm ≫ 2γ/R (Fig. 4.3). At

bubbles, 0.6 the same time, a high pg inhibits volatile exsolu- tion, causing supersaturated conditions. This is 0.4 referred to as “viscosity quench” (Thomas et al., 1994) and increases the potential for accelerated bubble nucleation and/or growth. olume fraction 0.2 V Diffusive limit 0 During decompression, the volatile concentra- 200 150 100 50 0 tion at the melt–vapor interface, based on the Ambient pressure, pm (MPa) assumption of local equilibrium, decreases. This 103 creates a concentration gradient for volatiles to Fragmentation threshold diffuse to this interface and exsolve. If τdec ≪ (1 MPa/φb) τ , volatile concentrations will remain close to 102 dif equilibrium throughout the melt. On the other hand, if τ /τ ≫ 1 (i.e., at high decompression 101 dif dec a) rates) the melt becomes supersaturated, which is a necessary condition for bubble nucleation (MP 0

m 10 (Lensky et al., 2004). Furthermore, in the diffusive – p

g limit, volatiles with different diffusivities may

p 0–1 1 fractionate (Gonnermann and Manga, 2005b; Gonnermann and Mukhopadhyay, 2007). 10–2 Solubility limit 10–3 200 150 100 50 0 During solubility-limited growth, bubbles are close to mechanical and chemical equilibrium, Ambient pressure, pm (MPa) that is, τdif/τdec ≪ 1 and τvis/τdec ≪ 1 (Lensky et al., Figure 4.3 Model results for diffusive bubble growth in 2004) so that both overpressure and supersatur-

rhyolite with initial conditions of bubble volume fraction ϕ0 = ation are small. Dissolved volatiles are near their 11 –3 0.001, bubble number density Nd = 10 m , 5 wt.% H2O, equilibrium concentrations throughout the 268 ppm CO2, T = 850 °C, magma pressure pm = 200 MPa, and melt and bubble radius R can be directly calcu- –1 decompression rate Δρ̇ = 0.1 MPa s . (a) Circles represent lated from mass balance, equilibrium solubility, cross-sections through bubbles along the modeled trend of ϕ b and an equation of state. Conditions favorable vs. p . For reference, the short-dashed black line represents m to solubility-limited bubble growth are low melt the equilibrium closed-system degassing trend. Initially ϕb remains lower than equilibrium because volatile diffusion is viscosity and/or low magma ascent rates.

slower than Δρ̇, and bubble over pressure pg – pm ≈ 2γ/R. As bubble radius R increases and bubble separation, S – R, 4.3.3 Coalescence decreases, diffusion starts to keep pace with decompression Coalescence takes place when bubbles come and ϕb approaches equilibrium values. (b) Bubble overpressure, into increasingly close proximity and the melt pg – pm, vs. pm (solid curve), with fragmentation threshold, film between bubbles ruptures. This may be Δ (dashed curve) of Spieler et al. (2004). At p < 30 MPa, ρ̇f m the consequence of (1) gravitational or capil- bubble growth becomes viscously limited, a consequence of lary drainage of interstitial liquid (Proussevitch H O exsolution and increasing melt viscosity. Consequently, 2 et al., 1993b); (2) coalescence-induced coales- magma pressure pm decreases more rapidly than bubble vapor cence, whereby the deformation of coalesced pressure pressure pg until the bubble overpressure reaches

the fragmentation threshold pressure, i.e., pg – pm = Δpf. bubbles by capillary forces induces a flow field

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that brings nearby bubbles into proximity the strain rate in the melt caused by buoyant (Martula et al., 2000); (3) bubble growth leading rise of the bubble or by magma flow. For small

to stretching and thinning of the liquid film Reb, deformation scales with the Capillary num-

that separates individual bubbles (Borrell and ber, Ca = η0ε̇R/γ, which characterizes the rela- Leal, 2008); and (4) advection and collision of tive importance of viscous stresses that tend to bubbles by buoyancy or by magma flow (Manga deform bubbles, and surface tension stresses and Stone, 1994). that act to keep bubbles spherical. If Ca exceeds

Gravitational and capillary film drainage, some critical value, Cacr, bubble elongation by as well as bubble collisions are probably most the flow becomes large enough that bubbles

important in low viscosity magmas. A positive will break up. The value of the Cacr depends on feedback between bubble coalescence, which the steadiness of the flow, melt viscosity, and increases bubble size, and bubble mobility due flow type. For the viscosities of silicate melts, 3 to buoyancy is expected (Section 4.3.5). Whereas Cacr should range from ~1 to >10 . In both types capillary drainage is dominant at bubble radii of flows, bubbles will become highly elongated <3 cm, gravitational drainage becomes dom- before breakup occurs. Capillary numbers will inant at radii >3 cm (Proussevitch et al., 1993). generally be large enough in conduits for large A characteristic velocity for capillary drainage deformation and breakup to occur. However, to can be obtained by balancing the capillary pres- achieve large deformation also requires large sure gradient, γ/R2, with viscous resistance to strains which may be limited to the sides of 2 flow, η0vf/R , where vf is the velocity in the film. conduits. It is thought that tube pumice is a Because the bubble–melt surface is assumed to manifestation of these conditions (Marti et al., be a free-slip surface, the characteristic length 1999). scale is the bubble radius R rather than film In basaltic magmas, inertial forces can no

thickness. The resultant scaling vf ~ γ/η0 high- longer be neglected once bubbles become larger lights the importance of melt viscosity. Whether than a few centimeters. In this limit, velocity and at what rate bubble collisions result in differences across bubbles can lead to breakup. coalescence depends on the Weber number, The relative importance of inertial forces and We = 2ρv2R/γ, where v is the velocity at which surface tension forces is characterized by We, two bubbles approach one another. based on the velocity difference across the melt

In high viscosity magmas bubbles remain that surrounds the bubble. For large Reb, breakup

essentially “frozen” in the melt and coalescence occurs if We exceeds a critical value, Wecr, which is principally a consequence of bubble growth. implies that there might be a maximum stable In this case, coalescence appears to create a per- bubble size (Hinze, 1955). Although the actual

meable network of bubbles due to relatively value of Wecr depends on the origin of the vel- persistent holes in the ruptured melt films (Klug ocity differences, it is generally between 1 and 5 and Cashman, 1996). Because the interfacial ten- for a wide range of flow conditions. sion between melt and crystals tends to be lower than between gas and crystals, coalescence in 4.3.5 Bubble mobility the presence of crystals is expected to occur at Bubbles are buoyant and their rise speed depends greater film thickness than in the absence of on size, volume fraction, and viscosity. If buoy- crystals (Proussevitch et al., 1993b). ant bubble rise is much slower than magma ascent, bubbles are dispersed throughout the 4.3.4 Breakup continuous melt phase and move passively with The breakup of bubbles into several smaller the flow. Such flows are called “dispersed” and bubbles originates with their deformation in the asymptotic limit of infinitesimally small by some combination of viscous stresses and dispersed bubbles the flow is termed “homoge- inertial forces. In silicic magmas, the bubble neous.” If the velocity of buoyant bubble rise is 2 Reynolds number, Reb = ε̇R ρ/η0, will always be similar to or greater than magma ascent veloci- small and inertia can be neglected. Here ε̇ is ties, bubbles are decoupled from the liquid phase

