Modeling heat transfer from a convecting ... - Wiley Online Library

Article Volume 12, Number 9 17 September 2011 Q09010, doi:10.1029/2011GC003612 ISSN: 1525‐2027

Modeling heat transfer from a convecting, crystallizing, replenished silicic magma chamber at an oceanic spreading center Lei Liu School of Earth and Atmospheric Sciences, Georgia Institute of Technology, 311 Ferst Drive, Atlanta, Georgia 30332, USA ([email protected]) Now at 540 Buckingham Road, Richardson, Texas 75081, USA

Robert P. Lowell Department of Geosciences, Virginia Polytechnic Institute and State University, 4044 Derring Hall, Blacksburg, Virginia 24061, USA ([email protected]) [1] Most hydrothermal systems at oceanic spreading centers are underlain by basaltic magma bodies;

however, some are underlain by higher‐silica magmas such as andesite or dacite. The different viscosity of the latter magmas, which results from their higher SiO2 content, lower liquidus and solidus temperatures, and higher water contents, affects the rate of heat transport from these magmas, and the behavior of the overlying hydrothermal system. We construct viscosity models for andesite and dacite melts as a function of temperature and water content and incorporate these expressions into a numerical model of thermal convective heat transport from a high Rayleigh number, crystallizing, replenished axial magma chamber (AMC) beneath a hydrothermal circulation system. Simulations comparing the time‐dependent heat flux from dry basalt, 0.1 wt.% H2O andesite, 3 wt.% H2O andesite, and 4 wt.% H2O dacite indicate that higher‐viscosity magmas convect less vigorously, resulting in lower heat flux, possibly lower vent temperatures, and a slower decay rate of the heat flux. Hydrothermal systems driven by unreplenished high‐silica melts may have slightly longer lifetimes than those driven by basalt; however, vent temperature and heat output decay on decadal time scales. Magma replenishment at a rate of ∼10−8–10−7 m/s across the base of the AMC can maintain relatively stable heat output between ∼107–109 Watts and typical hydrothermal vent temperatures. Such replenishment rates are not likely to result from buoyancy‐driven melt transport by porous flow through the lower crust, especially for high‐viscosity magmas such as andesite and dacite. Components: 8600 words, 13 figures, 1 table. Keywords: andesite; hydrothermal processes; oceanic spreading centers. Index Terms: 3001 Marine Geology and Geophysics: Back-arc basin processes; 3017 Marine Geology and Geophysics: Hydrothermal systems (0450, 1034, 3616, 4832, 8135, 8424); 3035 Marine Geology and Geophysics: Midocean ridge processes. Received 8 March 2011; Revised 22 July 2011; Accepted 22 July 2011; Published 17 September 2011. Liu, L., and R. P. Lowell (2011), Modeling heat transfer from a convecting, crystallizing, replenished silicic magma chamber at an oceanic spreading center, Geochem. Geophys. Geosyst., 12, Q09010, doi:10.1029/2011GC003612.

Copyright 2011 by the American Geophysical Union

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LIU AND LOWELL: HEAT TRANSFER FROM HIGH SILICA MAGMAS

1. Introduction [2] Hydrothermal systems at oceanic spreading centers play an important role in Earth’s thermal budget, geochemical exchanges between the ocean and crust, the formation of ore deposits, the support of microbial and macrofaunal ecosystems, and even for the development of life on early earth. As our understanding of hydrothermal processes at oceanic spreading centers has advanced over the last 30 years, the substrate composition, permeability of the oceanic crust, and the nature of the underlying heat source have become recognized as important factors influencing fluid‐rock interactions [Hajash and Chandler, 1981; Fouquet et al., 1993; Wetzel and Shock, 2000], hydrothermal vent chemistry [Von Damm et al., 1985; Von Damm, 1995], and the attendant microbial and macrofauna ecosystems [Tunnicliffe and Fowler, 1996; Juniper and Tunnicliffe, 1997; Kelley et al., 2002]. Axial magma chambers (AMCs) have now been found along sections of ridge axis over a full range of spreading rates and a variety of subsurface rock types [e.g., Detrick et al., 1987, 1993; Collier and Sinha, 1990; Canales et al., 2006; Van Ark et al., 2007; Singh et al., 2006; Sauter et al., 2004; Jacobs et al., 2007] in areas of active hydrothermal venting. These sill‐like magma bodies are typically tens to at most a few hundred meters thick and approximately 500 m to ∼2 km across axis. They may extend with some slight gaps for many tens of km along axis [Detrick et al., 1987; Canales et al., 2006; Singh et al., 2006]. Seismic reflection data thus reveals a close spatial association between hydrothermal venting and an underling relatively thin AMC at nearly all the high‐ temperature vent sites for which seismic imaging experiments exist. It is now generally accepted that the AMC represents the heat source that drives the hydrothermal circulation at oceanic spreading centers [e.g., Kelley et al., 2002; Maclennan, 2008; Lowell et al., 2008; Lowell, 2010]. [3] Since most vent fields are hosted by basalts, detailed experimental, theoretical, and field studies have been carried out on basaltic magma‐hosted hydrothermal activity. However, vigorous hydrothermal systems are hosted by a broad range of rock types. For example, the Rainbow vent field and others on the Mid‐Atlantic Ridge are hosted by peridotite [Kelley et al., 2001; Allen and Seyfried, 2003]. Pacmanus hydrothermal systems are hosted by dacitic volcanoes in the Eastern Manus back‐arc basin [Binns and Scott, 1993; Binns, 2003]. Some vent fields on the Eastern Lau Spreading Center (ELSC) (e.g., Valu Fa Ridge [Fouquet et al., 1993])

