Magmatic Processes: Physicochemical Principles © The Geochemical Society, Special Publication No.1, 1987 Editor B. O. Mysen

Physics of magma segregation processes

DoNALDL. TURCOTTE Department of Geological Sciences, Cornell University, Ithaca, New York 14853-1504, U.S.A.

Abstract-Partial melting occurs in the mantle due to pressure reduction or by volumetric heating. As a result magma is produced along grain boundaries at depths up to 100 km or more. Differential buoyancy drives this magma upwards by a porous flow mechanism. The small magma conduits on grain boundaries may coalesce to form rivulets of magma in the asthenosphere. The reduction in density due to the presence of magma may induce diapiric flows in the mantle. When the ascending magma reaches the base of the lithosphere it is likely to form pools of magma. Buoyancy driven magma fractures appear to be to the only mechanism by which this magma can penetrate through the lithosphere. How these fractures are initiated remains a subject of speculation.

INTRODUCTION mantle convection or above the core-mantle boundary for whole mantle convection (ALLEGRE PLATETECTONICSprovides a framework for un- and TURCOTTE,1985). Pressure release melting derstanding magmatic activity on and in the Earth. would produce the observed volcanism. However, However, many important aspects of magmatic in many cases, the magma must penetrate the full processes are poorly understood. A large fraction thickness of the cold lithosphere in order to produce of the Earth's volcanism is associated with mid- surface volcanism. ocean ridges. Over a large fraction of the ridge sys- We will consider three mechanisms for magma tem this is essentially a passive process. As the plates migration: (I) porous flow, (2) diapirism and (3) diverge mantle rock must ascend by solid state creep fracture. Aspects of the magma migration problem to fill the gap. Average upper mantle rock is sampled have been reviewed by SPERA(1980) and TUR- by this random, unsteady process. As the mantle COTTE(1982). rock ascends melting occurs due to the adiabatic decompression. A substantial fraction of the re- POROUS FLOW sulting magma ascends to near surface magma chambers due to the differential buoyancy. This Adiabatic decompression will lead to partial magma solidifies to form the oceanic crust. melting of the mantle on grain boundaries. Exper- Volcanism is also associated with subduction imental studies (WAFF and BULAU, 1979, 1982; zones. The origin of this volcanism is still a subject COOPERand KOHLSTEDT,1984, 1986; fuJII et al., of controversy. Friction on the slip zone between 1986) and theoretical calculations (VONBARGEN the descending lithosphere and overlying plate pro- and WAFF,1986) have shown that the melt forms vides some heat. However, as the rock solidus is an interconnected network of channels along triple approached, the viscosity drops to such low levels junctions between grains. Magma will migrate up- that direct melting is not expected (YUEN et al., wards along these channels due to differential 1978). Offsets in the linear volcanic chains asso- buoyancy. This configuration suggeststhat a porous ciated with subduction zones correspond with vari- flowmodel may be applicable for magma migration ations in the dip of the Benioffzones. This is strong in the region where partial melting is occurring. evidence that the magma is produced on or near Porous flow models for magma migration have the boundary between the descending lithosphere been given by FRANK(1968), SLEEP(1974), TUR- and overlying plate. This magma then ascends ver- COTTEand AHERN(1978a,b), and AHERNand TUR- tically 150 ± 25 km to form the distinct volcanic COTTE(1979). Implicit in these models is the as- edifices observed on the surface. How the magma sumption that the matrix is free to collapse as the collects, ascends and feeds the distinct centers is magma migrates upwards. The validity of this as- poorly understood. sumption has been questioned by MCKENZIE Volcanism also occurs within plate interiors. A (1984). Ithas been shown by ScOTTand STEVENSON fraction of this volcanism may be associated with (1984) and by RICHTERand MCKENZIE(1984) that ascending plumes within the mantle. These plumes the coupled migration and compaction equations may be the result of instabilities of the thermal have unsteady solutions (solitons). However, it was boundary layer at the base of the convecting upper shown approximately by AHERNand TuRCOTTE mantle, either at a depth of 670 km for layered (1979) and rigorously by RIBE (1985) that com- 69 70 D. L. Turcotte

