Beneath mid-ocean ridges, magma is thought to rise through a network of high porosity channels that form by reactive flow. Partial mantle melts travel rapidly through these channels to the surface, and retain the geochemical signature of their source rock. Global analyses of mid-ocean ridge lavas indicates that the mantle is chemically heterogeneous, but the consequences of this heterogeneity for reactive porous flow remain unclear. Using numerical models of coupled magma/mantle dynamics, we investigate the relationships between mantle heterogeneity, melting, and magmatic channelization. The models are based on conservation mass, momentum, energy, and composition in a system with two phases and two thermodynamic components in local thermodynamic equilibrium. One of these components is more fusible than the other. In this context, we find that heterogeneities enriched in the more fusible component can nucleate magmatic channels. To understand this result we consider an expression for the melting rate derived from the conservation principles. This expression quantifies the relationship of decompression, reactive flow, and thermal diffusion to the melting rate. With it, we assess their relative importance in the ambient mantle, channels, and enriched heterogeneities. In our models, heat diffuses into fertile channels and powers melting, in combination with reactive flow. These results suggest that thermal diffusion influences the dynamics of magmatic channelization.
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 For the geochemical identity of the source rock to be preserved in the erupted magma, melt transport must occur, to a large extent, without chemical re-equilibration or reaction with the ambient mantle. Chemically isolated flow is also required by observations of disequilibrium between MORB-forming magmas and residual mantle peridotites (rocks containing >40% olivine and >10% pyroxene). For example, residual peridotites (harzburgite and lherzolite) are saturated in orthopyroxene, whereas mid-ocean ridge basalts are not. It follows that MORB-forming magmas are undersaturated in orthopyroxene at Moho pressures. But at pressures greater than 8 kbar the same magmas are saturated in orthopyroxene [O'Hara, 1965; Stolper, 1980]. This pressure-dependent change in saturation indicates that basalt-forming liquids are chemically isolated from the solid, orthopyroxene-bearing mantle at pressures less than 8 kbar.
 Uranium series disequilibria measured in oceanic basalts adds the further, challenging, constraint that melt transport in the mantle must be rapid [Jull et al., 2002; Stracke et al., 2006]. To generate the observed disequilibria, magma must be able to flow from its point of genesis to the surface on a timescale similar to the half-life of230Th (75 kyr) [Turner and Bourdon, 2011].
 In general, then, global observations of MORBs require that magmatic flow in the mantle is rapid and chemically isolated, and that magma forms by melting of a chemically heterogeneous mantle. What, if any, is the relationship between these two aspects of magmatism? Is chemically isolated magmatic flow a consequence of mantle heterogeneity? Or can this style of melt transport occur independently of chemical heterogeneities in the source region?
 One currently popular model for rapid, chemically isolated melt transport invokes reactive porous flow of magma through a network of high porosity channels. This idea grew around field observations of the Oman ophiolite, which is thought to preserve the crust and uppermost mantle from a spreading ridge [Tilton et al., 1981; Alabaster et al., 1982; Nicolas, 1989; Kelemen et al., 1997]. The mantle section of the ophiolite contains a suite of anastomosing, tabular-shaped dunites (>90% olivine) [Kelemen et al., 2000]. The dunites replace, rather than intrude, mantle peridotites, and form when magma, migrating by porous flow, dissolves pyroxene out of the ambient peridotite [Kelemen et al., 1995]. In addition, the erupted melts preserved by the ophiolite are in chemical equilibrium with the dunites. The implication, therefore, is that dunites in the Oman ophiolite represent fossilized conduits, or channels, through which magma was transported to the surface. Similar conclusions have been drawn from recent observations made on unrelated ophiolites in Italy [Renna and Tribuzio, 2011].
 To understand the implications of the reactive infiltration instability for magmatic flow, Aharonov et al.  developed a numerical model of reactive melt transport in a column of chemically homogeneous mantle rock. Aharonov et al.  based their model on canonical equations for porous flow and compaction [McKenzie, 1984] and, to emulate the behavior of orthopyroxene in the mantle, assumed that the solubility of their model rock was greater at shallower depths. Magma that rose through their model domain consequently reacted with, and removed mass from, the solid. Provided that the solubility gradient was great enough, the flow was unstable, and localized into a network of high porosity channels.
 What, then, gives rise to magmatic channels? Hewitt showed that channels only form if the melting region is supplied with a sufficient, additional flux of magma. We hypothesize that this additional flux of magma is supplied by partially melting regions of more fusible rock. Provided that these regions are sufficiently fusible, this flux will nucleate and support high-porosity channels via a mechanism similar to the reactive infiltration instability.
 In this paper we examine the effect that chemical heterogeneities have on melting and melt extraction in an upwelling mantle. We use a thermodynamically consistent approach to model the coupled dynamics of magma and solid mantle rock in a thermally and chemically diffusive system. Using suites of numerical experiments, we investigate the effect that a single, isolated heterogeneity has on melting and magma flow, and we vary the physical and chemical characteristics of the heterogeneity between different experiments. Our aim is to elucidate the fundamental dynamics of the system, and so we accept simplifying assumptions such as thermodynamic equilibrium and a simple geometry for the heterogeneity.
 In section 2 we outline our numerical model. We describe the results of the model in section 3, and investigate its sensitivity to parameters that are either geologically or theoretically important. We explore the response of the system to the fertility, size, and shape of compositional heterogeneities, the upwelling rate of the mantle, latent heat of fusion and the thermal diffusivity. The results emphasize that the relationships between melting processes, compositional heterogeneities, and magma flow in the mantle are complex, and they underscore the importance of thermal diffusion to magmatic flow and melting in the presence of chemical heterogeneities.
