Spatial Variations in the Degree of Upper‐Mantle Depletion in a Mid‐Ocean Ridge–Transform Fault System

Partial melting beneath a mid‐ocean ridge creates a chemically depleted layer in the uppermost mantle. This chemical depletion lowers the density of the lithosphere compared with the unmelted mantle. Furthermore, dehydration associated with depletion leads to an increase in mantle viscosity that may affect the structure and dynamics of the oceanic plate. Previous studies have mainly considered the formation of this depleted upper‐mantle layer in a two‐dimensional mid‐ocean ridge setting, leaving the dynamics of mid‐ocean ridge–transform fault systems largely unexplored. In this study, spatial variations in the degree of depletion in the uppermost mantle are predicted for a mid‐ocean ridge–transform fault system using a three‐dimensional thermomechanical model. The degree of depletion generally increases with increasing half‐spreading rate. Less depletion is predicted beneath the transform fault and fracture zone compared with the surrounding mantle. Lateral differences in the degree of depletion in the ridge‐parallel direction are reduced when plastic yielding is considered. The degree of variation in the predicted depletion is related to the transform fault length especially at a low spreading rate, thereby suggesting that the large scatter in observed abyssal peridotite compositions with slow spreading rates could be partly attributed to the length of the fault.


Introduction
Partial melting occurs beneath mid-ocean ridges, where oceanic plates are formed.This upward-migrating melt is cooled near the Earth's surface to form new oceanic crust, whereas the residual mantle forms a chemically depleted layer just beneath the oceanic crust.This depleted layer has a lower density than the unmelted mantle due to the exhaustion of dense minerals and the extraction of iron into the melt (e.g., Lee, 2003).It may also be a dehydration layer because most of the water is partitioned into the melt.Therefore, this depleted layer is expected to have a higher viscosity than the unmelted mantle because the viscosity is reduced in the presence of water (e.g., Braun et al., 2000;Hirth & Kohlstedt, 1996).This depleted upper-mantle layer has been considered to play a critical role in the temporal evolution of the oceanic plates, including the initiation of subduction and onset of small-scale convection (e.g., Afonso et al., 2007;Lee et al., 2005;Ma & Dalton, 2019), due to its low density and high viscosity.
Previous studies have mainly considered the formation of the depleted layer in a two-dimensional (2-D) midocean ridge system (e.g., Braun et al., 2000;Plank & Langmuir, 1992;Pusok et al., 2022).However, the formation of this layer in the uppermost mantle is likely to be more complicated due to the presence of transform faults, which connect neighboring mid-ocean ridges.The thermal and flow structures of a mid-ocean ridgetransform fault system have been extensively investigated (e.g., Behn et al., 2007;Shen & Forsyth, 1992).Earlier studies have predicted a cold mantle beneath transform faults (e.g., Shen & Forsyth, 1992), although a later numerical study by Behn et al. (2007), which incorporated the effects of plastic yielding (PY), demonstrated that the temperature beneath the transform fault is higher than that of adjacent regions.The velocity component parallel to the ridge axis can be large (>20% of the half-spreading rate), reaching a maximum near the junction of the mid-ocean ridge and transform fault (e.g., Kuo & Forsyth, 1988;Shen & Forsyth, 1992).These previous studies have shown complex, three-dimensional (3-D) material trajectories in this tectonic setting.It is therefore expected that the degree of depletion can change significantly beneath the transform fault (and also the fracture zone) compared with the surrounding mantle.
Only a few studies have focused on the degree of depletion in a mid-ocean ridge-transform fault system (Morgan & Forsyth, 1988;Shen & Forsyth, 1992), with less depletion identified beneath the transform fault and fracture zone relative to the surrounding mantle.However, we have a limited understanding of the degree of depletion in this system for several reasons.First, these studies generally assumed relatively simple types of viscosity, including constant (Morgan & Forsyth, 1988), and temperature-and strain rate-dependent (Shen & Forsyth, 1992) viscosities.However, the effects of a more realistic rheology, such as PY, are required because the thermal structure that is obtained with this rheology may be more consistent with geophysical and geochemical observations in a mid-ocean ridge-transform fault setting (Behn et al., 2007).Second, Morgan and Forsyth (1988) and Shen and Forsyth (1992) reported few examples of the spatial distribution of the degree of depletion, such that the explored parameter range is limited.It is therefore necessary to investigate the effects of relevant parameters in a more systematic way to better constrain the chemical heterogeneity of the oceanic plate.Third, previous studies have only shown the degree of depletion for a few vertical cross-sections that were oriented perpendicular to the direction of plate motion (Morgan & Forsyth, 1988;Shen & Forsyth, 1992).It would therefore be useful to illustrate the 3-D distribution of the degree of depletion in a mid-ocean ridge-transform fault system.
This study aims to investigate the effects of rheology, the half-spreading rate, and transform fault length on spatial variations in the degree of upper-mantle depletion in a mid-ocean ridge-transform fault system.Only the cases of a single transform fault offsetting two mid-ocean ridges are considered in this study.