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Therefore, bubbles within a silicic melt have negligible mobility, even at very low eruption rates, whereas significant bubble mobility can be expected for mafic melts. At finite volume frac-

tions of bubbles, ϕb, hydrodynamic interactions

reduce Ut by a factor h(ϕb), the hindering func-

tion. Various formulations exist for h(ϕb) and a commonly adopted one is the Richardson–Zaki equation (Richardson and Zaki, 1954)

n (4.11) h()φφbb=1( − ) .

Here n is an empirical coefficient for the appro- priate range of bubble size and flow conditions, typically with values around 2.5 (Zenit et al., 2001). At flow regimes other than bubbly flow a different formulation for the relative velocity between gas and liquid phase has to be used Figure 4.4 Flow regimes observed in bubble column experiments (e.g., Wallis, 1969). The gas phase is shown in (Dobran, 2001). white and the liquid phase in black. 4.3.6 Bubbles and pressure loss In general, mass discharge rate depends on the and the flow is called “separated” (Brennen, total pressure drop between magma chamber 2005). During separated flow in laboratory and vent. Within the conduit the pressure gra- experiments the topology of the gas and liquid dient is to first order the sum of magma-static phases is associated with different flow regimes pressure loss, (dp/dz)p, and frictional pressure (Fig. 4.4), that depend primarily on the relative loss, (dp/dz)η. Magma-static pressure depends on volumetric flow rates of liquid and gas phase magma density and, hence, vesicularity as (Wallis, 1969). As this flux ratio increases to dp   (4.12) values of ~1, the flow transitions from “bubbly =1ρφgg≈−( bm)ρ .  dz  flow,” where bubbles are randomly dispersed, to ρ “slug flow,” where coalesced bubbles form rap- idly ascending gas slugs of diameter comparable For dispersed bubbly flow the frictional -pres to the conduit diameter. At flux ratios above sure loss can be approximated on the basis of ~10 the flow transitions to “annular flow,” established single-phase flow correlations using where the liquid phase forms an annular ring the appropriate mixture viscosity (Ghiaasiaan, surrounding a cylindrical core of rapidly ascend- 2008) ing gas. However, to what extent these different  dp 2 f flow regimes are applicable to volcanic erup- =,ρu (4.13)  dz  a tions remains controversial (Chapter 6). η The terminal velocity of a single bubble con- where a is conduit radius, u is magma velocity, taining vapor of viscosity ηg and density ρg, rising and f is the friction factor given by in an infinite liquid of viscosityη 0 and density ρm is (Batchelor, 1967) 16 ff= + . (4.14) Re 0 Rg2 ρρ− ηη+ U = ()mg 0 g . t 3 3 (4.10) η0 ηη+ Here Re = 2uρa/η, where η is the magma viscos- 0 2 g ity and f0 has values of about 0.002–0.02 (Mastin

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and Ghiorso, 2000). For highly viscous magmas equation (Carman, 1956) where permeability, k, the flow below the fragmentation level will, is given as under most conditions, be laminar (Re < 103) and β (4.15) f � 16/Re. Above the fragmentation level the flow k(φχbb) =.()φφ− bp will be turbulent and f � f0. For basaltic magma,

frictional pressure loss over a wide range of Here χ is an empirical constant and ϕbp is the ascent velocities is less than magma-static pres- volume fraction of bubbles that corresponds to sure loss and the flow dynamics will be more percolation threshold (Saar and Manga, 1999), sensitive to changes in the density of the ascend- with typical values of 2 � β � 4. An uncontro- ing magma than for eruptions where frictional versial permeability model for vesicular magma pressure losses are larger. Consequently, under remains elusive (Takeuchi et al., 2005; Wright separated flow conditions there is potential et al., 2009). Furthermore, it appears that the for feedbacks between discharge rate, gas-to- nature of magma deformation plays a critical melt flux ratios, and flow regime (Seyfried and role in creating permeability (Okumura et al., Freundt, 2000; Guet and Ooms, 2006). 2008; Okumura et al., 2010). Permeable outgassing can be modeled as 4.3.7 Permeable outgassing flow through a porous medium by combining The degassing history during ascent of an indi- the continuity equation for the exsolved volatile vidual magma parcel may be complex and phase with Darcy’s equation or, if the velocity in many cases gas may separate from rising of the gas phase becomes too large for inertial parcels of magma. We refer to this as outgas- effects to be neglected, Forcheimer’s equation sing, which may be a consequence of buoyant (Rust and Cashman, 2004). bubble rise, magma fragmentation, or perme- able gas flow. Chapters 6 and 8 discuss outgas- sing associated with separated flow and high gas-to-melt flux ratios during strombolian and 4.4 Crystal nucleation and perhaps hawaiian eruptions. In some silicic growth eruptions high melt-to-gas flux ratios are also suggestive of decoupled gas flow (Edmonds Crystals nucleate due to undercooling, which

et al., 2003). However, high viscosities and neg- is predominantly a consequence of H2O exsolu- ligible bubble mobility require that outgassing tion. Nucleation rates determine the crystal is associated with magma permeability, pre- number density and size distribution, which sumably a consequence of coalescing bubbles provide a record of the magma’s ascent history. that form a permeable network (Eichelberger During crystallization the volatile content of the et al., 1986; Klug and Cashman,1996) and per- residual melt phase increases because volatiles haps cracks and fractures produced by brittle preferentially partition into the melt phase. This magma deformation (Gonnermann and Manga, in turn affects bubble nucleation and growth, as 2003; Gonnermann and Manga, 2005a; Tuffen well as melt viscosity. et al., 2003). It has been suggested that if con- Undercooling, the thermodynamic driving duit walls are permeable, which is controver- force for crystallization, is the chemical poten-

sial (Boudon et al., 1998), then volatiles may tial of H2O in the melt and can be expressed as

also escape laterally from ascending magma �Te, the difference between actual temperature,

into the conduit walls and permeable outgas- T, and the liquidus temperature, Tl. Because sing has the potential to modulate eruptive the latter depends on composition and vola- behavior (Jaupart and Allegre, 1991; Woods and tile content, bubble nucleation and growth will

