Earth and Planetary Science Letters, 83 (1987) 137-152 137 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

[3]

Simple 2-D models for melt extraction at mid-ocean ridges and island arcs

Marc Spiegelman and Dan McKenzie

Department of Earth Sciences, University of Cambridge, Downing Street. Cambridge, CB2 3EQ (England)

Received August 15, 1986; revised version accepted February 1, 1987

A general set of 2-D equations for the conservation of mass and momentum of a two-phase system of melt in a deformable matrix is used to derive analytic solutions for the corner flow of a constant porosity melt-saturated porous medium. This solution is used to model the melt extraction processes at mid-ocean ridges and island arcs. The models indicate that flow of melt is controlled by pressure gradients induced by the Laplacian of the matrix velocity field and by the dimensionless percolation velocity which measures the relative contributions of buoyancy-driven flow to advection by the matrix. The models can account for many features of ridge and arc volcanism. Matrix comer flow at ridges causes melt to be drawn to the ridge axis enabling the extraction of small melt fractions from a wide melting zone while showing a narrow zone of volcanism at the surface. At zones melts do not percolate vertically but are drawn to the junction of the upper plate and subducting slab by corner flow in the mantle wedge. For subduction zones, if the dimensionless percolation velocity is below a critical value, slab-derived fluids will be carried down by the matrix and cannot interact with the mantle wedge. The geochemistry of island arcs will be controlled by the geometry of melt streamlines. This model is consistent with geophysical and geochemical data from the Aleutian arc.

1. Introduction melt-saturated medium [3,5], melting and compac- tion for simple 1-D upwelling models with ad hoc It is generally accepted that partial melting melting functions [6,7] and the existence and sta- with subsequent separation from the solid residue bility of 1-D porosity waves and solitons [4,5,8]. is the primary process governing the production of The purpose of this paper is to carry this work melt at oceanic spreading centers and subduction into two dimensions by using a general set of 2-D zones, yet little is actually understood of the equations for the conservation of mass and physical processes involved. As estimates of global momentum to derive analytic solutions to the magmatism indicate that oceanic magmatism simplest case of corner flow of a constant porosity accounts for approximately 75% of global output meh-saturated porous medium. These models are [1] it is important that work be done to quantify then used to illustrate the melt extraction processes these processes. which may occur at spreading centers and subduc- Until recently, however, this has been difficult tion zones. for lack of a comprehensive system of equations governing conservation of mass, momentum and 2. Governing equations energy for a two-phase system that takes into account the deformation of the solid, crystalline The general governing equations for the con- servation of mass and momentum of a low viscos- matrix. This has been remedied by several workers ity "melt" or "fluid" phase in a deformable "ma- [2-4] who have proposed several similar sets of trix" can be written [3]: equations. Here we will use McKenzie's formula- tion and notation. ~ (pfe~) + V'-(pfCv) = r (1) As these equations are rather complicated, it is necessary to investigate their properties systemati- ~ [p~(1 -~)] + V" [p.~(1 - O)l/] = -F (2) cally by looking at simple problems. The simplest at subset of problems are one dimensional and several -k~, workers have investigated 1-D compaction of a v- v= (3)

0012-821X/87/$03.50 ~ 1987 Elsevier Science Publishers B.V. 138 v'~= nv-'v+ (~" + n/3)v'(v', v) where Of is the density of the melt, 4, is the volume fraction occupied by the melt or porosity, + (1 - 4,) a0g/~ (4) v is the melt velocity, I" is the "melting function" k, - a24,"/c (5) which gives the rate of mass transfer from matrix

TABLE 1 Notation

Variable Meaning Value used Dimension a grain size 10 3 m c constant in permeability 1000 none d thickness of oceanic lithosphere km g acceleration due to gravity 9.81 m s 2 h thickness of oceanic crust km k ~ permeability m 2 k o = a 2q~3o/c permeability at porosity q'0 m2 I thickness of melting zone 30-60 km L = [rlUo/Apg] 1/2 length scale m L R length scale for ridge model m L.t length scale for trench model m p degree of partial melting 0.1-0.3 none piezometric pressure Pa ~R dimensionless piezometric pressure, ridge model none ~'r dimensionless piezometric pressure, trench model none r polar coordinate (distance from corner) m r" = r/L dimensionless polar coordinate none t time s U0 characteristic velocity: half spreading rate, ridge model subduction rate, trench model 10 ~°-10 9 m s V matrix velocity m s I v melt velocity m s - 1 V', v" = V, v/U o dimensionless velocities none Av specific volume change on melting m 3 kg 1 w 0 = k0(1 - (%)Apg/q~0p percolation velocity m s 1 x horizontal cartesian coordinate m z vertical cartesian coordinate m x', z" = x, z/L dimensionless cartesian coordinates none a wedge angle (ridge model) (10-40 o ) none fl angle of subduction (trench m(xlel) (30-80 ° ) none F = DvM/Dt (McKenzie [31) melting function kg m -3 s 1 ~" matrix bulk viscosity 10 ~s- 102~ Pa s matrix shear viscosity 10 Ix- 10 21 Pa s 0 polar coordinate none K thermal diffusivity of lithosphere 10 ..6 m 2 s- 1 melt shear viscosity 1 Pa s 0¢ mean density of oceanic crust 2800 kg m - 3 or density of melt 2800 kg m 3 P~ density of matrix 3300 kg m - 3 Ap = P~ -- Of 500 kg m -3 porosity none ~o characteristic porosity (constant) 0.006.0.04 none stream function m 2 s l ~ dimensionless melt stream function ridge model none ~ dimensionless melt stream function trench model none ~ dimensionless matrix stream function ridge model none ~r dimensionless matrix stream function trench model none 139

