Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (211) 186, doi: /j X x A high-order numerical study of reactive dissolution in an upwelling heterogeneous mantle I. Channelization, channel lithology and channel geometry Alan Schiemenz, 1,2 Yan Liang 1 and E. M. Parmentier 1 1 Department of Geological Sciences, Brown University, Providence, RI, USA. yan_liang@brown.edu 2 Division of Applied Mathematics, Brown University, Providence, RI, USA Accepted 211 April 28. Received 211 April 21; in original form 21 August 1 SUMMARY High-porosity channels are important pathways for melt migration in the mantle. To better understand the lithology and geometry of high-porosity melt channels, we conduct highorder accurate numerical simulations of reactive dissolution along a solubility gradient in an upwelling and viscously deformable porous column. In contrast to earlier studies, we assume the dissolution reaction to be at equilibrium, consider a finite soluble mineral abundance, and employ a high-order accurate numerical scheme. Using sustained perturbations in porosity and soluble mineral abundance at the inflow boundary, we explore the structure of steadystate high-porosity melt channels and their associated lithologies over a range of parameters, including soluble mineral abundance, solubility gradient, amplitude and lateral variation in inflow melt flux, melt fraction and upwelling rate. In general, high-porosity dunite channels are transient and shallow parts of pathways for melt migration in the mantle. The lower parts of a high-porosity channel are orthopyroxene-bearing dunite, harzburgite and possibly lherzolite. A wide orthopyroxene-free dunite channel may contain two or three high-porosity melt channels. The depth of dunite channel initiation depends on the solubility gradient, soluble mineral abundance, inflow melt flux, melt suction rate and upwelling rate. The amplitude and length scale of lateral variation in porosity and orthopyroxene abundance at the base of the upwelling column are important in determining the size and dimension of dunite channels, the strength of melt focusing and the melt suction rate, the presence of compacting boundary layer, as well as the number of high-porosity melt branches within a dunite channel. The spatial relations among the high-porosity melt channels, dunite and harzburgite channels documented in our numerical simulations may shed new light on a number of field, petrological and geochemical observations related to melt migration in the mantle. Key words: Numerical solutions; Self-organization; Mid-ocean ridge processes; Magma migration and fragmentation; Physics and chemistry of magma bodies. GJI Mineral physics, rheology, heat flow and volcanology 1 INTRODUCTION One fundamental geologic observation is that the major element composition of mid-ocean ridge basalt is not in chemical equilibrium with residual mantle at low pressure (e.g. O Hara 1965; Stolper 198; Elthon & Scarfe 1984; Klein & Langmuir 1987; Kinzler & Grove 1992). To preserve their geochemical signature developed at depth, melts must rise from depths of at least 3 km in the asthenospheric mantle to the surface without extensive reequilibration with their surrounding mantle at shallow depth. Focused melt flow Now at: Department of Earth and Environmental Sciences, Munich University, Munich, Germany. through melt-filled fractures or high-porosity melt channels or some combination of the two are possible mechanisms (Kelemen et al. 1997, and references therein). Petrologically, high-porosity melt channels are often identified as orthopyroxene-free dunite channels, though the spatial relationship between melt channel and dunite channel has not been examined until very recently (Liang et al. 21). Remnants of dunite channel have been observed as veins, tabular or sometimes irregular shaped bodies in the mantle section of ophiolites and peridotite massifs. According to Kelemen et al. (1997) the dunite dykes or veins make up 5 15 per cent of the mantle in the Oman ophiolite and their sizes range from tens of millimetres to 2 m in width and tens of metres to at least 1 km in length. The majority of the dunite dykes have sharp lithological contacts with their surrounding harzburgite or lherzolite matrix, forming a C 211 The Authors 641