Section 4 presents a derivation of the thermodynamically consistent melting rate, which accounts for thermal and chemical diffusion. The result of the derivation offers a structured framework in which to interpret the results. Using this derivation, we explore how different processes contribute to the melting rate. We conclude section 4 with a discussion of the limitations of our model.
2.1. Domain and Initial Condition
 The situation we consider is a two-dimensional column of upwelling rock that contains a compositional heterogeneity (Figure 1). The size and position of the domain are fixed: the top of the domain is aligned with the surface, where the lithostatic pressure is zero, and it extends 120 km into the mantle. Solid rock flows into the base of the domain at a rate W0. The rock has a known potential temperature , and has a known composition Cmin a two-component system that forms a solid solution series. One of the components is more fusible than the other. Regions that are enriched in the more fusible component have a lower melting temperature. We refer to these regions equivalently as ‘enriched’, or ‘more fusible’. In contrast, regions that are depleted in the more fusible component melt at a higher temperature. We refer to these regions as being ‘depleted’, or ‘less fusible’, although the use of these terms does not necessarily reflect concentrations of trace elements.
 The initial condition incorporates a circular chemical anomaly, or enclave, that is enriched in the more fusible component. The form of the anomaly is given by , where gives the degree of enrichment, r0 is the radius of the heterogeneity and r is the distance of a point from its center, and λ is a parameter that controls the sharpness of the edges of the heterogeneity. At some depth, the ascending mantle begins to melt. Since the enclave is enriched in the more fusible component, and the solidus temperature of the more fusible component is lower at all pressures, the enclave begins to melt at a greater depths than the surrounding, refractory mantle. Above this depth, the system contains two phases. The rock is modeled as a porous, compactible, solid that behaves as a fluid on geological timescales, and the magma is considered to be a buoyant liquid. At the surface, these interpenetrating fluids flow unrestrictedly out of the domain. The boundary conditions are described in more detail in section A1.
 Although our ultimate goal is to gain an insight into processes that might operate beneath a mid-ocean ridge, we do not attempt to model the physical setting of a mid-ocean ridge. At a spreading ridge the motion of the lithosphere introduces a horizontal component of motion into the solid mantle flow field. Since we aim only to understand how mantle heterogeneity influences melt transport in the mantle, we neglect the complicating factors introduced by plate motion. Furthermore, in this paper we will not attempt to infer whether our model can account for geochemistry of erupted magmas. To do so would require similar experiments to be run in a model configured to simulate a mid-ocean ridge. Examples of computations run with boundary conditions corresponding to a mid-ocean ridge are provided inKatz and Weatherley .
2.2. Governing Equations
 The foundations of the model are conservation statements for mass, momentum, chemical composition, and energy. We provide a comprehensive description of the equations in Appendix A, but discuss important features here.
 The set of equations on which the model is founded governs the mechanics and energetics of reactive porous flow in a compacting mantle. They are a reformulated version of the set derived by McKenzie , and later by others with minor differences [e.g., Fowler, 1985; Ribe, 1985; Scott and Stevenson, 1986; Bercovici et al., 2001]. In the upwelling column, large scale shear is neglected and the solid velocity vm can be written as vm = ∇U [Spiegelman, 1993], where U is a velocity potential. This potential is related to the compaction pressure , which is the difference between the magmatic pressure and the lithostatic pressure. Equations (A1) and (A2) govern and U respectively.
 Two more equations are needed to account for energy and composition. They are
Equation (1) represents the conservation of energy, and is cast as an equation for bulk enthalpy . In this equation is the volumetric bulk enthalpy, where ρ is density, L is the latent heat of fusion, and ϕ is the porosity, cP is the specific heat capacity, T is temperature, and T0 is the melting temperature for material of the reference composition C0 at zero pressure. is the phase averaged velocity, where vf is the velocity of the magma, and vm is the solid velocity. is the potential temperature, α is the coefficient of thermal expansion, k is the thermal conductivity, and z is depth. Equation (1) states that temporal changes in enthalpy arise from advection of latent heat, and advection and diffusion of sensible heat. Equation (2) represents the conservation of composition. Here, C = ϕCf + (1 − ϕ)Cm is the bulk composition, where Cf is the composition of the magma, and Cm is the composition of the solid, and is the coefficient of chemical diffusion in the magma. This equation states that temporal changes in composition result from advection of composition by the matrix and magma, and by chemical diffusion in the magma. These equations differ from those used by Hewitt  by allowing for thermal and chemical diffusion.
 To summarize, our model is constructed around four coupled partial differential equations, and we use these equations to solve for , U, , and C. These equations, however, depend on the additional variables ϕ, T, Cm, and Cf. We obtain values for ϕ, T, Cm and Cf through closure conditions provided by the Enthalpy Method.
2.3. Idealized Petrological Model
 The assumption of local thermodynamic equilibrium requires that the temperature of coexisting rock and magma is equal. We quantify this assumption with an idealized phase diagram shown in Figure 2. To simplify the thermal and chemical dynamics of the system, the solidus and liquidus surfaces are approximated to be linear. They are given by
where TS and Tl are the solidus and liquidus temperatures, Pl is the lithostatic pressure, and γ is the Clapeyron slope, which we take to be constant. MS and ML describe the slopes of the solidus and liquidus surfaces respectively. C0 is the composition of the ambient mantle at the onset of melting, ΔC is the compositional difference between rock and magma in equilibrium at the reference temperature T0. The solidus and liquidus surfaces relate the composition of the magma and matrix to the temperature and lithostatic pressure. Katz  describes how equations (3) and (4) are used to solve for ϕ, T, Cf, and Cm.