to melt in a frame fixed to the matrix (F= by equations (3) to (5). These general 2-D equa- DvM/Dt in McKenzie [3]). p~ is the density of tions are given in Appendix .A and are rather the solid matrix, V is the matrix velocity, k, is the complicated. We have not made much progress in permeability, ~ is the "piezometric pressure" or obtaining analytical solutions and further work the pressure in excess of "hydrostatic pressure" will require careful numerical investigation. How- (.~=P-pfgz), Ap=p~-pf and /~ is the unit ever, if porosity is constant and melting neglected, vector in the z direction with z increasing down- the problem becomes surprisingly simple and ana- wards. Equation (5) gives the permeability as a lytical solutions can be found that demonstrate non-linear function of the grain size a, porosity much of the significant physics of melt extraction. and a dimensionless coefficient, c. Estimates and The analytical solutions can also account for many explanations of parameters are given in Table 1. of the features of volcanism at mid-ocean ridges The first two equations govern the conservation and subduction zones. of mass for the melt and matrix respectively. Equation (3) is Darcy's Law which states that the separation velocity of melt from matrix depends 4. Simple corner flow models: constant porosity, no on the permeability and lies along gradients of the melting piezometric pressure. Equation (4) shows that the piezometric pressure has three contributing fac- Corner flow of a viscous incompressible fluid tors. The first term on the right-hand side is the has often been used to model the flow of the pressure gradient due to volume conserving shear mantle near spreading centers and subduction deformation of the matrix. It should be noted that zones [9-11]. As these regions of the mantle are only deformation with non-zero ~72V will cause probably partially molten a better approximation changes in the pressure. Uniform simple shear or is to examine the two phase corner flow of a melt pure shear will have no effect on the melt flow as saturated, deformable porous medium. The solid they have constant first derivatives. The second matrix behaves like the viscous fluid in the previ- term in equation (4) is the gradient due to volume ous models but the melt in the pores obeys Darcy's changes of the matrix on expansion and compac- law and flows in response to gradients in the tion and the third term is the gradient due to the piezometric pressure. In the simplest model the differential buoyancy between melt and matrix. porosity is constant and melting is ignored. This It is important to note that in one dimension necessarily rules out compaction. Mathematically, there is no distinction between shear deformation = ~0, F = 0 implies that: and compaction. In two and three dimensions, V" V=W.v=O (6) however, the difference is quite significant for even in the simplest case described below, where from equations (1) and (2), and therefore V and v compaction is neglected, the shear deformation can be written in potential form in terms of scalar controls the geometry of melt extraction. stream functions: V= WX (~J3 3. Governing equations in 2-D, constant density = v x (7)

If pf and p~ are constant, then in two dimen- where f is the unit vector perpendicular to the x, sions equations (1) to (4) represent 6 equations in z plane. As is shown in Appendix A, the problem 7 unknowns. If the melting function, F, is known then reduces to simply solving the biharmonic then equations (1) to (4) can be solved for if, .~P equation: and 2 components each of v and V. In two dimensions, however, the problem can be reduced V4¢ = 0 (8) to solving only three equations for the porosity for the matrix stream function subject to ap- and matrix velocity subject to appropriate propriate boundary conditions. boundary conditions. Once ~ and V are known, Batchelor [12] gives a similarity solution to the the piezometric pressure, permeability, and there- biharmonic equation with corner flow boundary fore the melt velocity are completely determined conditions. He shows that this problem can be 140 solved by inspection in polar coordinates with: ~= -- + + z(r, O) (17) r d03 V = ! ~+__i"~_ 0¢ G (9) r ~0 ~r +~ = rUof( O ) (10a) and: where: f(0)=C 1 sin0+C20sin0+C 3cos0 k°(1 - *°)APg (19) w 0 = q~0t~ + C40 cos 0 (10b) is the "percolation velocity", that is, the sep- where U0 is a characteristic velocity (i.e. the half- aration velocity between the melt and the matrix spreading rate for the ridge model and the subduc- given by Darcy's law in the absence of matrix tion rate for the trench model), r is the distance deformation. Note that w 0 depends on the square measured from the corner, 0 is the angle measured of the porosity. Equation (18) shows that the melt from an arbitrary zero axis and C~_ 4 are constants will follow the matrix stream lines only if the adjusted to match the two components of the dimensionless percolation velocity, wo/Uo=O, velocity on each of two straight boundaries. Given and that contrary to most simple petrological the matrix stream function, the piezometric pres- models, it will percolate vertically only in the sure and the melt stream function are completely absence of matrix deformation. determined and can be written: ~= -'qUo [ d3f + dr) 5. Ridge model: solution and results r ~ dO' d-O + z(r, O)(1 - qS0)Apg Boundary conditions for the ridge model are (11) shown in Fig. la. The wedge geometry is due to several workers [10,11] and is used to approximate # -k 0 [r/U0( dZf the thickening lithosphere near a ridge crest due to q~0P" t~d-~ " +f) cooling. Polar coordinates are defined by: x = r sin 0 (20) + x(r, 0)(1 - dpo)Aog I + ~k" (12) z = r cos O where in this case, the permeability is given by the with the 0 = 0 being the axis of symmetry. The Blake-Kozeny-Carman equation [3,13]: dimensionless boundary conditions for this geom- etry are ko = a2~ (13) ¢ at 0 = (~r/2- a): and x and z are standard cartesian coordinates. 1 0+~ _cosa --- sina Dimensional analysis shows that the piezomet- r 30 Or ric pressure is governed by a single length scale: at 0 = - (~'/2 - a):

L = (1 - q~0)Apg (14) - cos a - sin a (21) r 00 Or which is the length scale governing extraction of where a is the wedge angle. The dimensionless melt by shear in the matrix. This expression sug- matrix stream function is given by equations (10) gests a useful non-dimensionalization which re- and (16) as: veals the essential physics. Let: q~h = r(A sin 8- BO cos 0) (22a) (x, z, r)= (x', z', r')L where: +')= (lS) • -) +')'UoL 2 sln'a A= .~= ~'(1 - Oo)ApgL ~r- 2a - sin 2a Substituting and dropping primes yields: 2 B = (22b) ~P = rf(O) (16) ~r- 2a- sin 2a 141

2L. : 37 km

2L , : 37 km

b

Fig. 2. Melt streamlines (solid) and matrix streamlines (dashed) for slow-spreading ridges, U, = 1 cm yr-’ (a = 40 o ), matrix shear viscosity, q =lO *’ Pa s. Stippled region is the melt extraction zone containing all melt streamlines connecting the Fig. 1. (a) Boundary conditions for corner flow ridge model ridge axis to depth. h is the crustal thickness produced by the showing matrix streamlines and coordinate systems. (b) Defini- extraction zone. All figures are true scale. (a) Low porosity, tion of wedge angle (1. La is the characteristic length scale for w,,/U, = 1.56 (+c = l.O%), h = 0.5 km. (b) High porosity suffi- the ridge model, d is the depth to the base of the mechanical cient to produce - 6 km of crust, wc/.Y,, = 9.12 ($~a= 2.4%) boundary layer (equation (Bl)). (c) Boundary conditions for h = 6.5 km. corner flow trench model showing matrix streamlines and coordinate systems. 0 is defined differently from la.