2 642 A. Schiemenz, Y. Liang and E. M. Parmentier distinct dunite harzburgite or dunite harzburgite lherzolite sequence in the field. The formation of dunite channels in the mantle involves reactive dissolution when olivine normative basalts percolate through a partially molten harzburgite or lherzolite matrix (e.g. Quick 1981a; Kelemen 199; Kelemen et al. 1992, 1995a,b, 1997; Daines & Kohlstedt 1994; Asimow & Stolper 1999; Morgan & Liang 23, 25). Preferential dissolution of pyroxene and precipitation of olivine increases local porosity and grain size (Daines & Kohlstedt 1994; Morgan & Liang 23, 25; Braun 24) and hence dunite permeability, which accelerates melt flow and peridotite dissolution. Such positive feedbacks between dissolution and melt flow result in reactive infiltration instability (e.g. Daccord 1987; Ortoleva et al. 1987; Hoefner & Fogler 1988; Steefel & Lasaga 199; Aharonov et al. 1995; Spiegelman et al. 21). Linear stability analysis and 2-D numerical simulations of reactive dissolution in a stationary non-deformable matrix show that a planar dissolution front is unstable to large wavelength perturbations and reorganizes itself into scalloped geometry or fingers along the flow direction (Chadam et al. 1986; Daccord 1987; Ortoleva et al. 1987; Sherwood 1987; Hoefner & Fogler 1988; Steefel & Lasaga 199). The wavelength of the fingers is determined by a competition between chemical diffusion in the fluid, which is stabilizing for a binary system, and fluid flow, which is destabilizing. Aharonov et al. (1995), Spiegelman et al. (21) and Spiegelman & Kelemen (23) examined reactive dissolution in the context of dunite channel formation in the mantle by including matrix deformation and by considering dissolution along a linear solubility gradient in a binary system. Results from linear stability analysis and 2-D numerical calculations show that elongated high-porosity melt channels develop spontaneously in the presence of a solubility gradient that is positive along the incoming melt flow direction. These melt channels interact and coalesce in the upper part of the simulation domain forming interconnected high-porosity networks perhaps resembling the fractal tree model proposed by Hart (1993). Compaction reduces porosity of interchannel region giving rise to significant flow localization or focusing. The wavelength of the channels, as determined by the competition among dissolution, chemical diffusion and compaction, is smaller than the compaction length ( m in the mantle; Spiegelman et al. 21). In the presence of a prescribed background melting, Spiegelman et al. (21) noted that unstable dissolution channels become more difficult to form. The stabilizing effect of matrix melting on melt localization is further examined by Hewitt (21) who showed that a chemically uniform upwelling melting column starting from its solidus is stable relative to infinitesimal perturbation for the set of parameters explored in his study. Although significant progress has been made in understanding dunite channel formation as a dissolution instability during melt migration in the mantle, detailed structures of melt channel and dunite channel as well as their spatial relationships in an upwelling mantle column are still not well understood. It is not known, for example, if high-porosity harzburgite channels or orthopyroxenebearing dunite channels exist in an upwelling mantle, and how melt fraction within a dunite or harzburgite channel evolves as a function of time during mantle upwelling [Spiegelman & Kelemen (23) included mantle upwelling in a 2-D numerical model, but their study focused mostly on the variation and fractionation of incompatible trace elements between the channel melt and the matrix melt]. With only a finite amount of soluble mineral, mantle upwelling acts to sustain the amount of soluble mineral at any height in the melting column. To map out dunite channel distribution in a 2-D upwelling column, it is necessary to explicitly track the modal abundance of orthopyroxene (opx) in the simulation domain. The depth of dunite channel initiation in an upwelling mantle is another unresolved but important problem. It is not known if the relatively shallow occurrence of dunite bodies in the mantle sections of ophiolites is due to limited field exposure or processes related to dunite channel formation in the mantle. Spiegelman et al. (21) and Spiegelman & Kelemen (23) demonstrated the importance of matrix rheology in affecting the depth of melt channel initiation in an upwelling melting column. Asimow & Stolper (1999) showed the importance of melt focusing in determining the depth of dunite channel initiation. Other factors important in determining the depth of dunite channel initiation may include pyroxene modal abundance in the initial mantle, solubility gradient and mantle upwelling rate, though none of these factors have been examined before. To better understand the relationship between high-porosity melt channel and dunite channel and their bearings on melt migration in the mantle, we have formulated models of dissolution instability in an upwelling mantle. We consider an initial mantle composed of soluble (opx) and insoluble (olivine) minerals. As dissolution occurs, we explicitly track the amount of soluble mineral remaining. We treat dissolution with a local equilibrium formulation in which the amount of soluble mineral in the melt is determined by a saturation concentration that increases with decreasing pressure. The models are formulated with a high-order numerical method, the results of which will be reported in several papers. To provide new insights into the physical processes leading to melt channel and dunite channel formation during melt migration in the mantle, we conducted a detailed linear stability analysis of reactive dissolution along a linear solubility gradient with a bulk viscosity that is inversely proportional to porosity (Hesse et al., to be submitted to Geophys. J. Int., referred to as HSLP211s hereafter; see also Schiemenz et al. 211). In a 1-D steady-state upwelling column, the abundance of soluble mineral determines the maximum domain height through which reactive dissolution can take place. Increasing the mantle upwelling rate decreases the accumulated amount of matrix dissolution at the top of the column, reducing the growth rate of a local perturbation in porosity (i.e. upwelling is stabilizing). Strong interactions between compaction and dissolution lead to the emergence of compaction dissolution wave instability when the domain height is significantly larger than compaction length. When the domain height is comparable to the compaction length, fingering or channelling instability similar to those observed previously is found. In this study, we focus on the first-order characteristics of melt channel and dunite channel, given their closer link to the previous studies. A brief discussion of the melt channel and its associated lithologies was given in Liang et al. (21). A detailed numerical study of the compaction dissolution waves will be given in a companion paper. This paper is organized as follows. In Section 2, we present the governing equations and model setup. This is followed by a brief description of the numerical methods. In Section 3, we first examine the first-order features of melt and dunite channels. To better understand the dependence of these features upon model parameters, we then perform numerical simulations for varying parameter choices and heterogeneity structures. In Section 4, we analyse dunite channel geometry, determining factors that affect channel width and depth. Melt focusing, induced by lateral porosity variations, is shown to be a key factor. In Section 5, we first compare results from the kinetic and equilibrium formulations for the

3 Numerical studies of reactive dissolution 643 dissolution rate and then compare results from this study with those of previous studies. This is followed by a discussion of potential geological applications. In Appendix A, we derive a simplification of the momentum equation. In Appendix B, we outline the numerical method used in this study. Finally, in Appendix C, we derive an analytic expression for the depth of dunite channel initiation. 2 GOVERNING EQUATIONS AND SETUP We consider reactive dissolution along a solubility gradient in an upwelling and viscously deformable porous medium. For simplicity, we consider an effective binary system that consists of an interconnected melt network and a solid matrix comprised both of a soluble and an insoluble mineral. In the context of melt migration in the mantle, the soluble mineral may be identified as opx and the insoluble mineral as olivine (ol). Let Ɣ opx be the dissolution rate of opx and Ɣ ol be the precipitation rate of ol in the melt, measured as volume of mineral dissolved or precipitated per unit volume of the porous medium per unit time. Then Ɣ ol = βɣ opx,whereβ is the stoichiometric coefficient for opx dissolution. The rate of melt production due to dissolution is (1 β)ɣ opx. Mass conservation equations for the melt and total solid take the usual form (e.g. McKenzie 1984), φ f t + (φ f v) = (1 β)ɣ opx, (1) (1 φ f ) + [(1 φ f )V] = (1 β)ɣ opx, (2) t where φ f is the volume fraction of the melt. v and V are melt and solid velocities, respectively. To track the soluble mineral in the upwelling column, we use a similar mass conservation equation for opx, φ opx + (φ opx V) = Ɣ opx. (3) t The volume fraction of olivine is thus given by 1 φ opx φ f.in deriving eqs (1) (3), we assume that (i) the densities of the melt and the solid are the same (except in the buoyancy term given below) and variations in density are negligible (a Boussinesq approximation); and (ii) velocity of opx and ol is the same as the bulk solid velocity. Neglecting diffusion and dispersion in the melt and assuming constant and uniform mineral compositions, the mass conservation equation for the independent component in the melt for the effective binary system can be written as (φ f c f ) + (φ f vc f ) = ( c opx t ol) βc Ɣopx, (4) where c f, c opx,andc ol are concentrations of the independent component in the melt, opx and ol, respectively. The superscript indicates that the respective concentrations are independent of time and spatial coordinate. The velocities in eqs (1) (4) are determined by Darcy s law and a momentum or force balance equation (e.g. McKenzie 1984; Scott & Stevenson 1986; Fowler 199; Spiegelman 1993; Schmeling 2; Bercovici et al. 21; Ricard 29). When the solid flow field is irrotational and the shear viscosity of the solid is constant and uniform, Darcy s law can be written in terms of gradients of an effective pressure and melt buoyancy, φ f (v V) = κ φ μ [ p ρgn z], (5) where κ φ is the permeability, μ is the viscosity of the melt, p = p f p s is the pressure difference between the melt (p f ) and the solid (p s ), ρ = ρ s ρ f is the density difference between the solid (ρ s )andthemelt(ρ f ), g is the acceleration due to gravity, and n z is the unit vector, positive upward. In Appendix A, we outline a derivation of eq. (5). The permeability depends on porosity through the power-law relation ( ) n φ f κ φ = κ, (6) φ f where κ is the permeability at the reference melt fraction φ f.the exponent n takes on values of 2 to 3. In this study, we choose n = 3. The pressure difference between the melt and the solid is related to compaction rate of the matrix ( V) through the bulk viscosity ξ, p = p f p s = ξ V, (7) ξ = η φ f, (8) where η is the shear viscosity of the solid matrix. The inverse correlation between bulk viscosity and porosity has been suggested and/or used in a number of studies of melt migration in a viscously deformable porous medium (e.g. Scott & Stevenson 1986; Sleep 1988; Fowler 1985, 199; Schmeling 2; Bercovici et al. 21; Connolly & Podladchikov 27; Šrámek. et al. 27; Hewitt & Fowler 28; Ricard 29; Hewitt 21; Simpson et al. 21). The effective pressure can be solved from a Helmholtz equation obtained by combining eqs (1) and (2) and the divergence of eq. (5), [ κφ μ ( p ρgn z) ] = φ f p η. (9) Finally, Eq. (2) can also be written in terms of the effective pressure, φ f t + V φ f = (1 φ f ) φ f p η + (1 β)ɣ opx. (1) Eqs (5), (9) and (1) are essentially the same as those given by Spiegelman et al. (21, their eqs 1, 2 and 1), Connolly & Podladchikov (27, their eqs 6, 1 and 11) and Hewitt (21, his eqs. 1 and 28), although the definition for the effective pressure and derivations are slightly different. In the numerical simulations reported in this study, we solve for φ f, φ opx, c f, v and p from eqs (3), (4), (5), (9) and (1) subject to the dissolution rate, initial and boundary conditions discussed below. 2.1 Dissolution rate Mass transfer can be modelled by the first-order kinetic approximation (e.g. Steefel & Lasaga 199; Aharonov et al. 1995; Spiegelman et al. 21) ( eq ) Ɣ opx = αφ f φ opx c f (x) c f, <α<. (11) The coefficient of proportionality α is a rate constant, and φ f φ opx is a measure of surface area available for dissolution. For the effective binary system considered in this study, the equilibrium concentration c eq f varies only as a function of the vertical coordinate if the effect of temperature is small compared to that of pressure: c eq f (x) = c eq f (z) = c f + cf (z/l), (12) where F is defined below. At the lower boundary of our domain the solubility takes the value c f = c f,and c is the change in