Koyaguchi, 1994; Eichelberger, 1995). increase Tl and can result in sufficient undercool- Vesicularity-permeability measurements for ing for crystallization (Hammer, 2004). The rate volcanic rocks, and by inference magmas, are of homogeneous nucleation of critical nuclei, I usually analyzed using the Kozeny–Carman (m–3 s–1), is given by classical nucleation theory as

9780521895439c04_p55-84.indd 64 8/2/2012 9:36:26 AM DYNAMICS OF MAGMA ASCENT IN THE VOLCANIC CONDUIT 65

 *  ∆∆GG+ D (4.16) 4.5 Magma rheology II=.0 exp −   kTB  Silicate melts form a disordered network of

Here I0 is a reference nucleation rate, kB is the interconnected SiO4 tetrahedra where the

Boltzmann constant, and �GD is the activation self-diffusive motion of atoms, called structural energy for diffusion (James, 1985). �G* is the relaxation, results in a continuous unstructured free energy required to form a spherical nucleus rearrangement of the molecular structure. In of critical size and depends on the free energy, the presence of an applied stress, this molecular σ, associated with the crystal–liquid interface. rearrangement results in a directional motion

Therefore σ, nucleation rate increases with of SiO4 tetrahedra relative to one another and decreasing σ (Mueller et al., 2000). The rate of macroscopically manifests itself as viscous flow heterogeneous nucleation also follows Eq. (4.16), (Moynihan, 1995). but with a modified σ to account for the lower The intrinsic viscosity of silicate melts varies interfacial energies (Spohn et al., 1988). Because over orders of magnitude, even within a single σ is relatively poorly constrained, estimates of volcanic eruption. It depends on the degree of crystal nucleation rates are typically associated polymerization, a function of chemical com- with large uncertainties. Moreover, the validity position and volatile content. Within realistic of classical nucleation theory and its assump- ranges of compositional variability, eruptive tions that (1) the interface between nucleus and temperature, and volatile content, the viscosity melt is sharp; (2) the critical nucleus has the of basaltic melts may vary between about 10 and thermodynamic properties of the bulk solid; 103 Pa s, and between about 104 and 1012 Pa s or and (3) σ can be treated as a macroscopic prop- more for silicic melts (Fig. 4.5). Changes in vis- erty equal to the value for a planar interface, cosity during individual eruptions are especially remain controversial. pronounced at low pressures (depths <1 km),

Once a crystal nucleates its growth rate, Y, is where most of the H2O exsolves (Figs. 4.2 and given by (Spohn et al., 1988) 4.6). Magma rheology is discussed further in the context of lava flow emplacement in Chapter 5.

  ∆∆HT   ∆G  expee xp D (4.17) YY=10  −     −  ,   kTB lT    kTB  4.5.1 The effect of dissolved volatiles and temperature

where Y0 is a reference growth rate and �H is Dissolved water dissociates into molecular the change in enthalpy between melt and crys- water and hydroxyl ions, thereby depolymer- talline phases. If empirical parameters such izing the melt. In rhyolitic melts the effect of

as Tl and σ are known, the above equations water content can be tremendous and exceeds can be used to model magma crystallization. the effect of temperature (Fig. 4.5), but it is less Alternatively, parameterizations for I and Y can pronounced for mafic magmas (Giordano and be used to explore the dependence of eruption Dingwell, 2003). In contrast, viscosity is less

behavior on syneruptive crystallization (Melnik affected by CO2 (Bourgue and Richet, 2001). and Sparks, 1999). The prediction of crystal size Recent viscosity formulations applicable over a distributions is possible through the use of the range of compositions, temperature, and water Avrami equation (Marsh, 1998) contents are discussed in Hui and Zhang (2007), Zhang et al. (2007), and Giordano et al. (2008). 34 (4.18) φxv=1−−exp()kYtI. 4.5.2 The effect of deformation rate

Here a given crystal volume fraction, ϕx, is If the applied rate of deformation exceeds a cer- assumed to correspond to thermodynamic equi- tain threshold, the induced molecular motions librium after crystallization over time, t, and of the melt are no longer compensated by ran-

with kv as a volumetric factor. dom reordering of the melt structure. This is

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Figure 4.5 Dependence of rhyolitic melt viscosity on temperature and water content, after the empirical formulation of Hess and Dingwell (1996). See color plates section.

−3 G∞ ε st ∼ 10 . (4.19) η0

10 Here G∞ ~ 10 Pa is the shear modulus and η0 is the Newtonian melt viscosity, measured at

ε̇ ≪ ε̇st.

4.5.3 The effect of crystals Crystals increase magma viscosity (Lejeune and Richet, 1995; Stevenson et al., 1996; Arbaret et al. 2007; Caricchi et al., 2007; Lavallee et al., 2007).

For small crystal volume fractions, ϕx < 10 – 30%, and low strain rates, the viscosity of the crystal-

line magma relative to the crystal-free melt, ηr =

η/η0, can be approximated as

α ηφrx=(1/− φxcr). (4.20) Figure 4.6 Dependence of rhyolitic melt viscosity (Hess and Dingwell, 1996) on depth, assuming lithostatic pressure Here ϕ represents a critical volume fraction at and equilibrium volatile solubility. Note the wide range in xcr viscosity and the tremendous viscosity increase at shallow which crystals start to impede the ability of the suspension to flow and α � –2.5 (Llewellin and depths, due to the exsolution of H2O. Manga, 2005). At ϕx greater than some value, typically in the range of 10–50%, crystals may manifested in shear thinning, a decrease in vis- come into contact creating a framework that cosity from its “relaxed” or Newtonian value provides a finite yield strength. Shear thinning and will result in brittle failure if deformation and shear localization may occur as melt and rates exceed the onset of shear-thinning by crystals redistribute themselves to decrease the approximately two orders of magnitude (Webb, flow disturbance. It most likely occurs in a nar- 1997, and references therein). The strain rate at row zone near the conduit wall and may prod- which shear thinning is first observed can be uce a plug flow that can manifest itself at the defined in terms of the viscous relaxation time, surface as volcanic spine. A viscosity formula-

