Numerical simulation of convection and mixing in magma chambers replenished with CO2-rich magma

Numerical simulation of convection and mixing in magma chambers

replenished with CO2-rich magma

Antonella Longo,1 Melissa Vassalli,1,2 Paolo Papale,1 and Michele Barsanti3

Received 9 August 2006; revised 20 September 2006; accepted 27 September 2006; published 3 November 2006.

[1] Magma convection and mixing, and periodic refilling,commonly occur in magma chambers. We show here that thepresence of CO2 in the refilling magma is a very efficientmean of inducing buoyant-driven plume rise and large scaleconvection. Numerical simulations performed with anappositely developed code for the transient 2D dynamicsof multicomponent compressible to incompressible fluidsreveal several features of the processes of plume rise,convection and mixing in magma chambers associated withchamber refilling. A parametric study on CO2 abundance inthe refilling magma shows that progressively larger amountsof this volatile produce a shift from simple plume rise andspreading near the chamber top, to complex patterns of flowcirculation and large scale vorticity and mixing. Lowerchamber depth and lower magma viscosity largely enhancethe efficiency of mixing and convection, favoring theformation of multiple vortexes migrating with time.Citation: Longo, A., M. Vassalli, P. Papale, and M. Barsanti

(2006), Numerical simulation of convection and mixing in magma

chambers replenished with CO2-rich magma, Geophys. Res. Lett.,

33, L21305, doi:10.1029/2006GL027760.

1. Introduction

[2] Magma chamber refilling, convection and mixing arewidely recognized from the analysis of volcanic products[Snyder, 1997]. Such processes occur repetitively in magmachambers, and often trigger volcanic eruptions [Sparks etal., 1977; Pallister et al., 1992; Venetzky and Rutherford,1997; Coombs et al., 2000; Mashima, 2004]. New magmaentering a chamber is commonly less evolved and carriesless H2O and more crystals. Although it can be hotter, itsbuoyancy can be scarce or null [Phillips and Woods,2002].[3] We investigate the capability of CO2 to induce

buoyant-driven convection of the refilling magma. This isdone through numerical simulations of the 2D transientdynamics in magma chambers fed from their bottom,coupled with modeling of the H2O-CO2-silicate liquidthermodynamic equilibrium. The presence of CO2 in magmais expected to produce significant buoyancy forces, sinceeven small amounts of this largely insoluble componentproduce a decrease of the H2O saturation content, increasingthe gas volume in magma [Holloway and Blank, 1994]. Byparameterizing the CO2 content in the feeding magma, weevaluate CO2 efficiency in producing buoyant plumes and

large scale convection, and describe several aspects of theirdynamics. It is shown that increasing amounts of CO2 in therefilling magma produce a transition from simple plume riseto large scale convection and mixing, giving origin tocomplex flow patterns with the formation of stable ormigrating vortexes. Lower chamber depth and lower magmaviscosity largely enhance the efficiency of convection andmixing.

2. Physico-Mathematical Model

[4] First models of thermal-compositional convection inmagma chambers describe the mixing of incompressiblehomogeneous magmas [Oldenburg et al., 1989, and refer-ences therein]. The enthalpy formulation by Spera et al.[1995, and references therein] and Kuritani [2004] alsoaccounts for crystallization. Non-equilibrium multiphaseconvection of crystal-bearing magma associated with verti-cal density gradients was considered by Bergantz [2000].Withdrawal of incompressible magma driven by pressureforces due to chamber replenishment was modeled by Trialet al. [1992, and references therein]. The dynamics ofincompressible/compressible homogeneous bubbly magmadriven by exsolution of H2O and/or caldera subsidence weresimulated by Folch et al. [1999, 2001].[5] Our model describes the time-dependent 2D dynamics

of a compressible-to-incompressible homogeneous multi-component mixture made of liquid in equilibrium with anH2O+CO2 gas phase at local P-T-X conditions. The numer-ical algorithm is based on the finite element formulation byHauke and Hughes [1998]. This consists in a space-timediscretization with Galerkin least-squares and discontinuitycapturing terms, which allow high numerical stability. Thenumerical model is extended here to include a generalformulation for multicomponent fluids, making it particu-larly suitable for the investigation of the dynamics inmulticomponent systems where density changes, mixing,multiple volatile saturation and phase transitions play animportant role.[6] The mass, momentum and energy equations solved

by the model are the following:

@ryk@t

þr � rvykð Þ ¼ �r � rDkrykð Þ; k ¼ 1; . . . : :; n ð1Þ

@rv@t

þr � rv� vþ pIð Þ ¼

r � m rvþrvT� �

� 2

3m r � vð ÞI

� �þ rg

@re@t

þr � rveþ pvð Þ ¼ r � m rvþrvT� �

vþ krT� �

þ rg � v

ð3Þ

ð2Þ

GEOPHYSICAL RESEARCH LETTERS, VOL. 33, L21305, doi:10.1029/2006GL027760, 2006ClickHere

for

FullArticle

1Istituto Nazionale di Geofisica e Vulcanologia, Pisa, Italy.2Also at Dipartimento di Fisica, Universita di Bologna, Bologna, Italy.3Dipartimento di Matematica Applicata, Universita di Pisa, Pisa, Italy.

Copyright 2006 by the American Geophysical Union.0094-8276/06/2006GL027760$05.00

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where t is time, r is mixture density, yk is weight fraction ofkth component, v is the velocity vector, Dk is an effectivecomposition-independent diffusion coefficient of kth com-ponent, p is pressure, I is the identity matrix, m isNewtonian mixture viscosity, g is gravity acceleration, e ismixture total energy per unit mass, k is thermal conductiv-ity, and T is temperature. equation (2) implies that bulkviscosity is assumed to be negligible.[7] The physical properties of the mixture depend on the

local P-T-X conditions. Gas-liquid equilibrium is computedby the non-ideal multicomponent model of Papale et al.[2006]. The effect of dissolved H2O in changing liquid

density and viscosity, and the major effect of CO2 due todecrease of the H2O saturation content, are accounted for.[8] Numerical solutions of equations (1)–(3) are obtained

through the appositely developed finite element C++ codeGALES, which makes use of the OFELI (R. Touzani,OFELI–An Object Finite Element Library, 2005, www.ofeli.net) and Diffpack (Numerical Objects, 1997) libraries.The auxiliary material1 supplied with the present paperincludes the results of test cases which illustrate the capa-bility of the numerical code to simulate the time-dependent,1 or 2D, one- or multi-component, compressible to incom-pressible dynamics of fluids.

3. Numerical Simulations

[9] The simulated system consists of an elliptic non-deformable chamber 4 km wide and 2 km high, with itstop at 3 or 4 km depth, hosting trachytic magma. Themagma initially present in the chamber contains 5 wt% totalH2O, distributed among the liquid and gas phases accordingto the local magmastatic pressure, and no CO2. Magmadensity is calculated by the Lange [1994] equation of statefor the liquid phase, real gas properties, and standardmixture laws for multiphase fluids. Newtonian liquid vis-cosity is modeled as in the work by Misiti et al. [2006], andno-slip boundary conditions at the chamber walls are set.Non-deformable gas bubbles affect mixture viscosity as inthe work of Ishii and Zuber [1979].[10] Although crystals can be accounted for in our

homogeneous formulation, they are neglected for simplicity.However, in order to investigate a wider range of condi-tions, calculated viscosities have been arbitrarily increasedby up to three orders of magnitude in some simulations,corresponding to a crystal content of 40–60 vol% [Costa,2005].[11] At t = 0 new magma having same T, composition,

and total H2O content of the resident magma, but differenttotal CO2 content, enters the chamber from a bottomcentered inlet 200 m wide, at the constant velocity of0.01 m/s. Three cases are considered: in case A the refillingmagma has zero CO2 content as the resident one; in case Bit has a 0.5 wt% total CO2 content; finally, in case C therefilling magma has a large 3.5 wt% total CO2 content.Comparison between the numerical results allows thereforean evaluation of the role of CO2 in inducing buoyancy,convection and mixing in a magma chamber. In a fourthsimulation (case D) the same conditions as in case C areemployed with lower magma viscosity and lower chamberdepth (hence lower chamber pressure). Such changes concurto produce more efficient dynamics of plume rise andconvection, representing an end-member in the range ofthe simulated conditions.[12] Figure 1 shows the calculated H2O-CO2 saturation as

a function of pressure for the different total H2O-CO2 pairsin the simulations, and the resulting conditions in terms ofdensity contrast of the injected with the resident magma.Case A results in neutral buoyancy, while cases B–D areexpected to produce significant buoyancy and convection.