2.4. Numerical Methods
 We discretize the system of governing equations and boundary conditions using a semi-implicit, finite-volume scheme with a grid resolution of 0.5 × 0.5 km. To assist numerical solution, we split the governing equations into two groups (equations (A1) and (A2) and (A3) and (A4)). At every timestep, each group of equations is solved iteratively using an incomplete LU preconditioned Newton–Krylov method provided by the Portable, Extensible Toolkit for Scientific Computing (PETSc) [Balay et al., 1997, 2009].
 To preserve gradients in the system we used a total variation diminishing (TVD) flux limiting advection scheme. Šrámek et al. offer a useful overview of TVD schemes. TVD schemes are conservative and non-oscillatory. Normally, they are second order in space, but near sharp gradients, their order is reduced. The Superbee flux limiter [Roe, 1985] is used throughout this paper. Like Šrámek et al. , we found it to be the best flux limiter for minimizing dispersion and diffusion.
3.1. General Behavior
Figure 3 shows a time series of solutions from a single, representative experiment in terms of different variables. Figures 3a1–3f1 show porosity and Figures 3a2–3f2 show the solid composition. Since we do not map composition onto a real petrological system, we express it as ΔTC = MS(Cm − C0), which is the difference in solidus temperature due to deviation of the local composition from the reference composition C0. Figures 3a3–3f3 show the temperature and Figures 3a4–3f4 show the melting rate. To compute Γ we rearrange the conservation of mass equation for the fluid phase (equation (B1)) to give . Figures 3a5–3f5 show the the magma speed |vf|, and Figures 3a6–3f6 show the compaction pressure (equation (A2)). Negative compaction pressures indicate that the matrix is compacting, and that the magmatic pressure is less than the lithostatic pressure.
 The experiment in Figure 3 is initialized with a disk of material that is enriched in the more fusible component, with ΔTC = −30°C. The radius of the disk is 4 km and the initial depth of its center point is 100 km. For all of the other parameters we assign a preferred value, which is listed in Table 1.
Table 1. List of Parameters and Preferred Values
MS(C − C0)
1 × 10−6 m/s2
Upwelling rate for the matrix
Latent heat of fusion
4 × 105 J/kg
Slope of the solidus
Slope of the liquidus
Solidus temperature for reference mantle at 0 kbar
Reference mantle composition
Difference in composition between rock and magma in thermodynamic equilibrium
Coefficient of thermal expansion
3 × 10−5 K−1
Specific heat capacity
1 × 10−8 m/s2
Acceleration due to gravity
1 × 10−7 m2
Reference shear viscosity
1 × 1019Pa ⋅ s
 The complete time series of results in Figure 3 shows that an enclave of fertile rock, which is embedded into the upwelling mantle, partially melts and supplies the ambient melting region with an additional flux of magma. The additional flux of magma initiates reactive channelization.
 The left most column of Figure 3 (Figures 3a1–3a5) shows numerical solutions after a model time of 0.35 Myr. Throughout this paper a model time of zero is equal to the time at which the center point of the anomaly crosses the 100 km depth contour. At this time, the enclave is not partially molten and the ambient melting region is in steady state. Melting of the ambient mantle begins at a depth of 60 km. Above this depth the porosity (Figure 3a1), melting rate (Figure 3a4), magma speed (Figure 3a5), and ΔTC (Figure 3a2) increase toward the surface; the temperature (Figure 3a3) and magnitude of the compaction pressure decrease (Figure 3a6). These steady state results agree with the one-dimensional solutions found byMcKenzie ; Ribe ; Katz  and Hewitt .
 In Figure 3, the second column from the left shows results from our experiment after a model time of 1.15 Myr. Figure 3b1 shows that the enriched enclave has started to melt, and that encasing it is an annulus of unmolten mantle rock. The solidus temperature of rock within the enclave is approximately 20°C cooler than the reference mantle (Figure 3b2). Accordingly, the temperature of rock within the enclave is also lower than the surroundings. Heat diffuses into the enclave and powers melting. But diffusion removes heat from the surrounding mantle and suppresses melting, and so it gives rise to the annulus of subsolidus, impermeable rock that encases the enclave. This cool diffusion halo prevents magma flowing from the enclave into the ambient melting region.
 As the enclave and annulus continue to upwell, rock within the annulus melts by decompression, and establishes a magmatic pathway that connects the enclave with the ambient melting region. Figures 3c1–3c5show the system at an instant after the completion of the pathway. The figures show that magma flowing from the enclave has invaded a balloon-shaped region in the overlying mantle. Within this region the porosity (Figure 3c1) and melting rate (Figure 3c4) are larger than the ambient values. Two effects give rise to the larger porosities and melting rates. First, the fluid pressure of the initial magma that leaves the enclave is greater than the lithostatic pressure, and so (Figure 3c6). Thus the infiltrating magma mechanically increases the porosity in the overlying mantle. However, remains positive only for a relatively short period of time (approximately 50 kyr, Figure 3). Second, the flux of the infiltrating magma is greater than the ambient upwards flux of magma. Since the infiltrating magma is enriched, it must react with the ambient mantle rock to maintain thermodynamic equilibrium. By this reaction, the melting rate and porosity in the overlying mantle are increased.