τr = η0/G∞, as tion that is calibrated to a broad range of crystal

9780521895439c04_p55-84.indd 66 8/2/2012 9:36:28 AM DYNAMICS OF MAGMA ASCENT IN THE VOLCANIC CONDUIT 67

7

6 Small shear rate r 5 , η

4 Large shear rate

3 relative viscosity relative 10 2 log

1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Crystal content, φx Figure 4.8 The effect of bubbles at small and large Ca Figure 4.7 The effect of crystals on the relative mixture on the velocity profile in a cylindrical conduit for a given pressure gradient. Dashed curve is the bubble-free reference viscosity (ηr = η/η0), based on Eq. (4.20), with parameters case Ca = 1 (or equivalently ϕ = 0). Both solid curves are for ϕbcr = –0,066499 tanh (0.913424 log(ε̇) + 3:850623) b + 0:591806; ϕb = 0.6 and ϕbcr = 0.75 in Eq. (4.21). δ = –6.301095 tanh (0.818496 log(ε̇) + 2.86) + 7.462605; α = –0.000378 tanh (1.148101 log(ε̇) + 3.92) + 0.999572; β = 3.987815 tanh (0.8908 log (ε̇) + 3.24) + 5.099645. 1999; Stein and Spera, 2002). An expression for this dependence on Ca is given by Pal (2003) as

content and strain rates (Fig. 4.7) is provided by −4/5 −φbcr  11− 2/ηφ22Ca 5   rb=1 , (4.22) Caricchi et al. (2007) ηr  2   −   11− 2/Ca 5   φbcr 

 δ   φ x  =1 where ϕbcr is an empirical parameter that is ηη0  +      φ xcr    conceptually analogous to ϕxcr. The strain-rate −Bφ (4.21) β xcr dependent rheology of melt with suspended         π φx φx  bubbles and crystals is derived from a linear 1−+α erf  1    .   2α φ xcr   φ xcr        superposition of the individual dependencies (Thies, 2002). As illustrated in Figure 4.8, during laminar flow at a given pressure gradient, the Here B = 2.5 and the remaining empirical param- presence of bubbles can increase or decrease eters δ, α, and β are provided in Figure 4.7. mass discharge rate by almost one order of mag- 4.5.4 The effect of bubbles nitude. Chapter 5 further considers the com- bined effects of bubbles and crystals on magma The effect of bubbles on viscosity depends on rheology (Section 5.2.2). whether the bubbles are able to deform and

is a function of Ca = η0ε̇R/γ. If Ca ≪ 1, bub- bles remain nearly spherical and the viscos- ity is greater than that of the melt (Taylor, 4.6 Magma fragmentation 1932). In contrast, for Ca > O(1), bubbles can become elongated (Rust et al., 2003) and the During all explosive eruptions pyroclasts are viscosity decreases (Chapter 5; Bagdassarov and ejected from the volcanic vent. Here we focus Dingwell, 1992; Manga et al., 1998; Lejeune et al. on “dry” magma fragmentation, which does

9780521895439c04_p55-84.indd 67 8/2/2012 9:36:29 AM 68 HELGE M. GONNERMANN AND MICHAEL MANGA

not involve any interaction with non-magmatic Namiki and Manga (2005) suggested a poten- water. The type and efficiency of the fragmenta- tial energy criterion, based on the observation

tion process determines pyroclast size and how that during rapid decompression ϕb and �pf much magmatic gas is released per unit mass of determine the expansion velocity of a bubbly

magma. This, in turn, has implications for the liquid. Potential energy depends on both ϕb and

volcanic jet, column, and plume that are pro- �pf (Mastin, 1995), with fragmentation taking duced when the gas-pyroclast mixture exits the place above some threshold. volcanic vent (Chapters 8 and 9). Fragmentation Hydrodynamic or inertial fragmentation should is thought to occur when specific conditions be predominantly associated with mafic magma, are reached within the conduit. The proposed where rapid decompression results in rapid bub- criteria for fragmentation include: (1) a critical ble growth, inertial stretching, and hydrodynamic bubble volume fraction; (2) a stress criterion; (3) breakup (Shimozuru, 1994; Zimanowski et al., a strain-rate criterion; (4) a potential energy cri- 1997; Villermaux, 2007). A criterion for inertial terion; and (5) an inertial criterion. fragmentation is based on the Reynolds number,

The critical volume fraction criterion is Rev, and Weber number, Wev, of the expanding thought to arise from some form of instabil- magma (Namiki and Manga, 2008)

ity within the thin bubble walls, once ϕb ≈ 0.75 2 is reached (Verhoogen, 1951; Sparks, 1978). ρφ(1 − b)vL ROev = >1( ) (4.24) However, magma fragments have vesicularities 3η0 that range from 0 (obsidian) to >98% (reticulite) implying that a fragmentation criterion gov- and erned by a critical volume fraction cannot be 2 generally applicable (though very high vesicu- ρφ(1 − b)vL Wev =  1. (4.25) larities may reflect post-fragmentation growth γ of bubbles). The stress criterion is based on the view Here L is a characteristic length scale and v that fragmentation in high-viscosity magmas is the expansion velocity of the vesicular

takes place when volatile overpressure, �pf, magma, which can be obtained from the bubble exceeds the tensile strength of the melt and growth rate. ruptures bubble walls (Alidibirov, 1994; Zhang, 1999). Spieler et al. (2004) provide a formula- tion with good fit to a broad range of experimen- 4.7 Modeling of magma ascent tal data The past decade has seen much progress in 6 ∆p fb=10/Pa φ , (4.23) modeling magma ascent (Sahagian, 2005). In contrast to the modeling of effusive eruptions, that has been modified by Muelleret al. (2008) to explosive eruptions require a criterion to esti- account for permeable gas flow. mate the depth of magma fragmentation, and The strain-rate criterion is based on the merging of a model for the bubbly flow region observation that silicate melts can fragment if below the fragmentation level with one for deformation rates exceed the structural relax- the gas-pyroclast region above. Flow in the 2 ation rate of the melt at ε̇ ~ 10 ε̇st (Section 4.5.2). gas-pyroclast flow region is compressible and It is thought to be the consequence of a rapid is typically modeled as one-dimensional with a decompression that forces rapid bubble growth parameterization for wall friction and granular (Papale, 1999). Ittai et al. (2010) have argued that stresses during highly turbulent flow (Wilson both the stress criterion and the strain-rate cri- and Head, 1981; Dobran, 1992; Koyaguchi, terion are equivalent, because of their mutual 2005). dependency on shear modulus and magma Conduit models may be steady or rheology. time-dependent. The steady approximation is