4 644 A. Schiemenz, Y. Liang and E. M. Parmentier solubility over a compaction length L, where L is defined below. In the limit α, concentration tends to equilibrium value and the dissolution rate is obtained by combining eqs (1) and (4) with c f = c eq f : φ f (v n z ) c eq f / z Ɣ opx = copx I φopx,α =. (13) βc ol (1 β) ceq f There is no dissolution in the case when opx has been exhausted, and so we utilize the indicator function I φopx to enforce this: { 1, φopx (x) > I φopx (x) =, otherwise. (14) In the equilibrium limit, the opx dissolution rate (eq. 13) is proportional to the vertical melt flux and solubility gradient and is independent of opx abundance (cf. eqs 11 and 13). When the effect of temperature is also important, the equilibrium concentration c eq f in eq. (12) is a function of temperature and pressure or z. Itcan be easily shown then that the opx dissolution (or melting) rate is proportional to the linear combination of concentration and temperature gradients, similar to the gradient reaction of Phillips (1991). For equilibrium melting in a binary mono-mineralic system, Hewitt (21) showed that the melting rate is proportional to the Clapeyron slope. In this study, we focus primarily on isothermal dissolution with the dissolution rate given by eq. (13). In Section 5.1, we compare the effects of kinetic and equilibrium formulations on reactive dissolution. 2.2 Boundary conditions We consider a 2-D, rectangular domain (x, z) [, x max L] [, z max L], where x max and z max represent the number of compaction lengths in each dimension (see Fig. 2a; see also Section 2.3 below). Periodicity is assumed in the horizontal (x) dimension. Vertical boundary conditions are φ f (x,, t) = φ f (1 + G(x; A)), p() = p p ref, φ opx (x,, t) = φ opx (1 + G(x; p A )), z =, zmax c f () = c c f f + c c f,ref, z =. (15) zmax The term G(x; A) represents boundary perturbation at inflow with amplitude A < 1. Such perturbation may arise from the presence of heterogeneous lithology that starts to melt at a greater depth in the upwelling mantle column. In general this term may vary as a function of time; in this study we consider a simpler system with no time dependence. Proper values of c f,ref and p ref are selected such that the lower boundary layer in p and c f can be eliminated (HSLP211s; also see eq. 28 below). Given the boundary conditions, the rectangular domain considered here may be taken as the upper part of a melting column that lies directly beneath the axis of mid-ocean ridge, although the geometry and flow field of the latter are more complicated than those considered in this study. Melt flowing into the rectangular domain at z = is generated in the deeper part of melting column over a broader melting region than considered here. 2.3 Non-dimensionalized equations It is convenient to recast the governing equations in non-dimensional forms with the substitutions t = t t, x = Lx, φ f = φ f φ f, φ opx = φ opx φ opx, c f = c f + cc f, p = p p, v = w v, V = W n z + V V, (16) where φ f and φ opx are the background porosity and opx volume fractions at the inflow. The time, length, pressure and velocity scales are t = L κ η, L =, p = ηw W L, μφ f w = κ ρg φ f μ, V = φ f w (1 β) c copx βc ol (1 β). (17) c f The timescale t is the time needed for a matrix parcel to advect a compaction length via solid upwelling. L is the compaction length (McKenzie 1984). The pressure scale p follows from eq. (7). The fluid velocity scale w represents buoyancy-driven velocity, while V is the departure of the solid velocity from the upwelling velocity due to opx dissolution (deduced from eqs C3 and C5). For a list of numerical values we refer to Table 1. Substituting the non-dimensionalized variables (16) and scales (7) into (3), (4), (5), (9) and (1), disregarding (small) terms of order φ f and V /W (Table 1), and dropping primes, we arrive at the dimensionless governing equations φ f t φ opx t φ f c f t = φ f z + φ f p + δɣ opx, (18) = φ opx z φ f δ φopx (1 β) Ɣ opx, (19) = R (c f φ f v) + Ɣ opx, (2) [φ n f p] + φ f p = R φn f z, (21) φ f v = R 1 φ f n z φ n f (R 1 p n z ). (22) Eqs (18) (22) have four dimensionless parameters, φ f φ opx, δ = (1 β) c ĉ, Da = αĉlφ opx w, R = w W, (23) where ĉ = c opx βc ol (1 β)c f is a characteristic concentration. Typical numerical values of the dimensionless parameters are given in Table 1. The dimensionless opx dissolution rate is Ɣ opx = DaRφ f φ opx (F(z) c f ), Da <, (24) for the kinetic case and Ɣ opx = Rφ f (v n z ) F (z) I φopx, Da, (25) 1 δf(z) for the equilibrium case. Greater magnitudes of Ɣ opx imply greater rates of melt production and opx depletion. For purpose of illustration, we consider both linear and quadratic solubilities in the numerical simulations,