2L, = 60 km The dimensionless piezometric pressure and melt stream function are then: B,=(~+r)cose

+‘,=+(F+r)sinB+& 0 a The principal features of this simple model can be easily seen in Figs. 2 and 3 which show melt and matrix streamlines for reasonable combina- tions of spreading rate, matrix viscosity and poros- ity. Given U, and 9, the wedge angle (Y is ap- proximated using an argument from Skilbeck [lo] (Appendix B). The most noticeable aspect of this model is the convergence of melt streamlines towards the ridge b axis in contrast to the diverging matrix flow. This focussing effect is due to the first term in the Fig. 3. Melt and matrix streamlines for fast spreading ridges. Uc = 7.5 cm yr-’ (cy=13O). n =1021 Pa s. (a) Low porosity, piezometric pressure and arises from the pressure showing saddlepoint in the melt stream function, wc/Uc = 0.47 gradient due to shear in the matrix. Equation (23) ($c = 1.5%) h = 0.5 km. (b) High porosity, sufficient to pro- shows that this effect decays away from the corner duce - 6 km of crust, we/U, = 2.77 (+.e = 3.6%), h = 6.1 km. 142

as l/r with a characteristic length scale of L R = used to put constraints on the values of the vari- L(2B) '/2. Because of this decay not all melt at ous parameters. Dimensional analysis shows that depth is extracted at the ridge axis. In all cases the crustal thickness behaves as: there is a region of melt extraction, shown in Wo a2gp3[ ~Apg ] '/2 stipple, in which the melt streamlines from depth • (27) pass through the axis. Outside this region, melt is h~q'~Luo L--~-o J incorporated into the lithospheric wedge and may and therefore depends primarily on the mean solidify or generate off-axis volcanism. porosity and the shear viscosity of the mantle. The The absolute value of the bounding streamlines simple model also predicts that, all else being depends only on the dimensionless percolation equal, faster spreading ridges should produce velocity, w0/L %, and the wedge angle ~. If wo/U o thinner crust than slow ridges. The discussion will is small (e.g. porosity is small), then equation (24) deal with the probable ranges of ¢ and 7, their shows that the effect of the singularity will be implications for the mechanics of mid-ocean diminished, advection of melt by the matrix will ridges, and a possible explanation for the apparent be significant and the melt streamlines will ap- independence of oceanic crustal thickness from proximately follow those of the matrix (if wo/U o spreading rate. = 0, they will be identical). In this case there will be a point where the advection of melt by the 6. Subduction zone model: .solution and results matrix is balanced by the pressure gradient to- wards the singularity and the melt velocity will be Boundary conditions for the subduction model zero in a frame fixed to the ridge axis. Mathemati- are shown in Fig. lc. In this case it is mathemati- cally this corresponds to a saddlepoint in the cally convenient to define polar coordinates in a stream function and the bounding streamlines will slightly different way than the ridge model by: be the ones that pass through this point. This x = - r cos O saddlepoint is apparent in Fig. 3a, and is obtained z=r sin0 (28) numerically (Appendix B). As w0/Uo --* 0 the sad- dlepoint moves towards the origin and as wo/U o The dimensionless boundary conditions are --* ~ the position of the saddlepoint tends to at 0=0: r= L R, 0 = _+~r/2. If the computed saddlepoint 1 ~, 0 =0 lies outside the boundaries of the wedge, then the r 0e Or bounding streamline will be the one that is tan- (29} at 8=/3: gent to the boundary.. This situation is illustrated in Figs. 2a, b and 3b. 1 04`~- 0 4`~[ -1 =0 The dimensional mass flux of melt per unit r 00 Or length of ridge that can be extracted from this where /3 is the angle of subduction. The dimen- region is: sionless matrix stream function is given by equa- tions (10) and (16) as: dM R - 2/0fl~}0~..J,,L 14` f bound I (25) dt 4`~,=r[C(sinO-ScosS)+DSsinS] (30a) where ~Rboundif is the dimensionless value of the where: bounding streamline. A more useful measurement, C /3 sin /3 however, is the crustal thickness produced by this melt flux which is simply the area flux divided by /32 _ sin2fl (30b) .the spreading rate or: D = /3 cos/3 - sin l} ( D < 0) /32 _ sin2/3 h = Or~oL I 4'R~ h,,~d I (26) P~ The dimensionless piezometric pressure and melt stream function are then: where p~ is the average density of oceanic crust. The crustal thickness is perhaps the most im- •~i = -~2(CcosO-DsinO)+rsinO (31) portant result produced by this model and can be r 143

./ ...... L,,r= 31~ L r = 44~.- , , / / / ." i. / / n .~

4' /' • ." i: .';",;" .,"/" ./" ;' i" ,,' ,: •' . " ' ,.'-' ,;i ." I- ,' : / "3A LT . -/