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reasonable when the timescales over which dQ d ==()ρuA 0. (4.26) changes in eruption rate and geometry exceed dz dz ascent times (Slezin, 2003). Time-dependent behavior becomes important when ascent Here A is conduit cross-sectional area and ρ is the times are similar to the timescale over which bulk density of bubbly magma, given by ϕ ρ + boundary conditions change. This can be a con- b g (1 – ϕ )ρ . The value of ρ may be obtained from sequence of coupling between magma flow and b m g the ideal gas law, but use of another equation of (1) magma storage; (2) conduit wall elasticity; (3) state may be more appropriate at high pressures magma compressibility; (4) magma outgassing; (Kerrick and Jacobs, 1981). ρ can be obtained (5) viscous dissipation; and (6) magma rheology. m from empirical formulations (Lange, 1994), but A departure from one-dimensional models is in many cases the melt phase is assumed to be necessary if lateral variations in properties incompressible. The bubble volume fraction are important, for example due to shear-rate ϕ depends on ρ and the amount of exsolved dependent viscosity, shear localization, viscous b g volatiles. If it is assumed that bubble growth is dissipation, or permeable outgassing into con- limited by solubility, the amount of exsolved duit walls. volatiles per unit mass of melt is equal to the Dispersed two-phase flow below the frag- initial volatile concentration minus the equilib- mentation level is typically treated as homoge- rium solubility at the given pressure. However, neous, using an appropriate magma viscosity. in many eruptions bubble growthis viscously Above the fragmentation level, the homogenous and/or diffusively limited, requiring a concur- flow assumption is sufficient if fragments are rent calculation for bubble growth. If permeable � 10–4 m in diameter (Papale, 2001). However, outgassing is important, Eq. (4.26) may include treating the flow as separated for larger- frag a sink term based on a suitable porosity–perme- ments results in a more accurate characteriza- ability relation and Darcy’s law or Forcheimer’s tion of ascent velocity (Dufek and Bergantz, equation for radial gas flow to the conduit walls 2005). In the case of separated flow above or and/or upward gas flow to the vent (Jaupart and below the fragmentation level there are two Allegre, 1991). classes of models, those that explicitly incorpor- Equation (4.26) states that mass flux is con- ate interfacial momentum exchanges and those stant and volume expansion of the magma by that do not (Dobran, 2001). Separated flow mod- bubble growth must be balanced by acceleration els with no interfacial momentum exchange and/or by an increase in conduit cross-sectional treat the flow as single-phase, but incorporate area. Assuming a vertical conduit of cylindrical closure laws for the gas volume fraction and shape, conservation of momentum is governed frictional pressure loss of the two-phase mix- by (Mastin and Ghiorso, 2000; Dobran, 2001) ture (Wilson and Head, 1981). Separated models with interface exchange allow for different gas and liquid (pyroclast) velocities and flow direc- du dp  dp  dp dp 2 f ρρu ==−−  −   −−gu−ρ . tions. However, a priori assumptions about the dz dz  dz  ρη dz  dz a flow regime are required for these closure laws. (4.27) 4.7.1 Steady homogeneous flow in one Expanding Eq. (4.26) and substituting for du/dz dimension in Eq. (4.27) gives Below the fragmentation level homogeneous dp f ρu2 dA dρ flow models advect bubbles passively with the −+=.ρρgu2 −−u2 (4.28) dz a A dz dz melt. That is Ut ≪ Um and u = Um = Ub, where

Ub is the bubble velocity. Together with the assumption of constant mass flux throughout Equation (4.28) indicates that the change the conduit, Q, the equation of mass conserva- in magma pressure with respect to height tion becomes depends, from left to right, on magma-static

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pressure loss, frictional pressure loss, change in adjust the flow back to 1 atm at the vent exit. conduit diameter, and change in magma dens- If the exit pressure exceeds 1 atm, the flow is ity. Assuming isentropic conditions (denoted by “choked,” that is M = 1 and the exit velocity subscript S), that is an infinitesimal reversible is equal to c, the speed of sound in the gas– pressure change and negligible time for heat pyroclast mixture. transfer, then In many cases the assumption of isothermal conditions is reasonable; if not, an equation for

dρρ ∂  dp 1 dp ==. (4.29) conservation of energy has to be added (Mastin   2 dz  ∂p S dz c dz and Ghiorso, 2000; Dobran, 2001)

Here c is the speed of sound in the magma and dH ++udugdz =0. (4.32) using the Mach number, M = u2/c2, Eq. (4.28) becomes Here H is the specific enthalpy of the melt–gas mixture from which the magma temperature 2 dp f ρu dA can be calculated. −−1=M22ρρgu+− . (4.30) dz () a A dz 4.7.2 Steady separated flow in one Equations (4.26) and (4.30) are applicable to dimension homogeneous flow above or below the frag- As discussed in Section 4.3.5, the flow of the sep- mentation level and can be solved with stand- arated melt and gas phases below the fragmen- ard numerical methods (Press et al., 1992). There tation level may occur under different regimes. are two solution approaches: (1) set dp/dz to Criteria that determine the transitions between be constant, for example lithostatic, and then individual flow regimes, as well as formulations

dA/dz and u can be calculated, or (2) if A is known, for interphase drag forces, Fmg, need to be speci- then conduit entrance and exit pressures are fied. Existing models do not predict the occur- specified and dp/dz and u can be calculated rence of flow-regime transitions, but rather rely (Wilson and Head, 1981; Giberti and Wilson, on a-priori assumptions about the occurrence of 1990; Dobran, 1992; Mastin and Ghiorso, 2000). flow transitions, equivalent to the flow -transi For M < 0.3 the effect of magma compressibil- tion associated with magma fragmentation. For ity on the flow can be neglected, whereas for example, the model of Dobran (1992) is based M > 0.3 magma compressibility contributes to on the assumption of a constant mass flow rate the change in pressure of the ascending magma. through the conduit If p = 1 atm at the vent exit, then the right-hand d side of Eq. (4.28) has to go to zero to avoid a sin- (4.33) ()QQgg+ d =0 gular solution as M approaches 1. Therefore, the dz conduit flares upward with and

da 1  ga  = + f >0. (4.31) dz 2  u2  d Q − Q =0. (4.34) dz ()mgd This result can be derived by setting the right-hand side of Eq. (4.28) to zero with Here subscript g denotes the exsolved volatile A = πa2 (Mastin and Ghiorso, 2000). If Eq. (4.31) phase, gd the dissolved volatile phase, and m is satisfied at M < 1 and the conduit above con- the melt phase. As in the homogeneous flow tinues to flare, the magma will decelerate. On model it is possible to account for permeable the other hand if M = 1, the magma will accel- gas loss through an additional flux term on the erate to supersonic velocities and p will con- right-hand side of Eq. (4.36). The momentum tinue to decrease, potentially to values below equations for the exsolved volatile and the melt 1 atm. If that is the case, a shock wave will phases are, respectively

