Dunites in the mantle section of the Oman ophiolite represent conduits for chemically isolated melt transport through the shallow mantle beneath oceanic spreading centers. These dunite melt conduits exhibit a scale-invariant power law relationship between width and cumulative abundance, as measured over 4 orders of magnitude. We use this size/frequency distribution to assess several hypotheses for dunite formation and estimate the total melt flux that a dunite network can accommodate beneath an oceanic spreading center. Dunites, measured from one-dimensional lithologic sections and digital image mosaics at a variety of length scales, range in width from ∼3 mm to ∼100 m and follow a power law with a slope of ∼1.1. Extrapolation of the power law predicts that dunites as wide as 3.5 km may exist in the melting region beneath a mid-ocean ridge. Alternatively, perhaps the widest dunites we observe (∼100 m) represent a maximum size. Modeling of dunites as diffusive reaction zones around melt-filled hydrofractures cannot explain the existence of dunites wider than ∼10 m in Oman. Instead, dunites may represent high porosity conduits formed by reactive porous flow. Using the observed size/frequency relationship, the assumption that dunites form a coalescing network and the requirement that flux is conserved where dunites merge, we estimate the total flux through a porous dunite network and the fraction of that flux that remains chemically isolated. Our flux model predicts that the porosity in a dunite scales with the width. For maximum porosities of ∼1–4% in the widest dunites, a network of porous dunite conduits with the abundances observed in Oman can supply a sufficient flux of melt (of which > 95% remains chemically unequilibrated with shallow residual peridotites) to satisfy the observed mid-ocean ridge flux.
 Geophysical and geochemical observations suggest that melt transport beneath oceanic spreading centers is strongly focused toward the ridge axis and rapidly extracted from the shallow mantle predominantly via chemically isolating dunite conduits. In this study we examine the abundance and size distribution of dunite conduits preserved in the Oman Ophiolite to assess proposed mechanisms of dunite formation and place constraints on the melt flux that the dunite network can accommodate.
 Recent geophysical data indicate that the melting region beneath mid-ocean ridges is much larger than the seismically determined zone of crustal accretion at the ridge axis. Seismic data from the MELT experiment (17°S along the East Pacific Rise) indicate that melt is present for several hundred kilometers off-axis and to depths of 120–150 km [Forsyth et al., 1998]. However, seismic velocity data indicate the igneous oceanic crust attains its complete thickness within 5 km of the ridge axis [Dunn et al., 2000; Vera et al., 1990]. Geochemical studies show that lateral melt migration from a broad region of corner flow is required to reproduce the trace element characteristics of MORB [Spiegelman, 1996]. Therefore melts must be efficiently extracted from a broad melting region and transported to the ridge axis to be accreted within the narrow ridge axis. Although our understanding of the exact path of melt migration through the upper mantle is incomplete, several geochemical and morphological constraints can be placed on the nature of the melt transport system.
 Since mid-ocean ridge basalt (MORB) is not in chemical equilibrium with residual harzburgite in the shallow mantle, melt extraction must occur in chemically isolated conduits. Primitive MORBs are undersaturated in orthopyroxene (opx) with respect to the equilibrium cotectic melt composition at pressures commensurate with the base of the oceanic crust (1–2 kbar). Figure 1 shows the major element composition of ∼10,000 glasses from the global MORB database (blue dots), using the isomolar projections of Elthon . Liquids close to equilibrium with mantle olivine (i.e., molar Mg# (Mg/(Mg + Fe)) > 0.66, shown as red triangles) were last saturated in opx at pressures in excess of 8–10 kbar, in agreement with experimental data [e.g., Elthon, 1989; Elthon and Scarfe, 1980; O'Hara, 1965; Stolper, 1980]. The high dissolution rate of opx in undersaturated basalts [Brearley and Scarfe, 1986; Kuo and Kirkpatrick, 1985a, 1985b] suggests that these melts must be isolated from the surrounding harzburgite during migration to preserve disequilibrium. MORBs also preserve higher light/heavy rare earth element ratios than predicted for equilibrium with residual peridotites dredged from mid-ocean ridges (“abyssal peridotites” [e.g., Johnson et al., 1990]) and sampled in ophiolites [e.g., Kelemen et al., 1995a]. In addition, numerical models of melt migration suggest that the concentrations of trace elements in MORB can also be used to distinguish two-dimensional melt transport from strictly vertical ascent [Spiegelman, 1996] when sufficient data become available. Thus major and minor element data indicate that melts migrate through, at a minimum, the shallowest 25–30 km of the mantle with limited chemical interaction.
