The style of melt migration in the mantle is important to the interpretation of basalts erupted on the surface. Both grain-scale diffuse porous flow and channelized melt migration have been proposed. Through high-order accurate numerical simulations, we show that strong nonlinear interactions between compaction and dissolution in an upwelling mantle give rise to porosity waves and high-porosity melt channels that have well organized but time-dependent structures. Only the upper part of the channel is pyroxene-free dunite. The lower part is harzburgite. Transient melt flow in the wave regime results in significant lateral mixing and chromatographic fractionation even when mantle source compositions are independent of time. Caution must be exercised when inferring the geometry and spatial distribution of mantle heterogeneity based on spatial and temporal variations in isotopic ratios recorded in basalts.
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 High-porosity channels can be formed by reactive dissolution when olivine-normative basalts percolate through a partially molten harzburgite mantle [e.g., Kelemen et al., 1997]. Numerical simulations of reactive dissolution showed that only the upper part of the high-porosity channel is orthopyroxene-free dunite [Liang et al., 2010; Schiemenz et al., 2011]. The reaction feedback proposed for channelization can also lead to the spontaneous formation of compaction-dissolution waves [Aharonov et al., 1995; Hesse et al., 2011]. Figure 1 shows the conditions under which the two different instabilities are expected in an upwelling and viscously deforming column. Compaction-dissolution waves may be favored in the parameter regime expected beneath mid-ocean ridges [Hesse et al., 2011]. Given the strong evidence for localized melt migration, it is important to understand if high-porosity channels can also be formed in the wave regime. If the lower part of the high-porosity channel is orthopyroxene-bearing dunite and harzburgite, and/or if melt migration in at least part of an upwelling mantle is by means of compaction-dissolution waves, the geochemical consequences for transport of compositionally heterogeneous melts in the mantle must be reassessed. In this report, we present new results from high-resolution numerical simulations that capture the strong nonlinear interactions among compaction, decompaction, dissolution, and upwelling in a parameter regime appropriate for melt migration beneath mid-ocean ridges. We show that transient melt flow in the wave regime results in significant lateral mixing and chromatographic fractionation of elements of different incompatibility during melt migration in the mantle, which further complicates the interpretation of geochemical signatures recorded in basalts.
2. Model Setup
 We consider reactive dissolution in a 2-D rectangular domain that consists of a soluble mineral, an insoluble mineral, and an interconnected melt network in a gravitational field in which solubility of the soluble mineral increases along the vertical direction. In the context of melt migration beneath mid-ocean ridge, the soluble mineral may be identified as orthopyroxene (opx) and the insoluble mineral as olivine. The 2-D domain may be taken as an approximation for the upper most part of the upwelling mantle beneath the ridge axis where the solid flow field is nearly vertical, the rate of melting is small, and opx is not in chemical equilibrium with basalts erupted on the surface. We assume that the interstitial melt and minerals are in local chemical equilibrium. To maintain local equilibrium in the upwelling column, a melt percolating upward must dissolve opx. This setup is very similar to those used by Liang et al.  and Schiemenz et al. , except here we consider a larger domain of horizontal and vertical dimensions of 5.71 by 6.75 compaction lengths. The compaction length is similar as that defined by McKenzie  and is on the order of 102 to 104 m for melt migration in the mantle [Spiegelman et al., 2001]. For a detailed description of the governing equations, initial and boundary conditions, and numerical methods, a reader is referred to the auxiliary material.
Figures 2a–2d and 2e–2h show the evolution of the porosity and opx abundance in the simulation domain. Compaction-dissolution waves arise from infinitesimal perturbations of the steady 1-D solution [Hesse et al., 2011]. Well-organized porosity waves emerge after the solid has transited through the domain once (referred to as one solid overturn time; one solid overturn takes 6.75 time units in this example). The size of a wave remains approximately constant, whereas the amplitude grows with time (compare Figures 2a and 2b). At the end of the second solid overturn, compaction-dissolution waves are fully developed in the lower part of the domain, while high-porosity channels are initiated at the top and grow downwards along the nodal lines of the waves where the opx abundance has been reduced (Figure 2b). The localization of the melt into these channels dissolves the remaining opx, and opx-free dunite channels begin to emerge within the high-porosity channels (Figure 2f). At the end of the third overturn, six well-developed opx-free dunite channels are present along the nodal lines of the waves in the upper third of the domain (Figures 2c and 2g). These primary channels are accompanied by six pairs of opx-bearing secondary melt channels near the top. Within an opx-free dunite channel, the high-porosity melt channel bifurcates into two branches along channel boundaries. These are similar to those observed in the channel-only simulations reported earlier [Liang et al., 2010; Schiemenz et al., 2011], except melts are now transported through dunite channels as trains of wavelets.
