Some thermal problems associated with magma migration

Journal of Volcanology and Geothermal Research, 10 (1981) 267--278 Elsevier Scientific Publishing Company, Amsterdam -- Printed in Belgium

267

SOME T H E R M A L P R O B L E M S A S S O C I A T E D W I T H M A G M A MIGRATION

D.L. TURCOTTE

Department of Geological Sciences, Cornell University, Ithaca, N Y 14853 (U.S.A.) (Revised version accepted January 15, 1981)

ABSTRACT Turcotte, D.L., 1981. Some thermal problems associated with magma migration. J. Volcanol. Geotherm. Res., 10: 267--278. The mechanisms by which magma migrates from the point in the earth's interior where melting occu~ to the earth's surface are poorly understood. In this paper several aspects of this problem are examined. Magma can migrate upward due to its differential buoyancy on the scale of crystalline grains or as large diapirs. Magma transport is an effective means of heat transport. Magma transport at a rate of 0.15 cm/yr is equivalent to a heat flow of 10-6 cal/cm 2 s. If magma encounters country rock with a lower melting point the original magma is likely to solidify while melting the country rock. This would be an effective mechanism of purging silicic rocks and incompatible elements from the lower crust. Under some circumstances magma must penetrate up to 100 km or more of cold lithospheric rock. In order for this magma to reach the surface without solidification a heated path must be provided. The heating of this path requires the solidification of some magma. It is estimated that magma penetrates the lithosphere in about 5000 years and that the crack is lined by several hundred meters of frozen basalt.

INTRODUCTION

,

In general t e r m s the origin o f the m a g m a t h a t flows f r o m m o s t v o l c a n o e s on the e a r t h is r e a s o n a b l y well u n d e r s t o o d . A large f r a c t i o n o f surface volcanism can be directly related t o plate tectonics. As plates diverge at o c e a n ridges h o t m a n t l e r o c k ascends to fill the gap. Because the solidus t e m p e r a t u r e o f m a n t l e r o c k has greater d e p t h d e p e n d e n c e t h a n the adiabatic t e m p e r a t u r e , the adiabatic ascent o f m a n t l e r o c k can lead t o partial melting. The extensive basaltic volcanism t h a t f o r m s the o c e a n crust, at o c e a n ridges is a t t r i b u t e d t o this pressure-release melting ( V e r h o o g e n , 1954). T w o alternative h y p o t h e s e s have been a d v a n c e d t o explain the volcanism t h a t f o r m s the lines o f v o l c a n o e s t h a t lie parallel to o c e a n trenches. T h e first is the frictional h e a t i n g h y p o t h e s i s (McKenzie a n d Sclater, 1 9 6 8 ; O x b u r g h a n d T u r c o t t e , 1 9 6 8 ) . F r i c t i o n o n the slip z o n e b e t w e e n the d e s c e n d i n g plate at an o c e a n t r e n c h and the overlying plate heats the