 Some MORBs record melting and rapid extraction from even greater depths. Lu/Hf and Sm/Nd isotopic systematics [Salters and Hart, 1989] and 230Th/238U excesses [Beattie, 1993; Lundstrom et al., 1995; McKenzie, 1985] suggest the generation of small melt fractions in the presence of garnet, which is only stable in mantle peridotite at depths greater than ∼70 km [Hirschmann and Stolper, 1996; Koga et al., 1998]. To preserve the observed excesses of the shortest lived isotopes (i.e., 230Th), melt must be rapidly isolated from the depths where fractionation occurs and delivered to the surface without achieving secular equilibrium or reacting with the majority of the rock through which it passes (see Kelemen et al.  for a more complete discussion of melt velocity constraints).
 Mantle dunites (>90% olivine) may be the chemically isolated conduits for melt transport. Although residual harzburgites are not in equilibrium with MORB, both the major and trace element concentrations from clinopyroxenes (cpx) in dunites in the mantle section of the Oman ophiolite are in equilibrium with the lavas (very similar to MORB) that formed the overlying crust [Kelemen et al., 1995a]. In addition, spinel compositions from dunites exhibit high Cr/(Cr + Al) and high TiO2 [Augé, 1987; Kelemen et al., 1995a; Pallister and Knight, 1981] similar to those found in MORB [Dick and Bullen, 1984] and distinct from residual harzburgites [Allan and Dick, 1996; Arai and Matsukage, 1996; Dick and Bullen, 1984; Dick and Natland, 1996; Kelemen et al., 1997]. Dunites as observed in the mantle section of the Oman ophiolite, one of the largest, best exposed sections of oceanic crust and mantle, are generally tabular bodies with sharp contacts and orientations subparallel to the foliation in the surrounding harzburgite and the crust-mantle transition zone (paleo-Moho) [e.g., Boudier and Coleman, 1981; Lippard et al., 1986]. Contact relationships between dunite and harzburgite (summarized by, e.g., Kelemen et al. [1995a, Figure 2]) indicate that dunites replaced the surrounding peridotite, via reactions that dissolved pyroxene (and precipitated a similar yet smaller mass of olivine). If the harzburgite foliation developed as the result of deformation associated with “corner-flow” beneath a spreading ridge, then the dunites must have formed in the upwelling mantle beneath the ridge. Thus dunites accommodated chemically isolated transport of melt through the shallow mantle to the base of the crust.
 The extent to which dunites are the sole conduits for melt transport is debatable. However, replacive dunites are abundant in the shallow mantle and certainly represent regions of highly concentrated melt flux. As a result, the dunites embody a large part of the time-integrated history of melt migration beneath the spreading center. This study characterizes the abundance and distribution of dunites in the mantle section of the Oman ophiolite to assess (1) viable mechanisms for dunite formation and (2) the ability of a network of dunites to accommodate the observed melt flux to mid-ocean ridges.
2. Observations and Results
2.1. General Field Relations
 The Oman ophiolite consists of ∼13 large massifs which comprise a nearly continuous band of oceanic lithosphere more than 500 km long and 50–100 km wide. In this study we examine the distribution of replacive dunites exposed in the mantle sections of the Samail and Muscat massifs. Although there are a variety of interpretations of the structural data, detailed mapping efforts in the Samail massif and the adjacent Wadi Tayin massif indicate relatively little internal deformation since the initiation of obduction [e.g., Jousselin et al., 1998; Nicolas et al., 2000], and the peridotites represent residual mantle which passed beneath an oceanic spreading center.
 The mantle section of the Wadi Tayin massif preserves a stratigraphic thickness of up to 9–12 km, composed almost entirely of residual harzburgite and dunite [e.g., Boudier and Coleman, 1981]. The peridotite is overlain by a 5–7 km thick gabbroic crustal section [e.g., Pallister and Hopson, 1981] with a nearly continuous layer of sheeted dikes underlying pillow lavas with compositions very similar to MORB [e.g., Pallister and Hopson, 1981; Pallister and Knight, 1981]. It is inferred that these observations are applicable to the Samail and Muscat massifs as well, though the crustal section in parts of the Samail massive may have been thinner (∼4 km) [Nicolas et al., 1996].