Figures 3d1–3d5 show that 50 kyr after the magmatic pathway is established, a channel emerges from the enclave. The channel grows by the same feedback that is fundamental to the reactive infiltration instability: enriched magma flows into the channel, the porosity increases by reaction, and so the channel can accommodate an even greater flux of magma. Within the channel, however, the matrix compacts and works to stabilize the flow, but the rate of porosity generation by melting outstrips the rate of porosity destruction by compaction, and so the channel grows. The channel itself accommodates a higher porosity (Figure 3d1), melting rate (Figure 3d4), and magma speed (Figure 3d5) than the ambient mantle. It is enriched in the more fusible chemical component (Figure 3d2) and, as a consequence, the temperature within the channel is a minimum for a given depth (Figure 3d3). Heat, therefore, diffuses laterally into the channel, cooling the surrounding mantle and preventing it from melting. In this way, thermal diffusion encourages magmatic channelization at the expense of diffuse porous flow.
 Over the remainder of the experiment (Figures 3e1–3f5) the channel grows and delivers magma to the surface. Initially, channels grow at a rapid rate, but the growth rate slows with time. Figures 3d2, 3e2, and 3f2 show that the fusibilities of the residual mantle within the enclave and channel gradually decrease with time, but the porosity, magma speed, and melting rate remain significantly larger than the ambient values. The elevated melting rate within the channel is associated with strong compaction (Figure 3e6). Figures 3e1–3f1 show that new channels begin to form on the flanks of the existing channel. The mechanisms that control the location of these channels remain unclear.
 One feature of this and subsequent experiments is that the porosity generated within enclaves can exceed 20%. At porosities greater that 20% the matrix is likely to disaggregate [McKenzie, 1984], which violates our assumption of magma flowing through a porous matrix. Our aim, however, is not to fully capture the dynamics of magma flow within enclaves, nor the process of magma leaving the enclave. To do so would widen the scope of our models to include, amongst other processes, brittle fracture [Nicolas, 1986] and open fracture channels [Hewitt and Fowler, 2009]. Instead, we focus on the dynamics that arise when a partially molten enclave supplements the ambient melting regions with an additional flux of magma. Since the porosity within enclaves drops to reasonable background values immediately after the onset of channelization, we are confident that the large porosities generated within the enclaves do not adversely affect the trajectory of our experiments.
3.2. Thermodynamic Controls
3.2.1. Latent Heat
 The latent heat of fusion is an important control on melting, since it couples the rate of melt production to the energetic budget. We explore how L affects melting, and its subsequent influence on channelization, with an ensemble of five experiments (Figure 4). For these experiments we varied L over an order of magnitude between L = 9 × 104 J/kg and L = 9 × 105 J/kg. Experimental work by Kojitani and Akaogi  suggests that a reasonable value for the latent heat of fusion of mantle peridotite is 5 × 105 J/kg. All remaining parameters take their values from Table 1. In Figure 4 and each subsequent figure, the center column shows results from an experiment run with our preferred parameter values from Table 1.
Figures 4a1–4e1 show the results from the experiments in terms of porosity after a model time of 600 kyr. In each experiment, magma from the enriched enclave has not yet infiltrated the ambient melting region. Figures 4b1–4e1, which are taken from experiments where L ≥ 1.5 × 105 J/kg, show that the style of magmatic flow in the ambient melting region prior to the addition of melt from the more fusible enclave is stable, diffuse porous flow (Figures 4b1–4e1). When L < 1.5 × 105 J/kg, however, magmatic flow in the ambient melting regime is unstable; channels form independently of mantle heterogeneity (Figure 4a1). In this simulation, channels grow downward over time, and stop growing at a depth of approximately 30 km. Figures 4a2–4e3 show results from the same suite of experiments in terms of porosity and ΔTC at the later model time of 1.65 Myr. They show that when L is large, the fertility of enclaves decreases more slowly (Figures 4a3–4e3), and channels grow more slowly (Figures 4a2–4e2).
 Analysis of the results shows that the melting rate varies inversely with the latent heat. When L is small, enclaves melt more rapidly and supply a greater flux of enriched magma to their network of channels. Consequently, channels grow more rapidly when L is small. The inverse relationship between L and the melting rate also manifests itself in the properties of the ambient melting region. For example, when L is small and the melting rate is high, the porosity of the ambient mantle is large. The large porosity accommodates a greater upwards flux of magma. In the experiment where L = 9 × 104 J/kg the upwards flux of magma is sufficient to generate channels without requiring and additional flux of magma from a more fusible heterogeneity.
3.2.2. Thermal Diffusion
 To determine how the thermal diffusivity affects the dynamics of the system, we vary the thermal diffusivity between 10−7 m2/s and 10−5 m2/s within a suite of five different simulations. Every other parameter uses the default value given in Table 1.
Figure 5 shows results to these experiments after a model time of 0.91 Myr in terms of porosity (Figures 5a1–5e1) and ΔTC (Figures 5a2–5e2). For each experiment the same amount of time has elapsed, yet Figures 5a2–5e2 show that the enclaves in the most diffusive experiments are the least fertile (Figures 5a2–5e2). This implies that melting rates are higher in more thermally diffusive experiments. Consequently, partial melting of enclaves in more diffusive systems can supply their network of channels with magma at a faster rate. In such systems, channels tend to grow more quickly, have a higher porosity, and accommodate faster melting rates and magma speeds.