0377-0273/81/0000--0000/$02.50 © 1981 Elsevier Scientific Publishing Company

268

descending oceanic crust until it melts. A shear stress of a few kilobars is sufficient to provide the necessary heat. However, as the t em perat ure on the fault zone is increased to a substantial fraction of the melt temperature, the frictional resistance to sliding becomes so low that further heating is negligible. The conclusion is that it appears very difficult if n o t impossible to produce melting by steady-state, frictional heating on the slip zone (Yuen et al., 1978). An alternative hypothesis is that the descent of the plate at an ocean trench induces a secondary flow in the overlying mantle. T he interaction of the descending ocean crust with the hot overlying mantle produces partial melting of the crust. There are, however, difficulties with this hypothesis. The rocks of the overlying mantle must be below their solidus or melt would have been produced directly. These rocks could produce melt from the oceanic crust at a t e m per a t ur e below the dry basalt solidus. However, it is observed that a significant fraction of the volcanic rocks in these volcanoes are basalts and it appears very difficult to explain their production by this mechanism. Some authors argue that the presence of water can significantly depress the solidus but the petrology and high t em perat ure of many island arc magmas argue against this suggestion. A variation of this hypothesis is th at dehydr a t i on of the subducted crust induces melting in the overlying mantle (Anderson et al., 1976). The very strong geometrical correlation o f the volcanic line with geometry of the descending plate suggests that magma is produced directly from the descending slab. One possibility is that magma is produced by unsteady frictional heating. Thermal runaway associated with the coupling between frictional heating and an exponentially t e m p e r a t u r e - d e p e n d e n t rheology is one possible explanation. Alternative hypotheses are also available for intraplate volcanism. This volcanism may be the result of pressure-release melting in ascending mantle plumes (Morgan, 1971). An alternative explanation is that the magma reaches the surface from the asthenosphere due to tithospheric fractures under tensional stresses ( T ur c ot t e and Oxburgh, 1973). F r o m the point in the earth's interior where melting occurs the magma must migrate upwards tens and in some cases hundreds of kilometers in order to reach the earth's surface. This upward m o v e m e n t is driven by the differential b u o y a n c y of the lighter magma. Several mechanisms for magma migration have been proposed. Within the region where rock is at or above the solidus, i.e. the asthenosphere, the magma can migrate along grain ooundary intersections. The grains act as a matrix and the grain b o u n d a r y intersections act as channels for the migration of the magma. This flow can be modelled using flow in a porous medium (Frank, 1968; Sleep, 1974; T u r c o t t e and Ahern, 1978; Walker et al., 1978; A hem and T ur c ot t e , 1979). The force exerted on the matrix by the migrating magma can also induce an upward m o t i o n of the matrix. The flow of the matrix would be by a solid-state creep process such

269 as the motion of dislocations. The induced flow of the matrix could lead to mantle diapirism. The magma on grain boundaries could collect to form small magma bodies in much the same way that small rain drops collect to form large rain drops. In some cases such as ocean ridges the asthenosphere extends essentially to the earth's surface and no further mechanisms for magma migration are required. However, in other cases magma appears to penetrate a pre-existing lithosphere. One example would be Hawaii where magma continuously penetrates oceanic lithosphere. Several mechanisms have been proposed for this type of magma migration. One mechanism is the diapiric penetration of a magma b o d y (Marsh, 1976, 1978). Magma collects at the base of the lithosphere to form a large magma body. The b u o y a n c y force on the body drives it upward through the overlying cold lithosphere in much the same way that the gravitational instability of a salt layer leads to the formation of salt domes. A primary difficulty with this mechanism is the low velocity of penetration because large volumes of cold lithospheric rock must be displaced. As a result large volumes of magma are lost due to solidification. An alternative mechanism involves the propagation of a magma filled crack through the lithosphere (Weertman, 1971; Anderson and Grew, 1977). Liquid introduced into a solid at high pressure can induce a fracture. Presumably the differential buoyancy of the magma in an elongated, vertical cavity could lead to fracture at the top of the cavity and the upward migration of the magma. This mechanism is poorly understood and the thermal implications have not been considered. In this paper several aspects of the magma migration problem will be considered. POROUS FLOW MODEL When rock first begins to melt, the resulting magma forms on grain boundary intersections (Waft and Bulau, 1979). The resulting arcuate network of magma lineaments is expected to form a series of interconnected channels along which the magma can migrate. For simplicity this network of channels is modelled by a cubic structure of circular channels. The side length of the cubic structure b is equivalent to the grain size. The diameter 5 of the channels can be expressed in terms of the porosity X, defined to be the volume fraction of magma according to: X = 3~52/4b

2

(1)

A different geometrical model for the grain intersections would introduce a factor of order unity into the analysis: Because the magma has a lower density Pm than that of the residual crystalline solid Ps, the differential buoyancy will drive the magma upwards. In order to determine the velocity at which the magma rises, Vm it is ap-

270 propriate to use Darcy's law (Bear, 1972) in the form: k

~ XVm = - - (Ps 77m

(2)

Pm)g

where k is the permeability and ~)m the viscosity of the magma. For the porosity model that has been assumed, the permeability is given by (Bear, 1972): k = x6 2/96

(3)

Combining equations (1) to (3), the magma velocity can be related to the porosity by: Vm =

1

xb2(ps - P m ) g

24 n

~m

(4)