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and other back arc basins (e.g., Manus Basin) [Sinton et al., 2003] are hosted by andesitic rocks. Moreover, the occurrence of high‐silica andesitic and dacitic rocks near the 9°03′N overlapper on the East Pacific rise and at other sites along the mid‐ocean ridge system is reported by Wanless et al. [2010]. [4] Hydrothermal venting driven by andesite and dacite magmas is somewhat different from most mid‐ocean ridge vent fields that are hosted in basalts. For example, some high‐temperature hydrothermal fields at the Valu Fa Ridge of the Lau back‐arc basin [Fouquet et al., 1991] exhibit very low pH [Gamo et al., 1997] and high concentrations of trace metals (e.g., Zn, Cd, Pb, As) [Fouquet et al., 1993]. Primary gold is present in the accompanying mineral deposits [Fouquet et al., 1993]. Recent detailed exploration of Eastern Lau Spreading Center has discovered a number of other sites (i.e., ABE, Tu’i Malila, and Mariner) hosted on andesite [Tivey et al., 2005; Resing et al., 2008], and underlain by a magma chamber [Jacobs et al., 2007]. [5] The physical and geochemical properties of andesite and other high‐silica magmas are different from those of basalt. For example, compared to basaltic magma, high‐silica magmas have lower liquidus and solidus temperatures, greater viscosity, and lower density due to their higher water and SiO2 content [e.g., Spera, 2000]. These features affect the convective properties of the magma and correspondingly influence the magma heat output into the overlying hydrothermal system. [6] Since few studies have been conducted regarding high‐silica substrates, an analysis of hydrothermal systems driven by high‐silica magma chambers is needed. Here, we investigate convective heat transfer from cooling and replenished high‐silica andesite and dacite‐hosted magma chambers and its effects on the hydrothermal activity. The approach in this paper is similar to that of Liu and Lowell [2009] for basaltic magma sills; the principal difference here stems from the different thermodynamic and mechanic properties of andesitic and dacitic magmas. The rest of this paper is organized as follows. Section 2 describes dependence of magma viscosity on temperature, water content and crystallinity as a function of magma composition (SiO2) and presents the formulas for magma viscosity that we use in the simulations. Section 3 develops the mathematical formulation of magmatic heat transfer based on the proposed magma viscosity models and presents the results. Section 4 discusses the evolution of the magma chamber, the temperature of hydrothermal and magma systems, 2 of 17


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Table 1. Values of the Physical Parameters for High‐Silica Magma System Physical Meaning Thermal diffusivity of magma Vent field discharge area Horizontal magma area in chamber Grain size Specific heat of fluid Andesitic magma specific heat Basaltic magma specific heat The total thickness of magma chamber The initial thickness of magma chamber The thickness of liquid magma Hydrothermal heat flux Magmatic heat flux Magma replenishment heat flux Acceleration due to gravity The velocity of melt migration Permeability of hydrothermal discharge Permeability of matrix Latent heat of crystallization of andesite Latent heat of crystallization of basalt Rayleigh number Basalt liquidus temperature Basalt solidus temperature Dry Andesite liquidus temperature 3 wt.% Andesite liquidus temperature Dry Andesite solidus temperature 3 wt.% Andesite solidus temperature 4% dacite liquidus temperature 4% dacite solidus temperature Hydrothermal vent temperature Replenishment rate Hydrothermal specific discharge Thermal expansion coefficient of fluid Thermal expansion coefficient of basalt Thermal diffusivity Thermal conductivity Density of dry andesite Density of 3 wt.% andesite Density of basalt Density of dacite Density of fluid Density of solid Porosity Fluid kinematic viscosity Magma kinematic viscosity Volume fraction of crystals Critical crystal fraction The crystal content of input magma

Parameter

Value −7

m2/s m2 m2

8 × 10 104 106

b cf cm

1–10 5 × 103 1000 1400

Mm J/kg°C J/kg°C J/kg°C m

100

m

D0 Dm Fh Fm Fr g J k

M W/m2 W/m2 W/m2 m/s2 m/s m2

9.81

3 × 105

m2 J/kg

4.2 × 105

J/kg

Ra TbL TbS TaL

1200 1030 1100

‐ °C °C °C

TawL

1010

°C

TaS

970

°C

TawS

900

°C

TdwL TdwS Th u ud

950 800

°C °C °C m/s m/s

k L

af

−3

10

°C −5

am

5 × 10

lm ra raw rb rd rf rs vf nm c cc cin

8 × 10−7 2 2.5 × 103 2.4 × 103 2.7 × 103 2.2 × 103 103 2.9 × 103 1%–10% 10−7 variable 60%

and magma replenishment mechanisms. Section 5 concludes the paper.