 A cool diffusion-halo also forms around magmatic channels.Figure 5shows that the permeability of the rock within the diffusion-halo varies with the enrichment of the rock and magma within a channel. When rock within a channel is more fertile, and therefore cooler, the surrounding mantle is less permeable. This is because a greater amount of heat diffuses into more fertile channels to maintain thermodynamic equilibrium. Since thermal diffusion toward channels removes heat from the surrounding rock, material within the diffusion-halo has less enthalpy available for melting, and its porosity is lower.
3.3. Geophysical Controls
 We explore the effect that the enclave's size has on channelization using a suite of five different experiments (Figure 6). For these experiments the initial radius of the enriched enclave ranges between 2 km and 6 km. This range agrees with Helffrich , who estimated the characteristic size of mantle heterogeneity to be ca. 8 km. All other parameters take the values listed in Table 1. Figure 6 shows results from these experiments after a model time of 1.65 Myr. The top row shows the porosity, and the bottom row shows the composition, expressed in terms of ΔTC. At the time of the snapshots, the larger enclaves remain more fusible and support channels that have a longer total length.
Figure 6 shows that the larger enclaves, which initially contained a larger mass of the more fusible component, remain more fertile after the same model time has elapsed. It is difficult to assess, from the results presented in this paper, how the melting rate within an enclave varies with its size. However, Katz and Rudge relate the degree of melting to a Peclet number for the enclave, which represents the balance between diffusion-driven and adiabatic melting, and show that the melting rate decreases with increasing radius of the enclave. Since larger enclaves also contain a greater mass of the more fusible component, they can supply channels with enriched magma for a longer period of time. In consequence, their channels are more enriched and grow to greater lengths via reactive flow (Figure 6). Characteristics of more fertile channels include a higher melting rate, larger porosity, and a greater flux of magma.
Figure 7 examines the effect of the enclave's fertility on channelization through another suite of five experiments. The enclave in each experiment takes a different initial ΔTC, and the values of all other parameters are held constant at the values listed in Table 1.
Figure 7a illustrates the principal trend that emerges from this suite of experiments. It shows that the difference in time between the onset of channelization and the onset of melting within the enclave, Δt, is negatively correlated with ΔTC. Rather than describe the data with a single line, we tentatively suggest that the data lie on two branches. The first branch describes the data for ΔTC ≥ −30 K, and the second branch spans the data for ΔTC ≤ −40K. To better understand this trend we refer to the snapshots from each experiment that are shown in Figures 7b1–7f2.
Figures 7b1–7f1 show porosity fields at the onset of channelization. These figures show that more fusible heterogeneities begin to channelize at shallower depths. The figures also show a cool diffusion halo surrounding each enclave. The temperature gradient across the diffusion halos is steeper for more fusible enclaves. For material within the halo to melt, the more fusible enclaves need to upwell to shallower depths. It is only when a permeable pathway connects the enclave to the ambient melting region that a channel can begin to form.
 Branching of the data shown in Figure 7a is a consequence of the partially molten enclave's morphology. The heterogeneities shown in Figures 7e1 and 7f1 are each associated with two large pockets of melt. These pockets lie outside the perimeter of the original heterogeneity and are formed by partial melting of the surrounding mantle rock. This consumes latent heat and establishes a diffusive heat flux into the pocket, further chilling the surrounding mantle and delaying the onset of channelization.
 From Figures 7b1–7f1 it is clear that more fusible enclaves produce a larger volume of magma prior to channelization. In Figures 7b2–7f2, we show the same experiments at a later model time of 2 Myr to illustrate how these different volumes of magma affect channelization. They show that the enclaves for which the initial ΔTC ≥ −30°C give rise to channels with a relatively simple geometry (Figures 7b2–7d2), whereas the channels in Figures 7e2–7f2 are considerably more complex. By analyzing the full time series of results for each of the experiments, more trends emerge: channels rooted at more fertile enclaves are more enriched, accommodate higher porosities and faster fluid velocities, grow more quickly, and melt at a greater rate. However, our results suggest that no simple relationship exists between the melting rate within channels and the initial composition of the heterogeneity.
3.3.3. Upwelling Rate
Figure 8 shows results from a suite of five experiments that investigate the effect of the upwelling rate W0 on the system. For these experiments W0 ranges from 2 cm/yr to 6 cm/yr, and all other parameters take their values from Table 1.
Figure 8 shows results from the ensemble of experiments at 0.5 Myr after the onset of channelization. In general, the figures show a subtle increase in the melting rate with W0 for the ambient melting region. Correspondingly, the magma speed and porosity in the ambient melting region also increases with W0. Figure 8 shows that spatial variations in the melting rate correlate with spatial variations in the porosity and magma speed (Figures 8a2–8e3). This is an important result that we wish to emphasize. The melting rate is large in regions where the porosity and magma speed are also large. A similar result was found by Hewitt , who showed that the melting rate depends on the average upwelling rate .
 In general, the results in Figure 8 show that the upwelling rate affects the dynamics of the system principally through the melting rate. They suggest that when W0 is large, channels evolve more quickly to develop large porosities and melting rates. However, our experiments show no clear relationship between the upwelling rate and the geometry of channels.
 Above we investigated the relationship between channelization and heterogeneity by considering only circular shaped compositional anomalies. Since we aim to understand the most fundamental ways in which mantle heterogeneity influences magmatic flow, this simplified approach is justified. However, there is little doubt that heterogeneities in the real mantle have a geometry that is much more complex. In the real Earth, mantle flow will fold, stretch, and thin enclaves of recycled oceanic crust [Allègre et al., 1984]. Some of the enclaves will be boudinaged, and dissolution will blur the edges of enclaves.