paction is not important in mantle melting by de- The upward flux of magma can be related to the compression. The melting of the ascending mantle upward flux of the component to be melted by takes place sufficiently slowly that the crystalline matrix is free to collapse by solid state creep pro- vm=vf (2) 3 s cesses. It is appropriate to assume that the fluid pressure is equal to the lithostatic pressure. Any where Vm is the magma velocity, Vs is the upward difference between the hydrostatic (fluid) pressure velocity of the mantle prior to melting, andfis the and the lithostatic pressure will drive the buoyant fraction of the mantle that is melted. By using fluid upwards. The conclusion is that the unsteady Equation (1) to relate Vm and and takingf = 0.25, solutions are not relevant to magma migration in the appropriate values of Vm and are given in Fig- the mantle. ure I for specified values of 11mand Vs. For example, In order to quantify magma migration in a porous with Vs = I ern/year and 11m= 10 Pa s, we find that solid matrix, Darcy's law gives (TuRCOTTEand 7 = 0.003 and Vm = 10- m/s (300 cm/yr) as in- SCHUBERT,1982, p. 414): dicated by the solid dot in Figure I. b2t:..pg Caution should be used in the application of the t:..v=--- (1) porous flow model. The theory assumes a uniform 247l'l1m distribution of small channels and in fact the small where t:..v is the magma velocity relative to the solid channels may coalesce to form larger channels as matrix, b is the grain size, is the volume fraction the magma ascends in much the same way that of magma (porosity), t:..p is the density difference streams coalesce to form rivers. There is some ev- between the magma and the solid matrix, g is the idence that this occurs when ice melts. acceleration of gravity, and 11mis the magma vis- cosity. Taking b = 2 mm, t:..p = 600 kg/m" and g DIAPIRISM = 10 m/s2, the magma migration velocity is given Diapirism has long been considered as a mechanism as a function of the magma viscosity in Figure 1 for magma migration (MARSH,1978, 1982; MARSH for several values of porosity. and KANTHA,1978). Two types of diapirism must be considered. First, consider magma that has been '1m,Poise 10 10' 10' 10' segregated from the mantle by the porous flow 10" mechanism. The ascending magma is likely to pool 10' at the base of the lithosphere. The question is 10" whether the buoyancy can drive the pooled magma 10' through the lithosphere as a diapir. In order for this 10'" be 10' to occur, the country rock must displaced by vm 10.7 vm solid state creep so that the viscosity of the medium m through which the diapir is passing governs the ve- S 10' ~~ 10.8 locity of ascent. A second question is whether the 10 residual partial melt fraction in the asthenosphere 10-· can induce diapirism. In order to consider these problems we study the 10.10 idealized problem of a low viscosity, low density I 10 10' 10' fluid sphere ascending through a viscous medium "7 ,Pa s m due to buoyancy. The velocity of ascent for this FIG. 1.The solid linesgivethe magma migration velocity Stokes problem is given by (TURCOTTEand SCHU- Vm as a function of the magma viscosity '7rn for several BERT,1982,p. 267) values of the volume fraction of magma (porosity) t/>. This result from Equation (1) for the porous flowmodel assumes that the grain size b = 2 mm and density difference D.p (3) = 600 kg/m", The dashed lines give the magma migration velocity Vrn as a function of the magma viscosity '7rn for several values of the upward velocity of the mantle beneath where, r, is the radius of the sphere and, 11.,is the the melt zone VS' This result from Equations (I) and (2) viscosityof the medium. The ascent velocity is given assumes the melt fraction! = 0.25. The solid dot illustrates as a function of the viscosity in Figure 2 for several a particular example. Specifying '7rn = 10 Pa sand e, = 1 values of the sphere radius and melt fraction and ern/year determines the position of the dot which gives 7 t:..p 600 kg/m", Vrn = 10- mjs (300 cmjyear). The corresponding value = of the porosity is found from the intersecting solid line, Because the lithosphere certainly has a viscosity for this example t/> = 0.003. of at least 1024Pa s, the migration velocity spheres Magma segregation 71