.S" a

,, i , // rmm , / / / / / L ...... : ::s /

. i ._~;?4.aL, / i LT= 43 km ~ ~ . .. i| - . -~-..- .-~- L_~,~ ~-~':~f ~,~ Iq

.-1 .64 Lr

Fig. 4. Melt (solid) and matrix (dashed) streamlines for shallow Fig. 5. As for Fig. 4 with a steeper subduction angle, fl = 60 o angle of subduction, fl = 30 °. Subduction rate, U0 = 7.5 cm Uo=7.5cmyr I 7/=10 21 Pas.(a) High porosity, ~0=3.0%. yr- 1 rl = 10 21 Pa s. Stippled region is the melt extraction zone wo/Uo = 1.91, flux = 1.2 × 10- 3 km 2 yr- 1. (b) Medium poros- containing all streamlines from depth that connect to the ity, g'0 = 1.5%. The effect of downward advection by the matrix wedge corner. The melt flux is calculated through a circular arc is more pronounced for larger ft. wo/Uo = 0.47, flux = 1.5 x at rm,n = 21/2 L/4. Also shown is the length of slab sampled at 10 -.4 km 2 yr -1. (c) Low porosity, ~0=0.6%. Subcritical the primary arc. The mantle wedge is true scale but the slab wo/Uo = 0.07, shows saddlepomt. Back arc is entirely discon- thickness is schematic for illustration only. (a) High porosity, nected from the slab. Flux = 9.4×10 -6 km 2 yr -1, slab frac- ~o = 3.0%, wedge comer is entirely connected to slab. wo/Uo tion = 50%. =1.91, flux = 2.4×10 3 km 2 yr-l. (b)Medium porosity, ~0 = 1.5%. Lower permeability causes streamlines to "bulge" as melt flow is more affected by downwards matrix flow. The model, melt that lies in the extraction zone, shown streamlines, however, begin and end in the same positions as in stipple, is drawn to the wedge comer (r = 0) by (a). wo/Uo = 0.47, flux = 2.9×10 -4 km 2 yr -1. (c) Low poros- a singularity in the piezometric pressure at the ity, ~o = 0.6%. wo/Uo = 0.07 is less than the critical value and a saddlepoint exists in the melt stream function causing the corner. Any melt outside this region follows the slab to partially disconnect from the comer. The length of slab streamlines until it intersects the lower boundary sampled at the arc is less than in (a) and (b). Flux = 1.8 × 10-5 of the plate above the sinking slab (8 = 0) and km 2 yr- 1 slab fraction = 79%. then percolates vertically, either to the surface or until it becomes incorporated into the upper plate. The surface expression of this behaviour will be marked by a narrow region of primary magmatism --w0 [2(C sin 8 + D cos 8) - r cos 8] lP'rl = U0 tr directly above the wedge comer followed by a narrow quiet zone of width LT=L(-2D) 1/2, +q~r (32) where the melt moves downwards and volcanism The principal features of the trench model can should be absent, succeeded by a broad region of be seen clearly in Figs. 4 and 5. As in the ridge diffuse magmatism. If the region of primary 144

magmatism above r = 0 represents the volcanic 1.o i i t ..... i ...... arc then it is important to note that the melt extracted at island arcs is not representative of the o.8 ~- mantle directly beneath the arc but of a much larger region of melting. The behaviour of this extraction zone is surprisingly complex given the g .... ~.oO,O6 simplicity of the model. The shape of the extrac- tion zone is controlled by the angle of subduction and the dimensionless percolation velocity. The oo ~-i ..t 1 I A " size, of course, is controlled by L. the only length o 30 ~o 9o scale in this problem. Subduction AngJ° (de9) Analysis shows (Appendix B) that the points at Fig. 6. Solid line is the critical dimensionless percolation which the bounding streamline intersects the up- velocity (wo/L~) as a function of subduction angle. For wo/&~ greater than critical, the slab is entirely connected to per plate (0 = 0, r = LT) and the subducting slab the surface. The slab becomes partially disconnected for wo/&~ (0 =/3, r = r~) depend only on the angle of sub- less than critical, duction and L: therefore, the width of the quiet zone and the maximum length of slab that can be sampled by the arc magmatism are independent of porosity. This result can be seen by comparing to the arc will be the streamline that passes through Fig. 4a to 4b and 5a to 5b which show similar the saddlepoint (Figs. 4c, 5c). Any melt from the geometries but different porosities. Though the slab outside this last connected streamline will be ends of the bounding streamline are fixed by /3 carried down with the matrix. More importantly and L, the path is controlled by w

If wo/U o is greater than critical, the slab frac- for many of the major features of the geometry of tion is clearly 100%; however, if wo/U o is less ridge and subduction related volcanism. than critical the slab fraction is given by: In the case of mid-ocean ridges, melt is ex- truded at the surface in narrow belts no more than q~T sad - ~ r~ I S t = (34) 5 km wide, yet the most reasonable mechanism for I f melt production, adiabatic pressure release, sug- gests that a broad partially molten zone should where ~rT sad is the value of the melt stream func- tion at the saddlepoint. exist beneath ridges to depths of - 60 km [3]. The As was remarked before, the behaviour of even constant porosity ridge model reconciles both ob- this simplest model for melt extraction at subduc- servations by permitting the extraction of small tion zones is surprisingly complex and shows that melt fractions from a large region while at the many of the ad hoc assumptions of the structure same time extruding it at the surface in a narrow of subduction zones will have to be reevaluated. band. This model is also consistent with the ob- Furthermore, this model can be tested quantita- servation that ridges appear to be passive features tively and can be used to place constraints on uncoupled to any deep mantle structure [14]. The important parameters such as the porosity and melt suction mechanism proposed here is strictly a matrix shear viscosity. The discussion will deal local effect that depends only on the mantle defor- with these tests and their implications for the mation near a ridge crest and does not require the processes of melt extraction acting at subduction presence of a deep mantle plume. zones. As for subduction related magmatism, the most notable feature is that nearly all major volcanic arcs lie in a narrow band 100-150 km above the 7. Discussion seismically active region of the subducting slab [15]. While it is possible that significant partial Before discussing these models it is worth re- melting occurs in this depth range, an alternative viewing the assumptions made and their implica- explanation is afforded by the subduction zone tions. Because of the primary assumptions of con- extraction model. Figs. 4 and 5 show that, regard- stant porosity and no melting, these models of less of melt source, the principal fraction of melt melt extraction are not complete models of melt- in the mantle wedge will be drawn to the intersec- ing and segregation. They indicate how melt will tion of the upper plate and the subducting slab. flow if it is present, but cannot predict its distribu- This junction should occur at the base of the tion. When melting is included in a 2-D model, mechanical boundary layer of the upper plate at there will necessarily be porosity gradients (Ap- depths of 90-100 km and the seismic zone in the pendix C) and these could lead to instabilities or subducting slab should be somewhat deeper. The other unforeseen behaviour. Nevertheless, if melt- simple model shows that there should always be a ing does not seriously perturb the matrix flow volcanic arc above the slab junction; however it then the most significant feature of the simple also indicates that the nature of the melts ex- models, extraction and concentration of melt by tracted can vary enormously depending on local matrix shear, should carry through. The physical conditions. Thus this model can qualitatively ex- reason for this statement is clear. The focussing of plain both the constancy of volcanic arc geometry melt towards the corner depends on the ~7 2 V term and the variety of island arc geochemistry. in equation (4) and therefore depends only on the While the qualitative similarities are striking, matrix velocity field. On the length scale of this the models can also be tested quantitatively. The problem, 10-100 km, mass balance indicates that most important results of these models are that the real flow of the matrix near a ridge crest or they give us a length scale, L, and a velocity scale subduction zone should approximate the ideal w 0, for melt extraction. With these, constraints corner flow of the simple models and one should can be placed on the two most poorly known expect the melt to move in a manner similar to parameters in the model, the matrix shear viscos- that illustrated in Figs. 2-5. If this is the case, ity which controls L and the porosity which then these models offer some simple explanations dominates w 0. 146