 Although the spreading rate during the formation of the Oman crust is unknown, the presence of the continuous igneous crustal layer [e.g., Nicolas and Boudier, 1995; Nicolas et al., 2000] suggests a half-spreading rate greater than ∼0.05 m/yr. Mid-ocean ridges with spreading rates less than ∼0.05 m/yr exhibit extensive exposures of peridotite on the seafloor, especially near fracture zones [e.g., Cannat, 1996; Dick, 1989]. In addition, there is little evidence for crystal plastic deformation in Oman gabbros [Nicolas and Ildefonse, 1996], whereas plastically deformed gabbros are commonly observed along the Mid-Atlantic Ridge [Cannat et al., 1997]. We infer that the Oman ophiolite represents the igneous crust and residual peridotite generated at an intermediate to fast spreading oceanic spreading center.
 The majority of dunites in the mantle section of the ophiolite occur as tabular veins, intersecting each other at small angles, with sharp contacts against the surrounding harzburgite. In this study we have chosen exposures that are planar over the length scale of the image and orthogonal to the intersection lineation formed by the planar dunites. Outcrops oriented orthogonal to the ones imaged show no intersections. The widths of dunites range from less than 1 cm to nearly 100 m, and lengths range from less than a meter to several kilometers. To avoid confusion with the large, partially cumulate, dunites formed at the base of the crust, we consider only dunites that are overlain by harzburgite. In some cases, our chosen map areas lie more than 1 km below the crust-mantle transition zone, though this is not possible to verify for outcrops in the Muscat massif.
2.2. Photomosaics and Image Processing
 We have measured dunite widths over 4 orders of magnitude (10−2 to 102 m). Data were collected using one of several methods depending on scale: one-dimensional (1-D) measured sections, outcrop mosaics for small dunites, hillside/mountainside photomosaics for intermediate scales, and airphoto analysis and chain and compass surveying for the largest dunites.
 For the highest resolution, one-dimensional sections were collected in Wadi Lufti (Samail) along a chain line perpendicular to the strike of the predominant dunite orientation/peridotite foliation (335°/65°SW). Seven sample lines ranged in length from 7 to 66 m, for a composite section of 208 m, with every contact along the sample line measured to millimeter precision (Figure 2a).
 Dunite distributions over larger length scales are derived from two-dimensional image mosaics. Photomosaics of individual outcrops (Figures 3a and 3b) were constructed from images of 0.5 by 0.5 m areas along a grid superimposed on the outcrop. Lithologic contacts in each image were then retraced and reassembled into the mosaic using Adobe Photoshop. This is an improvement upon the method used by Kelemen et al. , exploiting greater overlap, better camera positioning, and less image distortion. The prepared outcrops ranged in size from 3 × 3.5 m to 16 × 5 m, with a final precision of ∼3 mm (determined by the width of the marker and the grain size of the rock). Where possible, one-dimensional sections traversed previously analyzed two-dimensional outcrops.
 At length scales larger than individual outcrops (Figures 4 and 5), images were gathered and processed in a slightly different manner. For each location, a series of scaled photographs, taken from a fixed point at a known distance from the hillside were digitized and mosaiked, again using Adobe Photoshop. The mosaics were printed to large sheets, over which the facies contacts were hand traced, then redigitized. This same process was used for airphoto analysis (Figure 6) to fill in gaps in the ground survey. All mosaics are converted into scaled bitmap (binary) images containing just dunite (or harzburgite) objects for image processing.
2.3. Statistical Analysis
 To evaluate the proposed mechanisms of dunite formation and to estimate the total potential flux through dunites, the abundance and distribution of dunites in the melt network must be determined. Kelemen et al.  show that the abundance of dunites exhibits a power law dependence on dunite width in the Ingalls ophiolite. In addition, Kelemen et al. extrapolate this dunite size/frequency relationship to predict the abundance of large dunites in the Wadi Tayin massif in Oman.