Before obtaining specific results, it is appropriate to discuss some of the assumptions implicit in this simple model. In the form given above, melting and freezing of magma have not been considered. It is relatively easy to extend the model to include these effects (Ahem and Turcotte, 1979). Also, the velocity of the matrix has been neglected. Since the velocity of magma migration is usually large compared with velocities of sea-floor spreading and, therefore, mantle convection, this assumption should not affect the conclusions obtained. Implicit in writing equation (1) is the assumption that the matrix or lithostatic pressure is equal to the magma or fluid pressure. The assumption is that the matrix is free to collapse as magma is withdrawn. Because the temperature necessary to have magma present is relatively high, thermally activated creep processes are effective in deforming the matrix. Quantitative calculations (Ahern and Turcotte, 1979) show that, in fact, the matrix will collapse on geological time scales. It is of interest to relate the magma velocity and porosity to the rate at which magma is being supplied. If magma is being supplied at a velocity Vm0 at the base of the zone of partial melting, then conservation of magma requires: l

(5)

3 VmX = Vm°

Combining equations (4) and (5) gives: [ Vm(~b2(p.s - Pm)g Vm =

4/rlTm 72nVm0~m

X =

] V2

b~sC~m)g

] 1/2 (6)

271

As typical values of the parameters given above, we take b = 0.2 cm, Ps P m = 0.5 g/cm 3, g = 10 ~ cm/s 2, and 77m = 100 poise. The value of the magma viscosity can vary widely depending upon the temperature and the chemical properties of the magma (Kushiro et al., 1976). The dependence of Vm and × on Vm0 for these values of the parameters is given in Fig. 1. It is seen that the fraction of liquid present is very small, being of the order of 1%. The magma migration velocity is about three orders of magnitude larger than the magma supply velocity. It is concluded that as soon as magma is produced it will migrate to the top of the zone where the temperature is at or above the solidus. Large par tial melt fractions of magma within this region are unstable. This precludes " b a t c h " melting under many circumstances. 2

Vm

x

x 10 3

I 0 "2

crn yr

0 0

I

2

3

4

5

v too, c m / y r

Fig. 1. Dependence of velocity of magma rise (Vm) and porosity (volume fraction of magma, ×), on the rate at which magma is being supplied (Vm0). THE HEAT-PIPE MECHANISM

The isothermal (or near isothermal) transport of magma in the earth's interior is an effective means of transporting heat. If two phases of the same material flow in opposite directions under isothermal conditions, heat is transported due to the latent heat of the phase change. In the engineering literature this is known as the heat-pipe mechanism. Heating equivalent to the latent heat of formation is required to produce the magma. When the magma reaches a cooler region such as the base of the lithosphere, the magma will solidify releasing the latent heat of formation. The resultant heat transfer will thin the lithosphere. The a m o u n t of heat transported by magma migration qhp is simply related to the magma supply velocity Vm0 and the latent heat of fusion of the magma L by: qhp

= VmoPmL

(8)

Taking L = 90 cal/g and Pm = 2.7 g/cm 3, we find that magma supplied at a rate Of 1 c m / y r corresponds to a heat flow of 6.86 ~cal/cm 2 s. It is clear

272 t h a t quite c o n s i d e r a b l e q u a n t i t i e s o f h e a t can be t r a n s p o r t e d b y the heatpipe m e c h a n i s m . RATE OF LITHOSPHERIC THINNING In o r d e r to d e t e r m i n e the rate at w h i c h the l i t h o s p h e r e can be t h i n n e d by the h e a t - p i p e m e c h a n i s m , we c o n s i d e r the p r o b l e m o f the semi-infinite half-space to w h i c h h e a t is a d d e d at a c o n s t a n t rate qhp at the m e l t t e m p e r ature T m. As the m a t e r i a l is h e a t e d to T m it b e c o m e s p a r t of the h e a t pipe, t h e r e f o r e the h e a t i n g f r o n t m o v e s u p w a r d at a v e l o c i t y urn. We a s s u m e for s i m p l i c i t y t h a t the half-space is initially at a u n i f o r m t e m p e r a t u r e T L. Actually, the t e m p e r a t u r e of the l i t h o s p h e r e decreases as the h e a t i n g f r o n t migrates u p w a r d s , b u t o u r results will be a p p r o x i m a t e l y valid. We wish to solve the e q u a t i o n : dT -Um

dy

d2T = ~

dy 2

(9)

in a c o o r d i n a t e s y s t e m m o v i n g w i t h the m e l t i n g f r o n t . We require T = Trn at y = 0 and T ~ T L as 3' -~ oo. T h e s o l u t i o n t h a t satisfies these b o u n d a r y c o n d i t i o n s is: T = To + (Tm - TL) e -umy/<