Units

am Ad Am

D

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°C

2. Magma Viscosity [7] Magma viscosity can vary by many orders of magnitude within a single environment, and depends on a variety of factors such as chemical composition [Liebske et al., 2003], temperature [Bottinga and Weill, 1972; Shaw, 1972], pressure [Kushiro et al., 1976; Scarfe et al., 1987], volatile content (mainly H2O) [Richet et al., 1996; Whittington et al., 2000], and crystal content [Marsh, 1981; Lejeune and Richet, 1995]. The viscosity of naturally occurring silicate melts varies from 10−1 Pa s to 1014 Pa s in response to changes of magma temperature and melt composition [Dingwell, 1996; Giordano et al., 2004, 2008]. Because the viscosity of magma exerts a fundamental control over many geological processes, including the segregation of melt from source regions, ascent and emplacement of magma, mixing in magma chambers, and eruption styles, magma chambers hosted by different magmas exhibit different rates of convective heat transfer and temporal evolution, and thus affect the overlying hydrothermal system in different ways. In this paper we construct a viscosity model for an andesitice or dacitic AMC that incorporates the effects of water content, temperature and crystal content of the melt. Because MORBs are essentially dry, we assume that the viscosity of basalt is a function of temperature and crystal content only. [8] For the dependence of magma viscosity on crystal content we use the classical Einstein‐ Roscoe approach [Roscoe, 1952] and write vm ¼ vm0 ð1 =c Þn

ð1Þ

−1 −1

m2/s Wm/°C kg/m3 kg/m3 kg/m3 kg/m3 kg/m3 kg/m3 ‐ m2/s m2/s ‐ ‐ ‐

where vm denotes the kinematic viscosity of the melt‐crystal mixture, c denotes the volume fraction of crystals, cc is critical fraction value at which crystals effectively solidify the suspension, resulting in n m approaching infinity, vm0 denotes the viscosity of the liquid magma, and n is an adjustable constant exponent, with the range from 1.5 to 5. A full list of variables and parameters is given in Table 1. Based on the experimental and empirical evidence of Marsh [1981], the parameters cc = 60% and n = 2.5 are widely adopted as typical values [Shaw, 1980, Marsh, 1981]. Correspondingly, the magma viscosity in equation (1) becomes vm ¼ vm0 ð1 1:67Þ2:5

ð2Þ 3 of 17


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We then note that magma viscosity depends indirectly on temperature through the relationship between magma temperature and crystal content. In the absence of a more precise formulation, we follow Hort [1997] and assume that the crystal content c is linearly related to magma temperature Tm over the melting range. That is, ðTm ðt ÞÞ ¼

TL Tm ðt Þ TL Ts

ð3Þ

where TL and Ts denote the liquidus and solidus temperature, respectively, and t is time. [9] To obtain the prefactor, n m0 in equation (2) for

different melt compositions, we use the formulations of Vetere et al. [2006] and Whittington et al. [2009], who relate the magma viscosity to temperature Tm in °C and water content w in wt % for andesite and dacite, respectively. That is, for andesite: log10 va0

¼ 4:86 þ

8198 6060 ðTm 257Þ ðTm 300Þ w log10 a 1:1673 ðw 1:1952 þ 0:0056 Tm Þ ð4Þ

and for dacite: log10 vd0 ¼ 4:43 þ

7618:3 17:25 log10 ðw þ 0:26Þ Tm 679:1 þ 292:6 log10 ðw þ 0:26Þ

log10 d

ð5Þ

where ra and rd represent the density of andesite and darcite (kg/m3), respectively. For dry basalt, we use the fit to experimental data shown by Spera [2000], vb0 ¼ v0 ðTL =Tm ðtÞÞ

8:5

ð6Þ

[see Liu and Lowell, 2009]. The general viscosity formulation from equation (2) is then written as vc ðTm ðtÞÞ ¼ vc0 ½1 1:67ðTm ðt ÞÞ2:5

ð7Þ

where c is either “a,” “b,” or “d” to denote andesite, basalt, or dacite, respectively. The prefactor vc0 is given by equations (4), (5), or (6), and c(Tm(t)) denotes the crystal content from equation (3). [10] For demonstration purposes, we show the rela-

tionship between magma viscosity and crystal content, water content, and temperature, respectively. Four different magmas are taken into account based on their water content, including dry basaltic magma, 0.1 wt.% H2O andesitic magma, 3 wt.% H2O andesitic magma, and 4 wt.% H2O dacitic magma. The solidus temperature TS and the liquidus temperature TL are selected to be 1030°C and

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1200°C for dry basaltic magma, 970°C and 1100°C for 0.1 wt.% H2O andesitic magma (dry andesitic magma), 900°C and 1010°C for 3 wt.% H2O andesitic magma, and 800°C and 950°C for 4 wt.% H2O dacitic magma, respectively. First, we show the relationship between magma viscosity and crystal content in Figure 1, assuming the prefactor in equation (7) is constant. Figure 1 shows that the viscosity monotonically increases as a function of crystallinity for all types of magma. For the same crystal content, dacitic magma has the highest viscosity, and both andesitic and dacitic magmas have the higher viscosity than the basalts due to the higher silica content. Moreover, viscosity increases nonlinearly with increasing crystal content, especially when the crystal content approaches the critical value of 60%. Magma convection stops when the crystal content approaches 60%. [11] The dependence of viscosity on water content