'1s,Poise MAGMA FRACTURE '8 22 10'" 10 10'0 10 A third mechanism for magma migration is fluid fracture. This is probably the only mechanism ca- pable of transporting magma through the cold lithosphere. The common occurrence of dikes is direct observational evidence for the existence of magma fractures. The dynamics of pressure-driven fluid fractures has been studied by SPENCEand TuRCOTTE(1985) and EMERMANet al. (1986). These authors showed that dike propagation is re- stricted by the viscosity of the magma and by the fracture resistance of the media, but in most geo- logical applications, the fracture resistance can be neglected. The vertical transport of magma over large dis- tances (10-100 km) is almost certainly driven by FIG. 2. The ascent velocity of a low viscosity, spherical the differential buoyancy of the magma. One ofthe diapir, Vd, as a function of the viscosity of the medium most spectacular examples of buoyancy driven through which it isrising, '7" for several values of the sphere magma fracture is a kimberlite eruption; the re- radius, r, and porosity t/>. This result from Equation (3) assumes D.p = 600 kg/nr'. quired velocities are estimated to be 0.5-5 m/s (PASTERIS,1984). Buoyancy-driven fluid fractures as a mechanism for magma migration have been studied by WEERTMAN(1971), ANDERSONand with a radius less than 100 km is extremely small. GREW(1977), ANDERSON(1979) and SECORand The diapir will solidifybefore it can migrate through POLLARD(1975). The dynamics of a buoyancy a cold lithosphere. If the path through the litho- driven fluid fracture have been given by SPENCEet sphere was heated so that the viscosity was lowered, at. (1986). These authors found that the stress in- then significant migration may occur. But what is tensity factor (fracture resistance) plays an essential the mechanism for heating if magma cannot pen- role in the solution. etrate the lithosphere? The conclusion is that dia- The propagation of a buoyancy-driven fluid pirism is not a viable mechanism for the migration fracture is governed by the equations for the fluid of magma through the lithosphere. flow in the crack; the equation governing the de- One possibility for this type of migration is that formation of the elastic medium, and the equation heat transported from the magma or heating by vis- relating the empirically derived stress intensity fac- cous dissipation, or both, softens the medium tor to the tip curvature of the crack. Mathematical through which the diapir is passing. Studies of this details for the two-dimensional problem have been problem (RIBE, 1983; EMERMANand TURCOTTE, given by SPENCEet al. (1986). These authors find 1984; OCKENDONet al., 1985) show that these a steady-state solution for a propagating crack. A mechanisms do not significantly help in the pene- universal shape is obtained in terms of non-di- tration of cold lithosphere. mensional variables. The non-dimensional crack Results given in Figure 2 can also be used to de- width 2H is given as a function of the non-dimen- termine whether the melt fraction in the astheno- sional distance from the crack tip in Figure 3. The sphere can induce diapirism. Asthenosphere vis- non-dimensional variables are related to actual 18 21 cosities may be as low as 10 Pa s although 10 variables by Pa s is probably a better estimate. With 11. = 1021 Pa s, = 0.01, and r = 100 km, we find Vd = I cm/ pg H='!!:_, X=([I-V]t:.. )1/2X (4) yr. Under these conditions, diapirism due to partial hoo hoo µ. melt is relatively unimportant. However, with 11. = 1018 Pa s, the velocity may approach 103 cm/yr. where 2hoo is the width of the tail, µ. the shear mod- Thus it is not possible to make a definitive conclu- ulus, v Poisson's ratio, t:..pthe density difference and sion on the relative role of diapirism within the as- g the acceleration of gravity. One of the important thenosphere. To treat this problem satisfactorily, results of the analysis is the requirement that the the mantle convection problem must be solved si- crack must have an infinitely long constant width multaneously with the melting and porous flow mi- tail. The flow in the tail is a balance between the gration problems and this has not been done so far. buoyancy driving force and the viscous resisting 72 D. L. Turcotte

