High-porosity channels for melt migration in the mantle: Top is the
dunite and bottom is the harzburgite
Yan Liang1, Alan Schiemenz1,2, Marc A. Hesse1,§, E. Marc Parmentier1, and Jan S. Hesthaven2
1
Department of Geological Sciences
Brown University, Providence, Rhode Island 02912
2
Division of Applied Mathematics
Brown University, Providence, Rhode Island 02912
§
Present address: Department of Geological Sciences
University of Texas, Austin, Texas 78712
Orthopyroxene-free dunite dikes, veins, or irregular shaped bodies have been
frequently observed in mantle sections of ophiolites where mantle rocks are exposed on the
surface. These dunite bodies make up 5-15% of the mantle in Oman ophiolite and their
sizes range from tens of millimeters to ~200 meters in width and tens of meters to at least
10 km in length1. Several lines of evidence suggest that orthopyroxene-free dunites
represent remnants of high-porosity melt channels through which basaltic magmas
generated in the deep mantle were extracted to the surface1,2. The formation of dunite
channels involves dissolution of pyroxene and precipitation of olivine when olivinenormative basalts percolate through a partially molten mantle2-10. It has been suggested
that an interconnected coalescing network of high-porosity orthopyroxene-free dunite
channels allow basaltic melts to efficiently segregate from their source region while
preserving their geochemical signatures developed at depth1-3. However, the depth of dunite
channel initiation in an upwelling mantle remains controversial. It is not known if the
relatively shallow occurrence of dunite bodies in the mantle section of ophiolites is due to
limited field exposure or processes related to dunite channel formation in the mantle. It is
also not known if high-porosity harzburgite channels or orthopyroxene-bearing dunite
channels exist in the mantle, and how melt is distributed within a dunite or harzburgite
channel. Is dunite channel synonymous with high-porosity melt channel? Here we show,
through high-order accurate numerical simulations of reactive porous flow in an upwelling
deformable mantle column, that dunite channels are the shallow part of high-porosity melt
channels that extend significantly deeper into the upwelling mantle. Deep harzburgite
channels overlain by shallow dunite channels are the natural consequence of melt-rock
reaction during melt migration in the mantle.
Melting occurs in an upwelling mantle when a mantle parcel crosses its solidus. Depending
on potential temperature and amount of volatile elements in the mantle, the depth of initial
melting of lherzolite ranges from 65 km to 110 km in the mantle11,12. In general, melt generated
in the deeper part of the melting column is not in chemical equilibrium with the overlying mantle
and has a strong tendency to dissolve pyroxenes at shallow depth8,13-17. A positive feedback
between dissolution and melt flow may result in flow localization5-7,18-19. To better understand the
first order characteristics of the high-porosity melt channel and dunite channel in an upwelling
mantle we consider a highly simplified problem of reactive dissolution in a two-dimensional
upwelling column that consists of a soluble mineral, an insoluble mineral, and an interconnected
melt network in a vertical gravitational field. The soluble mineral may be identified as
orthopyroxene (opx, and possibly diopside) and the insoluble mineral may be taken as olivine.
For simplicity, we treat olivine + opx + melt as an effective binary system in which the melt and
minerals are in local chemical equilibrium. Further, we assume that the solubility of opx
increases upward in the vertical direction. To maintain local chemical equilibrium in the
upwelling column, a melt percolating upward, therefore, must dissolve opx. To map out the
2
spatial distribution of opx-free dunite in the upwelling column, we explicitly track the abundance
of opx in the simulation domain by solving a mass conservation equation for opx. A brief
summary of the model setup and numerical methods is given in the Methods.
Figures 1 and 2 show two examples of the calculated porosity and opx abundance in the
computational domain after each system has approached quasi steady state. Figure 1 is a case of
a short column fed by two sustained perturbations of different amplitudes. Figure 2 is a case of a
long column fed by one sustained perturbation. Four melt channels and two dunite channels are
observed in Fig. 1, whereas six melt channels and three dunite channels are observed in Fig. 2.
Key observations and conclusions from our numerical simulations are summarized below.