is shown in Figure 2 for andesite and dacite. Figure 2 shows that water content has a significant effect on viscosity and that the higher the silica content, the greater the effect of small changes in water content on the viscosity, i.e., water decreases the viscosity of the dacitic magma more than for andesites. Therefore, the determination of the water content of magma is critical for understanding the dynamic behavior of melts. In addition, Figure 2 shows that viscosity increases sharply for dacites when water content is below 3 wt. % and that a further increase in the amount of water above 3 wt.% has little influence on the viscosity (see equations (4) and (5)). [12] Last, the relation between the magma viscosity

and temperature is shown in Figure 3 for different magmas, including the effect of crystallization as the temperature decreases. Although different magmas have different temperature ranges during convection, the viscosity increases with a decrease of magma temperature between their respective liquidus and solidus temperatures, because cooling generates more and more crystals and correspondingly increases viscosity. Also, we observe that the range of viscosity of andesites and dacites are greater than that of basalts due to the high SiO2 content. In the following sections, we will discuss heat transfer from a magma chamber based on the proposed viscosity models.

3. Mathematical Formulation and Results [13] To better understand the evolution of heat

flux from the convecting, crystallizing magma to 4 of 17


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Figure 1. The relationship between magma viscosity and crystal content in the crystals suspended case for different magmas. The vertical dashed line shows that the viscosity becomes infinite when crystal content reaches 60%.

hydrothermal system, Liu and Lowell [2009] adapted an approach by Huppert and Sparks [1988] and developed numerical models of magmatic heat flux for basaltic magma. In this section, we extend this work to the investigation of the hydrothermal system behavior driven by high‐silica magmas and compare the characteristics of different hydrothermal systems driven by basalt, andesite and dacite with different water content. [14] The mathematical formulation of the problem

is essentially identical to that of Liu and Lowell [2009]; consequently, we refer the reader to that paper for most of the details concerning the model-

ing rationale and discussion of the equations. For convenience, a number of key equations are given in Appendix A. As in the work of Liu and Lowell [2009], we consider convective magmatic heat transfer both with and without magma replenishment. We also show results for end‐member cases in which (1) crystals stay suspended in the melt and (2) crystals settle rapidly to the floor of the AMC. Liu and Lowell [2009] argued that for basaltic AMCs, the crystals settling cases appeared to be more realistic. We consider the crystals suspended cases here, because crystal settling rates may be lower in high‐viscosity silicic magma chambers and the results provide a useful end‐member scenario.

Figure 2. Viscosity of andesitic and dacitic magma at a fixed magma temperature of 970°C plotted as a function of wt% H2O. The greater the silica content of the magma, the greater the effect of H2O on viscosity, especially when H2O is less than 3 wt.%. 5 of 17


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Figure 3. Viscosity versus magma temperature with varying wt.% H2O for basaltic, anhydrous and 3 wt.% andesitic, and 4 wt.% dacitic magmas in crystals suspended case. The temperature affects the melt viscosity significantly, especially at low temperatures.

3.1. Convective Heat Transfer Without Replenishment [15] Figures 4 and 5 show the total heat output from

a 100 m thick sill, with a surface area of 106 m2, for the crystals suspended and crystals settling scenarios, respectively. The magmas have a range of compositions, and water contents; and viscosity is determined from equation (7). We assume that the heat transferred from magma chamber is carried solely by hydrothermal convection, i.e., no heat loss through surrounding rock by conduction occurs. Because the estimated range of the integrated heat fluxes in hydrothermal vents at oceanic spreading centers is between 107 and 109 Watts

[Baker, 2007; Ramondenc et al., 2008], we assume that hydrothermal circulation effectively stops when the magmatic heat output drops below 107 Watts. The horizontal line at 107 Watts in Figures 4 and 5 denotes this value. [16] Figure 4 shows that the initial total heat output

and decay rate (or effective convective lifetime) are both functions of magma viscosity. As shown, for example by the basalt and andesite heat output profiles versus time, the lower the viscosity, the higher the initial heat output. This is essentially a function of Rayleigh number. On the other hand, the higher the viscosity, the more slowly heat output decays and the longer the effective con-

Figure 4. Total heat output from basaltic, andesitic and dacitic magma plotted as a function of time in crystals suspended case, without magma replenishment. 6 of 17


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Figure 5. Total heat output from basaltic, andesitic (anhydrous and with 3 wt.% H2O), and dactic with 4 wt.% H2O magmas plotted as a function of time in crystals settling case, without magma replenishment.

vective lifetime. For example, Figure 4 shows that dacitic magmas have the slowest decay rate followed by andesites and basalts, because dacites have the highest SiO2 content and the correspondingly highest viscosity. For the same type of magma, for example andesite in Figure 4, the heat output from magma with 3 wt.% H2O decays faster than the one with 0.1 wt.% H2O. As a result of the slower decay in heat output, the effective convective lifetime increases as viscosity increases. While a basalt‐driven hydrothermal system lasts less than 10 years, a hydrothermal system driven by 0.1 wt. % H2O andesitic magma lasts for 30 years before the total heat output drops below 107 watts. [17] A comparison between Figure 5 and Figure 4 shows that under similar conditions the effective convective lifetime for the different magmas is greater for crystals settling than for crystals suspended models. This occurs because of the rapid increase in viscosity and decrease in Rayleigh number when crystals are suspended. Even for the basaltic magma, the lifetime more than doubles for the crystals settling versus crystals suspended models. Systems with higher initial viscosity and lower initial heat output such as the 4 wt.% H2O dacitic magma have the longest convective lifetime. This occurs because the lower the initial heat output, the lower the overall rate of decay.