 To explore whether the general behavior of our experiments holds for enclaves of different shapes, we initialized two experiments with ellipse-shaped enclaves (Figure 9). In one of the experiments (Figures 9a1 and 9a2), the ellipse is aligned with its long axis parallel to the vertical; in the other, the long axis is rotated anticlockwise from the vertical by 45°.
Figures 9a1 and 9b1 show that, in contrast to the previous experiments, for which the initial shape of the enclave was circular, the pocket of magma can be wider than the enclave. Figures 9a2 and 9b2show that the ellipse-shaped enclaves also produce channels. The geometry of these channels is arguably more complex than in the previous simulations. To illustrate, the enclave inFigure 9a2 hosts two widely spaced channels that are growing in synchrony, and in Figure 9b2, the large channel incorporates lateral deviations along its length. Nevertheless, the basic behavior of the channels and their relationship with the surrounding mantle is consistent with the experiments described in the previous sections.
3.5. Summary of Numerical Results
 We investigated how the dynamics of magmatic channelization varies in response to changes in latent heat L, thermal diffusivity κ, the size R, shape, the fusibility of the heterogeneity, and the upwelling rate of the mantle W0. Our numerical experiments indicate that the evolution of high-porosity channels is coupled to composition, heat flow, and fluid dynamics of the system. To summarize, channels grow faster, and are depleted more rapidly whenL is small and κis large. Fast-growing channels tend to have higher porosities, faster magma speeds, and greater melting rates. These properties are also apparent in channels that evolve in systems where the heterogeneities are larger and more fusible (ΔTC is more negative), and the upwelling rate is faster.
 Many of the trends apparent in each of the suites of experiments above are best understood in terms of the melting rate. We showed that melting rate Γ depends on a range of thermodynamic and geophysical parameters. Specifically, the melting rate increases with the upwelling rate (Figure 8), magma speed (Figures 3 and 8), porosity (Figures 3 and 8), and the thermal diffusivity (Figure 5). They also show, in Figure 4, that Γ scales inversely with L. In the next section we quantify the effect that the upwelling rate, thermal diffusivity, latent heat and composition have on the melting rate, and explore the processes that give rise to spatial variations of Γ.
4.1. Melting Rate
 The governing equations ((A1)–(A4)) can be combined to give a thermodynamically consistent expression for the melting rate. The derivation is presented in Appendix B; here we emphasize the most important findings. For a diffusive system that is not necessarily in steady state, and for which MS = ML = M, the melting rate is
where and are dimensional constants
Equation (5) shows that the melting rate is a constant multiple of contributions from the phase averaged upwelling rate, thermal diffusion, and chemical diffusion in the liquid phase. The multiplying constant (equation (6)) depends on the reference density ρ, the ratio of the latent heat of fusion and the specific heat capacity, and ΔCM, which is the temperature difference between the solidus and liquidus at a fixed composition. Using values from Table 1, (L/cP)/(ΔCM) ≈ 10. As such, the multiplying constant depends more strongly on the ratio of the latent heat of fusion and the specific heat capacity than on properties of the phase diagram. Furthermore, L determines how strongly the melting rate is coupled to the energetics of the system. Taking L → 0 removes the principal coupling between the melting rate and the energetics of the system. In this case, our expression for the melting rate is similar to that for the reactive flow models of Liang et al.  and Schiemenz et al. , who neglect conservation of energy.
 The first term in equation (5) shows that Γ depends on a constant multiple of the phase averaged upwelling rate. Because Γ depends on, and increases with, the upwelling rate of the magma, our expression for the melting rate (equation (5)) models reactive flow. Unlike the other two terms in equation (5), is prefactored by an additional constant . The constant depends on the Clapeyron slope and incorporates a small correction for adiabatic cooling. The product of these constants is closely related to standard expressions for the productivity [Asimow et al., 1997; McKenzie, 1984]. The first term in equation (5) is identical to the melting rate derived by Hewitt , who solved similar governing equations but neglected diffusion. He derived an approximate stability condition for the melting rate and found that the thermodynamically consistent mechanism that drives the reactive infiltration instability is a melting rate that increases with the melt flux. The increase in porosity caused by the melt flux is countered by compaction, but the rate of compaction also increases with the melt fraction. For this reason, compaction tends to stabilize small perturbations. In the non-diffusive caseHewitt  found that the system is unstable only if the rate of porosity generation by melting exceeds the rate of destruction by compaction.
 The second and third terms in equation (5) demonstrate that the melting rate depends on thermal and chemical diffusion in the magma. They show that heat diffuses into partially molten, enriched heterogeneities and increases the melting rate, whereas chemical diffusion blurs compositional gradients and reduces the melting rate. Using the nondimensional quantities outlined in Appendix A, the second term in equation (5) for thermal diffusion scales as κδT/l2, where l is a diffusive length scale, and the third term in equation (5) for chemical diffusion scales as . Since δT = MδC, the relative size of these scales is . In other words, the contribution to the melting rate from chemical diffusion relative to that from thermal diffusion is negligible.
 Given this scaling, we neglect chemical diffusion and rewrite the melting rate as
Equation (8)emphasizes that thermal diffusion powers melting in cooler regions. Since the assumption of thermodynamic equilibrium explicitly relates temperature and composition, the melting rate can be re-expressed in terms ofCf by combining equations (4) for the linearized liquidus and (8) to give
Equation (9) shows that compositional variation gives rise to spatial variations in the melting rate. Importantly, though, the dependency of the melting rate on the second derivative of composition is through the thermal diffusivity. Equation (9) also implies that the melting rate within channels is related to the composition of the heterogeneity at which it is rooted. However, this relationship is not straightforward, since the prior reaction history will have modified the composition of magma and rock at any point in a channel. The composition of the magma will be further modified by melts that are drawn into the channel from the ambient mantle.