(10)

T h e r e q u i r e m e n t t h a t q = qm at y = 0 p r o v i d e s an e x p r e s s i o n for the velocity: um =

qhp

PsCp(Tm - TL)

(11)

This is a simple h e a t balance t h a t states t h a t t h e f r o n t will m i g r a t e u p w a r d s at a rate n e c e s s a r y t o h e a t the r o c k f r o m its initial t e m p e r a t u r e T L to T m . S u b s t i t u t i o n o f the h e a t flow f r o m e q u a t i o n (8) yields: um =

VmoPmL PsCp(Tm TL)

(12)

-

This is the v e l o c i t y of t h e heating f r o n t as it m i g r a t e s u p w a r d s t h r o u g h t h e l i t h o s p h e r e . T a k i n g Ps = 2.9 g / c m 3, Cp = 0 . 2 5 cal/g, (Tm - TL) -- 500°C a n d o t h e r p a r a m e t e r s as given above, it is f o u n d t h a t Urn/Vmo = 0.6. T h e h e a t i n g f r o n t migrates at a sizeable f r a c t i o n o f the v e l o c i t y at w h i c h m a g m a is supplied. H o w e v e r , it follows t h a t this ratio is also t h e r a t i o o f t h e t o t a l migration d i s t a n c e to t h e t h i c k n e s s o f m a g m a supplied. A sill o f m a g m a 1 k m t h i c k will h e a t 6 0 0 m o f l i t h o s p h e r e if t h e l i t h o s p h e r e is initially at a ternp e r a t u r e o f 500°C. T h e c o n c l u s i o n is t h a t v e r y large v o l u m e s o f m a g m a are r e q u i r e d to thin the l i t h o s p h e r e significantly. T h e r e a s o n f o r this is t h a t t h e l a t e n t h e a t o f fusion o f m a g m a is n o t p a r t i c u l a r l y large.

273 We have discussed the heat-pipe mechanism in terms of the porous flow model for magma migration because this is the usual context in which the mechanism is applied. The liquid magma rises upward through the solid matrix which is displaced downward. This counterflow of two phases is a necessary condition for the heat-pipe mechanism. Heat will in general be transferred by the upward migration of magma. However, what happens to the rock displaced by the magma is not usually discussed. GEOCHEMICAL IMPLICATIONS Magma supplied to the base of the crust has the capability of partially melting a thickness of the crust of the order of the thickness of the magma supplied. The very simple analysis given above assumes a single c o m p o n e n t with a well-defined melt temperature. In fact, all rocks have melting ranges and the rock solidus varies strongly with composition. Let us consider qualitatively what might happen if this mechanism were operative within the continental lithosphere. As the heating front moved through the mantle, the liquid component would most likely be basalt. However, as the heating front encountered the lower crust, more silicic rocks with a lower melting point might be expected to be present. The basaltic magma would freeze while melting the silicic rocks, and the silicic rocks would become the liquid fraction that would continue the upward migration of the heating front. As rocks of changing composition were encountered by the freezing front, further melting and freezing would be expected. The freezing of the basaltic rocks in the lower crust would continue until all the silicic rocks were driven upwards by this process. The lower part of the crust would have a near basaltic composition, and heat would be transported by the migration of a small a m o u n t of basaltic melt. Along with the silicic melt fraction, most of the incompatible elements, including the heatproducing elements, would be driven upwards. This process would continue to be operative in the crust as long as the magma addition to the base of the crust continued. Eventually, silicic rocks of near uniform composition would be concentrated near the surface. Magma migration through these rocks could create the exponentially decreasing concentration of radioactive elements which explains the linear relation between surface heat flow and surface radioactive heat production. In some cases the flow rate of the mantle produced basaltic magma would be sufficiently high that it would reach the surface but would be contaminated by the melting of crustal rocks (Taylor, 1978; Hawkesworth and Vollmer, 1979). LITHOSPHERIC HEATING In the previous sections we have been primarily concerned with the migration of magma through a region that has been heated to the solidus temper-