shows that to support axial valleys tens of kilome- ters in width requires shear viscosities of 10 2o- 10 2~ Pa s. This argument is certainly valid for slow-

>, 2o ] spreading ridges that show prominent axial val- leys, but is more problematic for fast spreading ridges without axial valleys. 2 ' ~'o ~'~" • For estimates of the mean porosity of the melt- ing zone beneath ridges, a simple scaling argument (Appendix C) shows it must be small - 2-4% if 1 2 4 6 8 lO 12 reasonable assumptions are made about melting Porosity (%) and melt extraction. This argument estimates the Fig. 7. Values of" ~7, ~o, and ~ that will produce 6 km of mean porosity that can be supported by a given oceanic crust as calculated from the ridge model. Contours are melting rate against compaction due to melt ex- half spreading rates in cm yr - i. traction. The results are shown in Table 2 and show two important features. First for reasonable degrees of total melt generation (10--20% by For the ridge model, the crustal thickness pro- volume) the porosity at any time is small, 2-3%. duced by the extraction zone is most useful for This is supported quantitatively by the I-D melt- constraining r/and ft. Physically, the simple model ing models [6,7]. Second, Table 2 implies that implies that the prerequisite 6 km of oceanic crust faster spreading ridges should have slightly higher can either be produced by extracting small melt porosities. Physically this also m'a_kes sense. If the fractions from a wide area or larger melt fractions primary melting mechanism is adiabatic pressure from a narrow extraction zone. Fig. 7 shows the release [3] then the melting rate should be propor- combinations of 77 and q~ for a given half spread- tional to the upwelling velocity and faster melting ing rate that will produce 6 km of crust as calcu- rates can support higher porosities. lated from the constant porosity model. As The implications of these two results are clear. spreading rates are known, if limits can be placed First, if ¢, is small, the simple model requires a on either viscosity or porosity, then the other large extraction length to produce 6 km of crust. parameter can be determined. Fortunately, there Inspection of Fig. 7 shows that this implies a large are at least two independent arguments for viscos- matrix viscosity, consistent with the shape of axial ity and porosity and both faw)r the combination valleys. Second, if the porosity increases with of high viscosity, 10z°-10 2~ Pa s, and small poros- spreading rate it is possible to reconcile the results ity, 2-4%. of the simple model with the observation that The simplest explanation for axial ridge valleys oceanic crustal thickness is apparently indepen- is that they are dynamically supported, and their dent of spreading rate. Equation (27) and Fig. 7 width then depends on the matrix viscosity both imply that to produce a constant 6 km of [10,11,17]. Under these conditions they should be crust requires a small increase in the porosity, supported by the same stresses responsible for - 1% for an order of magnitude increase in the melt extraction and therefore the length scale given spreading rate, which is comparable to the in- by the width of axial valleys should be related to crease in porosity predicted by the simple scaling the length scale of melt extraction. Skilbeck [10] argument. This argument is self consistent and fits

TABLE 2 Expected mean porosity (%) as a function of the degree of partial melting and spreading rate

Total degree Half spreading rate (cm yr- l ) of partial melting 1.0 2.5 5.0 7.5 10.0 10% 1.2 1.6 2.0 2.3 2.5 20% 1.6 2.1 2.6 3.0 3.3 147 both qualitatively and quantitatively what little is -4 i t i i i 1 i t known of the melt extraction processes at mid- ocean ridges. / If little is known about melt production and extraction at ridges, even less is known about the processes governing subduction related volcanism. ~04 ~ 0.3 Fortunately, the subduction zone model is insensi- ~ ~" 0.2 Q~ tive to the nature of the "melt" or "fluid" derived ~o -6 - ~ °~ from the slab or to how it moves out of the slab. -- ' Wo Uo:O,t~ As long as a low viscosity phase is present at the surface of the slab, the simple model indicates -7 how it should flow through the mantle wedge.