 For the Oman ophiolite, the data from the composite 1-D section in Wadi Lufti show that a robust power law relationship between dunite width and frequency does exist over at least four orders of magnitude in width (Figure 2b). We collected additional dunite width data from images over a wide range of scales using an “intercept width” technique, similar to that used by Kelemen et al. . Dunite widths are measured along parallel lines perpendicular to the dominant dunite strike for each image mosaic. All widths are then corrected for apparent thickness based on field measurements of the true dunite orientation and camera position. In addition, several large dunites were measured using standard chain and compass techniques to further constrain the mosaics. The same width analysis is carried out for the harzburgite images, and therefore the percentage of dunite at each location can be determined by combining the dunite and harzburgite images. To account for the higher number of observations in larger areas, all frequency data are normalized by the total intercept length used in each image. Using cumulative frequency per meter allows for direct comparison of size/frequency data collected at different scales over different size areas. (Using frequency per meter also allows for extrapolation to larger length scales, as will be discussed in section 3.1.)
 For each data set, a cumulative distribution function and probability density function can be established from histograms of the width data. The cumulative distribution function (CDF) is defined as the number of objects per meter greater than a given width, w. For a power law, the CDF has the form
where D is the power law exponent and a is related to the proportion of dunite in the image. The probability density function (PDF) is an estimate of the number of dunites of a given width per meter. The PDF is the derivative of the CDF and for the power law above has the form
 Ideally, the PDF could be determined directly from the width/frequency histogram. However, the values of both D and a are sensitive to the bin sizes used in creating the histogram. Therefore we exploit the mathematical relationship between the distribution functions to calculate the PDF from the more robustly determined CDF.
 If the true distribution of dunite widths is indeed scale-invariant over some range of widths, then the abundances of sampled dunites should fall along a common line (Figure 7). However, any sampling of the true population is subject to truncation and censorship errors. Even samples derived from ideal power law distributions will exhibit deviations from perfect power law behavior. Truncation errors occur at the smallest scales since dunite width is limited by the resolution of the image. At the larger scales, object width is limited by the size of the area of interest. In addition, censorship can systematically diminish the sampling frequency. At the grain scale (0.5–3 cm) it is very difficult to distinguish the matrix of olivine around pyroxenes in the harzburgite from a discrete dunite vein. Even though all widths are subject to this precision limit, small-scale features are most affected, since the error is a larger fraction of the true width. At the largest scale, the widths of dunites with one boundary outside the field of view are also underestimated. This type of censoring can occur for any object but is more likely for larger objects and more important for rare observations. For a combination of these reasons, the number of the largest and smallest dunites are underestimated. Several corrections have been developed to account for these errors in cumulative frequency plots [e.g., Barton and Zoback, 1992; Laslett, 1982]. However, the effects of the corrections are minimal if the relationship is measured from the central linear portion of the distribution [Pickering et al., 1995], which comprises the majority of the data. We define the central portion of the distribution as data that lie within the 80th percentile of absolute range from the median.
 Values of D and a are obtained from the culled CDF data using a linear regression with data weighted by the studentized residuals to eliminate outliers with high leverage. We initially fit the linearized system, logF = loga–Dlogw, using a least squares method. Outliers are culled based on the value of their studentized residuals, and the retained data are refit. In all cases, the retained data (Figure 8, open circles) represent ∼80% of the total data set. For a more complete description, see Weisburg [1985, p. 109–117]. Values of the fitted parameters for individual locations, as well as their average (D = 1.11, a = 0.088), are shown in Table 1. For a given dunite size, there exists 10D, or ∼13 times as many dunites of that size compared to the number of dunites that are an order of magnitude wider. The CDFs for the data collected in Oman fall on a common distribution line spanning ∼4 orders of magnitude. (Figure 8, gray band) Because the areas mapped in detail have variable percentages of dunite (18–68%), individual data sets may fall above or below the average line, reflected in the value for a. For this reason we use the mean values of D and a for extrapolation to larger length scales.
Table 1. Fit Parameters Derived From Each Dunite Image Presented in This Study, Including the Least Squares Correlation Coefficient, R2a
For each image, the corresponding figure number and percentage of dunite in the image are also listed. The details of the parameter fitting are described in the text.
Lufti 1-D composite
Lufti o/c #10
Lufti o/c #18
 Dunite melt conduits as observed in the shallow mantle section of the Oman ophiolite range in size from less than 1 cm to nearly 100 meters. In addition, the abundance of these dunites exhibits a power law dependence on width as measured over 4 orders of magnitude. In the section 3.1, we use size/frequency distribution of dunites as measured in Oman to assess the proposed mechanisms for the formation of replacive dunites and subsequently estimate the total unequilibrated melt flux a dunite network can accommodate beneath an oceanic spreading center.