3.2. Convective Heat Transfer With Magma Replenishment [18] Although the convective heat output from

high‐silica andesite and dacite magmas decays more slowly than that from lower viscosity basalt,

the results of the previous section show that heat output decays on decadal timescales. Thus, following the methodology of Liu and Lowell [2009], we investigate whether heat transfer from high‐ silica magmas can also reach quasi‐steady state with reasonable rates of replenishment from the underlying mush zone. With u(t) as the velocity of magma replenishment, the rate of heat replenishment Fr(t) is expressed by Fr ðtÞ ¼ ½m cm Tin þ m Lð1 in ÞuðtÞ

ð8Þ

where Tin denotes the replenished magma temperature, and cin is the corresponding crystal content, which can be obtained from equation (3) with Tm(t) replaced by Tin. In addition, the latent heat remaining in the newly added magma depends linearly on the fraction of liquid magma being added, i.e., L(1 − cin). As in the previous section, the initial thickness of magma is set to 100 m, and we assume that during replenishment the thickness of the AMC increases as described by equation (A8) in Appendix A. Because seismic evidence indicates that the thickness of AMCs is less than a few hundred meters, we run the simulations until the thickness of the magma chamber doubles. [19] Figure 6 shows the total magmatic heat output

in the crystals suspended case with a constant replenishment rate of 5 × 10−8 m/s for the four different magmas considered previously. Figure 6 shows that with magma replenishment, the magma chamber size doubles after 65 years and the total heat output reaches a quasi‐steady state between 107 and 109 W. For the same initial chamber size and constant rate of replenishment, heat output 7 of 17


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Figure 6. Total heat output as a function of time for four types of magma in the crystals suspended case with a constant magma replenishment rate at 5 × 10−8 m/s. Initial sill thickness is 100 m and simulations stop when the thickness of magma chamber doubles.

decays more slowly for high‐silica magmas than for basaltic magma. As in the case of without replenishment, this occurs because the high‐silica magmas have higher viscosity, which results in lower Rayleigh numbers. Thus, convection in high‐ silica magmas is less vigorous, and provides lower heat output than otherwise equivalent basaltic magma. In addition, the steady state magmatic heat outputs of high‐silica magmas drop below that of basaltic magma. For replenished magmas, heat output represents a balance between cooling and heat input from new magma. Because heat output from basaltic magma chambers initially decays more rapidly than from more silicic ones, and the rate of heat input is the same, basaltic magmas reach equilibrium more quickly than high‐silica magmas. [20] With the same simulation setup as in Figure 6,

Figure 7 shows the total heat outputs for different magmas in the crystals settling case. Similar to the crystals suspended case, the thickness of all the magma chambers doubles after 65 years, because the growth of magma thickness depends only on the magma replenishment rate. Also, for each magma type, the heat output decays more slowly than for the corresponding case of crystals suspended (see Figures 6 and 7, respectively), because the viscosity in the crystals settling case is lower than in the crystals suspended case.

4. Discussion [21] The mechanism of heat extraction by hydro-

thermal circulation along mid‐ocean ridges is

fundamentally the same among fast, intermediate, and slow spreading centers and back‐arc basins. Thus, the variations in AMC properties have implications for both the evolution of the AMC and for the hydrothermal heat flux. Below we discuss the evolution of liquid magma thickness, magma and hydrothermal temperature, and the physics of magma replenishment.

4.1. Liquid Magma Thickness in Crystals Settling Model [22] In this part, we discuss the evolution of liquid

magma thickness. For demonstration purpose, we only consider the crystal settling case. Figures 8 and 9 show the time‐varying liquid magma thickness during convection for different magmas with and without magma replenishment, respectively. For the case without replenishment, and an initial magma chamber thickness of 100 m, the thickness of liquid basalt quickly drops to 40 m; in conjunction with rapidly decreasing heat output and the end of the convection (see Figure 5). In contrast, the thickness of liquid, high‐silica magma decreases more slowly as a result of the slow decay of the heat flux (see Figure 5). With magma replenishment, all the magma chambers grow for 65 years, corresponding to a doubling of the initial magma thickness at the replenishment rate of 5 × 10−8 m/s. The liquid magma thickness is greater than 40 m at steady state for all types of magmas. The thickness of basaltic liquid magma layer decreases rapidly during the first decades of magma chamber cooling and crystallization. A 100 m thick melt lens thins 8 of 17


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Figure 7. Total heat output as a function of time for four types of magma in the crystals settling case with a constant magma replenishment rate at 5 × 10−8 m/s. Initial sill thickness is 100 m and simulations stop when the thickness of magma chamber doubles.

to 45 m (55%), because there is more vigorous convection in basaltic magma so that the convective heat loss exceeds the rate of heat input from replenishment. After the heat flux from magma replenishment balances the convective heat loss, the thickness of liquid magma remains stable over time. Even though the thickness of liquid basaltic magma is less than that of liquid high‐silica magmas, the total heat output of basalt system is higher than that of high‐silica magma systems (see Figure 7). This is a result of the lower viscosity of basalt.