 From equations (8) and (9) it is clear that the melting rate depends on several different variables. To better understand these dependencies, we decompose the melting rate into three terms:
Here, is the contribution from the matrix velocity, is the contribution to the melting rate from reactive flow, and is the contribution from thermal diffusion.
 To determine the relative importance of each of the terms in equation (10), we applied the decomposition to the representative calculation presented in Figure 3. Figure 10 shows the contributions from ΓW, ΓR, and ΓT, expressed as a fraction of Γ. Figures 10b–10d show that in the ambient melting region, at distances far from the enclave and channel network, decompression accounts for the majority of the total melting rate. At the base of the ambient melting region, ΓW accounts for over 95% of the total melting rate. The ambient porosity and the magma speed increase toward the surface. Consequently ΓW/Γ decreases and the contribution from reactive flow, ΓR, becomes more important. Close to the top of the ambient melting region, ΓW accounts for approximately 80% of the total ambient melting rate, ΓR supplies the remaining 20%, and the contribution from thermal diffusion is negligible.
 Within the fertile enclave ΓW and ΓT each account for approximately 45% of the total melting rate, while ΓR supplies the remaining 10%. These results show that diffusion of heat into a fertile enclave increases melting by a factor of 2 over that from adiabatic upwelling. Figure 10d shows that thermal diffusion suppresses melting in the mantle that surrounds the heterogeneity to values lower than that for the ambient mantle at equal depths. The positive contributions to melting, therefore, are decompression and reactive flow in the approximate ratio 10:1.
Figure 10ashows that the greatest melting rates occur within the channel. The characteristic features of channels that give rise to large melting rates are large porosities, fast magma speeds, and a discrete, steep-sided composition anomaly across which the ∇2Cf is large and positive. Figures 10b–10d show that thermal diffusion provides the most significant contribution to the melting rate within channels. In this case, ΓT accounts for approximately 60% of the total melting rate within the channel, ΓR accounts for approximately 30%, and ΓW supplies the remaining 10%. Thermal diffusion, therefore, powers much of the melting within channels. But in the surrounding mantle, it suppresses melting to such an extent that the Γ < 0.2ΓW.
 The results in Figure 10 show that the relative importance of ΓW, ΓR, and ΓT to the melting rate varies with the flow regime. In the ambient melting region where magmatic flow is in steady state, the decompression rate of the matrix provides the most important contribution to the melting rate. In regions where magmatic flow is channelized, thermal diffusion provides the greatest contribution to the melting rate; reactive flow accounts for much of the remainder.
4.2. Limitations of the Model
 Results presented in this paper indicate that magmatic channelization is a consequence of melting in a chemically heterogeneous mantle. The model is necessarily simplified in order to address the fundamental dynamics of the system. An important assumption made by the model is that local thermodynamic equilibrium holds everywhere. Estimates for the Damköhler number of the mantle, which measures the relative importance of advection and reaction, are, in general, poorly constrained [Aharonov et al., 1995; Spiegelman et al., 2001; Spiegleman and Kelemen, 2003], but are large enough for thermodynamic equilibrium to be a judicious approximation.
 Another important assumption inherent in the petrological model is that melting is congruent throughout the top 100 km of the mantle. Chen and Presnall , however, showed that in the pressure range spanned by the melting region beneath mid-ocean ridges, the mode of orthopyroxene melting switches from congruent at high pressure to incongruent at low pressure. Since incongruent melting replaces fusible orthopyroxene with inert olivine, the melting rate in dunite channels will be lower than congruently melting channels under similar conditions.
 In general, the nature of mantle heterogeneity is poorly constrained. Almost certainly, heterogeneities occur across a range of length-scales with a variety of fusibilities and complex topologies. Using the same model as presented in this paper,Katz and Weatherley suggest that kilometer-scale heterogeneities of relatively low fusibility (ΔTC ≈ −5K) are capable of nucleating magmatic channels. However, to investigate how the system responds to heterogeneities on a sub-kilometer scale requires more challenging simulations at higher resolution.
 A persistent feature of the experiments reported here is that magmatic channels evolve to be just one grid cell wide. Grid scale features seem to be a hallmark of existing numerical experiments that use the Enthalpy Method. Hewitt  found similar behavior in his calculations of magma flow in the mantle, as did Katz and Worster  in their calculations of chimney formation in mushy layers. These features may stem from the numerical methods adopted by these studies. Given this, the reason for grid scale channels may be that the fluid pressure, at any constant depth, is lowest in the center of a channel, and melt consequently moves toward the grid cell that corresponds to the center of the channel. Irrespective of the resolution, magma always appears to localize to this central cell. Only if the porosity there reached 100% would we expect the channel to occupy a wider space. In this limit, however, magma flow cannot be described by Darcy's law, and the study would require a different parameterization, such as that used by Hewitt and Fowler .
 Alternatively, higher-order numerical methods, such as those used byLiang et al.  and Hesse et al. , may improve the resolution of channels, and resolve further instabilities in the system. For example, Richter and Daly  suggested that melting in a heterogeneous mantle will give rise to porosity waves. Liang et al.  and Hesse et al.  verified this and showed that nonlinear interactions between compaction and dissolution give rise to porosity waves and magmatic channels. Their results offer an additional mode of channel formation, but it remains to be shown that these additional instabilities feature in systems that conserve energy.