5 Numerical studies of reactive dissolution 645 Table 1. List of symbols and their typical values. Variable Description φ f Porosity or melt fraction φ opx Volume fraction of opx c f Concentration of independent component in melt p Pressure difference between melt and solid (= p f p s ) v Melt velocity V Solid velocity Parameter Description Typical value(s) c f Concentration of melt at inflow (eq. 12).47 c ol Concentration of independent component in olivine.5 c opx Concentration of independent component in opx.55 Da Damköhler number (eq. 23) 1 2 g Acceleration due to gravity 1 m s 2 L Compaction length (eq. 17) 1 1 km n Porosity exponent in permeability 2 3 p Pressure scale (eq. 17) R Ratio of buoyancy-driven melt velocity to solid upwelling rate (eq. 23) t Upwelling timescale (eq. 17) 1 Ma w Inflow melt velocity (eq. 17) 1 1 m yr 1 W Solid upwelling rate (eq. 17) 1 1 mm yr 1 α Dissolution rate constant (eqs 11, 39) yr 1 β Ratio of opx dissolution rate to olivine precipitation rate.5 δ Dimensionless solubility gradient (eq. 23) c Change in equilibrium melt concentration per compaction length (eq. 12) 1 6 ρ Density difference between solid and fluid 5 kg/m 3 V Velocity reduction of solid due to dissolution (eq. 17) mm yr 1 Ɣ opx opx dissolution rate (eqs 11, 13, 24, 25) κ φ Permeability (eq. 6) ξ Bulk viscosity (eq. 8) φ f Inflow porosity (eq. 15).2 φ opx Inflow opx fraction (eq. 15).15 ĉ Characteristic concentration.65 η Shear viscosity of solid Pa s κ Reference permeability m μ Viscosity of melt 1 1 Pa s F(z) = az 2 + (1 az max ) z, (26) where z max is the height of simulation domain and a is a constant. For convenience of linear stability analysis and verification of numerical schemes, HSLP211s and Schiemenz et al. (211) used a dissolution time [L/(w δ)] to scale the governing eqs (3), (4), (5), (9) and (1). (Note the pressure scale is also slightly different.) Since the primary focus of the present study is steady-state distributions of the melt channels and their associated lithology in an upwelling column, it is more convenient to choose a timescale that is closely related to solid upwelling in our numerical simulations (e.g. L/W ). With the solid upwelling timescale, steady or quasi-steady state solutions are typically achieved by a terminal solution time on the order of the vertical column length, so that the original solid has completely cycled through the domain (referred to as one solid overturn time below). 2.4 Parameters, boundary conditions and numerical method Linear stability analysis of our model demonstrates two regimes of instability, as well as a stable regime (Schiemenz et al. 211; HSLP211s). The unstable regimes are referred to as the channel and wave regimes. In the former case, unstable solution modes for porosity are monotonic, whereas in the latter case the unstable solution modes are travelling waves. Fig. 1 is a regime diagram showing the two kinds of instability separated by a stable region in the parameter space of relative upwelling rate (R 1 ) versus dimensionless solubility gradient (δ). The exact boundaries of the three regions are functions of the abundance of soluble mineral entering the domain from below. In the limit of no solid upwelling (R ), channelling instability prevails (e.g. Aharonov et al. 1995; Spiegelman et al. 21). Additionally, a third parameter, the mantle fertility φ f /φ opx,is demonstrated in HSLP211s to strongly affect stability. Analysis in HSLP211s is conducted by selecting a column height of 75 per cent the total distance to opx exhaustion. For many of the case studied in this paper we instead select a column height that is approximately 56 per cent of the distance to opx exhaustion. This slightly increases the stable region of parameter space, as shown in Fig. 1. The focus of this study is the basic properties of melt channel and dunite channel as well as their interaction, rather than the time-dependent effects of porosity waves. To systematically study the first-order features of the melt channel and dunite channel, we first focus on the behaviour of a single channel, induced by a sustained perturbation in porosity at the inflow boundary of the domain. We take the boundary condition to be a Gaussian pulse added to unity, φ f (x,, t) = 1 + A exp[ 1(x x max /2) 2 ], }{{} A >. (27) G(x;A)

6 646 A. Schiemenz, Y. Liang and E. M. Parmentier R 1 1 Unstable Waves δ Stable 75% 56% Unstable Channels Figure 1. Regime diagram showing the stability fields of the wave regime and channel regime as a function of relative upwelling rate (R 1 = W /w ) and solubility variation [δ = c/ (ĉ (1 β))]. Mantle fertility is φ f /φ opx =.2/.15. The dashed black line marks the stability boundary for a column height of 75 per cent H, whereas the dashed red line marks the boundary for a column height of 56 per cent H; H is the maximum column height (i.e. the distance to opx exhaustion for the 1-D problem). Parameters used in most of the numerical simulations reported in this study are shown as a star that is in the stable regime for a column height of 56 per cent H. Other parameters selections used in this study are given by circles. The regime diagram is constructed following the linear stability analysis and numerical method outlined in Schiemenz et al. (211) and HSLP211s, for a case of linear solubility gradient, F(z) = z. A basic setup is shown schematically in Fig. 2. For an initial condition we extend (27) across the entire vertical column, although results presented in this study are independent of the initial conditions. Similarly, we may also consider a perturbation of this type applied to the opx fraction with amplitude A. The resulting dimensionless boundary conditions are given as δ (1 + R) φ f (x,, t) = 1 + G(x; A), p(x,, t) = 1 + (nr), 1 φ opx (x,, t) = 1 + G(x; A p ), z =, (x,zmax,t) c f (x,, t) = 1 Da, c f z =. (x,zmax,t) (28) The boundary conditions for c f and p are chosen to eliminate a boundary layer at inflow; the boundary conditions for c f are moreover only needed in the kinetic formulation. The values are derived by computing the outer solution to the steady 1-D problem (HSLP211s). The non-dimensionalized eqs (18) (28) are solved numerically using a high-order accurate method, consisting of the tensor product of a finite difference method in the horizontal dimension and a nodal discontinuous Galerkin method in the vertical dimension. This method is especially suitable and computationally effective for the advection dominated problem explored here. Time integration is performed with an explicit, fourth-order Runge-Kutta method. Appendix B summarizes the numerical method used in this study. 3 NUMERICAL RESULTS A total of 62 numerical simulations were conducted in this study. Of these, 46 were run using the parameter values R = 1, δ = 1 2, φ f /φ opx =.2/.15. This selection is denoted by a star in Fig. 1. Other tests were run by varying the inflow perturbation and one of the three parameters while fixing the other two to the standard case, R ={1, 25, 5, 2, 5, 1}, δ ={.5,.5,.1} and φ f /φ opx ={.3,.75}. In this section, we first examine the detailed structures of the melt channels and their associated lithology by considering a one-channel Figure 2. Model setup. (a) Initial porosity distribution and boundary conditions. Initial porosity is unit except for a central region where a Gaussian perturbation in porosity is added (eq. 27). (b) Cross section of porosity perturbations for the initial condition and sustained boundary condition used in the simulations. (c) Linear and parabolic solubility functions used in the simulations (see eq. 26).