However, if that phase can be identified by its 10 30 50 70 90 isotopic "slab signature" and can maintain that Subductlon Angle (deg) signature throughout its movement, the simple Fig. 8. Volcanic arc melt flux (per length of arc, km2 yr -1) model presented here offers several quantitative normalized by L T, as a function of subduction angle and internal checks for validity. The isotopic ratio we wo/Uo. The bold line shows flux/LT for critical values of will use for determining slab signature is 87Sr/86Sr wo/Uo. Values plotting above the critical line have slabs that as it records sea alteration of the oceanic connect to the surface. Values plotting below have discon- crust by hydrothermal circulation through the nected slabs. The stippled region shows data from the Central ridge flanks [21]. Altered oceanic crust should and Eastern Aleutians and indicates that the Aleutians are near critical and should show little to no slab signature in the show a higher 87Sr/S6Sr ratio than unaltered crust. back-arc region. This geochemistry forms the basis for tests of the simple model, which we-will first outline and then apply to the central-eastern Aleutians and show melt flux, subduction angle and subduction rate that this model can account for the observations are known then wo/U o can be read off the graph. in a consistent manner. The final check for consistency is geochemical. If The subduction zone model offers three inde- 13 and wo/U o are known then Fig. 6 (or 8) pre- pendent methods for estimating L and wo/U o. dicts whether or not the slab should be connected The first and most obvious test gives a direct to the back arc. If wo/U o is near the critical value measure of the length scale of extraction. The for a given subduction angle then the back arc simple model predicts that the primary arc should volcanism should show little to no slab signature. show signs of shallow melting with some slab Therefore, to test this model the following data signature while the first melts to reach the surface are needed for an arc with subsidiary back arc behind the arc should show signs of deeper melt- volcanism: (1) angle of subduction near the plate ing or no slab signature at all. The distance behind junction, (2) geochemical data as a function of the arc where this change should occur is roughly position behind the arc, (3) the subduction rate r = L T and depends only on the angle of subduc- U0, and (4) the flux per unit length of arc. One tion and the length scale of extraction. Therefore, region where all these measurements have been to find L requires a carefully sampled traverse carried out is the central and eastern Aleutians, across an arc of known subduction angle showing especially the Cold Bay region with the second arc secondary back arc magmatism from - 3-100 km volcano at Amak [18]. While the geochemical data behind the arc. Any abrupt change in chemistry for the secondary volcanism is sparse it is still will give the length scale of extraction. If L can be sufficient to constrain the model and explain some determined by this method and the volume flux of the features of this region. Fig. 9 shows per length of trench is known, then constraints 87Sr/86Sr [18] plotted against distance for the can be placed on wo/U o as was done for the ridge primary arc, the Frosty peak volcanoes (- 3 km), problem. Fig. 8 shows the melt-flux/L- r plotted Mount Baldy ash flow (- 10 km), the second arc against subduction angle for contours of wo/U o volcanoes Amak (42 km) and Bogoslof (- 50-60 as calculated from the simple model. If L T, the km) and the back-arc islands of St. George (300 148

• Cenlral Islands -'~ Cold Bay youngvolcanlcs • Second Arc Islands L,rt 10 km~ i.'. Back Arc Islands

.70350 1 i i

l I-+,o .70330

.70310

m

.70290

LZ t-.

,70270 I I10 A Fig. 10. Melt and matrix streamlines for the two end member 0 100 1000 models of the Aleutian arc. B = 65 o, Uo = 7 cm yr- i. Length kms. behind first arc of slab sampled at the primary arc in both cases is - 50 krn. Fig. 9. ~Sr/~rSr as a function of distance behind the primary (a) Short length scale, L r = -10 km ('0 = 1.1 × 10 2° Pa s); volcanic arc for the Central and Eastern Aleutians (data from wo/Uo=0.41 (~=1.35%), flux = 3.2 ×10 -5 km2 yr -t. (b) von Drach et al. [18]). Distance scale is logarithmic and data Long length scale, LT = --50 km (r/=3.0×1021 Pa s); are shown for 0, 3, 9, 42, 60, 300 and 600 km. The sharp drop w0/U0=0.14 (g,=0.8%), flux = 3.4 ×10 -5 km2 yr -1, slab in 87Sr/86Sr at - 10 km is consistent with values at - 50 km fraction = 59%. and behind. Far back-arc samples are for reference and should show no slab contribution. region are probably sufficiently indirect that the slab contribution is minimal. While clearly ob- km) and Nunivak (600 km). The last two are for servations from other arcs are required to say reference as they should be far enough removed anything conclusive, this first simple test shows no from the subduction zone as to only record mantle glaring inconsistencies. melting in the back arc region. Fig. 9 shows a sharp drop in 87Sr/86Sr at 10 km 8. Conclusions which is consistent with the values at 50 km and back and is representative of typical unaltered These models indicate that for reasonable ma- Pacific MORB levels [18]. As the subduction angle trix viscosities, 102°-10 21 Pa s, the movement of is fl= 60-70 ° [19] this gives a range of length melt beneath mid-ocean ridges and island arcs is scales from 10 to 50 km (rl - 102°-3 × 10 21 Pa s). controlled by non-zero V 2V shear in the matrix. Melt flux per km of arc for the central Aleu- The pressure gradients generated by corner flow tians is calculated to be 3.5 × 10 -5 km 2 yr -] [1]. of the matrix at spreading centers cause melt to Flux/L-r and fl are plotted (Fig. 8) for the Aleu- migrate toward the ridge axis enabling the extrac- tians with subduction rate 7 cm/yr [20] and give tion of small melt fractions from a wide melting wo/Uo- 0.15 for L T- 50 km and wo/Uo- 0.41 zone yet producing a narrow zone of volcanism at for L v- 10 kin. Melt streamlines for these two the surface. Similarly corner flow in the mantle end members are shown in Fig. 10. For the longer wedge beneath island arcs causes melt to flow to length scale the slab should be entirely discon- the slab-plate junction and can account for the nected from the secondary arc which is consistent concentration of volcanism in regions where earth- with the uniformly low 87Sr/S6Sr values in the quakes in the subducting slab reach a depth of back arc. For L v - 10 km, while the entire slab is 100-150 km. The subduction zone model also connected, the melt streamlines for the back arc implies that the geochemistry of island arc 149 volcanism will be strongly influenced by the conditions. Once these are known, V.~, k,, and therefore v geometry of melt streamlines and the connectivity are completely determined by equations (3) to (5). of the slab with the overlying mantle wedge. Equations with constant porosity and no melting

Acknowledgements if q~=% and F=0, equations (A1) to (A3) reduce to a very simple form. Equations (AI) and (A2) become, respec- Many thanks go to R.H. Hunter for useful tively: advice and to G. Wasserburg for the Aleutian W" V = 0 (A4) data. M.W. Spiegelman is supported by a Marshall v2(vx v) = 0 (AS) Scholarship. This research forms part of a general and (A3) vanishes identically. By writing V in potential form investigation of mantle melting supported by (equation (7)) then (A4) is satisfied explicitly and (A5) reduces NERC. to the biharrnonic equation which is easily solved for corner flow boundary conditions. Appendix A--General equations in two dimensions Appendix B--Analysis: ridge and trench models The simplest set of equations for the conservation of mass and momentum in two dimensions plus time are easily derived Ridge model from the general equations (1) to (5). If Pf and p~ are constant, equation (2) becomes: Estimation of wedge angle a. A straightforward definition of a is shown in Fig. lb and derives from an argument from Dv#, i)~ F + V. WO = (1 - q~)V'V+ -- (A1) Skilbeck [10]. If we assume that the actual shape of the Dt ~t Os lithospheric plate near a ridge axis is controlled by the thermal Taking the curl of equation (4) gives: history of the plate then a reasonable approximation for the wedge angle is: "qV 2( W V) 1--- (a2) x =-Apgox - 8x g tan-, [ d(LR) ] ~= LR ] (ma) and adding equations (1) and (2) and substituting in (3) and (4) yields: where