 Results from our energetically consistent model of magma flow and congruent melting indicate that channelized melt transport is a consequence of melting in a heterogeneous mantle. Channels arise when magma from partially molten heterogeneities supplies the ambient mantle with an additional, and sufficient, flux of magma. The results are best understood in light of the melting rate, which is derived from conservation principles. The melting rate is coupled to the energetic budget dominantly through the latent heat. For values close to the estimated latent heat of the mantle, the mode of melt transport in the absence of heterogeneities is diffuse porous flow. Selecting a small value for the latent heat decouples the melting rate from the energetic budget; under these conditions, the model of melt transport in a chemically homogeneous mantle is channelized flow.
 The melting rate is a linear combination of contributions from decompression, reactive flow, and thermal diffusion. In the ambient melting region (i.e. not within, or in the vicinity of, high-porosity channels), melting is largely the response of adiabatic decompression. Within channels, however, thermal diffusion provides the most significant contribution to the melting rate. Since heat diffuses into channels, the adjacent mantle is starved of energy for melting. The permeability of these rocks is low, and magma upwelling from deeper depths is focused into high-porosity regions. Consequently, thermal diffusion provides a positive feedback on channelization, and could discourage mixing between magmas flowing through channels and magmas in the adjacent mantle. Our results predict, for the first time, that thermal diffusion is important to the genesis and dynamics of magmatic channels, and underscore the need to use energetically consistent models in studies of coupled magma/mantle dynamics.
Appendix A:: Governing Equations
 The models used in this paper are based on conservation statements for mass and momentum for the solid rock and liquid magma, conservation of chemical composition within those phases, and conservation of energy. These statements, first derived by McKenzie , were reformulated by Katz et al.  to make them more amenable to numerical solution. Katz  provides a comprehensive derivation of the equations for conservation of energy and composition. In this study, the equations that govern the mechanics of the system are simplified, in that matrix shear is neglected and the fluids are assumed to be Boussinesq. The set of partial differential equations that we use to model the coupled dynamics of magma and mantle rock are
Equations (A1) and (A2) conserve mass and momentum for the mantle rock and magma. Here, K is the permeability, and is related to the porosity ϕ by
where k0 is a reference permeability. μ is the viscosity of the magma, and ξ = ζ + 4/3η0, where ζ is the bulk viscosity and η0 is the reference shear viscosity. We define ζ = η0/ϕ, in agreement with Simpson et al. . Δρ is the difference in density between solid rock and magma, and g is gravity. is the pressure that the matrix exerts on the magma when rock compacts, and ∇U = vm describes the matrix velocity vm in terms of a compaction potential U. We solve equations (A1) and (A2) for and U respectively.
Equation (A3) relates temporal changes in enthalpy to advection of latent heat and sensible heat and diffusion of heat. In this equation, defines the volumetric bulk enthalpy , where ρ is the density, L is the latent heat of fusion, cP is the specific heat capacity, and T is temperature; T0 is the solidus temperature at zero pressure and Cm = C0. Additionally, t is time, is the potential temperature of the mantle, α is the coefficient of thermal expansion for the rock, and κ is the thermal diffusivity.
Equation (A4) conserves the bulk composition C = ϕCf + (1 − ϕ)Cm, where Cf is the composition of the magma, and Cm is the composition of the solid rock. is the chemical diffusivity, and vf is the velocity of the magma, given by
 The boundary conditions for material flowing into the domain (Figure 1, bottom boundary) are as follows:
They specify that solid rock flows into the domain at a rate W0 (equation (A9)). It has a bulk composition C0, and a total enthalpy given by equation (A7), where zb is the depth of the bottom boundary. Since the rock is solid the compaction pressure is zero (equation (A10)).
 The outflow boundary conditions (Figure 1, top boundary) are designed to allow material to flow unrestrictedly out of the domain. We impose ∂ϕ/∂z = 0 and , then specify
These conditions prescribe the vertical derivatives of C, U, and to be zero on the boundary. The enthalpy of the outflowing material is given by equation (A11), where zt is the depth of the boundary.
 The following characteristic scales are used to nondimensionalize the governing equations ((A1)–(A4))
where H is the height of the domain, and w0 = k0Δρg/μ is the reference magma velocity. To nondimensionalize temperature and composition, we use the following variables
where is the compaction length, the Stefan number , is the adiabatic constant, and PeT = Hw0/κ and are the thermal and chemical Peclet numbers respectively.
Appendix B:: Melting Rate
 The melting rate is derived by combining conservation statements for energy (equation (1)) and composition (equation (2)) with equations that conserve mass for the liquid and solid phases. The equations for conservation of mass are
For this derivation, we express the equation for conservation of energy in terms of the actual temperature, rather than the potential temperature. With this modification, the conservation of energy is
where k is the thermal conductivity.
 We combine equations (2) and (B3) with equations (B1) and (B2), and substitute in the definition to express the statements conserving energy and composition in terms of the melting rate Γ:
where we have simplified the adiabatic term by taking . Next, we use equations (3) and (4) to rewrite equation (B4) in terms of Cf, and to eliminate Cm from equation (B5) . By making these modifications and combining the results, the melting rate is given by
To arrive at the expression for the melting rate given by equation (5), we take R = 1, which renders , and .
 We would like to thank Ian Hewitt for useful discussions that improved this manuscript, Yan Liang and Dave Stevenson for their helpful and constructive reviews, and Joel Baker for handling the manuscript. We gratefully acknowledge Oxford Supercomputing Centre for providing compute time, HPC resources, and technical support.