7 Numerical studies of reactive dissolution 647 system produced by a single sustained Gaussian perturbation in porosity at the bottom of the simulation domain. We then explore factors that control the geometry of the melt channel and the dunite channel: abundance of opx available for dissolution, variation in inflow melt flux and solubility gradient and the shape of porosity perturbation. We restrict ourselves to the equilibrium formulation (Da = ), as this is more relevant to melt migration beneath midocean ridges (see Section 5.1). 3.1 Detailed structures of melt channel and dunite channel As a reference case, we consider reactive dissolution in a short domain of size (x, z) [, 1] [, 2] and use the dimensionless parameter values δ = 1 2, R = 1 2 and φ f /φ opx =.2/.15. Short domain and low opx abundance (15 per cent) may be realized in the upper part of a long upwelling melting column. With these parameter values, a column height of two compaction lengths is approximately 56 per cent of the total distance to opx exhaustion. Linear stability analysis shows that this reference case is actually stable to random perturbation in porosity (star in Fig. 1). To produce a stable high-porosity channel system, we impose a timeindependent porosity perturbation at the bottom of the simulation domain (A =.2 in eq. 27, see also Fig. 2b) and evolve the system to a steady state. Figs 3(a) (f) display the calculated porosity and opx volume fraction, and Figs 4(a) (c) show the compaction rate, opx dissolution rate and ratio of horizontal to vertical components Figure 3. Steady-state distributions of porosity φ f (a) and the soluble mineral opx φ opx (d) in an upwelling column. Close-up views of the porosity and opx fields in the highlighted areas are shown in (b) and (e), respectively. (c) Variations of porosity at the top of the domain. (f) Variations of opx at the top ofthe domain. Regions between AB and A B are compacting boundary layers. Dark blue regions in (d) and (e) are opx-free dunite channel. White curves in (b) and (e) are streamlines of the melt. Steady state is established approximately after one solid overturn (t 2). Parameters used in this simulation are δ = 1 2 and R = 1. Porosity and opx abundance are normalized to their respective background values (2 per cent and 15 per cent, respectively) at the bottom of the upwelling column. A movie showing the development of the melt channel system (a) and (d) can be found in the Supporting Information (Movie S1). Figure 4. Steady-state distributions of compaction rate φ f p (a), dissolution rate δɣ opx (b), and relative melt flux (φ f v n x )/(φ f v n z ) (c) corresponding to the simulation shown in Fig. 3. For diagram clarity, relative melt flux near the opx-free dunite channel in the highlighted region in (b) is shown in (c). For reference, region of opx-free dunite is outlined by the black dashed line in (c). Oscillations along dunite channel boundary in (c) are numerical artefacts introduced by the horizontal discretization and are therefore disregarded.

8 648 A. Schiemenz, Y. Liang and E. M. Parmentier of melt velocity in the upwelling column. Results are given at time t = 3, when a steady state is well established (a movie showing the temporal evolution of porosity and opx fields, can be found in the Supporting Information (Movie S1). The sustained higher influx of melt in perturbed region around the centre of the lower boundary (x =.5) results in a higher dissolution rate (eq. 25, Fig. 4b) and the development of a high-porosity melt channel that extends all the way to the top of the upwelling column (Fig. 3a). The sustained influx of melt accumulates in the vertical direction, ultimately depleting all available opx for dissolution, resulting in an opx-free dunite channel in the upper part of the column (Fig. 3d). The lower part of the high-porosity melt channel is opx-bearing dunite and harzburgite. Upon exhaustion of opx, melt production ceases in the opx-free dunite region, and the main melt channel splits into two smaller channels due to compaction within the dunite channel (Figs 3b and 4a). A compacting boundary layer (regions between A and B or A and B in Figs 3c and f) is observed outside the main channel structures, where melt fraction locally decreases and opx fraction increases relative to the far-field solutions (Figs 3b and e). As shown in Figs 3(d) (f), the interface between the dunite channel and its surrounding harzburgite is very sharp near the top of the simulation domain, whereas the lithological contact is diffuse near the bottom of the dunite channel. The development of the high-porosity melt channel and opx-free dunite channel is best illustrated through the animation which is available online. The melt and dunite channels initiate at the top of the simulation domain and grow downward through time. Splitting of the melt channel happens only after opx is exhausted in the dunite channel. Steady state is established after approximately one solid overturn time (t = 2). Superimposed melt velocity streamlines in Figs 3(b) and (e) demonstrate the effects of focusing and isolation of melt from the adjacent matrix, in response to the presence of the compacting boundary layer. As shown in Fig. 4(c), melt is focused from the surrounding matrix into the high-porosity opx-bearing dunite/harzburgite channel in the lower part of the upwelling column and also around the tip of opx-free dunite channel. Compaction within the opxfree dunite channel, however, forces melt to localize towards the dunite harzburgite boundary in the upper part of the dunite channel. Compaction within the compacting boundary layer (AB or A B in Fig. 3f) also drives melt to the adjacent matrix, resulting in a slight local enrichment of melt near the out-edge of the compacting boundary layer (region near A or A in Fig. 3c, see also Fig. 11j below). The extent of melt focusing, however, depends on the lateral porosity variations. Fig. 5 displays a series of simulations as a function of the width of inflow perturbation while keeping the amplitude fixed at 15 per cent. For narrow perturbations, which corresponds to steep lateral porosity gradient, the dunite channels are deep with strong, opx-enriched compacting boundary layers. The melt Figure 5. Effect of inflow perturbation width on the steady-state distributions of porosity (top panels) and soluble mineral opx (bottom panels) in the upwelling column. The five Gaussian perturbations in porosity at the inflow (same amplitude of A = 15 per cent) are shown as red curves at the bottom of the respective porosity fields. Narrow perturbation implies greater horizontal perturbation gradient, which leads to more melt focusing and stronger compacting boundary layers, as highlighted by the melt streamlines in the opx field (thin white lines).

