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Geochemistry, Geophysics, Geosystems
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Supporting Information for
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A high-order numerical study of reactive dissolution in an upwelling heterogeneous
mantle: II. Effect of shear deformation
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Conroy Baltzell1, E.M. Parmentier1, Yan Liang1, Seshu Tirupathi2, 3
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1. Department of Earth, Environmental and Planetary Sciences, Brown University,
Providence, RI 02912, USA
2. Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
3. Now at IBM Research, Dublin, Damastown Industrial Park, Mulhuddart, Dublin
15, Ireland
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Contents of this file
Supplementary Text 1-9
Figures S1 to S9
Table S1
Captions S1 to S16
Additional Supporting Information (Files uploaded separately)
Movies S1-S6
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Introduction
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The following text and supporting figures give variations in orthopyroxene (opx) and
melt fraction for varying parameters and specific cases that support the main text as well
as more detail than strictly necessary on some of the points in the text for readers who are
interested. In addition, there is a table that gives the raw data from the numerical
experiments used to determine Eqs. 14 and 15 in the text. Lastly, there are movies
depicting the full time evolution for some of the figures used in the text.
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Supplementary Material
1. Steady State vs. Initial Conditions
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For simulations conducted in the stable regime, the steady-state solutions reported in
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this study are independent of the initial condition of the channel and instead depend only
39
on the bottom boundary condition of the channel. Figure S1 compares opx and melt
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fraction at 1 transit or overturn time, i.e., the time it takes for matrix at the bottom of the
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domain to move out the top (which take 2 dimensionless time units here) for two choices
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of shear rates. The first case (Figs. S1a-S1d) shows an initial condition corresponding to
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steady state for γ = 0.1 (Figs. S1a and S1b), which is then sheared for 1 overturn time
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with a shear rate γ = 0.5 (Figs. S1c and S1d). The second case (Figs. S1e-S1h) shows an
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initial condition corresponding to steady state for γ = 0.5 (Figs. S1e and S1f), which is
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then sheared for 1 overturn time with shear rate γ = 0.1 (Figs. S1g and S1h). Figs. S1a
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and S1b are equivalent to Figs. S1g and S1h and Figs. S1c and S1d to Figs. S1e and S1f.
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Since the bottom material moves all the way through the domain in 1 overturn time, the
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steady state is dependent on the bottom boundary condition, but not the initially state of
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the channel. Thus, if a high porosity region forms a bifurcated channel and then is
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subsequently sheared, the behavior will not be different from an initially high porosity
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region as used in this numerical experiment as long at the high porosity is maintained on
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the bottom boundary.
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Figure S1. Distribution of soluble mineral opx (left column) and porosity (right column)
at initial and steady-state showing the independence on initial conditions. (a,b) The
steady state evolution for 𝜸 = 0.1 and is used as the initial condition for (c,d), which has
𝜸 = 0.5. (e,f) Takes the steady state of the same shear rate as (c,d), 𝜸 = 𝟎.5 and is used as
the initial condition for (g,h), which has 𝜸 = 0.1. There is no qualitative difference
between (a,b) and (g,h), and between (c,d) and (e,f).
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2. V-Shaped Channels
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As the seed channel at the inflow increases its width, the shape of the dunite channel
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changes from parabolic to more linear (Figs. S2a), as described in Schiemenz et al.
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[2011b]. For linear or V-shaped channels the behavior changes and bands do not form as
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readily. Fig. S2 compares opx fraction and porosity as a function of shear rate and inflow
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geometry. The wider region of opx dissolution inhibits the formation of dunite (due to
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less melt focusing), further reducing the formation of a low porosity region (Fig. S2c) and
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the subsequent bifurcation. The asymmetry in the opx is maintained disrupting the dunite
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formation (Figs S2a, S2c, and Fig. S3).
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Similarly, the lack of low porosity region formation prevents decompaction on the
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downwind side. (Fig. S2d). For sufficiently permeable channels with a large enough
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shear rate (A = 0.28, γ = 1.0), porosity bands can form for a wide channel. However,
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these bands have a weak amplitude and have little effect on the melt velocity.
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Figure S2. Steady-state distributions of soluble mineral opx (left column) and porosity
(right column) for V-shaped channels with R = 100,  = 0.01, and a channel width of 1
compaction length (w=5 in Eq. 13). (a,b) opx fraction and porosity for 𝜸 = 𝟎.1. (c,d) opx
fraction and porosity for 𝜸 =1. The contour lines correspond to an opx fraction of 0.5.
The nearly vertical white lines in (b,d) are melt streamlines and the thicker white lines in
(a,c) are solid streamlines. The Gaussian perturbations in porosity at the inflow are shown
as black curves at the bottom of the porosity field (20% and 28% above background in ab and c-d, respectively).