V'[~(~v2V+(I~+~/3)V(V'V)+(I-(p)Aogk] d(LR) - 2( ~r"LR ],/2 (Bib)

FAO = 17'. V - -- (A3) is approximately the depth a coofing pulse in an infinite half PfP~ space would have traveled in time t = LR/U o, and therefore is Equation (A1) is the conservation of mass for the matrix approximately the thickness of the lithosphere at a distance L R and states that the change in porosity in a frame fixed to the from the ridge axis. K is the thermal diffusivity of the mantle matrix is the difference between compaction (Div V < 0) and and LR is the characteristic length scale for the ridge. Unfor- melting. (A2) shows that the vorticity of the matrix is tunately, inspection of Figs. 2 and 3 shows quite clearly that maintained by horizontal porosity gradients and in general by the characteristic length scale for melt extraction at ridges is: horizontal density gradients. Equation (A3) is the most com- L R = L(2B) '/2 (B2) plicated of the three but the physics is quite straightforward. The left-hand side is simply the net melt flux through a small where L and B are defined as before and B depends on the volume of matrix and the right-hand side is the sum of wedge angle a. Therefore, (B2) yields a transcendental equa- compaction and the volume change on melting. Therefore. the tion for a which was solved using a simple iterative scheme amount of melt that can be extracted from a small volume starting from a = 0 in (B2). depends on how much is squeezed out by compaction and how much is expelled by the increased volume of the melt. Pre- Solution for the bounding streamline. Saddlepoints exist at val- liminary work indicates that (A3) will be dominated by the ues of r and 0 for which: non-linear behaviour of the permeability with respect to small v= WX (Lkkf) = O (B3a) changes in porosity. Furthermore, if the volume changes are relatively small and the buoyancy term is large (third term on or: left-hand side of (A3)) then the compaction will be controlled by the extraction of melt by gravity and less by the viscosity of +1 cos0+.4cos0-B(cos0-0sin0)=0 the matrix. vo If the permeability and the melting function are known (or (B3b) presupposed) then (AI) to (A3) can be solved for porosity and the two components of V, subject to appropriate boundary ~--1 sin0+,4sin0-B0cos0=0 (B3c) 150

Eliminating r 2 terms, saddlepoints exist where then be found using (B6) and (B9) with 0 = ft. This gives: ga(O) = BOcos2O+ ,--+--A{ w° B ] sin2O=O (B4) rtl = L [ ( - 2 D ) ' /2 + [ 2 c°s ~ ( C]cos,8 sin B + D c°s ~ ) - 2 D ] ' /2 ~, 2 !

If gR( ~/2- ~) > 0 no saddlepoints exist within the boundaries. (BI0) In this case the bounding streamline is the one that is tangent which is also independent of wo/U o. to the boundaries at 0 = ~/2- a or:

Solution of saddlepoint: subduction zone model. In the subduc- ve( r, ~r/2- a) = - ~p~ = 0 (B5a) tion zone model the melt stream function will develop a "dr O ~/'2 ,, saddlepoint if wo/U o is less than the critical value. As before that is: the saddlepoint exists where v = gr × (q.,~.f) = O or solving for 0 only, the saddlepoint exists where: d rta n = 2(w°/U°)B( wo ) 71/" (BSb) ~,(0) = dO[(C sin 0 + D cos 0)a(0)] = 0 (Bll)

B(."r/2- a) sin a + -Ia~o - A cosa The last streamline to connect the subducting slab to the wedge corner is the streamline that passes through the saddle- point. The point where this streamline intersects the subduct- Subductton zone model ing slab at 0 = 13 is solved for numerically.

Criucal value of wo/U o. A given streamline is found by plot- Appendix C -- A scaling argument to estimate ting: porosity in the presence of melting a(0)r 2 - ,/,!, r -2~(CsinO+DOcosO)=Ow(, (B6a) In steady state, the porosity is governed by: for a fixed value of 4,~r where: I" w 0 v. v,~, = (I- ~,) v,- v + - (ci) a(O)=C(sinO-OcosO)+ DOsinO+--cosO (B6b) P, which states that the equilibrium value of the porosity is Whether or not the slab is wholly connected to the wedge controlled by the balance of compaction and melting. Now, if corner depends only on the zeros of a(O). For 1/, = 0, where the results of the I-D models are applicable to two dimensions. a = 0. r -~ o¢. If wo/U o is greater than the critical value then then compaction is primarily the result nf melt extraction by a has no zeros. If wo/U o is less than critical, a has two zeros differential buoyancy and equation (A3) can be approximated and the zero streamline can never connect. The critical value of by: wo/U o occurs when a has only one zero. When this happens, the zero streamline is connected but only as r --+ ~. Therefore. IT.V--V.[~(1-qJ)A#gkl+l,z, (("2) the critical value of wo/U o as a function of the subduction angle fl can be found by solving the simultaneous equations: where At, is the volume change on melting and /~ now points a(0) =0 upwards. Equation (C2) shows that where the porosity is near da/dO = 0 (B7) zero, and therefl)re separation is negligible, the matrix must expand to take up the volume increase on melting. Beneath This was done numerically and the results are shown in ridges, this region will be near the lower boundary of the Fig. 6. melting zone. In this region the porosity will grow rapidly until the permeability is high enough to permit separation by Solutton for the bounding streamhne. Inspection of Figs. 4 and 5 buoyancy. Once separation becomes significant, however, the shows that the bounding streamline that connects the mantle matrix must compact and porosity will grow more slowly. This wedge to the wedge corner is the one that is tangent at the is supported by the I-D melting models [6,7 I. Schematically upper boundary (O = 0). As in the ridge model, this implies this is shown in Fig. 11. An important consequence of equa- that % = 0 at O = 0. The tangent point is then easily computed tions (C1) and (C2) is that the porosity can never be indepen- to be: dent of position in steady state unless F -= 0. More importantly, these equations can be used to estimate L.) =L(-2D) I/2 (D<0) (B8) the melting rate required to support a mean porosity q'0 and is independent of wo/U o. The value of the bounding against compaction, or conversely, they can be used to estimate streamline is then simply the characteristic porosity produced by a constant rate of