9 Numerical studies of reactive dissolution 649 streamlines above the tip of dunite channel are subparallel to the boundaries of the dunite harzburgite contact. Hence, the melt in the dunite channel is drawn mostly from the underlying matrix in this case. For wider perturbations, corresponding to smaller lateral porosity gradient, the effect of melt focusing is smaller. Consequentially, we observe shallow and wide dunite channel with weaker (or no) opx-enriched compacting boundary layers. The melt streamlines, in this case, cut across the dunite harzburgite boundaries. Lateral porosity variation, therefore, has a strong effect on the depth of melt extraction through the dunite channel. This point will be further illustrated in Section 3.3. In general, opx-free dunite channel is only part of high-porosity melt channel. Depending on the clinopyroxene abundance in the mantle, the lower part of the high-porosity melt channel is opxbearing dunite, harzburgite and possibly lherzolite. For convenience, we refer to the high-porosity melt channels and their associated lithologies as the high-porosity melt channel system. 3.2 Importance of soluble mineral abundance in reactive dissolution As shown in Figs 3(a) and (d), abundance of the soluble mineral opx plays a key role in determining lithology of the melt channel, extent of melt generation and compaction in and around the melt channel. The importance of the soluble mineral opx on melt and dunite channel formation can be further illustrated by comparing a set of simulations in which we impose sustained perturbations of variable opx abundance at the bottom of the simulation domain. As with porosity boundary perturbations, such variations can be regarded physically as representing mantle heterogeneity. It has been demonstrated experimentally that reaction between pyroxenite-derived melt and lherzolite can result in a hybrid mantle that is enriched in opx abundance (e.g. Yaxley & Green 1998; Takahashi & Nakajima 22). Fixing the porosity perturbation amplitude at 2 per cent under the Gaussian pulse perturbation given in eq. (27), we utilize the same function to define the inflow opx boundary condition φ opx (x,,t) (eq. 28). Results of five simulations are shown in Fig. 6. Larger amplitudes in the opx fraction at the inflow correspond to shallower and narrower dunite channels, as the amount of dissolution needed to exhaust the opx increases. As the opx-free dunite channel becomes narrower, bifurcation of the melt channel becomes less obvious. In our equilibrium formulation, the melt fraction and compaction rate do not depend upon opx fraction, so long as the opx fraction remains positive. Only then does the opx affect porosity, as its depletion causes the local melt channel structure to split into two smaller ones. Therefore the porosity fields in Fig. 6 show less variation in comparison to later simulations shown in Figs 7 and 8, where parameters directly affecting melt production are varied. Figure 6. Effect of inflow perturbed opx abundance on the steady-state distributions of porosity (top panels) and opx (bottom panels) in the upwelling column. The five Gaussian perturbations in opx abundance at the inflow (amplitude A ={,.1,.2,.4,.8}) are shown as black curves at the bottom of the respective opx fields. Porosity perturbations are the same for the five simulations with an amplitude of A = 2 per cent. Larger opx perturbation at inflow leads to shallower and narrower dunite channel.

10 65 A. Schiemenz, Y. Liang and E. M. Parmentier Figure 7. Effect of inflow melt flux in perturbed region on the steady-state distributions of porosity (top panels) and soluble mineral opx (bottom panels) in the upwelling column. The widths of perturbation are fixed. Amplitudes of the five Gaussian perturbations in porosity at the inflow are A ={.5,.1,.2,.4,.8}. Larger perturbation at inflow implies greater lateral porosity gradient and stronger melt focusing, leading to deeper and wider dunite channel. 3.3 Effects of inflow melt flux and solubility gradient In the equilibrium limit, opx dissolution rate is directly proportional to the product of the vertical melt flux and the solubility gradient (eqs 13 and 25). In Figs 7 and 8 we show numerical experiments conducted with differing values of inflow melt flux and solubility function, accomplished by varying the amplitude A of the porosity perturbation in eq. (27) and the quadratic coefficient a in the solubility function F in eq. (26). Figs 2(b) and (c) show the porosity perturbations and solubility functions used in our simulations. As shown in Fig. 7, larger amplitude in porosity perturbation leads to greater melt focusing and hence melt flux, which in turn increases the magnitude of opx dissolution rate. The increased melt production exhausts opx faster, thereby lowering the depth of the opx-free dunite channel initiation and resulting in deeper and wider dunite channel with stronger melt channel bifurcation. Similarly, Fig. 8 shows that greater magnitudes of dissolution, and thus deeper dunite channels, occur as the solubility gradient at the inflow increases. This is explained by the dependence of the dissolution rate on the gradient of the equilibrium concentration at a given depth (eq. 25). At inflow, this gradient increases as a approaches z 1 max. Relative to the linear case a =, negative values of a result in greater melt production near the base of the domain. However, for the upper half domain z > z max /2, dissolution is diminished relative to the linear case. Hence we observe in Fig. 8 for the most extreme case of a =.5 that the width of the dunite channel appears constant as a function of height near the top of the domain. This contrasts with Fig. 7, where dunite channel width is always increasing with height. In Section 4.2, we will present a simple model that links the aforementioned parameters to the depth of dunite channel initiation. 3.4 Effects of multiple heterogeneities Thus far our numerical experiments have considered the effects of a single sustained boundary perturbation in porosity or opx fraction. When two or more sustained boundary perturbations are present, the two melt channel systems induced by the perturbations may interact with each other, depending on the amplitudes and separation distance of the two perturbations. As an illustrative example, we consider two sustained porosity boundary perturbations of the form φ f (x,, t) = 1 + A 1 exp( 1(x x 1 ) 2 ) +A 2 exp( 1(x x 2 ) 2 ). (29) Results from numerical simulations are given in Figs 9 and 1 and are discussed below. In Fig. 9 we consider a fixed amplitude A 1 =.4 of the first perturbation (centred at x 1 =.425), and a variable amplitude A 2 of the second perturbation (A 2 =.1 to.4, centred at x 2 =.575). The relatively large amplitude of the first perturbation gives rise to a deep opx-free dunite channel with strong melt channel bifurcation. When the amplitude of the second perturbation is small (A 2.12), the melt flux that arises from the perturbation is also small.

11 Numerical studies of reactive dissolution 651 Figure 8. Effect of inflow solubility gradient on the steady-state distributions of porosity (top panels) and soluble mineral opx (bottom panels) in the upwelling column. Same porosity perturbation at inflow (A = 1 per cent). Solubilities used in the comparison are shown in Fig. 2(c) with non-linear coefficient for the solubility function (eq. 12) a ={.5,.25,,.25,.5}, from left to right. The widths of perturbation are fixed. Deeper channel corresponds to greater solubility gradient c eq f / z at the inflow. Consequently, only a shallow and narrow opx-bearing dunite channel is developed above the second perturbation, with little or no interaction observed between the two melt channel systems. When viewed at the top of the simulation domain, we observe an opx-free dunite channel juxtaposing a harzburgite on the left and a narrow opx-bearing dunite or opx-depleted harzburgite on the right. A rapid transition occurs for.12 A 2.16, as the smaller dunite channel is absorbed into the larger channel (Fig. 9d). For larger amplitude of the second perturbation (cases of A 2 =.25 and.4 in Figs 9a and b), the two melt channel systems interact strongly, giving rise to one wide opx-free dunite channel and three melt channels of variable porosity at the top of the simulation domain. In Fig. 1 we examine the effect of separation distance x 2 x 1 on the geometry of the melt and dunite channels by fixing amplitudes (A 1, A 2 ) = (.4,.15) of the two porosity perturbations and location of the first perturbation (x 1 =.3) while varying coordinate x 2 of the second perturbation. For small separation distance x 2 x 1 between the two perturbations we observe the melt and dunite channels coalescing into one, similar to the large-amplitude cases from Fig. 9, except the right branch of the melt channel appears slightly deflected. As the separation distance increases, however, we observe the depth of the right-most dunite channel increases until finally, the two channels are nearly independent of one another. This result is summarized in Fig. 1(d), where the depth of the shallower (right-most) dunite is plotted as a function of x 2 x 1.As the separation distance shrinks, the depth approaches the maximum domain height of two compaction lengths, indicating that the shallower melt channel has been absorbed into the larger one. A dashed vertical line is shown at the distance x 2 x 1 =.275; this denotes the width of the compacting boundary layer for the larger channel (measured at the top of the column) from the centre of the dunite to the edge of the compacted region. Hence, the compacting boundary layer width is closely related to the minimum distance for which distinct sources of heterogeneity act independent of one another. For a comparison, a dashed horizontal line is given at the depth 1.33; this denotes the depth of the shallower channel in a numerical simulation that has been conducted without the deeper channel present. 3.5 Non-Gaussian perturbation So far we have only considered Gaussian perturbations of variable amplitudes and width. Nevertheless, the general observation and conclusions drawn from these simulations are independent of the exact form of the perturbation. Fig. 11 shows four simulations using a linear porosity perturbation with standard parameter values (R 1 = δ = 1 2, φ f /φ opx =.2/.15), computed over a square domain of two compaction lengths. We show the porosity (a d) and opx fraction with overlain fluid velocity streamlines (e h), and use