9780521895439c04_p55-84.indd 70 8/2/2012 9:36:33 AM DYNAMICS OF MAGMA ASCENT IN THE VOLCANIC CONDUIT 71

dug dp  ∂uzr∂u  ρφggu b = −−φρbmFFgw−−gbφ g (4.35) τηrz =.−  +  (4.41) dz dz  ∂r ∂z  and 2 du dp Here λη= − K , where η is the viscosity of the ρφu (1 −−)=m (1 −−φ ) FF− 3 mm b dz bmdz gwm (4.36) magma. K is called the bulk viscosity, which −−ρφmb(1 ).g accounts for the compressibility of the two-phase magma mixture (Massol et al., 2001) and is often Here F and F are the wall drag forces for the wg wm assumed negligible. Various simplifications of phase that is assumed to be in contact with the Eqs. (4.37) and (4.38) can be employed. If the flow conduit wall. For example, Dobran (1992) used is laminar (Re < 103), which is usually the case F = (dp/dz) , F = 0 below the fragmentation wm η wg below the fragmentation level, the second and level and F = 0, F = (dp/dz) above. wm wg η third terms on the left-hand side can be neglected. Furthermore, if the flow is steady the time deriv- 4.7.3 Two-dimensional flow atives are zero. Assuming isothermal conditions, If magma flow is radially non-uniform, a the momentum equations are complemented by two-dimensional modeling approach can provide an equation for the conservation of mass additional insight. Cases where two-dimensional modeling has been important involve ∂ρ 1 ∂ ∂ + (ρρrurz) + ()u =0. (4.42) non-Newtonian magma rheology and shear heat- ∂tr∂r ∂z ing, as well as permeable outgassing and heat Examples of recent two-dimensional conduit loss through the conduit walls. models are given by Collier and Neuberg (2006); The direct approach is to solve the Navier– Costa et al. (2007a,b); Hale (2007); Hale and Stokes equations in two cartesian or axisymmet- Muehlhaus (2007). ric cylindrical coordinates. For axisymmetric If vertical flow is a reasonable approximation homogeneous flow, conservation of momentum (u = u , u = 0), Eqs. (4.37) and (4.38) become is given by (Bird et al., 1960) z r ∂ ∂p (4.43) ∂u ∂u ∂u ()rrτ rz =,− ρρr + u r + ρu r ∂r ∂z ∂t r ∂r z ∂z (4.37) ∂p 1 ∂ τ rz = − − (rτ rr ) − and upon integration ∂rr∂r z ∂u rp∂ and τη==− z − . (4.44) rz ∂r 2 ∂z ∂u ∂u ∂u ρρz + u z +ρu z ∂t r ∂r z ∂z The relative ease with which Eq. (4.44) can be (4.38) ∂p 1 ∂ ∂τ zz solved makes it easier to model more complex = −−ρτg − ()r rz − , ∂zr∂r ∂z magma rheologies (Costa and Macedonio, 2005; Mastin, 2005; Vedeneeva et al., 2005), and to include with stress tensors subgrid scale calculations of bubble growth, where the model for bubble growth is discretized ∂u 1 ∂ ∂u r  z  (4.39) at a smaller scale than the conduit model itself τηrr =2− − λ (rur ) +  , ∂rr ∂r ∂z  (Fig. 4.9; Gonnermann and Manga, 2003, 2007). To investigate the role of shear heating

∂uz  1 ∂ ∂uz  or magma cooling near the conduit walls, an τηzz =2− −λ (rur ) +  , (4.40) ∂zr ∂r ∂z  energy equation has to be included. Neglecting thermal diffusion in the vertical direction, the and energy equation is (Bird et al., 1960)

9780521895439c04_p55-84.indd 71 8/2/2012 9:36:36 AM 72 HELGE M. GONNERMANN AND MICHAEL MANGA

0 (a) (b) (c)(d) (e)

1

2

∆pf 3

4 Depth (km)

5

6

7

(pg-pm) 8 0 100 200 00.5 110–2 100 102 104 106 108 10–3 10–1 101 –1 pm (MPa) φb ∆p (MPa) η0 (Pa s) dT/dt (K s )

Figure 4.9 Illustrative model for rhyolite magma rising in a cylindrical conduit of 30 m diameter using Eq. (4.43) 4.7.4 Coupling the conduit and the with sub-grid scale diffusive bubble growth at a constant magma chamber 15 –3 The boundary conditions for the momentum bubble number density of 10 m , H2O-content-dependent non-Newtonian rheology, viscous dissipation, and a constant equations at the bottom of the conduit can be mass discharge rate of 107.5 kg s–1. The vertical axis is a specified magma flux or specified pressure.

depth on all plots. (a) The change in magma pressure pm Either may be transient because of changes in due to magma-static and frictional pressure loss below chamber pressure, in turn a consequence of the fragmentation depth. Above the fragmentation depth magma withdrawal or replenishment, as well as pressure loss is assumed to be linear (Koyaguchi, 2005). compressibility of the vesicular magma and sur- (b) Bubble volume fraction ϕ at the center of the conduit. rounding rock. A simple equation for chamber (c) Bubble overpressure pg – pm (solid line) and fragmentation pressure is obtained by combining conservation threshold Δpf (dashed line). Note the large increase in bubble

overpressure at shallow depths until pg – pm = Δpf, when of mass with an equation of state (Denlinger fragmentation is predicted to occur (Spieler et al., 2004). and Hoblitt, 1999; Wylie et al, 1999)

(d) Newtonian melt viscosity η0 as a function of depth. Note the rapid viscosity increase at shallow depths due to H2O dp 1 exsolution, resulting in viscously limited bubble growth. = ()QQin − out . (4.46) dt ρβeV (e) Viscous heating at the conduit wall becomes significant as viscosity increases, principally within a few hundred meters below the fragmentation depth. Here V is the volume of the magma chamber or

conduit, Qin is mass recharge rate, Qout is mass discharge rate, and β is the effective compress- e ∂T κ  ∂2T 11∂T   ∂u  ibility. The latter incorporates both magma = z ,  2 +  − τ rz  (4.45) ∂tcρρpp ∂rr∂rc  ∂r  compressibility and wall-rock elasticity, and is usually much smaller than the compressibil-

where cp is the specific heat andκ is the thermal ity of vesicular magma (Huppert and Woods, conductivity of the magma. The first term on the 2002; Woods et al., 2006). Typically it is the feed- right-hand side accounts for diffusion of heat, backs between magma chamber and conduit and the last term represents viscous heating. that result in complex and unsteady eruptive

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Figure 4.10 Summary of processes and mechanisms that govern magma flow in the volcanic conduit, based on modeling results and relations discussed in this chapter. Neglecting unsteady effects, discharge rate and viscosity control which processes have sufficient time to become dominant and affect eruption style.