Figure 1. (a) Saturation contents of H2O and CO2 for thethree different total H2O and CO2 pairs adopted in thiswork, calculated with the model of Papale et al. [2006]employed in the present simulations. (b) Composition of thegas phase at equilibrium. In Figures 1a and 1b, letters oncurves refer to simulation cases in Table 1. (c) Fields ofdenser and lighter conditions for the refilling magma withrespect to the resident magma, in terms of total H2O andCO2 content in the refilling magma. Magma characteristicsare the same as those employed in the numerical simulationsand entering the chamber of cases A–C (solid line) or D(dashed line). The lines connect pairs of total H2O and CO2

resulting in a magma density equal to that of residentmagma at chamber bottom. The symbols indicate theconditions pertaining to the simulations A–C (black circles)and D (ringed black circle). Note that the total H2O contentfor the simulated cases is slightly different, reflectingdilution due to the presence of CO2.

1Auxiliary materials are available in the HTML. doi:10.1029/2006GL027760.

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[13] Table 1 reports the parameters used in the simula-tions. The use of same T and composition (apart from CO2)for the resident and refilling magmas implies no use ofequation (3) and simpler conditions with gas-liquid equilib-rium only. Although real cases involve more complexreactions with formation of solid phases, the present sim-plification allows focusing on the role of CO2 in inducingbuoyancy-driven convection.[14] The simulations are executed with 2D cartesian

coordinates and translational symmetry along the third axis.The computational domain represents one half of thechamber, under the assumption of symmetry with respectto its vertical median plane. The discretization grid isprogressively denser toward the symmetry plane and theinlet area, with a total of 1500 nodes 5–140 m apart. Thisresolution implies that mixing here refers to a macroscopicscale, with no reference to the actual molecular scale ofmixing.[15] The time step is also non-uniform, starting with 10�4 s

and progressively increasing to 10�1 s according to localsolution residuals and speed of convergence.

4. Numerical Results

[16] Figure 2 refers to case Awith no CO2 in the refillingmagma. Shown are the evolution of pressure and velocity(Figures 2a–2c), and the distribution of resident and refill-ing magmas (Figures 2d–2f) up to 4 hours after thebeginning of magma injection. At this time a bulge ofnew magma representing 3.7% of the total mass in thechamber has piled up above the chamber inlet to a height of150 m. The dynamic state of magma inside the chamber ispoorly affected, with maximum velocities at chamber inlet.On the contrary, the magmatic pressure (and the gasvolume) change over the entire chamber, due to increasedmass in it. After about 4 hours the pressure has grown byabout 20 MPa, an amount larger than the tensile strength ofmost rocks [Chau and Wong, 1996; Roche and Druitt, 2001;Zhang, 2002] and likely to produce wall rock deformation,fracturing, and dyke injection. The gas volume, which wasinitially 3.3% at the chamber top, is now zero everywhere.[17] Figure 3 shows case B with 0.5 wt% total CO2 in the

refilling magma. At chamber bottom the resident magma isinitially undersaturated (zero gas volume), while the dis-solved H2O content is 4.4 wt% in the refilling magma, with200 ppm dissolved CO2. Accordingly, the gas volume in theinjected magma is 5.7%. The density difference between

resident (2180 kg/m3) and refilling (2100 kg/m3) magmasinduces the formation of a buoyant plume after about 1h300,when the chamber pressure has grown by about 6 MPa.Once formed, the plume rapidly accelerates, since its riseimplies pressure decrease, further volatile exsolution andgas expansion, density decrease and enhanced buoyancy.The plume takes only about 150 to reach the chamber top,then it spreads laterally. This causes the formation of avortex centered at about half chamber height, 3–400 mfrom the chamber axis. After about 2h300 the magmatic

Table 1. Parameters Adopted in the Present Simulations

Resident/Refilling Total H2O,a wt% Total CO2, wt% Depth,b km T, K Compositionc Viscosityd Inlet v, m/s

Case A resident 5 0 4 1220 trachyte 103

refilling 5 0 1220 trachyte 103 0.01Case B resident 5 0 4 1220 trachyte 103

refilling 4.97 0.5 1220 trachyte 103 0.01Case C resident 5 0 4 1220 trachyte 103

refilling 4.82 3.5 1220 trachyte 103 0.01Case D resident 5 0 3 1220 trachyte 1

refilling 4.82 3.5 1220 trachyte 1 0.01aProgressive decrease of H2O content in cases B–D with respect to case A reflect dilution due to the presence of CO2.bDepth of magma chamber top.cOxides reported as composition ‘‘AMS_B1’’ in Table 1 of Romano et al. [2003].dFactor multiplying viscosity as calculated from Misiti et al. [2006], to which the effect of non-deformable gas bubbles is added as explained in text.