12 652 A. Schiemenz, Y. Liang and E. M. Parmentier Figure 9. Interaction of two sustained porosity boundary perturbations: one with a fixed amplitude (A 1 =.4) and the other with variable amplitude A 2 ={.12,.14,.25,.4}, per eq. (29). Steady-state distributions of porosity (a), opx abundance (b), and inflow porosity (c). (d) Variation of total width of dunite at outflow as a function of perturbation amplitude A 2. Open circles represent the four cases shown in (a) and (b). Smaller solid circles are additional simulations. A rapid transition occurs in the region.12 A 2.16, as the smaller dunite briefly emerges, initially disjoint from the larger dunite, and then finally the two coalesce as one large feature. a piecewise-linear inflow perturbation with varying slopes (i) for the four cases. Outflow porosity and opx fractions are shown in (j) and (k), respectively. For larger inflow porosity gradient (e.g. Fig. 11d), the fluid streamlines indicate greater melt focusing into the dunite channel from below. Consequentially, a low-porosity, high-opx compacting boundary layer is formed adjacent to the dunite channel similar to that shown in Fig. 5 for the cases of Gaussian perturbation. The compacting boundary layer greatly affects the channel geometry by isolating fluid flow from the adjacent matrix. Stronger compacting boundary layers lead to more vertically shaped dunite channel, whereas a weaker compacting boundary layer implies a dunite channel with greater curvature. For smaller inflow porosity gradient, there is less melt focusing into the dunite channel. Fig. 11(e) shows that the melt streamlines cut across the relatively diffuse dunite harzburgite boundary,

13 Numerical studies of reactive dissolution 653 Figure 1. Interaction of two sustained porosity boundary perturbations: fixed amplitudes but variable separation distance x 2 x 1 ={.125,.15,.2,.42}. Steady-state distributions of porosity (a), opx abundance (b), and inflow porosity (c). (d) Variation of depth of shallower dunite channel as a function of separation distance between the two perturbations. Open circles represent the four cases shown in (a) and (b). Smaller solid circles are additional simulations. A vertical line denotes a compacting boundary layer width (centre of melt channel to edge of compacting region) of approximately.275 for the deeper channel. Outside this region, the shallower melt channel appears nearly independent of the deeper channel. A horizontal line denotes the depth of the shallowerchannel when the deeper channel is not present. similar to the case of the wider Gaussian perturbations shown in Fig. 5. Figs 11(j) and (k) compare the porosity and opx variations at the top of the simulation domain with compacting boundary layer for the four simulations. Points A, B and C mark the simulation corresponding to Figs 11(a) and (e). The greater variabilities (than those shown in Fig. 5) in porosity both in and outside the dunite channel may have important implications for the rheology of a melt channel system and trace element distributions across a dunite harzburgite sequence. The temporal evolution of the melt channel system is also very interesting. It demonstrates the importance of solid upwelling in determining the steady-state distribution of the melt channel system. A movie showing the development of the porosity field in Fig. 11(b) and opx field in Fig. 11(f) can be found in the Supporting Information (Movie S2; note the strong time-dependency of the two melt channels in the opx-free dunite channel in this simulation).

14 654 A. Schiemenz, Y. Liang and E. M. Parmentier Figure 11. Examples showing the effect of lateral porosity variation at the inflow on the steady-state distributions of porosity (a) (d) and opx abundance (e) (h) in the upwelling column. (i) Porosity at the inflow. (j) Porosity at the outflow. (k) opx fraction at the outflow boundary. Simulations are conducted on a square domain of 2 compaction lengths, but only the middle portion (.5 x 1.5) is shown in (a) (h). The remaining area is essentially far-field solution. Thin white curves are melt streamlines. Parameters used in the simulations are R 1 = δ = 1 2 and φ f /φ opx =.2/.15. A movie showing the development of the melt channel system (b) and (f) can be found in the Supporting Information (Movie S2). 4 DUNITE CHANNEL GEOMETRY Dunite exposed in the mantle sections of ophiolites is one of the few physically measurable quantities in the field that may be used to test melt migration models. Although dunite bodies observed in the field may have experienced various extent of deformation, it is important to understand the factors that control the geometry of dunite channel under simple settings. In this section, we consider the 2-D geometry of an idealized dunite channel, its depth of initiation and its width as a function vertical coordinate in the upwelling column. We also show that channel geometry is dependent upon lateral variation of the inflow porosity perturbation. 4.1 Geometry of dunite channel Let (x, z ) be the coordinate of the tip of an opx-free dunite channel (Fig. 12). We make the ansatz that the channel width is described