 Downward growth of high-porosity channels strongly perturbs local melt flow field, giving rise to significant lateral flow in the mid to upper part of the domain (compare melt streamlines in Figures 2f and 2g). Continuous nonlinear interactions between compaction-dissolution waves in the lower part and the channels in the upper part of the domain lead to a 50% reduction of the horizontal wavelength of both (compare Figures 2b and 2d). The transition in wavelength is complete after the fourth solid overturn and remains unchanged until the simulation is terminated after seven overturns. However the melt flow field remains time-dependent throughout the whole simulation. This cascade of different melt migration styles occurs during a period of 5 ∼ 50 Ma, depending on compaction length, and highlights the close relationship between compaction-dissolution waves and high-porosity channels. Additional simulations in the wave regime using several choices of model parameters (porosity/opx volume fraction at the inflow, nonlinear solubility curve, random vs. sustained perturbation in porosity at the inflow, and column height) have confirmed the four-stage evolution of the wave-channel system described here. Details of this expanded study will be presented elsewhere.
4. Geochemical Implications
 The strong time-dependent melt streamlines, their complex relationships to lithological boundaries, and the superposition of waves and channels have important geochemical consequences. If a chemically heterogeneous melt were fed into the upwelling column from below, one would expect chromatographic fractionation among elements of different incompatibility in the pyroxene-bearing region of the mantle in a manner similar to those described previously [e.g., McKenzie, 1984; Navon and Stolper, 1987; Richter and Daly, 1989; Bodinier et al., 1990; Watson and Spiegelman, 1994; DePaolo, 1996; Liang, 2008]. However, such fractionation would not be expected if a spatially heterogeneous but time-independent mantle source were fed into the column and the melt and solid flow fields are independent of time. The geometry of the mantle sources, which may be described as a bundle of long vertical strings (each with a unique composition), can then be inferred from composition and spatial distribution of basalts erupted on the surface.
 However, such a simple relationship breaks down for melt migration in the wave regime. As an illustrative example, we consider 87Sr/86Sr and 143Nd/144Nd isotope variations in the melt produced by melting and melt migration in the upwelling mantle that initially consists of two isotopically enriched sources imbedded in a depleted background mantle: one narrow heterogeneity feeding into the column along a nodal line of the wave (at x = 1) and the other wider heterogeneity entering between two nodal lines (centered at x = 3.5). To highlight the effect of waves on tracer transport, we assume that composition and width of the two heterogeneities entering the simulation domain from below are independent of time and in the form of Gaussian functions in x. Geometrically, this is a special case of the bundle of long vertical strings model described above. In the absence of the wave field, distributions of the enriched melts follow the vertical melt streamlines in the simulation domain and 87Sr/86Sr and 143Nd/144Nd ratios of the sources are perfectly correlated in melts sampled along the top of the domain (see Figure S4 in Text S1 of the auxiliary material).
 In the presence of compaction-dissolution waves, fields of the enriched melts are quickly distorted by the differential flow upon entering the column from below, resulting in expansion, contraction, stretching, folding, and mixing in the mid to upper part of the column (Figures 3a–3f). The spatial distribution of enriched melts is highly time-dependent in the upper third of the domain where presence of dunite channels further complicates the melt flow field. Only a part of the enriched melts originating from the two sources was transported through the channels. Figures 3a–3f show that isotope compositions of the melt sampled at top of the upwelling column are highly variable in time, in contrast to the time-independent source compositions used in our simulation. These time-dependent variations are broadly similar to the 1-D results of DePaolo  and Liang  for trace element and isotope fractionation during melting and melt migration in a heterogeneous mantle column in which melt and solid flow fields are constant and uniform, but composition and size of mantle sources are time-dependent. Caution therefore must be exercised when inferring the geometry and spatial distribution of mantle heterogeneity based on spatial and temporal variations in isotope ratios recorded in basalts.