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Figure S3. Variations in opx fraction across the top of the domain for different shear rates
(𝜸) and channel geometries (A and w in Eq. 13) for V-shaped channels.
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3. Steady State for Different Shear Rates and Channel Geometry
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Figure S4 shows the opx and fluid fraction for a shear rate of 0.5 and 1 with different
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channel geometries. A shear rate of 0.5 results in fewer porosity bands (cf. Fig. S4b and
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Fig. 3h) and a more concentrated region of opx depletion (cf. Fig S4a and Fig. 3g)
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Varying the channel width and amplitude while maintaining the same shear rate
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results in no qualitative difference, though a quantitative difference in amplitude (cf. Fig.
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S4d and Fig. 3d). The region of opx dissolution for the wider channel is spread over a
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larger region. Thus the minimum opx value is greater for the wider channel and the
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asymmetry in opx gradient around the dunite is reduced.
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Figure S4. Steady-state distributions of soluble mineral opx (left column) and porosity
(right column) for varying shear rates and channel geometries. (a,b) opx fraction and
porosity for 𝜸 = 𝟎.5, a channel width of 0.6 compaction lengths. (c,d) opx fraction and
porosity for 𝜸 =1, a channel width of 0.8 compaction lengths. The contour lines in (a,c)
correspond to a normalized opx fraction of 0.5. The nearly vertical white lines in (b,d) are
fluid streamlines and the thicker white lines in (a,c) are solid streamlines. The timeindependent Gaussian perturbations in porosity at the inflow are shown as magenta
curves at the bottom of the porosity field (20% and 28% above background in a-b and cd, respectively). This compares to Figure 3.
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4. Opx Depletion as a Function of Shear Rate, Channel Geometry, and Time
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Figure S5 shows minimum opx fraction with a melt to matrix velocity ratio, R = 100
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for different shear rates and channel geometries. The minimum opx fraction in the
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domain increases with increased shear rate and channel width but decreases with
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decreased channel porosity, consistent with the argument that a higher shear rate and
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wider/less porous channel inhibit the formation of dunite (that is complete Opx
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dissolution). Following the solid lines in Fig. S5a, it is apparent that increasing the shear
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rate inhibits the formation of dunite. For no shear and γ = 0.5 (red and blue respectively),
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the variation is insignificant to the point that the red line is completely covered by the
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blue. As the shear rate increases dunite forms later, and for γ = 0.25 (magenta), dunite
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only forms at 1 over turn time. As γ increases further, dunite ceases to form.
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Figure S5.Variations in minimum and average opx fraction for different shear rates (𝛾)
and channel geometries (A and w in Eq. 13). (a,b) Minimum opx value in the domain at
steady state. Variations in color, point style and line style represent variations in 𝛾, A, w,
respectively. Simulations in (a) vary shear rates, 𝛾, w and A in such a way as to keep the
melt flux at the bottom boundary constant and equivalent to w = 20 and A = 0.2.
Simulations in (b) vary 𝛾, w, and A without normalizing the channel geometry such that
the melt flux at the bottom boundary is constant. (c) The difference between the average
opx fraction at varying times and the initial average opx fraction as a function of height.
Initial and final differences are shown as solid lines for varying shear rates 𝛾 = {0.0, 0.05,
0.25, 0.5, 1.0}. The dashed red lines show the average opx fraction under no shear
conditions in increments of 0.2 time units. The maximum difference in opx fraction at a
given time between different shear rates is 0.05.
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does not keep opx dissolution constant. Note the pairs of blue lines and black lines, each
A comparison of different line styles in Fig. S5 reveals maintaining a constant flux
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has the same shear rate and the same melt flux at the base of the channel and yet has
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different minimum opx values with time.
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Fig. S5b shows that widening the channel inhibits the formation of dunite. For γ = 0.5
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and a more permeable channel, opx-free dunite fails to form (black solid line); whereas a
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narrower channel reaches opx-free dunite (black dash-dotted). Similarly, for γ = 0.05 a
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wide channel fails to form opx-free dunite (dashed blue), whereas a narrow channel
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reaches opx-free dunite (solid blue). This also reveals the strong dependence of inflow
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channel parameters: as can be seen in the solid blue and dash-dotted black line, doubling
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the channel porosity amplitude and reducing the channel width by a factor of
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comparable effect (achieves opx-free dunite at the same time) to increasing γ by an order
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of magnitude.
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has a
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Fig. S5c shows the difference between the average opx fraction at varying times and
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the initial average opx fraction as a function of height. There is little difference (~0.05) in
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opx fraction for different shear rates, supporting the observation that the total opx
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dissolution is independent of shear rate. With time, the average opx fraction decreases
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due to dissolution. Since the dissolution rate increases with height, the opx fraction
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decreases with height.