15 Numerical studies of reactive dissolution 655 P-shaped channel, and dunite channel with straight walls (b = 1) as V-shaped channel. The cases were conducted under identical parameter settings (R 1 = δ = 1 2, φ f /φ opx =.75, column height 4.6), but with differing inflow perturbation width. Clearly, greater melt focusing leads to deeper dunite channel initiation (i.e. smaller z ), because higher melt fraction leads to higher opx dissolution rate (Ɣ opx φ 3 f, cf. Figs 4b and c). Furthermore, melt focusing can be shown to also affect dunite channel geometry by influencing the width of the compacting boundary layer. The relative amount of melt focusing can be measured by considering the ratio of horizontal to vertical melt flux. From eq. (22) we have φ f v n x φ f v n z = φ 3 f p/ x Rφ 3 f φ 3 f p/ z + φ, (32) f where n x and n z are the unit vectors along the x and z directions, respectively. Noting that R is large ( 1) compared to all other terms, and assuming that porosity does not approach zero, we can expand the ratio as a Taylor series, φ f v n x φ f v n z p/ x R ( 1 + p/ z φ 2 f R ). (33) Figure 12. Streamlines of melt dividing the upwelling column into three distinct regions: the far field where solutions are the same as 1-D steady-state solution (region F), compacting boundary layer (C), and the melt channel system (D), the top of which is dunite. The tip of dunite channel is marked by the filled circle with coordinate (x, z ). Melt suction rate in eqs (37) (38) is evaluated along the dashed red line. by the power-law relationship x x (z z ) b, (3) l where b depends upon the model parameters and heterogeneity present, and l is the width of the inflow perturbation. Using data from our numerical simulations we obtain a discrete representation of the boundary of the opx-free dunite channel. To determine the value of b we examine the log log plot of (z z )versus x x /l.these results are given in Fig. 13(a) for numerical simulations conducted over a large range of model parameters and inflow perturbation structures, and suggest an exponent 1 b 1. (31) 2 Oscillations in the height versus width comparisons in Fig. 13(a) are numerical artefacts introduced in the fitting of the narrow channel boundary and are therefore disregarded. In general, the geometry of the lower part of a dunite channel can be described as parabolic, as b 1/2 for most of the cases examined (Fig. 13a). Dissolution fingers with parabolic geometry have been observed in the numerical study of Steefel & Lasaga (199) and laboratory dissolution experiments of Kelemen et al. (1995b). Similar parabolic shapes have also been frequently observed in viscous fingers formed by flow of fluids of differing viscosity or composition (e.g. Homsy 1987). Figs 13(b) and (c) compare two simulations, representing typical cases for b = 1/2 and 1. For convenience, we refer to dunite channel having a parabolic shape (b = 1/2) as Fig. 4(c) shows an example of the distribution of the relative melt flux in a steady-state column. Thus, it is the horizontal gradient in the effective pressure p = p f p s, induced by the prescribed gradient in inflow porosity, that generates melt focusing and the compacting boundary layer (cf. Figs 4a and c). This observation suggests further study into the relationship between compaction rate and porosity. 4.2 Depth of dunite channel initiation As documented in our numerical simulations, the depth of dunite channel initiation depends on a number of factors, including but not limited to, solubility gradient, inflow melt flux and melt focusing. Here, we quantify these relationships by considering a simple 1-D model and by considering lateral melt transport or melt focusing. The derivation of the analysis is given in Appendix C. We first consider a purely 1-D steady-state problem, for which an analytical solution exists. Integrating over the vertical dimension of the steady-state governing eqs (1) (4), we obtain an exact expression for the solubility at depth z at which opx is completely consumed (φ opx = ). In dimensional form, we have F(z /L) = c 1 φ opx W ( c opx βcol (1 β) ) c f φ f w + (1 β) ( φopx W ). (34) The depth z can be found then by inverting the prescribed solubility function F (eq. 26). For most cases considered in the present study this is simply the linear function F(z) = z. Eq. (34) therefore dictates that the following will lead to deeper dunite channel (smaller z ): (i) larger influx of melt at the base of the column (φ f w ), (ii) larger solubility variation ( c), (iii) smaller rate of upwelling (W ), (iv) smaller abundance of soluble mineral opx (φ opx ). This is in excellent agreement with the numerical simulations showninfigs6 8.DefiningQ as the vertical component of the melt flux φ f v and accounting for perturbations in the inflow porosity and

16 656 A. Schiemenz, Y. Liang and E. M. Parmentier Figure 13. Variations of dunite channel height z z as a function of normalized dunite channel width x x /l, wherel is the standard deviation of the probability density function corresponding to the chosen perturbation. Data is collected from 47 simulations conducted over a range of model parameters and Gaussian-shaped inflow perturbations. Oscillations are numerical artefacts introduced in fitting of the narrow channels, and are therefore disregarded. For reference, lines with slope b = 1 and b = 1/2 are shown as green and red dashed lines, respectively. (b) A simulation with a narrow porosity perturbation and a parabolic geometry (P-shaped channel). (c) A simulation with a broad porosity perturbation and straight-walled geometry above the tip of the dunite channel (V-shaped channel). Simulations were conducted on a domain size x max 2.86, z max 4.6, but only the middle 5 per cent of of the column is shown (the remaining parts are essentially the 1-D solution). Dashed white lines outline the boundaries of opx-free dunite channels. Solid yellow lines represent inflow porosity perturbation. opx fields, eq. (34) can be written in dimensionless form as F(z φ opx () (1 β) ) = [ ]. (35) φ δ f RQ() + (1 β) φ φopx opx () Here, we use the maximum inflow values for φ opx () and Q() (i.e. those corresponding to the centre of the Gaussian perturbation). Fig. 14 compares the predicted depths from this 1-D analysis (crosses) to those measured from the 2-D numerical simulations. Although the predictions generally follow the correct trend, a systematic deviation clearly persists as most of the predicted depths of dunite channel initiation are shallower than the measured values. One important factor that has not been taken into consideration in the 1-D model is lateral transport of melt from lower porosity matrix into the high-porosity channel. This focusing effect will obviously increase the local melt flux directly beneath the dunite channel, increasing the depth of dunite channel initiation. Hence to improve the predictions in eqs (34) (35) we introduce the melt suction rate S, definedas S = (φ f v n x ). (36) x Although the melt suction rate is inherently a 2-D term, we evaluate S only along the vertical column directly below the dunite tip (see Fig. 12), so it is purely a function of z. In dimensional form, we have a revised expression for the solubility at which opx is completely dissolved (φ opx = ) F(z /L) = c 1 φopx W ( c opx βcol (1 β) ) z c f + SF(z/L) dz φ f w + (1 β) ( φopx W ) z. + Sdz (37) Below the dunite tip, melt flow is focused inwards (Fig. 4c; see also fluid velocity streamlines in Fig. 12), so the melt suction rate is positive and will decrease z. While the integrated S appears in both the numerator and denominator, it is only a significant term in the denominator. The dimensionless form of eq. (37) is given as F(z ) = (1 β) φ opx () + δr φ f z SF(z) dz φopx [ φ δ f RQ() + (1 β) φ φopx opx () + φ f R ]. (38) z Sdz φopx Unlike the purely 1-D case given in eqs (34) and (35), eqs (37) and (38) yield implicit representations for z. Nonetheless, the depth is easily computed by finding the intersection of the left- and righthand sides of the equations. A quasi 1-D approximation to the melt suction rate in eq. (38) is calculated by evaluating eq. (36) from simulation data in the vertical column directly below the dunite tip (i.e. the centre of the porosity perturbation). In Fig. 14, we give results for this quasi 1-D approximation. It is observed that the melt suction rate correction nearly perfectly aligns the predicted depths to those measured from numerical simulations, underscoring the