W O melting over a melting depth /. This estimate is based on the ~k!, b,,una = 2--U(-) ( - 2D ) '/2 (B9) following assumptions: (1) I" = I"o over the melting range and the intersection of this streamline with the lower slab can (2) q~ - q~(z) porosity changes are primarily in the z direc- 151

Surface ~-0o-- // melting rate and higher melting rates can support higher mean / porosities. / / / References / / 1 J.A. Crisp, Rates of emplacement and volcanic / / output, J. Volcanol. 20, 177-211. 1984. / 2 N.H. Sleep, Segregation of magma from a mostly crystal- L / line mush, Geol. Soc. Am. Bull. 85, 1225-1232, 1974. / 3 D. McKenzie, The generation and compaction of partially / molten rock, J. Petrol. 25, 713-765, 1984. / 4 D.R. Scott and D.J. Stephenson, Magma solitons, Geophys. / Res. Left. 11, 1161-1164, 1984. 5 F.M. Richter and D. McKenzie, Dynamical models for oo melt segregation from a deformable matrix, J. Geol. 92, Base of 729-740, 1984. I I I I n',en in g Zone 2 4 6 8 10 6 N.M. Ribe, The deformation and compaction of partial molten zones, Geophys. J. 83, 487-501, 1985. Porosity (%) 7 F.M. Richter, Simple models for trace element fractiona- Fig. 11. Schematic diagram for scaling argument (Appendix C) tion during melt segregation, Earth Planet. Sci. Lett. 77, showing 1-D porosity profile as a function of height above the 333-344, 1986. base of the melting zone. Dashed line shows porosity profile 8 D.R. Scott, D.J. Stephenson and J.A. Whitehead, Observa- for constant melting rate and no separation. Solid curve is the tions of solitary waves in a viscously deformable pipe, porosity profile when separation of melt from the matrix by Nature 319, 759-761, 1986. buoyancy is significant (see [6,7]). ep0 is the characteristic 9 D. McKenzie, Speculations on consequences and causes of porosity with separation. / is the thickness of the melting zone. plate motions, Geophys. J. 18, 1-32, 1969. 10 J.N. Skilbeck, On the fluid dynamics of ridge crests, in: Notes on the 1975 Summer Study Program in Geophysical tion. Fluid Dynamics at the Woods Hole Oceanographic Institu- (3) (1 - ,~) - 1 mean porosity is small tion, Vol. 2, pp. 113-122, Woods Hole Ocean. Inst, Ref. (4) k, - a2~"/c No. 75-58, 1975. (5) At, is negligible 11 A.H. Lachenbruch, Dynamics of a passive spreading center, Substituting these into (C1) and (C2) yields: J. Geophys. Res. 81, 1883-1902, 1976. 12 G.K. Batchelor, An Introduction to Fluid Dynamics, Cam- bridge University Press, 1967. -- w+ Aog --- (c3) ,3z p~ 13 F.A.L. Dullien, Porous Media Transport and Pore Struc- ture, Academic Press, New York, N.Y., 1979. Now, W- Uo and inspection of Figure 11 shows that: 14 G. Houseman, The deep structure of ocean ridges in a ¢ - % convecting mantle. Earth Planet. Sci. Lett. 64, 283-294, a,t, % (c4) 1983. 15 B.L. Isacks and M. Barazangi, Geometry of benioff zones: az / lateral segmentation and downwards bending of the sub- where 9o is the characteristic porosity. To first order then, ducted lithosphere, in: Island Arcs, Deep Sea Trenches and (C3) becomes: Back Arc Basins, M. Talwani and W.C. Pitman, eds., pp. 99-114, American Geophysical Union. Washington, D.C., P - ~o(1 + n w° (C5a) 1977. 16 B. Parsons and D. McKenzie, Mantle convection and the where: thermal structure of plates, J. Geophys. Res. 83, 4485-4496, 1978. p =/lo/p,Uo (C5b) 17 N.H. Sleep, Sensitivity of heat flow and gravity to the is the "total degree of partial melting", that is, the volume mechanism of sea floor spreading, J. Geophys. Res. 74, fraction of matrix melted by passing through a constant melt- 542-549, 1969. ing zone of length I at speed U0. Physically (C5a) is reasonable 18 V. von Drach, B.D. Marsh, and G.J. Wasserburg, Nd and as it shows that in the absence of separation (w 0 = 0) the Sr isotopes in the Aleutians: multicomponent parenthood porosity equals the degree of partial melting whereas if sep- of island-arc , Contrib. . Petrol. 92, 13-34, aration is important, the mean porosity must be smaller than 1986. p. (C5) also shows that for a fixed degree of partial melting, 19 E.R. Engdahl, Seismicity and plate subduction in the central fast spreading ridges should show a slightly higher porosity. Aleutians, in: Island Arcs, Deep Sea Trenches and Back This is also reasonable as (C5b) shows that to produce the Arc Basins, M. Talwani and W.C. Pitman, eds., pp. 259-271, same degree of partial melting at fast ridges requires a higher American Geophysical Union, Washington, D.C., 1977. 152

20 K.H. Jacob, K. Nakamura, and J.N. Davies, Trench volcano 21 R.K. O'Nions, Isotopic abundances relevant to the identifi- gap along the Alaska-Aleutian arc: facts and speculations cation of magma sources, Philos. Trans. R. Soc. London, on the role of terrigenous sediments for subduction, in: Ser. A 310, 591-603, 1984. Island Arcs, Deep Sea Trenches and Back Arc Basins, M. Talwani and W.C. Pitman, eds., pp. 99-114. American Geophysical Union, Washington, D.C., 1977.