First of all, contrary to conventional wisdom, dunite bodies are not synonymous with highporosity melt channels, although there is a strong correlation between them (cf. Figs. 1a and 1d,
Figs. 2a and 2b, Figs. 3a-c and 3d-f). The melt channels extend all the way to the bottom of the
computational domain (Figs. 1a and 1b), whereas opx-free dunites (the dark blue region in Figs.
1d and 2b) emerge at the middle to upper part of the simulation domain. Depending on the
diopside abundance in the mantle, the lower part of the high-porosity melt channel is opxbearing dunite, harzburgite, and possibly lherzolite. Furthermore, porosities of the opx-bearing
dunite and harzburgite/lherzolite channels are higher than their surrounding matrices, whereas
the porosity of the opx-free dunite channel is highly variable. A melt channel in the lower part of
the domain bifurcates into two high-porosity branches within the wider dunite channel (Figs. 1
and 2). In the absence of opx dissolution (hence porosity generation), compaction in the central
part of the larger dunite forces melt to localize along the boundaries of the dunite channel.
Porosity of the upper central part of the wider dunite in Fig. 2 is actually lower than that of the
3
far-field harzburgite matrix. High-porosity opx-free dunite channels are therefore transient and
shallow parts of pathways for melt migration in the mantle.
Secondly, low-porosity compacting boundary layers form around high-porosity melt
channels (deep blue regions in Figs. 1a and 2a). The abundance of opx in the compacting
boundary layer is higher than that in the surrounding matrix (Figs. 1d and 2b), as a result of
reduction in the local melt flux. The presence of a low-porosity compacting boundary layer
hinders lateral melt transport between the channel and the far-field matrix, reducing melt flow
into the high-porosity channel in the upper part of the melting column. Melt in the dunite channel
is drawn mostly from a broader region below the dunite that has a width comparable to the
distance between the ends of the two compacting boundary layers on either side of the dunite
channel (Figs. 1a and 1d). Individual melt channels, therefore, may be fed by geochemically
distinct reservoirs, irrespective of their spatial separation.
The differences in the size and dimension of the two dunite channels in Fig. 1d are due to a
difference in the amplitude of the boundary perturbation in porosity. The boundary perturbation
is also responsible for the formation of the wider and deeper primary dunite channel in Fig. 2b.
Small perturbations in melt fraction may be realized by the presence of chemically and/or
lithologically heterogeneous materials that have lower solidi than the background mantle.
However, the two smaller secondary dunite channels in Fig. 2b emerge only after the formation
of the primary channel. Figures 3a-3f display the porosity and opx abundance in the highlighted
region in Fig. 2 at early times (t = 2, 3, 4) when the system has not reached a quasi-steady state
(Fig. 2 at t = 7). The two secondary channels were excited by melt reorganization in and around
the compacting boundary layer surrounding the primary channel. The melt has a significant
4
horizontal flow component in the lower part of the inter-channel region (Fig. 2b), permitting at
least partial communication between the primary and secondary channels.
Finally, the depth of dunite channel initiation depends strongly on the total melt flux entering
the channel from below. The total melt flux at the bottom of the dunite channel depends on the
total incoming melt flux across the perturbed region feeding into the overlying high-porosity
channels in the dunite, the solubility of opx, and to a lesser degree, the mass flux of opx entering
the domain. Numerical simulations using several choices of opx solubility and upwelling rate
show that the total melt flux coming out of the dunite channel at the top of the domain is
inversely proportional to the solubility of opx at the bottom of the dunite channel. Hence, given
the functional form of opx solubility, it may be possible to constrain the depth of dunite channel
initiation in the mantle in the future.