­behavior (Melnik and Sparks,1999; Macedonio discharge rates on the order of 106 kg s–1 (sub- et al., 2005). plinian) and 106–108 kg s–1 (plinian), sustained over several hours (Chapter 8). Syneruptive vola- tile exsolution and crystallization can lead to 4.8 What conduit models viscosity-limited bubble growth, large frictional pressure losses, and large decompression rates have taught us (Papale and Dobran, 1993, 1994), resulting in gas overpressure, supersaturation, and bubble nucle- Figure 4.10 shows the relationship between ation (Massol and Koyaguchi, 2005), as well as magma composition (in essence a proxy for fragmentation (Papale, 1999). viscosity), eruption rate (or equivalently ascent Whether fragmentation takes place depends rate), and eruption style. The boundaries shown on the conditions of magma ascent. However, are not “hard” in that uncertainties in thresholds fragmentation also feeds back on eruption for fragmentation, unsteady flow, variation in dynamics. For example, the presence of pum- volatile content, conduit geometry, two-dimen- ice fragments after fragmentation, as opposed sional flow, and crystallization are not addressed. to complete fragmentation to ash, means that Overall these boundaries are meant to provide not all compressed gas contained in bubbles insight into the conditions under which certain may be released, resulting in ascent velocities processes are expected to dominate and to illus- and pressures that are lower than if all gas were trate their relationships with eruption style. released. Incomplete outgassing of coarse frag- ments also increases the density of the erupting 4.8.1 Subplinian and plinian eruptions gas–pyroclast mixture, with consequences for Subplinian and plinian eruptions are character- the dynamics of the eruption column (Chapter 8, ized by brittle magma fragmentation at mass Papale, 2001). Furthermore, the granular stress

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induced by >~1 cm-size pyroclasts decreases and Martini (2009); James et al. (2009); Pioli exit velocities and enhances the lateral inhomo- et al. (2009). geneity in clast-size distribution throughout the conduit (Dufek and Bergantz, 2005). 4.8.3 Hawaiian eruptions The strong dependence of viscosity on tem- Hawaiian eruptions are characterized by gas-rich perature in silicic magmas is of considerable fountains of basaltic magma sustained for hours importance for shear heating near the conduit to days at mass discharge rates of the order of 104 walls (Rosi et al., 2004; Polacci et al., 2005). The to >106 kg s–1 (Houghton and Gonnermann, 2008). resulting viscosity reduction will change the Fountains reach heights that exceed hundreds of flow profile toward a more plug-like flow and meters producing ash and pyroclasts, presumably reduce frictional pressure loss. Consequently, through hydrodynamic fragmentation (Namiki the discharge rate increases for a given pressure and Manga, 2008). Microtextural studies of pyro- drop between chamber and surface, and hence clasts indicate less open-system gas loss than for affects the depth of fragmentation and allows strombolian clasts (Polacci et al., 2006). for multiple stable discharge rates (Costa and Conduit models by Wilson and Head (1981) Macedonio, 2002; Mastin, 2005; Vedeneeva et al., predict that magma ascent in hawaiian eruptions 2005; Costa et al., 2007). is homogeneous, with dispersed bubbles, so that

Eruption intensity may correlate with pre- rapid decompression and shallow H2O exsolution

eruptive H2O content and magma composition lead to expansion-driven acceleration and the for- (Papale et al., 1998; Polacci et al., 2004; Starostin mation of a sustained magma fountain above the

et al., 2005). However, changes in CO2 content vent (Parfitt et al., 1995; Parfitt, 2004). The tran-

can have the opposite effect of H2O (Papale and sition from hawaiian to strombolian behavior is Polacci, 1999; Ongaro et al., 2006). The transition thought to occur at magma rise speeds of less from explosive to effusive eruptions in silicic than about 1 m s–1, with a correlation between magmas is thought to occur as magma ascent required rise speed and initial magmatic volatile rates decrease, which allows more time for per- content (Parfitt and Wilson, 1995). Alternatively, meable outgassing and bubbles grow at near based on analog laboratory experiments it has equilibrium conditions (Jaupart and Allegre, been suggested that bubbles accumulate to form 1991; Woods and Koyaguchi, 1994; Gonnermann a magmatic foam, whose collapse produces and Manga, 2007). an annular gas-melt flow within the conduit (Chapter 6; Jaupart and Vergniolle, 1988). 4.8.2 Strombolian eruptions Typical strombolian eruptions (Chapter 6) 4.8.4 Effusive eruptions consist of prolonged periods of impulsive Large volumes of basaltic magma often erupt explosions that typically eject 0.01–10 m3 of effusively over prolonged periods of time, at pyroclastic material and volcanic gases at about rates up to 104 kg s–1 (Wolfe et al., 1987). Silicic 100 m s–1. Mass balance considerations imply magmas of high viscosity may erupt effusively to that much of the gas is derived from a larger vol- form lava domes and coulees with time-averaged ume of magma than is erupted, a consequence eruption rates ranging from 10–1 to 104 kg s–1 of slow magma ascent rates (< 0.01–0.1 m s–1). (Pyle, 2000). The resulting low ascent rates This permits significant bubble coalescence to (� 0.01 m s–1) are thought to facilitate permeable form gas slugs and/or accumulations of bubbles, outgassing (Eichelberger et al., 1986; Jaupart which decouple from the melt and rise to the and Allegre, 1991; Woods and Koyaguchi, 1994; surface where they burst. Aside from thermal Massol and Jaupart, 1999). Sparks (1997) attrib- modeling (Giberti et al., 1992), some numerical uted the occurrence of time-dependent behav- conduit models that have dealt directly with ior during dome-forming eruptions to feedbacks details of strombolian eruptions are Wilson and associated with volatile-dependent magma vis- Head (1981); Parfitt and Wilson (1995); James cosity, magma degassing, outgassing, and crys- et al. (2008); O’Brian and Bean (2008); D’Auria tallization. Conduit models suggest that large