3.1. Extrapolation to the Melting Region
 By extrapolating the power law size frequency distribution measured in this study, we can estimate the size and abundance of dunite melt conduits over larger length scales (i.e., the melting region beneath mid-ocean ridges). The CDF data are normalized per unit of observation length to enable comparison of data from regions of different size. This normalization also allows for easy extrapolation, simply by multiplying the distribution function by the desired length scale, L. Therefore the number of dunites wider than w predicted in L meters of observation perpendicular to the general strike of the tabular dunites is defined as
 Extrapolation in this manner adequately predicts the abundance of the largest dunites observed in individual massifs in Oman (L = 10 km) as reported by Kelemen et al.  (Figure 9). Estimating dunite abundance over length scales commensurate with the melting region beneath mid-ocean ridges requires 1 order of magnitude of additional extrapolation (L = 100 km). This approach suggests that there may be at least one dunite as wide as ∼3.5 km in the melting region beneath an oceanic spreading center (Figure 9). A dunite of this width, even with a 3–5% steady state melt fraction, is as yet undetectable by seafloor broadband seismic networks [Hung et al., 2000]; so unfortunately, this hypothesis cannot be tested. Alternatively, there could be a smaller maximum size limit. The widest dunites observed in the Oman mantle section (∼100 m) could be the largest present beneath mid-ocean ridges.
3.2. Mechanisms of Dunite Formation
 Dunites have been hypothesized to form by (1) diffusive reaction around melt-filled hydrofractures [Nicolas, 1986, 1990; Suhr, 1999], (2) random merging of growing reaction zones around several hydrofractures [Kelemen et al., 2000], and (3) a reactive infiltration instability resulting in high porosity conduits. [Aharonov et al., 1995; Kelemen and Dick, 1995; Kelemen et al., 1995b; Spiegelman et al., 2001] (Figure 10a–10c). For all three hypotheses, replacive dunites are formed by reaction between ascending MORB melts and the surrounding peridotite. As pressure decreases, the solubility of silica-rich phases (opx and cpx) in basalt increases [Dick, 1977; Quick, 1981; Kelemen, 1990]. Therefore ascending melts dissolve pyroxene and precipitate olivine, increasing the liquid mass and, where the time integrated melt flux is large enough, creating dunite in their wake [Daines and Kohlstedt, 1994; Kelemen, 1990; Kelemen et al., 1995b]. Kinetic studies show that these reactions can occur rapidly [Brearley and Scarfe, 1986; Kuo and Kirkpatrick, 1985a, 1985b; Zhang et al., 1989]. The discriminating difference between hypotheses is the melt transport mechanism. In hypotheses 1 and 2, pyroxene-undersaturated melt within hydrofractures forms dunite along the margins as the melt reacts with the wall rock. In hypothesis 3, there is no crack, and high porosity dunite channels form as a result of a dissolution instability. The increased porosity resulting from the dissolution reaction increases the local permeability, which in turn increases the melt flux to the area. The increased flux draws in more undersaturated melt, resulting in further dissolution.
3.3. Dunites as Diffusive Reaction Zones
 Dunite growth as reaction zones around melt filled hydrofractures is limited by silica diffusion in the melt. Whether in hydrofractures or in the surrounding porous rock, buoyancy-driven melt transport is predominantly vertical, whereas, dunite margins grow in the direction orthogonal to melt transport. Since the kinetics of pyroxene dissolution are fast, silica must be continually moved from the reaction front back to the hydrofracture in order to drive continued dissolution. Because the melt transport velocity in the host peridotite is small (∼1 m/yr [e.g., Kelemen et al., 1997]), hydrodynamic dispersion will also be very small [Freeze and Cherry, 1979]. Therefore dunite growth is limited by the lateral transport of silica by diffusion.
 This is a limiting case with no horizontal advection of melt. Focused flow networks require that melt, initially formed along grain boundaries uniformly distributed in a porous source rock, flows laterally into channels of focused transport. This lateral advection into the channels (10−1–10−4 times the vertical velocity of ∼1 m/yr in porous models [Spiegelman et al., 2001]) may be generally faster than the diffusive transport velocity of SiO2.