4.2. Evolution of Magma and Hydrothermal Temperature [23] Based on the rate of magma replenishment in

equation (8), different magma replenishment temperatures may lead to different magma chamber evolution. Assuming that latent heat is released uniformly between the liquidus and solidus temperatures, magmatic heat is linearly proportional to its temperature, which can be obtained from equation (A2) by replacing Tm with Tin. In addition, the latent heat is linearly dependent on the liquid fractionation of the added magma, i.e., L(1 − cin).

Figure 8. The thickness of liquid magma as a function of time for different magmas for crystals settling cases without magma replenishment. Initial sill thickness is 100 m. 9 of 17


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Figure 9. The thickness of liquid magma as a function of time for different magmas in the crystals settling model with magma replenishment at 5 × 10−8 m/s and an initial sill thickness of 100 m.

We consider two scenarios of the replenished magma at temperature Tin. One is that Tin is less than liquidus temperature TL. In this case, there has already been a certain amount of latent heat lost from the incoming melt. The other is that the original magma has less initial latent heat, i.e., its initial temperature Tm(0) is less than TL. In this case, we set Tin to be TL, which is larger than Tm(0). Figures 10 and 11 show the total heat output of the magma chamber for these two cases, respectively. For both cases, we observe that the lower the replenishment temperature, the faster the decay of the heat flux. However, the effect of the temperature of added magma on the total heat output behavior of the magma chamber is slight.

[24] Liu and Lowell [2009] assume that magmatic

heat supply is directly coupled to hydrothermal heat flux and vent temperature. Here we employ the same relationships to investigate this link for different AMC magma compositions. We assume that no heat loss occurs during the heat transfer between the magma and hydrothermal fluid, and that the variation of hydrothermal temperature Th and heat flux Fh instantly reflect the heat flux from the magma chamber. Thus, we have [Liu and Lowell, 2009] Fh ðt ÞAd ¼ Fm ðt ÞAm ðt Þ

ð9Þ

where Ad is the area of hydrothermal discharge zone. For a hydrothermal system venting at tem-

Figure 10. Total heat output as a function of time for different magmas in the crystals settling case with magma replenishment rate of 5 × 10−8 m/s. Tin is less than TL, which means replenished magma has less latent and sensible heat than the original magma. 10 of 17

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Figure 11. Total heat output as a function of time for different magmas in crystals settling case with magma replenishment rate of 5 × 10−8 m/s. We assume added magma is at its liquidus temperature TL, but the original magma in magma chamber had a temperature less than TL, which means the original magma had lost some crystals before emplacement into the sill.

perature Th(t), the heat flux of the hydrothermal venting is Fh ðt Þ ¼ f cf ud ðt ÞTh ðt Þ

ð10Þ

where rf is the density of hydrothermal fluid, cf is the specific heat of the fluid, and ud(t) is the Darcian upflow velocity. Combining equations (9) and (10), we obtain the temperature of hydrothermal venting as Th ðtÞ ¼

Fm ðt ÞAm vf Ad f cf f gk

1=2 ð11Þ

Figure 12 shows hydrothermal temperatures resulting from magmatic heat supply for different magma systems with a constant replenishment rate. For illustration purpose, we choose the replenishment rate of 10−8 m/s. The value of permeability k and the area of discharge zone Ad used to calculate the hydrothermal temperature are 10−13 m2 and 104 m2, respectively. Hydrous high‐silica magmas have lower initial hydrothermal temperatures because the hydrothermal temperature is proportional to the magmatic heat output to the 1/2 power as shown in equation (11). In addition, the hydrothermal temperature exhibits a correspondingly slower decay rate, which is consistent with the slow decay rate of magmatic heat output in Figure 7. Nevertheless, hydrothermal temperatures of basaltic and anhydrous andesitic magma decrease significantly on decadal timescales during the initial cooling phase of magmatic heat transport, but stabilize as the heat input from magma replenishment balances convective

heat loss. Hydrothermal temperatures resulting from convection in the more viscous dacitic and wet andesitic magmas are relatively more stable than for similar low‐viscosity magmas.

4.3. Physics of Magma Replenishment [25] In the paper, we have shown that a convecting

AMC requires magma replenishment for the stable heat flux. Based on our analysis, the replenishment rate between 10−7 and 10−8 m/s tends to stabilize the heat output on the decadal timescale for both basaltic and high silica magma chambers. Also, in our comparison of different magma systems, the replenishment rates are assumed to be the same in different magma systems. In reality, however, magma replenishment rates may depend on the physical processes in the lower crust and upper mantle that depend upon the properties of magma. Several different mechanisms may be involved in magma transport in the lower crust. [26] As magma initially at its liquidus beneath a

colder upper boundary represented by the hydrothermal system begins to crystallize, its volume shrinks. As the internal pressure in magma chamber declines new magma may be driven upward from the reservoir of relatively crystal‐rich partial melt that occupies the lower crust [e.g., Mainprice, 1997; Dunn et al., 2000]. Magmas from the partial melt zone may also ascend because of the buoyancy force resulting from the density difference between the melt and crystals in the partial melt as a combination of buoyant porous flow and compaction 11 of 17