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changes in discharge rate at relatively small subaerial flow, and incorporation of bubble changes in boundary conditions are the conse- nucleation. quence of such feedbacks (Melnik and Sparks, (2) The dynamics of magma flow within the 1999, 2005). They also demonstrate how non- volcanic conduit during strombolian and linear feedbacks produce different steady dis- hawaiian eruptions remains controversial. charge rates for the same boundary conditions The main challenge here is the modeling of but different initial conditions, perhaps explain- the separated two-phase flow, where the top- ing how transitions between effusive and explo- ology of the flow can change drastically and sive behavior occur (Melnik et al., 2005; Clarke rapidly in both space and time. et al., 2007). (3) Magma flow in the volcanic conduit is For long-lived dome-forming eruptions, affected by processes that occur over a magma recharge is often treated as constant wide range of length scales, from the fluid over part of the eruption. However, transient dynamics of magma ascent over kilometers eruptive behavior can ensue as a direct con- to crystal/bubble nucleation and growth, sequence of compressibility of the vesicular as well as fragmentation over millimeter magma and/or surrounding wall rock (Meriaux to micrometer scales. Accounting for these and Jaupart, 1995; Denlinger, 1997; Huppert different scales requires the development and Woods, 2002; Costa et al., 2007a). Feedbacks of subgrid-type models capable of coupling may be such that pressure build-up drives the explicit modeling of the various small-scale system into a state where discharge rate exceeds processes with fluid dynamical modeling of recharge rate. With time, pressure decreases magma flow. until the cycle repeats or is dynamically damp- (4) Eruption triggering is perhaps of foremost ened toward a stable state (Barmin et al., 2002). importance in terms of hazard mitigation. High shear stress at the conduit walls, in con- Integration of geodetic and remote-sensing junction with shear-thinning rheology, may observations with models of magma flow, result in shear localization and brittle deform- supply, and storage is thus highly desirable. ation and this may in turn enhance permeabil- A major challenge is the incorporation of ity and outgassing. Conduit models have also geological (tectonic and structural) com- explored how brittle deformation may be asso- plexities, which may require a departure ciated with stick-slip behavior at the conduit for idealized one- or two-dimensional model walls and shallow seismicity, especially for mag- geometries, as well as coupling between mas with high crystal content (Denlinger and tectonically produced stresses and stresses Hoblitt, 1999; Goto, 1999; Collier and Neuberg, within the magmatic system. 2006; Iverson et al., 2006; Hale, 2007). 4.10 Notation 4.9 Summary a conduit radius (m) A cross-sectional area of conduit (m2) Conduit models of magma ascent help us under- B empirical constant stand which mechanisms and processes are c speed of sound in magma (m s–1) important for a broad range of effusive and dry c specific heat (J kg–1 K–1) explosive volcanic eruptions (Fig. 4.10). Some p C concentration of dissolved CO (ppm) important open questions are: c 2 Ci mass fraction of dissolved volatile i

(1) What causes the observed unsteady behavior Cw concentration of dissolved H2O (wt.%) during sustained explosive eruptions? This D diffusivity of dissolved volatile in melt will require models capable of exploring (m2 s–1)

time-dependent dynamical feedbacks within Dc diffusivity of CO2 dissolved in melt the conduit, coupling between conduit and (m2 s–1)

9780521895439c04_p55-84.indd 75 8/2/2012 9:36:38 AM 76 HELGE M. GONNERMANN AND MICHAEL MANGA

Di diffusivity of dissolved volatile species i Qin magma chamber mass recharge rate (m2 s–1) (kg s–1)

Dwb diffusivity of H2O dissolved in basalt Qout magma chamber mass discharge rate melt (m2 s–1) (kg s–1 –1 Dwr diffusivity of H2O dissolved in rhyolite Qm mass flow rate of melt phase (kg s ) melt (m2 s–1) r radial distance from center of bubble or f friction factor for flow in a cylindrical from center of conduit (m) pipe R bubble radius (m)

Fwg wall drag force of the gas phase (N) S radial distance from bubble center to

Fwm wall drag force of the melt phase (N) midpoint of between adjacent g acceleration due to gravity (m s–2) bubbles (m)

G∞ shear modulus of melt (Pa) t time (s)

�GD activation energy for diffusion (J) T temperature (K) * �G free energy for nucleus formation (J) Tl liquidus temperature (K)

h hindering function for Richardson–Zaki �Te undercooling of melt (K) equation u magma velocity (m s–1) –1 –1 H specific enthalpy (J kg ) ug gas velocity (m s ) –1 �H change in enthalpy between melt and um melt velocity (m s )

crystal (J) ur radial component of magma velocity I homogeneous crystal nucleation rate (m s–1) –3 –1 (m s ) uz vertical component of magma velocity –1 I0 reference homogeneous crystal (m s ) –3 –1 –1 nucleation rate (m s ) Ub bubble ascent velocity (m s ) –3 –1 –1 J bubble nucleation rate (m s ) Um ascent velocity of the melt phase (m s ) 2 k permeability (m ) Ut terminal rise velocity of a single bubble –1 kv volumetric factor for Avrami equation in infinite liquid (m s ) –1 kB Boltzmann constant (J kg ) v approach velocity of two bubbles or K bulk viscosity (Pa s) expansion velocity of vesicular magma L characteristic length scale for inertial (m s–1)

fragmentation (m) vf velocity in the melt film between n empirical coefficient bubbles (m s–1)

Nd number of bubbles per volume of vr radial velocity of the melt surrounding melt (m–3) a growing bubble (m s–1)

p magma pressure (Pa) vR radial velocity of the melt-vapor –1 pc partial pressure of CO2 (MPa) interface of growing bubble (m s ) 3 pm pressure of the melt (Pa) V volume of magma chamber (m ) –1 –1 pṁ melt decompression rate (Pa s ) Y crystal growth rate (m s ) –1 pg pressure of the vapor inside Y0 reference crystal growth rate (m s ) bubbles (Pa) z vertical coordinate (m)

pw partial pressure of H2O (MPa) α empirical constant

�pf critical gas overpressure for brittle β empirical constant for Kozeny–Carman fragmentation (Pa) equation 2 –1 �ps supersaturation pressure of volatile in βe effective compressibility (m N ) melt (Pa) γ vapor–melt surface tension (N m–1) Q mass flow rate of magma (kg –1s ) δ empirical constant –1 –1 Qg mass flow rate of gas phase (kg s ) ε strain rate (s )

Qgd mass flow rate of dissolve volatile phase ε̇f critical strain rate for structural failure (kg s–1) of the melt (s–1)

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ε̇st strain rate at which shear thinning the manuscript. Both authors were supported by begins (s–1) NSF while preparing this review. η magma viscosity (Pa s)

η0 Newtonian melt (liquid) viscosity (Pa s)

ηg viscosity of vapor inside a bubble (Pa s) References

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