-4 -3 -2 -I o 4 When the critical stress intensity factor is specified, the half-width of the crack is obtained from Equa- tion (8). The propagation velocity is then obtained from Equation (5) for laminar flow and Equation (6) for turbulent flow. The critical stressintensity factor, Xc, is a measure of the fracture toughness of the material. Cracks can propagate at low velocities (_10-4 m/s) by the mechanism of stress corrosion; in this range K < Xc. However, when K = Xc, crack propagation becomes catastrophic and the propagation velocity may rap- idly accelerate to a large fraction of the speed of sound. The values of K; for a variety ofrocks have been obtained in the laboratory. This work has been reviewed by ATKINSON (1984). Measured values for granite range from K; = 0.6 to 2.2 MN/m3/2 and 4 from K; = 0.8 to 3.3 MN/m3/2 for basalts. A measured value for dunite is K, = 3.7 MN/m3/2. These values were obtained at atmospheric pressure. The influence of pressure on the stress intensity fac- 5 tor is difficult to predict. SCHMIDT and HUDDLE (1977) found a factor of four increase at a pressure of62 MPa for Indiana limestone. Thus, it is difficult to specify values of the critical stress intensity factor ~------~~6~------~ for regions of the crust and mantle where buoyancy- FIG.3.Shape ofan upward propagating magma fracture. driven magma fracture is occurring. H The non-dimensional half-width is given as a function By taking µ. = 2 X 1010 Pa, V = 0.25, t:..p = 300 of the non-dimensional distance from the crack tip X. kg/m? and g = 10 m/s2 the half-width of the tail The non-dimensional variables are defined in Equa- tion (4). from Equation (8) is given as a function of the crit- ical stress intensity factor in Figure 4. Propagation velocities from Equations (5) or (6) are given as a force. Thus the velocity of propagation of the crack, function of the critical stress intensity factor in Fig- c, is related to the half-width of the tail hoo by ure 5. Results are given for magma viscosities 11m gh~t:..p = 0.1, 3,100 Pa s. We use the laminar result equa- c=-- (5) 311m tion (5) for Reynolds numbers Re = PmChoo/l1m less than 103 and the turbulent result Equation (6) for for laminar flow in the crack and Reynolds numbers greater than 103. The corre-

(6) for turbulent flow. For the tail of the crack, the non-dimensional half-width is Hoo = 1. The maximum crack half- width is H = 1.8975 and this occurs at X = 1.152. A steady-state solution to this problem is found only for a singlevalue of the non-dimensional stress intensity factor, A, defined by

x = ~1~ V))3/4 x; ((1 - (7) 21/2 µhoo (gt:..p)I/4 = 1.3078 10 102 where K; is the critical stressintensity factor. Solving Kc -3/2 for the half-width of the tail gives MNm FIG.4. The half width of the crack tail hoc as a function K~/3(1- v) of the critical stress intensity factor K; from Equation (8) hoo = 0.440 µ.(gt:..p)I/3 . (8) by assuming µ = 2 X 1010Pa, v = 0.25, D.p = 300 kg/rrr', Magma segregation 73

m. If the length of the crack is 1 km the volume flux through the lithosphere is 95.5 m3/s.