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Figure 12. Hydrothermal temperature from equation (11) plotted as a function of time for different magma systems in the crystals settling case with magma replenishment at 10−8 m/s. Permeability k = 10−13 m2 and the area of discharge zone Ad = 104 m2 were used to calculate the hydrothermal temperature. The horizontal line at Th = 250°C denotes the lower limit for observed black smoker vent temperatures. The result shown is for illustration purposes to compare the hydrothermal temperature as a function of magma viscosity. The hydrothermal Th for basalt is unrealistically high in this case, and it would be even higher if we used a higher replenishment rate. On the other hand, it can be lowered by using a higher value of k or Ad.

[Turcotte and Ahern, 1978; McKenzie, 1984, 1985], or as buoyant crack propagation [Shaw, 1980; Lister, 1990; Lister and Kerr, 1991]. Melt may also be transported by dike propagation from a deeper lying magma body [Sim, 2004; Kelemen et al., 1997]. [27] Porous flow models of upward magma migra-

tion assume that after pressure release melting in the mantle, interconnected melt along solid grain boundaries generates permeability such that the melt rises at a constant velocity J through the residual solid [Turcotte and Ahern, 1978]. The differential buoyancy of the liquid and solid is responsible for the pressure drop with drives the liquid flow out of the solid [Frank, 1968]. The velocity of melt in the permeable‐porous matrix is given by [Turcotte and Ahern, 1978] u ¼ J ¼

k ðs m Þg vm m

ð12Þ

where k is the permeability, is porosity, rs is the density of solid, respectively. There are many ways to relate the matrix permeability to the porosity, but most formulations suggest that k / 3b2, where b is an effective grain size. For example, for a set of planar, thin vertical cracks of width d and spacing b, = d/(b + d) and k ¼

3 b2 12

ð13Þ

[28] The permeability is proportional to the cube

of the volume fraction of liquid magma. For simplicity, we neglect compaction [McKenzie, 1984] and apply this model to the lower partially molten zone of the oceanic crust. Figure 13 shows the relationship between magma viscosity and magma replenishment rate for porosity = 1% or 10% and grain size b = 1 or 10 mm. Porosity has more effect on magma replenishment rate than grain size, since it enters the permeability equation (13) as a cubic rather than a square. The vertical lines in Figure 13 show the initial values of viscosity for different crystal free magmas at their liquidus. Figure 13 shows that the higher the viscosity, the lower the velocity of melt through the interconnected porosity (see also equation (12)). Figure 13 shows that the magma replenishment rate in almost all cases is much less than the replenishment rate of ∼10−8 m/s needed to stabilize convective heat output between ∼107–109 Watts. For the highest porosity and grain size, the velocity is within the needed range only for basaltic magma. These results are generally consistent with the conclusions of Korenaga and Kelemen [1998], and suggest that some other mechanism is needed to replenish the AMC on decadal time scales.

5. Conclusions [29] Heat transfer from a vigorously convecting

crystallizing and replenished magma chamber 12 of 17


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Figure 13. The relationship between magma viscosity and magma replenishment rate for basaltic, andesitic and dacitic magma with different values of porosity. and b denote the porosity and an effective grain size, respectively.

overlain by a hydrothermal circulation system depends upon magma viscosity, which is a strong function of temperature, chemical composition and water content. In unreplenished magma chambers in which crystals are assumed to settle quickly to the floor, numerical simulations of convection and crystallization in a ∼100 m thick sill, corresponding to an AMC beneath an oceanic spreading center suggest that the maximum rate of heat transfer and its rate of decay both decrease significantly as the magma viscosity increases. Assuming a heat output cutoff of 107 Watts for the hydrothermal system, the results suggest that higher viscosity magmas such as andesite or dacite may drive longer‐lived but lower heat output hydrothermal systems than their basaltic counterparts. Heat output and vent temperature would still decay on decadal time scales, suggesting the need for magma replenishment on these time scales. [30] Simulations with magma replenishment at a

velocity of ∼5 × 10−8 m/s stabilize heat output and corresponding hydrothermal vent temperatures independent of magma viscosity. As in the case without replenishment, the resulting heat outputs and hydrothermal temperatures tend to decrease as the viscosity increases. Unfortunately data to support this result are not yet available. Idealized models of magma flux as buoyancy‐driven porous flow in the lower crust show that the rate of replenishment depends inversely on magma viscosity. In this model, the magma flux is not sufficient to maintain stable hydrothermal heat output of ∼107–109 Watts except possibly for basaltic magmas. Then the porosity (i.e., melt fraction)

must be ∼10% and the grain size (or crack spacing) must be ∼10 mm.