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5. Fourier Transform
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Figure S6 shows the spatial frequency at steady state corresponding to the greatest
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power following a Fourier Transform as a function of shear rate. The porosity for
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different channel geometries as well the pressure for a channel of 0.6 compaction lengths
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and porosity of 20% above background are plotted. As shear rate increases, the frequency
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tends to decrease, though not linearly. The pressure (blue line), matches the
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corresponding porosity (back line) well.
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Figure S6. Frequency vs. shear rate for different channel geometries. Porosity is shown in
solid lines. Pressure corresponding to a channel of 0.6 compaction lengths and porosity of
20% is shown with the blue dashed lines.
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6. Melt Travel Time
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The effects of the alternating porosity pattern on the melt velocities are illustrated in
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Figure S7 which shows the time it takes for melt to move from the bottom to the top of
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the domain following the fluid streamline of minimum time as a function of non-
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dimensionalized time (run to steady state) for a melt to solid upwelling velocity ratio R =
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100. A dashed black line at the top of the plot shows the travel time far away from the
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channel where the porosity is the background porosity. Since shearing creates low
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porosity regions, it increases travel time through the domain. While the fluid streamlines
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are deflected towards the high porosity regions and away from the low porosity regions,
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they are not sufficiently deflected to only stay within a high porosity region (as in the
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case of no shear). The greater the shear rate, the more compacted region the fluid has to
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pass through, reducing fluid speeds.
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Figure S7. Time for melt to travel through the domain following the fastest streamline for
varying conditions of shear rates, 𝜸, and channel geometries, A and w, as a function of
non-dimensionalized time. The dashed black line is the time for a streamline in the
background, without the influence of the high porosity channel. The channel permeability
was normalized by the width to keep the flux in all cases a constant corresponding to a
porosity of 20% above background (A = 0.2 in Eq. 13) and a channel width of 0.6
compaction lengths (w = 20).
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In general, however, while shearing disrupts the porous pathways, compaction-
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dissolution still increases the porosity such that the travel time decreases with time.
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Furthermore, the influence of shearing is dependent on the channel width wherein for a
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narrower channel the disruption of the porous path is more significant. For sufficiently
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narrow channels (0.4 compaction lengths wide and a porosity of 28% above background),
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even though compaction-dissolution begins strongly decreasing the travel time with
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channel evolution time, the disruption from shearing becomes significant enough to
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increase the travel time from the initial, un-sheared, conditions.
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The melt travel time goes as 1/R. For a constant R and physically reasonable
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variations in shear rate and channel width/porosity, melt travel time can vary by as much
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as a factor of 2. For R = 100, the non-dimensionalized times are around 0.01, which
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gives dimensionalized times of 0.02L/Vz to pass through the domain. For matrix
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upwelling rates of 10 mm/yr melt velocities are approximately 1-2 m/yr, giving a time to
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travel 50 km of 25-50 kyr. Since the half life of 230Th about 30 kyr, a travel time change
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of this magnitude may be detectable in MORBS [Kelemen et al. 1997].
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7. Non-zero Shearing at the Base of the Domain
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To explore the effects of varying shearing at the bottom of the domain a modified
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matrix velocity, 𝑉𝑥 = 𝛾(𝑧 + ℎ0 ), was used for h0 ≠ 0. Figure S8 displays opx and melt
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fractions as a function of different conditions of shear, domain heights H, and ℎ0 for the
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same inflow boundary conditions as Figure 1. For a constant height, the behavior of the
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melt fraction is dependent on the integral over the domain of 𝑉𝑥 :
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𝐻
𝐼𝑧 = ∫0 𝛾0 (𝑧 + ℎ0 )𝑑𝑧 .
(S1)
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Figure S8: Steady-state distributions of soluble mineral opx and porosity under three
different conditions of shear rate (𝜸), domain height (H), and horizontal matrix velocity at
the bottom of the domain (𝒉𝟎 ) with a channel porosity of 20% above background and a
width of 0.6 compaction lengths. The black and white contours correspond to pressure.
The black horizontal line in the opx field is a contour for a normalized opx fraction of
0.25. The white line in the opx field is the solid streamline. Nearly vertical white lines in
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the porosity field correspond to melt streamlines. Parameters used in the simulations are
given in each panel. Iz is 1 for (a) and (b) and 2.25 for (c)-(f).
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however, the fluid fraction changes even for constant 𝐼𝑧 . For 𝐼𝑧 = 2.25, when the domain
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height is constant, H = 2.5, the behavior is the same even for different 𝛾 and ℎ0 (Figs.
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S8e and S8f). However as the domain height changes, so does the behavior (Fig. S8d).