274 ature. Large quantities of magma are unstable in this region. Magma will rapidly migrate upward when the magma fraction reaches a few percent. However, we know that magma that reaches the t op of an asthenospheric region can penetrate the cold overlying lithosphere. Examples are the oceanic islands such as Hawaii where magma has penetrated a thick oceanic lithosphere. As discussed in the introduction the mechanism or mechanisms for this penetration are poorly understood. It is clear that a heated path must be provided or the magma would solidify at depth. In this section an idealized analysis of this heating problem is presented. It is assumed that magma migration through the lithosphere can be modelled using a simple crack model. A pre-existing crack is assumed to penetrate the lithosphere. Magma is forced up this crack. The walls of the crack are initially at a temperature T0 which is below the melt temperature of the magma Tin. The magma is assumed to have a well defined melt temperature and is assumed to be injected at this melt temperature.

yc] vc

iY

J ip:

;.~

Fig. 2. Coordinate system adopted in the analysis of the problem of lithospherie magma migration. The coordinate system is illustrated in Fig. 2. The magma f r o n t is migrating upwards along the crack at a velocity re. To simplify the analysis ve is taken to be constant. The magma f r ont penetrates the thickness of the lithosphere YL in a time r L = Y L / V c . The distance from the surface to a point on the crack is y and the distance of this point from the magma f r o n t is y'. The d ep th of the magma f r ont Ye is given by: Ye = YL

- Vet

(13)

where the time t is measured from the initiation of magma migration. Magma freezes to the side of the crack. The horizontal velocity at which freezing occurs uf is given by the solution of Stefan's problem (Carslaw and Jaeger, 1959): u f = ~ ( K / t ' ) '/2

(14)

where: t' = y ' / V c

(15)

275 is the time since solidification started and ~, is given by Carslaw and Jaeger {1959, fig. 38). Since: (YL -Y)/Vc

t'= t-

(16)

(14) can be written: [

K

uf = X

] 1/2

t - ( Y L - Y)/Vc

(17)

And the thickness of solidified magma xf is given by: t

1~

YL -Y )/Vc

Uc

The flow of magma into the base of the crack that solidifies on the walls of the crack per unit length of crack qf is given by: YL --Vct ufdy qf = 2 f YL YL--Vct [ f

= 2k

KVc ~Yc +~ c t

YL

~ yL

] 1/2 dy

(19)

= 4~vc(Kt) 1/2

if Vct < YL and: 0

f u f d y = 4~,Vc((~t) 1 ~ - [ K ( t - yL/Vc)] 1/2) YL

qf = 2

(20)

if Vct >~ YL. As a typical magma supply rate we consider magma flow when the magma front just reaches the surface: qf0 = 4X(KVcYL) 1/2

(21)

The total amount of magma lost by solidification on the walls of the crack prior to the magma front reaching the surface Qf0 is given by: Qfo =

YL/Vc f 0

8 q f d t = ~ ky L

~ KYL t ~ \ Vc ]

(22)

We next consider some specific examples. As in previous sections we take L = 80 cal/g, Cp = 0.25 cal/g °C and T m - To = 500°C and find from Carslaw and Jaeger (1959) that k = 0.7. It is also assumed that YL = 50 km.