Appendix A: Summary of Mathematic Formulation [31] The mathematical framework used here to

model convective heat transfer from a high Rayleigh number magma chamber vigorously convecting magma chamber is discussed by Liu and Lowell [2009]. The model is based on that of Huppert and Sparks [1988] to describe melting and convection of a granitic magma body underplated by a basaltic sill. In this work the granitic layer is replaced by a convecting hydrothermal system. The convective heat transfer from the magmatic sill uses the classical scaling between the dimensionless Nusselt number Nu and Rayleigh number Nu ∼ Ra1/3, which leads to the expression [Turner, 1973]: Fm ðt Þ ¼ 0:1m cm m ga2m =vm ðt Þ

1=3

ðTm ðt Þ TS Þ4=3

ðA1Þ

where rm is the magma density, cm is the magma specific heat, am is the coefficient of thermal expansion, g is the acceleration due to gravity, am is the thermal diffusivity, and vm(t) is the kinematic viscosity. In our analysis, vm(t) is modeled in equation (7), representing andesite, basalt, or dacite. In using equation (A1), we have assumed that convective heat transfer and the effective Rayleigh number Ra is controlled by the temperature difference between the mean melt temperature Tm(t) and the solidus temperature Ts rather than by an amount of undercooling ∼Tm – Ti [e.g., Kerr et al., 1989; Worster et al., 1990, 1993]. This assumption also 13 of 17


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neglects the growth of solid at the upper boundary and the presence of a mushy zone associated with a propagating cooling front [Kerr et al., 1989; Worster et al., 1990, 1993]. Incorporating these factors would decrease the effective Rayleigh number and the rate of convective heat flux from the magma; but since Nusselt number Nu ∼ Ra1/3, the effect of these simplifications is somewhat muted. As a result of this simplification, the rate of heat loss from the convecting magma is overestimated and provides an upper bound on hydrothermal heat flux. The effect of this assumption is also less important in cases of magma replenishment, which is a main focus of this paper (see section 3.2).

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liquid AMC is a constant value D0, which leads to the differential equation: dTm ðt Þ Fm ðtÞ ¼ dt m LD0 ′ðTm ðt ÞÞ m cm D0

ðA4Þ

where Fm(t) is given by equation (A1), and c′(Tm(t)) is found from the derivative of equation (3). In the crystals settling case, the thickness Dm(t) of the liquid magma decreases as crystals form so that Dm ðt Þ ¼ D0 ½1 ðTm ðt ÞÞ

ðA5Þ

[32] The heat content of the AMC of thickness

Dm(t) and area Am is given by: H ðt Þ ¼ Dm ðtÞAm ½m cm Tm ðt Þ þ m ð1 ðTm ðt ÞÞL

ðA2Þ

Substituting equation (A5) into equation (A2), and the result into equation (A3) we eventually obtain the equation for the crystals settling case

dTm ðt Þ Fm ðt Þ ¼ dt m D0 ′ðTm ðtÞÞ½cm Tm ðt Þ þ Lð1 ðTm ðt ÞÞ þ D0 m ½1 ðTm ðt ÞÞð′ðTm ðtÞÞL cm Þ

where c(Tm(t)) is the volume fraction of crystal content at the magma temperature Tm(t), and L is the latent heat of the crystallization of the magma. When magma replenishment is absent, energy conservation is expressed by equating the time derivative of the heat content in the magma body to the rate of heat loss through the top boundary dH ðtÞ ¼ Fm ðtÞAm dt

ðA3Þ

We solve equation (A3) under two end‐member scenarios: (1) crystals suspended [e.g., Sparks et al., 1984; Solomatov and Stevenson, 1993], where crystals remain suspended and well mixed with the liquid magma as they crystallize, and (2) crystals settling, where crystals instantly settle to the floor of the magma chamber during cooling [e.g., Martin, 1990; Martin and Nokes, 1988; Worster et al., 1990; Liu and Lowell, 2009]. In the crystal‐ suspended scenario the thickness of the effectively

ðA6Þ

When magma replenishment occurs, energy conservation is then given by dH ðt Þ ¼ Fm ðt ÞAm þ Fr ðt ÞAm dt

ðA7Þ

Where Fr(t) (from equation (8) is the rate of heat input into the AMC across the area Am. In addition, the added magma leads to growth of the AMC thickness, since we assume that Am is constant (Liu and Lowell [2009] discuss the case of D remaining fixed and Am is a function of time) dDðtÞ ¼ uðt Þ dt

ðA8Þ

Assuming magma replenishment at a constant velocity u0, we obtain dTm ðtÞ Fm ðtÞ m uðtÞðcm ðTin Tm ðtÞÞ þ ððTm ðt ÞÞ in ÞLÞ ¼ dt Dðt Þm ð′ðTm ðtÞÞL cm Þ

ðA9Þ

for crystals suspended and

dTm ðt Þ Fm ðtÞ Fc þ m uðtÞ½ð1 ðTm ðt ÞÞÞðcm Tm ðtÞ þ ð1 ðTm ðt ÞÞÞLÞ ðcm Tin þ Lð1 in ÞÞ ¼ dt Dðt Þm ′ðTm ðt ÞÞðcm Tm ðt Þ þ Lð1 ðTm ðtÞÞÞÞ þ Dðt Þm ð1 ðTm ðt ÞÞÞð′ðTm ðt ÞÞL cm Þ

ðA10Þ

for crystals settling, respectively. 14 of 17


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Acknowledgments [33] The comments of the editor, Joel Baker, the Associate Editor and an anonymous reviewer of the manuscript are gratefully appreciated. This work was supported in part by NSF grant 0926418 to R.P.L.

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