 If dunites are diffusive reaction zones, then the time required for formation via Fickian diffusion can be estimated by
Assuming the melt composition in the hydrofracture (i.e., degree of undersaturation) remains constant, the dunite width, w, is proportional to the square root of the formation time, t. The effective diffusivity, Deff, of a particular species in the two phase system is defined as [Crank, 1975]
where ϕ is the interconnected porosity and Dmelt and Drock are the diffusivities of silicon in the melt and solid phases, respectively. Given a moderate porosity of 1–3%, Dmelt ≈ 10−11 m2/s, and Drock ≈ 10−15 m2/s [Lesher et al., 1996], the time required to form the largest observed dunites in Oman (∼100 m) is 250–750 Ma. However, the lifetime of solid material rising through the melting region beneath a spreading center is only ∼1–10 Ma for spreading rates of 0.12–0.01 m/yr, respectively. Therefore the widest dunites that could be formed by diffusion around melt-filled hydrofractures are ∼3–20 m (depending on porosity and spreading rate) (Figure 11). If the spreading rate during the formation of the igneous crust in Oman was at least 0.05 m/yr, then dunites wider than ∼3–10 m could not be formed as the result of diffusion alone.
3.4. Dunites as Merging Reaction Zones
 Large dunites (>10 m wide) are present in the Oman ophiolite in the abundances predicted by the observed power law (∼10/km). However, diffusion around individual melt-filled hydrofractures cannot explain their formation. In modeling dunite formation as reaction zones around a static distribution of cracks, Kelemen et al.  suggested that wide dunites may be the result of random merging of reaction zones around individual hydrofractures.
 To test this hypothesis, we examined a series of forward models in which reaction zones around dunites are allowed to grow and merge and compared the results to the distribution of observed dunite widths in Oman. These models start with an initial, spatially random distribution of five parallel cracks. Reaction zones around each crack grow with the square root of time. In addition, five new dunites nucleate at each time step (unlike Kelemen et al. ). With time, the merging of reaction zones creates increasingly wider dunites, and a weak power law size/frequency distribution begins to emerge. However, the merging process fills space rapidly. After 200 time steps, the largest dunites are only ∼20 times wider than the initial width, but the dunites occupy ∼ 42% of the total area (Figure 12a). By the time very wide dunites (>103 times the initial width) form, more than 94% of the model space is filled with dunite and the power law has degraded (Figure 12b).
 We examined a range of growth and nucleation rates, which affect the time required to fill space with dunites, but in all cases, the range in dunite size never spanned more than 1.5 orders of magnitude before dunites filled more than 50% of the available space. Additional model parameters could be added to modify the fracture mechanics or diffusive growth as a function of dunite width (i.e., new dunites may preferentially nucleate near larger dunites, or larger dunites may grow faster). Such feedback effects may produce a power law size/frequency distribution over several orders of magnitude in width, by analogy with the feedback mechanisms examined for purely porous conduits in the next section of this paper. However, there is as yet no evidence to suggest either of these is necessary. Preliminary size/frequency data for harzburgites (the space between dunites) in the images presented here exhibit exponential distributions, suggesting the random spatial distribution of dunites within the areas we mapped in detail. (We have concentrated on areas with high dunite density and suspect that on the massif and ophiolite scale, dunites may be clustered in a nonrandom fashion.) Regardless of the nature of the spatial distribution, the diffusion equation (4) specifies that the growth rate for any given dunite should decrease with time, preventing larger dunites from growing faster.
3.5. Dunites as Porous Conduits
 Dunites may represent high porosity melt channels formed as a result of the reactive migration of melt in a solubility gradient [Aharonov et al., 1995; Kelemen and Dick, 1995; Kelemen et al., 1995b; Spiegelman et al., 2001]. As melt migrates upward, it becomes progressively undersaturated in pyroxene. The resulting reaction between the melt and harzburgite dissolves pyroxene and precipitates olivine while increasing the liquid mass [Daines and Kohlstedt, 1994; Kelemen, 1990; Kelemen et al., 1995b]. The increased porosity increases the local permeability, which in turn increases the melt flux to the area. The increased flux draws in more undersaturated melt, resulting in further dissolution. Numerical models of this positive feedback, or reactive infiltration instability, indicate that a self-organized network of high porosity dunite conduits, coalescing downstream, can rapidly form within the melting region [Aharonov et al., 1995; Spiegelman et al., 2001]. Modeled widths of these porous conduits, measured in terms of the width over which the soluble solid phase has been completely removed by dissolution, are ∼1 km wide when scaled to the size of the melting region beneath mid-ocean ridges.