Although simplified in the model setup, the spatial relations among the high-porosity melt
channel, dunite channel, harzburgite and possibly lherzolite channels documented in our
numerical simulations may shed new light on a number of field, petrological, and geochemical
observations related to melt migration in the mantle. The close proximity among dunite bodies of
different width in the mantle sections of ophiolites and peridotite massifs20,21 may be in part due
to differences in their depth of initiation and the presence of secondary channels. Compaction in
and around a dunite channel and interaction between compaction and dissolution play a central
role in distributing melt in the dunite channel. A wide opx-free dunite channel may contain two
or more high-porosity melt channels. Flow rate in the narrower melt channel is higher than that
in a wider channel, expediting the delivery of short-lived isotopes to the surface. In situ
crystallization in and focusing of melt from the high-porosity region near a thermal boundary
layer may provide a possible explanation for the occurrence of clinopyroxene mega-crystal trails
5
and at least some clinopyroxenite, websterite, and orthopyroxenite veins or dikes within the opxfree dunite in the mantle sections of ophiolites2,22-25. These pyroxenite crystals or veins have often
been observed near the boundaries of dunite dikes or bodies. Localization of melt along the
boundaries of opx-free dunite channel weakens local rheology and may promote deformation and
melt relocalization in the shallow mantle26. Fossilized dunite channels have only been observed
in the top 10-20 km of the mantle section of ophiolites, yet the very depleted nature of the
incompatible trace element in diopside in abyssal peridotites and olivine hosted melt inclusions
implies near fractional melting27,28 and hence deeper melt channels. If high-porosity harzburgite
channels are present in the lower part of an upwelling mantle column, one must reassess the
geochemical consequences of channelized melt flow under such a setting29-32.
METHODS SUMMARY
We solve the mass and momentum equations for flow and dissolution in a deformable porous
medium in a 2-D domain using a high-order accurate numerical method that consists of the
tensor product of a finite difference method in the horizontal dimension and a discontinuous
Galerkin method in the vertical dimension. This method is especially suitable and
(computationally) effective for the advection dominated problem explored here. Time integration
is performed with an explicit, fourth-order Runge-Kutta method. We assume periodicity in the
horizontal dimension, free flow at the top, and prescribed solid and melt flow at the bottom of
the computational domain.
Full methods are available in the online version of the paper.
6
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8
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migration on Uranium series disequilibra. Geochim. Cosmochim. Acta 66, 4133-4148 (2002).
31. Spiegelman, M. & Kelemen, P. B. Extreme chemical variability as a consequence of
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Acknowledgments
The authors wish to thank Yinhua Xia and Greg Hirth for helpful
discussions. This work was supported by a CMG grant from the US National Science Foundation
(DMS-0530862).
Author Contributions Y.L., E.M.P, and J.S.H. secured funding for this project; Y.L. planned
and wrote the paper; M.A.H., Y.L., and E.M.P. contributed to the formulation and simplification
of the governing equations; M.A.H. conducted a linear stability analysis and provided 1-D
steady-state solutions. A.S. and J.S.H. designed the numerical scheme; A.S. conducted the
numerical simulations and made all the figures; M.A.H., Y.L., E.M.P. and A.S. contributed to the
interpretation of the results.
9
Author Information The authors declare no competing financial interests. Correspondence and
requests for materials should be addressed to Y. L. ([email protected]).
10
FIGURE CAPTIONS
Figure 1. Steady-state distributions of porosity (a) and the soluble mineral opx (d) in a short
upwelling column. b. Close-up view of the porosity field in the highlighted area near
the top of the wider channel. c. Close-up view of the opx field in the highlighted region
near the bottom of the wider dunite channel. Dark blue regions in d mark the opx-free
dunite channels. White curves in b and c are streamlines of the melt. Melt is focused
into the dunite channel from below. The streamlines in c are sub-parallel to the opx-free
dunite boundary, suggesting very small lateral melt transport across the dunite channel.
The two dunite channels were initiated by two small but sustained Gaussian
perturbations in porosity at the bottom of the upwelling column. Steady state is
established after one solid overturn (t ≥ 2). Porosity and opx modal abundance are
normalized to their respective background values (2% and 15%, respectively) at the
bottom of the upwelling column.
Figure 2. Distributions of porosity (a) and opx (b) in a long upwelling column at time t = 7. Dark
blue regions in b mark the opx-free dunite channels. Close-up views of the porosity and
opx fields in the highlighted region are also shown. The white lines in b are melt
streamlines. The deeper melt and dunite channels along the center plane were initiated
by one small but sustained Gaussian perturbation in porosity at the bottom and are
referred as primary channels in the text. The two smaller dunite channels are referred as
secondary channels. Note porosity of the central part of the primary dunite channel near
the top of the upwelling column is lower than the porosity of the far-field matrix.