CONCLUSIONS

Pressure release melting produces magma on c grain boundaries. Magma on triple junctions be- ill 5 tween grains produce an interconnected network of channels. Under mantle conditions the magma drains rapidly upwards due to differential buoyancy and the residual solid matrix collapses due to solid state creep. Magma that is produced over a depth 10-8L.. j__ ...L... --' range of approximately 50 km mixes during the I 10 102 103 vertical ascent to produce the magma reaching the K MNm-3/2 c base of the lithosphere. FIG. 5. Crack propagation velocity, c, as a function of At mid-ocean ridges there is no lithosphere and the critical stress intensity factor, K, for several values of the ascending magma can form directly the oceanic the magma viscosity, '7rn. Laminar flow from Equation (5) crust. However, off the axis at ocean trenches and and turbulent flow from Equation (6). at intraplate volcanic centers, the magma must penetrate through the lithosphere. Magma fracture is a mechanism for the rapid transport of magma through the lithosphere. A theory for steady state sponding values of the volumetric flow rate V magma fracture through the lithosphere is given. A = 2choo per unit crack length are given in Figure 6. typical fracture width is 2 em and fracture velocity CARMICHAEL et al. (1977) have shown that the is 5 m/s. Under these conditions relatively little entrainment of xenoliths in basaltic flows implies magma is lost by solidification during ascent. The velocities, c, of at least c = 0.5 m/s. A variety of existence of dikes is evidence that magma fracture studies indicate that basaltic magma migrates up- is a pervasive mechanism. The extensive systems ward at velocities in the range c = 0.5 to 5 m/s. A of dikes in deeply eroded terranes is evidence that typical viscosity for a basaltic magma is 11m = 0.1 dikes are the dominant mechanism for the ascent Pa s. From Figures 4 and 5 the corresponding range of magma through the crust. Kimberlite eruptions of the stress intensity factor is K; = 20 to 100 MN/ are direct evidences that magma fracture can trans- m3/2 and the range of tail half-widths is b-. = 5 port magma through the lithosphere. to 50 mm. From Figure 6 the corresponding range How magma fractures initiate remains a matter of flow rates per unit length is V = 0.0025 to of speculation. It may be possible to form magma 0.25 m2/s. fractures as magmas perculating in small channels Studies of the Kilauea Iki eruption on an island of Hawaii during 1959-1960 give flow rates of 50 to 150 m3/s (WILLIAMS and McBIRNEY, 1979, pp. 232-233). Taking a flow rate of 100 m3/s, the range of flow rates per unit length given above, V = 0.0025 to 0.25 m2/s, correspond to crack lengths between 400 m and 4 km. These appear to be reasonable 10-2 lengths for the crack feeding Kilauea Iki. Although V, 10-4 the above comparison appears reasonable, there is m' conflicting evidence whether the flow of magma at 5 10-6 the surface represents the flow through the litho- sphere. The role of shallow magma chambers in storing magma is poorly understood. As a specific example of a magma fracture that transports magma through the lithosphere we take: 10-12 µ. = 2 X 1010Pa, v = 0.25, t:..p = 600 kg/m", g = 10 I 10 102 3 Kc,MNm -3/2 10 m/s2, 11m = 0.1 Pa s, and K; = 50 MN/m3/2. We find that hoo = 0.0167 m and c = 2.86 m/s. To FIG.6 Volumetric flow rate per unit crack length, V, as convert the shape given in Figure 3 into actual dis- a function of the critical stress intensity factor, K, for tances, we require h = 0.0167H m and x = 272X several values of the magma viscosity, '7m. 74 D. L. Turcotte coalesce to form larger channels as suggested by MARSH B. D. and KANTHA L. H. (1978) On the heat and FOWLER (1985). Alternatively, magma may pool at mass transfer from an ascending magma. Earth Planet. the base of the lithosphere. When the pool reaches Sci. Lett. 39, 435-443. OcKENDON J. R., TAYLOR A. B., EMERMAN S. H. and a critical size the buoyancy forces may initiate a TURCOTTE D. L. (1985) Geodynamic thermal runaway magma fracture that subsequently drains the pool. with melting. J. Fluid Mech. 152,301-314. PASTERISJ. D. (1984) Kimberlites: complex mantle melts. Acknowledgements-The author would like to acknowl- Ann. Rev. Earth Planet. Sci. 12, 133-153. edge the many contributions of D. A. Spence to our un- RlBE N. M. (1983) Diapirism in the Earth's mantle: Ex- derstanding of the magma fracture problem. This research periments on the motion a hot sphere in a fluid with was supported by the Division of Earth Sciences, National temperature-dependent viscosity. J. Volcanol. Geo- Science Foundation under grant EAR-8518019. This is thermo Res. 16,221-245. contribution 830 of the Department of Geological Sciences, RlBE N. M. 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