 The primary requirement of the melt transport process is that it provides a sufficient flux of unequilibrated melt to the ridge axis to form the oceanic crust. We can assess the ability of a high-porosity dunite network to accommodate this flux using the dunite size/frequency relationship established from our field observations. The total melt flux to the ridge can be estimated as the product of the crustal thickness, half-spreading rate, and ridge length. Assuming a constant global crustal thickness of 6 km, the estimated melt flux to the ridge is 60–900 m3/yr per meter of ridge segment along axis for slow and fast spreading ridges (0.01–0.15 m/yr), respectively.
 The total flux through the dunite network can be estimated by integrating the predicted flux in a dunite of a given size with the observed size/frequency data. The 2-D flux (volume per unit time per unit length) in an individual dunite, Jw, can be defined as
which is the product of the dunite width, w, and the Darcy flux
where Δρg is the driving pressure gradient due to melt buoyancy, η is the melt viscosity, and the permeability, k, is a function of porosity, ϕ, grain size, d, and geometry, C:
For an interconnected porous network around tetrakaidecahedral grain edges, n = 2 and C = 1600 [von Bargen and Waff, 1986]. The grain size, d, is assumed to be ∼4 mm, consistent with the observations from dunites in Oman [Boudier and Coleman, 1981]. The driving pressure gradient is derived from the density difference, ∼500 kg/m3, between basaltic melt and olivine. The total flux of the system, Jtot, is the integral of the product of the flux function, Jw, and the dunite PDF, fw, over the range of dunite sizes predicted for the given length scale, L, where
 To estimate the total flux through all dunites, the minimum bound for the integral, wmin, is equivalent to the width of the smallest observed dunite (∼1 cm). The upper integration bound, wmax, is determined from the extrapolation of the power law at the length scale of the melting region. However, as discussed above, only dunites greater than the length scale of diffusive equilibration can preserve chemical disequilibrium. Therefore the total flux of unequilibrated melt, Juneq, occurs only through dunites wider than 3–10 m.
 In addition, we assume that the dunites form a network of channels that coalesce downstream. As discussed by Kelemen et al. , this is consistent with, though not required by, the observed power law relationship between dunite width and frequency. In such a network, flux must be conserved where porous conduits merge. In systems where flux is conserved at binary intersections the number of channels is inversely proportional to the flux they carry, or fw ∝ Jw−1 [e.g., Hart, 1993]. In its simplest form, at each intersection, two channels each carrying one unit of flux join to form one channel carrying twice the flux.
 Using a series of stochastic models, we show that this relationship between the number of channels and the flux they carry holds regardless of the initial distribution of flux or the order in which channels merge. In each case, ∼4100 channels are initially assigned a flux according to a particular probability density function, continuous uniform, exponential, or gaussian, with a fixed maximum allowable flux ranging from 1–100 times the smallest initial flux (Figures 13a, 13c, and 13e, respectively). The channels randomly merge in pairs until all the flux is in one channel. This process is repeated 100 times for each class of initial flux distribution. For any given maximum initial flux, the total flux through the system is constant, regardless of the shape of the initial distribution. In all cases, the number of channels is inversely proportional to the flux they carry as measured over ∼4 orders of magnitude (Figures 13b, 13d, and 13f). This consequence of flux conservation is also observed in theoretical [Aharonov et al., 1995] and numerical [Spiegelman et al., 2001] models of reactive porous flow, where fw ∝ Jw−1.