276 TABLE 1 The velocity of the magma front uc, the flow into the crack per unit length when the crack just reaches the surface q[(,, the amount of magma solidified on the walls of the crack per unit length Q%, and the mean thickness of the solidified magma as a function of the time TL that it took the magma front to penetrate a lithosphere 50 km thick 7L

t¥

qf¢~

(~)L,

Xf~

(years)

(cm/yr)

(km:/yr)

(km:)

(kln)

l0 ' 1 10 10: 103 104 10 s 106 10"

5× 5 × 5× 5x 5× 5× 50 5 0.5

2.5 0.78 0.25 7.8× 2.5 × 7.8 X 2.5 X 7.8× 2.5 ×

0.17 0.52 1.7 5.2 17 52 170 520 1700

0.0017 0.0052 0.017 0.052 0.17 0.52 1.7 5.2 17

10 106 l0 s 10 ~ 103 10:

10 : 10 -" 10 ' 10 ~ 10 ~ 10.4

A r a n g e o f 7 L f r o m 1 0 - ~ t o 107 y e a r s is c o n s i d e r e d . T h e c o r r e s p o n d i n g v a l u e s o f Vc, qf0, Qf0, a n d t h e m e a n v a l u e o f t h e t h i c k n e s s o f s o l i d i f i e d m a g m a a t t h e t i m e t h e m a g m a f r o n t r e a c h e s t h e s u r f a c e ~f0 are g i v e n in T a b l e 1. I f t h e m e l t i n g f r o n t r e a c h e s t h e s u r f a c e r a p i d l y l a r g e m a g m a f l o w rates are required but the total amount of magma lost to the walls of the c r a c k is r e l a t i v e l y s m a l l . I f t h e p r o p a g a t i o n s p e e d o f t h e m a g m a f r o n t is s m a l l r e l a t i v e l y l o w f l o w r a t e s a r e r e q u i r e d b u t a large a m o u n t o f m a g m a is l o s t by solidification to the walls of the crack. We c o m p a r e t h e s e r e s u l t s w i t h t h e d e r i v e d m a g m a f l o w r a t e s f o r t h e v o l c a n i s m o f t h e H a w a i i a n I s l a n d s . T a k i n g t h e age o f t h e H a w a i i a n - E m p e r o r bend to be 40 Myr the mean magma flow rate required to construct the g l a n d s o f t h e c h a i n is 0 . 0 2 1 k m 3 / y r ( S h a w , 1 9 7 3 ) . W e a s s u m e t h a t a t a given t i m e m a g m a f l o w s s t e a d i l y u p a c r a c k 2 0 0 k m l o n g . I f t h e c r a c k is propagating at a rate of 10 cm/yr this corresponds to a steady-state period of volcanism of 2 Myr. The resultant flow up the crack per unit crack l e n g t h is q f = 1 0 -4 k m ~ / y r . In e x a m i n i n g T a b l e 1 t h i s a p p e a r s t o b e a n u n reasonably low flow rate since it corresponds to a penetration time greater t h a n 1 0 0 M y r . T h e c o n c l u s i o n is t h a t t h e a s s u m p t i o n o f u n i f o r m f l o w u p a c r a c k 2 0 0 k m l o n g is n o t v a l i d . A n a l t e r n a t i v e a p p r o a c h t o t h e v o l c a n i c s o f t h e H a w a i i a n I s l a n d s is t o c o n s i d e r t h e s h o r t - t e r m c o n s t r u c t i o n a l v o l c a n i s m a t K i l a u e a . I t is e s t i m a t e d ( S w a n s o n , 1 9 7 2 ) t h a t t h e s u p p l y r a t e o f m a g m a t o t h i s v o l c a n o is 0 . 1 1 k m ~ / y r . I f t h i s m a g m a w a s f l o w i n g u p a c r a c k 10 k m w i d e t h e n q f = 0 . 0 1 1 k m 2 / y r . A s s e e n f r o m T a b l e 1 t h i s c o r r e s p o n d s t o a 7L o f 5 0 0 0 y e a r s . This would appear to be a reasonable time for magma to penetrate the lithosphere. The amount of magma lost during ascent through the lithos p h e r e is n o t e x c e s s i v e . O n c e a m a g m a p a t h h a s b e e n p r o v i d e d b y h e a t i n g

277

magma can continue to flow to the surface as long as new magma reaches the base of the lithosphere. It should be emphasized that the mechanism for the upward movement of the magma has n o t been specified in this analysis. If this mechanism was understood the velocity of magma migration could be determined. ACKNOWLEDGEMENTS

The research was partially supported by the Division of Earth Sciences of the National Science Foundation under grant EAR-76-82556. This paper is Contribution 677 of the Department of Geological Sciences, Cornell University.

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