Porosity and opx modal abundance are normalized to their respective background
values (2% and 40%, respectively) at the bottom of the upwelling column.
Figure 3. Distributions of porosity (a-c) and opx (d-f) in the highlighted region in the long
upwelling column shown in Figs. 2a-b at three selected times. Note the two secondary
melt channels near the top of the upwelling column are opx-bearing dunite in b, but
opx-free dunite in c. Also note the rotation of the secondary melt and dunite channels in
b, c, Fig. 2a and e, f, Fig. 2b. Although the porosity and melt flow fields are still timedependent (in the form of compaction-dissolution waves), the geometry of the melt
channels and dunite channels are practically independent of time after one solid
overturn (t ≥ 6.75).
11
Methods
We consider reactive dissolution in an upwelling and compacting porous column in a 2-D
domain. The governing mass and momentum equations are similar to those given in ref. 7. New
features included in our numerical studies are a mass conservation equation for the soluble
mineral opx, a porosity dependent bulk viscosity ( η0 φ f ), and a local equilibrium treatment of
opx dissolution. The new equations are
(
∂ ρsφopx
∂t
) + ∇ ⋅ (ρ φ
s opx
)
Vs = −Γ *opx ,
(1)
,
where
(2)
is the density of the solid; p is the pressure difference between the melt and the solid;
and
are volume fractions of the melt and soluble mineral opx, respectively;
is a
is the solid velocity; and Γ *opx is opx dissolution rate. Under the
reference bulk viscosity;
assumption of local chemical equilibrium, the interstitial melt composition is a function of the
vertical coordinate z. The opx dissolution rate is completely determined for the effective binary
system,
Γ
*
opx
where
=
(ρ φ V ⋅ n )
f
f
f
z
dC eq
f
Copx − β Col − (1 − β ) C eq
dz
f
( )
I φopx ,
(3)
is the melt flux along the vertical direction;
and
are compositions of
the independent component in the soluble mineral and insoluble mineral (ol), respectively; C eq
f , a
function of z, is the solubility of opx in the upwelling column;
( )
is the rate of olivine
precipitation relative to the rate of opx dissolution; and I φopx is the indicator function and it
12
equals 1 if
is positive and zero otherwise. Equation (1) applies to regions where
and is essential in identifying the opx-free dunite. Equation (2) relates the pressure difference
between the melt and the solid to the compaction rate and has been used in a number of melt
migration studies (refs. 33-35). Equation (3) was obtained by treating opx dissolution as a
gradient reaction in a vertical solubility field36. It can be shown that the kinetic formulation used
in previous studies of reactive dissolution along a vertical solubility gradient (refs. 6, 7, 31)
recovers our equilibrium case when local chemical equilibrium is established. For melt migration
in the mantle, it is reasonable to assume local chemical equilibrium between the crystals and
interstitial melt for the major elements.