Figure 2. Distribution of (a–c) pressure and velocity field,and of (d–f) mass fraction of resident and refilling magma,for case A (no CO2 in the refilling magma) at three timesafter the beginning of magma injection. The diffusivecorona in Figures 2d–2f depends on the use of a diffusioncoefficient (in this case, 10�11 m2/s), necessary fornumerical stability. In the present case, however, diffusionhas no physical meaning since the two resident and refillingmagmas are the same. At the higher Re pertaining to casesB–D in Figures 3, 4 and 5 diffusion is a second order effect,and its numerical value does not affect the dynamicssignificantly.

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pressure has grown by 9.3 MPa, and the mass of newmagma is 2.3% of total mass.[18] Figure 4 shows case C with 3.5 wt% total CO2 in the

refilling magma. The dissolved H2O and CO2 in the refillingmagma at time zero are 3.4 wt% and 330 ppm, respectively;the gas volume is as large as nearly 19%, and the bulkdensity is 250 kg/m3 less than that of the surroundingresident magma. This large density contrast results inenhanced buoyancy, so that a plume starts rising after onlyabout 450. The plume is initially 100 m displaced from thechamber axis, and it soon forms a vortex which moves upwith the rising plume, forcing it close to the chamber axis.The plume reaches the chamber top after 1h50, implying anaverage plume ascent velocity of 1.7 m/s. The progressiveup-rise of the vortex and the acceleration imposed by therising plume cause the detachment of a CO2-rich region,which remains trapped at the vortex center. AdditionalCO2-rich magma is instead transported to the chamber top,then moved laterally forming a large vortex at the chamberscale. After 1h160 the pressure and total mass in the chamberhave grown by about 5 MPa and 0.12%, respectively.[19] Cases A, B and C involve high-viscosity magma and

chamber top at 4 km depth (Table 1). In case D (Figure 5)the same volatiles as in case C are considered, but theviscosity is three orders of magnitude lower, and thechamber top is at 3 km depth (Table 1). The combinationof lower pressure, thus enhanced volatile exsolution, andlower viscosity results in enhanced buoyancy and muchfaster dynamics. Only after 304000 a buoyant plume hasreached the height of 250 m above the chamber bottom(Figure 5b). The range of Re along the rising plume (basedon local plume width and average properties and velocity) is102–103, compared with values 1 in cases B and C. Such

Figure 3. Distribution of density and velocity field forcase B (0.5 wt% CO2 in the refilling magma) at six differenttimes after the beginning of magma injection.

Figure 4. Distribution of density and velocity field forcase C (3.5 wt% CO2 in the refilling magma) at six differenttimes after the beginning of magma injection.

Figure 5. Distribution of density and velocity field forcase D (3.5 wt% CO2 in the refilling magma, lowerviscosity and chamber depth with respect to cases A–C,see text) at six different times after the beginning of magmainjection.

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a high Re (still in the range of dominantly laminar motion[Bergantz, 2000]) induces the formation of a strong vortex afew hundred meters above chamber inlet (Figure 5c), andflow instabilities which fragment the plume into separatedbatches of rising magma. After 604000 the plume reaches thechamber top (Figure 5d). At this time two vortexes arepresent. The largest one, centered at a height of about1.25 km, 300 m from chamber axis, is associated with themain plume. The second vortex, centered at about 650 mfrom the bottom, 500 m from chamber axis, partly re-circulates magma which descends from the chamber top.Bulk density in the plume top is 1650 kg/m3, 7% less thanthe initial density at same level. After 802000 the large vortexhas moved away from the chamber axis (Figure 5e), whilethe second vortex has been adsorbed into the main plume.Magma re-circulates from the vortex into the rising plume,then again in the vortex, involving the whole chamber. After1104000 the total mass in the chamber has grown only byabout 0.02%. The main vortex has moved down along thechamber border and a number of small vortexes around itproduce complex patterns of magma flow and re-circulation(Figure 5f).

5. Discussion and Conclusions

[20] To simplify the problem, we have assumed that therefilling magma carries the same total H2O as the residentmagma. However, the efficiency of CO2 in producingmagma convection is not limited to such cases. In fact, asshown in Figure 1c, buoyant conditions may occur with arefilling magma having total volatile content lower than thatof the resident magma. This is due to the much lowersolubility of CO2 with respect to H2O, and to its large effectin reducing the H2O saturation content (Figure 1a).[21] Two main processes concur to change the total