Fig. S8b is equivalent to Fig. S4a (𝐼𝑧 = 1 for both). As the domain height varies,
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As γ increases the maximum opx dissolution still follows the solid streamline, but
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another peak of high opx dissolution emerges (as seen in the opx contours in Fig. S8c).
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With increased γ and ℎ0 the second peak becomes stronger reaching the point of two
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regions of maximum opx dissolution. This results in a new pattern of asymmetric opx
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distribution, as illustrated in Figure S9 below.
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Figure S9. Variations in opx fraction across the top of the domain for different shear rates
(𝜸) and shearing at the bottom of the domain (𝒉𝟎 ).
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8. Time Evolution
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Movies S1-S3 show the temporal evolution of opx fraction, porosity, and pressure,
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respectively, associated with Figs. 1a-1c. The scale bars, streamlines, and contour plots
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are the same as in Fig. 1 and each cycle corresponds to 0.02 nondimensional time units.
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Movies M4-M6 show the temporal evolution of opx fraction, porosity, and pressure
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respectively associated with Fig. 3. The scale bars, streamlines, and contour plots are the
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same as in Fig. 3 and each cycle corresponds to 0.02 nondimensional time units.
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Movie S1. Temporal evolution of opx fraction. Streamlines and contours are the same as
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in Figure 1d. Each cycle corresponds to 0.02 nondimensional time units.
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Movie S2. Temporal evolution of porosity. Streamlines and contours are the same as in
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Figure 1e. Each cycle corresponds to 0.02 nondimensional time units.
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Movie S3. Temporal evolution of pressure. Streamlines and contours are the same as in
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Figure 1f. Each cycle corresponds to 0.02 nondimensional time units.
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Movie S4. Temporal evolution of opx fraction. Streamlines and contours are the same as
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in Figures 3a, 3d, and 3g. Each cycle corresponds to 0.02 nondimensional time units.
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Movie S5. Temporal evolution of porosity. Streamlines and contours are the same as in
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Figures 4b, 4e, and 4h. Each cycle corresponds to 0.02 nondimensional time units.
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Movie S6. Temporal evolution of pressure. Streamlines and contours are the same as in
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Figures 3c, 3f, and 3i. Each cycle corresponds to 0.02 nondimensional time units.
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Table S1. Data used to construct Eqs. 14 and 15.
Trial
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38*
39*
γ
δ
R
0.75
0.75
0.75
0.75
0.75
0.5
0.5
0.5
0.5
0.75
0.75
0.75
0.75
0.5
0.5
0.25
0.25
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.5
0.5
0.5
0.5
0.25
0.3
0.4
0.5
0.6
0.75
0.9
1
1.25
0.3
0.4
10
25
50
75
125
25
50
75
125
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
H
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.003
0.005
0.008
0.013
0.015
0.018
0.005
0.008
0.013
0.015
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
2
2
2
2
2
2
2
2
1.5
2.5
3
3.5
2.5
3
3.5
2.5
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
Angle(α) Spacing(δsp)
49.2
1.11
39.3
0.702
34.2
0.552
32.0
0.481
28.4
0.411
35.8
0.801
31.0
0.633
27.5
0.556
30.1
0.430
29.2
0.439
30.1
0.439
30.1
0.439
30.0
0.439
24.7
0.480
24.7
0.480
24.7
0.480
22.7
0.595
23.7
0.595
32.6
0.616
30.9
0.517
31.0
0.474
30.1
0.411
28.4
0.383
30.1
0.386
28.4
0.586
26.6
0.528
27.4
0.455
27.4
0.419
19.8
0.595
19.8
0.550
24.6
0.516
25.6
0.488
26.5
0.464
29.2
0.439
31.8
0.424
33.4
0.417
35.7
0.378
-0.544
-0.492
15
261
262
263
264
265
266
267
268
269
270
40*
41*
42*
43*
44*
45*
46*
0.5
0.6
0.75
0.9
1
1.25
1.5
100
100
100
100
100
100
100
0.01
0.01
0.01
0.01
0.01
0.01
0.01
2
2
2
2
2
2
2
------38.6
0.481
0.460
0.426
0.419
0.413
0.407
0.377
List of the parameters, and resulting band spacing and band angle used for the numerical
experiments as plotted in Fig. 6. The shear rate (𝜸) ranges from 0.25-1.5, the ratio of fluid
to solid upwelling rate (R) from 10-125; the dimensionless solubility gradient (𝜹) from
0.0025-0.0175; and the domain height (H) from 1.5-3.5. The depth of the low porosity
band was taken to be the bottom of the contour corresponding to 95% of the background
for most of the measurements (and high porosity 105%), though the measurements with
asterisks had depths at 85% and 115%. There is no significant difference between the two
measurement methods. To avoid double counting, the band angles were not repeated for
the second measurement method, and hence those are left dashed.
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