 When viewed in the 143Nd/144Nd vs. 87Sr/86Sr isotope diagram, the original perfect linear correlation between the depleted and enriched sources appears significantly scattered (Figure 4a). The lack of correlation between 143Nd/144Nd and 87Sr/86Sr ratios results from chromatographic fractionation, as Sr is more incompatible than Nd during mantle melting. In general, chromatographic fractionation between Sr and Nd is absent in a steady-state column fed by time-independent mantle sources. However, during their transport in the wave regime, fields of the enriched melts are non-uniform in the upwelling column (Figures 3a–3f), which then serve as time-dependent sources for the overlying mantle. As demonstrated previously [Richter and Daly, 1989; DePaolo, 1996; Liang, 2008], chromatographic fractionation of trace elements of different incompatibility is inevitable during melting and melt migration in a mantle column where crystal and melt are in local chemical equilibrium and when composition of the mantle source is time-dependent. The preceding example demonstrates that diffuse porous flow followed by channelized melt migration in the wave regime is not an effective mechanism in preserving spatial variations in incompatible trace element and isotope signatures of the mantle source region.
 One way to eliminate or reduce the chromatographic effect is through chemical disequilibrium. This may be achieved by transport through melt-filled hydro-fractures or grain-scale porous flow with limited extent of crystal-melt re-equilibration [e.g., Spiegelman and Kenyon, 1992; Kelemen et al., 1997]. It conceivable that hydro-fractures may develop locally in the asthenospheric mantle where pore pressure in the melt exceeds the strength of the solid matrix [e.g., Nicolas, 1986; Sleep, 1988; Rubin, 1998; Phipps Morgan and Holtzman, 2005]. Another way is to allow opx-free dunite channels to form in the deeper part of the upwelling mantle [Lundstrom, 2000; Jull et al., 2002; Liang, 2008; Liang and Parmentier, 2010]. Since the depth of dunite channel initiation depends primarily on vertical melt flux and horizontal melt suction rate [Asimow and Stolper, 1999; Schiemenz et al., 2011], it is difficult to form pyroxene-free dunite channels near mantle solidus where the melt flux is small and melt suction effect is weak. Although diffusion and dispersion in the melt help to reduce concentration gradients, their effects are relatively small for melt migration in the mantle because the rate of melt flow is significantly larger than the rates of diffusion and dispersion in the melt [Liang, 2008]. Yet another way to reduce the chromatographic effect is through mixing in a magma chamber or conduit [DePaolo, 1996]. Figures 4a and 4b show the importance of spatial and temporal averaging, via magma mixing, in reducing the scatterness and anti-correlations between 143Nd/144Nd and 87Sr/86Sr in melt collected at top of the domain. Time averaging (Figure 4b) is especially effective, as the vertical melt flux is significantly larger than the horizontal melt flux. Scattered-correlations, such as the one shown in Figure 4b, have been widely observed in MORB and OIB [e.g., Hofmann, 1997, 2003; White, 2010] and may result, in part, from processes related to melt extraction in the crust and upper mantle.
 Results presented in this report demonstrate the importance of melt migration processes in affecting the distribution of geochemical variations in basalts erupted on the surface. Interpretations of the spatial and temporal distributions of mantle heterogeneities based on basalt compositions therefore must proceed with caution. As a significant fraction of the variations may be of dynamic origin, systematical studies of high-resolution spatially correlated mantle samples, such as those from mantle sections of ophiolite, may help to constrain melt flow patterns and melt migration mechanisms.
 We wish to thank Richard Katz and an anonymous reviewer for their comments and suggestions. This work was supported in part by NSF grants EAR-0911501 and DMS-0530862, and the Seventh Framework Program by the European Commission (QUEST project).
 The Editor wishes to thank Richard Katz and an anonymous reviewer for their assistance evaluating this paper.