volatile content of evolving magmas. These are crystalliza-tion, which implies volatile concentration with respect to theliquid phase, and open system degassing, which leadsinstead to volatile depletion. While crystallization doesnot produce a change of the total CO2/H2O ratio (exceptin cases where a volatile enters the crystal lattice), opensystem degassing invariably results in a remarkable deple-tion of the less soluble CO2 component [Papale, 2005]. Asa consequence, deep magmas which have undergone lowerextent of gas loss likely contain less H2O and more CO2

than shallow, degassed magmas. This excess CO2 maybe sufficient to trigger buoyancy of deep into shallowmagma, producing complex patterns of chamber dynamicsillustrated in this work.[22] As a summary, the numerical simulations presented

here show the following: (i) the presence of CO2 in therefilling magma is a very efficient mean of producingbuoyant plume rise, large-scale convection, and magmamixing; (ii) larger amounts of CO2 translate into fasterdynamics of convection and more efficient magma mixing;(iii) complex flow patterns characterized by multiple vor-texes, either stable in position or migrating with time, candevelop as a consequence of chamber refilling by CO2-richmagma; (iv) increasing CO2 contents, as well as decreasingmagma viscosity and chamber depth, produce a shift fromsimple plume rise and spreading at the chamber top, to earlyvortex formation and large-scale convection. Such a variety

of behaviors can be synthesized by means of the Archi-medes number Ar = gd3 rDr/m2, where d is a characteristicdimension (200 m for magma entrance in the presentsimulations). With reference to the conditions characterizingmagma entrance in the chamber and plume formation, case Dcharacterized by very fast convection dynamics and migratingvortexes corresponds to Ar � 105. Cases B and C, whereplume rise occurs on a time scale of less than one to nearlytwo hours, correspond to Ar � 0.5–5. Finally, the lowdensity core in the vortex of case C, for which no buoyancyoccurs, corresponds to Ar � 10�2.[23] The major limits of the present study are represented

by the followings: (i) the inlet velocity is not allowed tovary with time. This implies that the confidence in thenumerical results decreases as long as the pressure insidethe magma chamber increases, since this is expected toproduce a parallel decrease of the inlet velocity; (ii) me-chanical separation between gas and liquid phases is notallowed. As noted before [Phillips and Woods, 2002], gas-liquid decoupling can reduce the efficiency of convection,since the degassing magma would progressively increase itsdensity and decrease buoyancy. The present results aretherefore more appropriate in cases where inertia is smalland Ar relative to gas bubbles is �1, or where inertiadominates over gas bubble buoyancy, i.e., Ar/Re2 � 1. Inthe present case, efficient separation of gas bubbles wouldoccur for bubble diameters of the order of 10�1 m; (iii) theonly chemical reactions allowed to occur are those related tomulticomponent gas-liquid equilibrium. In real cases itshould be expected that the resident and refilling magmashave different composition, temperature, and crystal contentbesides having different volatile content. Therefore, mixingand convection would be accompanied by heat exchange,complex patterns of chemical diffusion, and chemical reac-tions involving gas-liquid-solid equilibria. The effects ofthese complex patterns should form the object of futureinvestigation with more sophisticated modeling; (iv) thepresent simulations are performed in 2D with the assump-tion of translational symmetry along the third dimension,which is therefore implicitly assumed to be much longer.Neglected 3D effects might affect both the plume shape andvortex patterns.[24] Taking in mind the above limits in the present

analysis, as long as CO2 is present in appreciable quantitiesin the feeding magma, and magma chamber evolution isaccompanied by open system degassing, convection andmixing are expected to be common occurrences followingthe injection of deep magma into a magma chamber. Sincethe efficiency of convection and mixing increases withincreasing difference in CO2 content between the refillingand resident magmas, and since this difference increaseswith increasing amount of volatiles lost from the magmachamber, it is expected that more efficient chamber degass-ing or longer periods with no chamber refilling be followedby more efficient convection and mixing dynamics with thenext episode of magma ingression into the chamber.

[25] Acknowledgments. This work was funded by MIUR-FIRBproject RBAU01M72W, and by the Italian Dipartimento della ProtezioneCivile in the frame of the 2004–2006 Agreement with Istituto Nazionale diGeofisica eVulcanologia– INGV.AntonioCosta and an anonymous reviewerare greatly acknowledged for their useful comments and suggestions.

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�����������������������M. Barsanti, Dipartimento di Matematica Applicata, Universita di Pisa,

Via Buonarroti 1, I-56127 Pisa, Italy.A. Longo, P. Papale, and M. Vassalli, Istituto Nazionale di Geofisica

e Vulcanologia, Sezione di Pisa, Via della Faggiola 32, I-56126 Pisa, Italy.([email protected])

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