For simplicity, we assume small porosity ( φ f  1), and ignore matrix shear and density
differences between the solid and the melt, except in the buoyancy term. These simplifications
are similar to those adopted in the numerical studies of reactive dissolution in refs. 7, 31. The
independent variables were scaled by the reaction time and compaction length so that the
dimensionless equations are given by37,38
∂φ f
∂t
+
∂φopx
∂t
(
∂φ f
+
∂z
= pφ f + δΓ opx ,
(4)
⎡ φ 0f δ
⎤
= −⎢
Γ ,
0 ⎥ opx
∂z
⎢⎣ (1 − β )φopx ⎥⎦
∂φopx
)
∇ ⋅ φ 3f ∇p − pφ f = R
∂φ 3f
∂z
(5)
,
(6)
n ⎞
⎛
⎛1
⎞
φ f ⎜ Vf − z ⎟ = −φ 3f ⎜ ∇p − n z ⎟ ,
⎝
⎝R
⎠
R⎠
Γ opx
(7)
( Rφ V ⋅ n ) dC I (φ ) ,
=
1 − δC
dz
f
f
z
eq
f
eq
f
(8)
opx
13
where
(= 0.01) is a dimensionless solubility across a compaction length; R (= 100) is the ratio
between buoyancy driven melt percolation rate and solid upwelling rate;
(0.02) and
(0.15
for Fig. 1 and 0.40 for Figs. 2 and 3) are volume fractions of the melt and soluble mineral opx
entering the simulation domain from below. Hence for a given compaction length scale, the more
the soluble mineral (larger the
) is available, the taller the simulation domain will be. For the
two numerical examples given in this paper, we used a linear solubility gradient (
set
) and
. The horizontal boundary conditions are periodic. The boundary conditions in the
vertical dimension are
(
)
2
φ f ( x, 0,t ) = 1 + ∑ A j exp ⎡ −b j x − x j ⎤ ,
⎣
⎦
j =1
N
, p ( x, 0,t ) = −δ (1 + R )
3R
,
1 + 3R
,
(9)
where A j and bj determine the amplitude and width of the jth perturbation located at (xj,0),
respectively;
is the height of the computational domain. For the example shown in Fig. 1,
we used A1 = 0.1 and A2 = 0.2 . For the example shown in Figs. 2 and 3 we used A1 = 0.15 . 1-D
steady state solutions were used as the initial conditions. Linear stability analysis of reactive
dissolution in an upwelling column has shown that the growth rate of high-porosity channel
decreases as the rate of upwelling increases (Hesse et al., Gradient reaction infiltration instability
in a compacting porous medium, to be submitted to Journal of Fluid Mechanics). To facilitate
melt channel formation we introduce small but sustained Gaussian perturbations in melt fraction
at the base of the simulation domain. Small perturbations in melt fraction may be realized by the
presence of chemically and/or lithologically heterogeneous materials that have lower solidi than
the background mantle. The sustained perturbation leads to a quasi steady-state solution that
14
allows us to examine the first order characteristics of the melt channel and the dunite channel in
an upwelling column.
The nondimenzionalized equations 4-8 were solved numerically in a 2-D rectangular domain
of either 2x1.5 or 6.75x2.85 (vertical x horizontal, all scaled by the compaction length) using a
high-order accurate method, consisting of the tensor product of a finite difference method in the
horizontal dimension and a nodal discontinuous Galerkin method in the vertical dimension. This
method is especially suitable and (computationally) effective for the advection dominated
problem explored here. Time integration is performed with an explicit, fourth-order Runge-Kutta
method. For the vertical dimension we used 20 discontinuous Galerkin elements for the shorter
domain and 24 elements for the longer domain, using a 6th order polynomial representation. For
the horizontal dimension we used 240 grid points for the shorter domain and 480 grid points for
the larger domain. Numerical experiments have shown the selection of sixth-order polynomials
to be robust under varying parameter selections and domain sizes, achieving high-order accuracy
while allowing for a reasonably large explicit time step to be taken. Therefore we need only
adjust the number of elements to resolve simulations of different column height. For a complete
description of the governing equations and numerical methods, a reader is referred to the refs. 37,
38.
33. Sleep, N. Tapping of melt by veins and dikes. J. Geophys. Res. 93, 10255-10272 (1988).
34. Bercovici, D., Richard, Y. & Schubert, G. A two-phase model for compaction and damage 1.
General theory. J. Geophys. Res. 106, 8887-8906 (2001).
35. Hewitt, I. J. & Fowler, A. C. Partial melting in an upwelling mantle column. Proc. R. Soc. A
464, 2467-2491 (2008).
15
36. Phillips, O. M. Flow and Reactions in Permeable Rocks. Cambridge University Press (1991).
37. Schiemenz, A. R. Advances in the discontinuous Galerkin method: Hybrid schemes and
applications to the reactive infiltration instability in an upwelling compacting mantle. Ph. D.
Thesis, Brown University (2009).
38. Schiemenz, A. R., Hesse, M. & Hesthaven, J. Modeling magma dynamics with a mixed
Fourier collocation – discontinuous Galerkin method. Submitted to Communications in
Computational Physics (2010).
16