 Our dunite size/frequency observations from Oman indicate that fw ∝ w−(D + 1). Equating these relationships for fw suggests that flux is nonlinearly proportional to dunite width, or
Using this width-flux relation, we can define a porosity scaling as a function of dunite width. After substitution of equation (6) into the above relation, it follows that
Canceling w from both sides and substituting equation (8) into equation (7) yields the following relationship between porosity, ϕ, and channel width, w:
Assuming grain size, d, melt viscosity, η, grain boundary geometry, C, or the driving pressure gradient, Δρg, do not vary significantly between dunites of different sizes, the only free parameter is the interconnected porosity. Therefore, by specifying the maximum porosity, ϕmax, in the largest dunite, wmax, the porosity in any given dunite can be written as
After substitution of equation (11) into equation (9b), if flux is nonlinearly proportional to dunite width, the total flux can be expressed as
 Adopting the spreading rate constraint for Oman, we assume that all flux through dunites wider than 5 m will preserve disequilibrium between migrating melt and the surrounding harzburgite. For length scales commensurate with mid-ocean ridge spreading centers (L = 100 km), the largest dunites in the network may be as wide as ∼3500 m. Given a maximum porosity in the widest dunites of 1–3%, the total flux through the dunite network is comparable to the observed ridge flux (Figure 14a). More importantly, the flux of unequilibrated melt accounts for greater than 99% of the total melt flux. Alternatively, if the largest dunite widths in the melting region are limited to the widest dunites observed in Oman (wmax = 100 m), this “nonlinear flux model” (Figure 14b) still supplies a sufficient unequilibrated melt flux (∼95% of the total flux) to satisfy the mid-ocean ridge constraints at reasonable porosities (ϕmax < 4%). These results indicate that melt transport through dunites wide enough to preserve disequilibrium with the shallow mantle can be accommodated entirely by porous flow, and thus there is no requirement for transport through melt-filled cracks.
 Flux conservation is a necessary constraint on the melt flux model. A “linear melt flux” model that assumes a constant Darcy flux (e.g., integration of equation (9b) assuming the porosity in channels is constant and therefore independent of dunite width), does not reproduce the observed fluxes of unequilibrated melt at mid-ocean ridges. The total ridge flux in the “linear melt flux” model can be accommodated if all dunites have porosities of 1–2%. However, the unequilibrated melt flux accounts for only ∼34% of the total flux, if the maximum dunite width is 3578 m (Figure 15a), and only 22% of the total flux if the maximum dunite width is 100 m (Figure 15b). Such volumes of equilibrated melt would result in an andesitic composition for the igneous oceanic crust, corresponding to a liquid composition in equilibrium with olivine, pyroxene, and spinel in the shallow mantle immediately below the base of the crust (∼2 kb), very different from the observed basaltic bulk composition of oceanic crust and primitive MORB (Figure 1). Thus dunite channels can only accommodate the flux of unequilibrated melt through the shallow mantle if the porosity within the dunites is proportional to their width.
 Dunite widths as measured in the mantle section of the Oman ophiolite exhibit a scale-invariant power law size/frequency distribution over 4 orders of magnitude. The power law slope (D ≈ 1.1) predicts that there exist ∼13 times more dunites for every order of magnitude decrease in dunite width. Extrapolation to larger length scales is consistent with field observations at the massif scale and suggests that dunites as wide as 3.5 km may exist in the melting region beneath oceanic spreading centers. Alternatively, dunites may never exceed 100 m, equivalent to the width of the largest dunites we observe in the Oman mantle section.
 Diffusive reaction zones around individual melt-filled hydrofractures cannot explain the formation of wide dunites, as the time required to form the largest observed dunites is greater than the transport time of solid material through the melting region. For the inferred minimum spreading rate of the spreading center that formed the Oman ophiolite, dunites more than ∼3–10 m wide could not have been formed by this process. Forward models for the formation of wide dunites by the random merging of reaction zones rapidly fill space and do not generate the power law relationship observed in the field. Instead, our data are most consistent with the formation of dunites as a network of high porosity dissolution channels.
 We use the dunite width/frequency statistics and diffusive equilibration length constraints to estimate the total flux of melt through porous dunite channels to the ridge axis. In a system of coalescing channels that conserves flux, porous flow in dunite dissolution channels can accommodate the observed melt flux to the ridge over a wide range of spreading rates and moderate porosities and can preserve the observed disequilibrium between ridge basalts and the shallow, residual mantle.
 On the basis of our observations of dunite shape, size, abundance, and distribution, we offer the following schematic illustration of the melt migration network beneath a spreading center (Figure 16). In the melting region beneath a spreading center, ascending melts have the potential to dissolve pyroxene and precipitate olivine, leaving dunite in its wake. These high porosity dunite channels coalesce upward and toward the ridge axis. Merging increases dunite width and porosity while decreasing the number of conduits, thus focusing melt transport toward the ridge axis.
 We would like to thank Greg Hirth, Marc Parmentier, Jack Whitehead, and Maria Zuber for a thorough examination and insightful comments regarding this work, as well as Günter Suhr, Kazuhito Ozawa, and Susumu Umino for their thoughtful reviews. This work was funded by a NSF Graduate Research Fellowship and NSF grants OCE-0118572 and OCE-9819666.