Numerical Approach
The degree of melting is predicted in a mid-ocean ridge-transform fault system by first predicting the steady-state 3-D temperature and solid rock velocity field, and then finding the maximum degree of melting that is experienced at each target location.The details of these two steps are given below.

Step 1: Temperature and Solid Rock Velocity
The Boussinesq approximation is applied to the equations that describe the conservation of mass, momentum, and energy, which are written as follows: respectively, where v is the velocity vector, P d is the dynamic pressure, η is the shear viscosity, ε is the strain rate tensor, which is given by: T is the temperature (in Kelvin), κ is the thermal diffusivity (=7.2727 × 10 7 m 2 s 1 ), L is the latent heat of melting (=400 kJ kg 1 ), Γ is the melting rate, ρ m is the reference density (=3,300 kg m 3 ), and c p is the specific heat (=1,250 J K 1 kg 1 ).The transpose operation is represented by the subscript T. See Table 1 for a description of model parameters.These equations are solved to obtain the 3-D temperature and solid rock velocity fields.Equation 2 describes passive corner flow beneath the ridge axis and transform fault (see Spiegelman and McKenzie (1987) for the 2-D geometry).Equation 3 describes the steady-state temperature profile, which was also obtained using time-dependent calculations (see Supporting Information S1).
Figure 1a shows the model domain and boundary conditions, with the key model parameters indicated.The transform fault length, d, is one of the key model parameters investigated in this study.Mantle flow is driven by Note.The values of most parameters are from Morishige (2015), with the exception of those associated with plastic yielding, which are from Behn et al. (2007).
the imposed half-spreading rate, V p , with the rock that enters the system through the bottom boundary eventually leaving the system through the boundaries in the x-direction.The surface velocity near the mid-ocean ridge is assumed to increase linearly from zero at the ridge axis to a given half-spreading rate over a 6-km distance in the xdirection to reduce the impact of stress singularity (e.g., Morishige & Honda, 2011, 2013;van Keken et al., 2002).
The melting rate in Equation 3 is given by: where F is the degree of melting.Melt freezing (i.e., negative Γ) is ignored in Equation 5 because the location and rate of melt freezing depend on the details of melt migration, which are neglected here, and the effects of melt freezing are only important near the base of the lithosphere (and will therefore have a limited impact on the degree of depletion, which is the main focus of this study).The degree of melting for a given set of temperature-pressure conditions is calculated based on the parameterization of Katz et al. (2003).The pressure at each location is calculated using assumed oceanic crust and mantle densities of 2,950 and 3,300 kg m 3 , respectively, and an oceanic crustal thickness of 7 km.
Two types of mantle rheology are considered, deformation via diffusion and dislocation creep, and deformation via diffusion, dislocation creep, and PY.The shear viscosities that correspond to these three deformation mechanisms (diffusion, dislocation creep, and PY) are written as follows: The degree of depletion is not predicted above 10 km depth because this region corresponds to the oceanic crust.The locations below 74 km depth are only considered when the parameterization of hydrous mantle melting is used (Katz et al., 2003).
respectively, where η diff is the shear viscosity for diffusion creep, A diff is a constant for calculating the viscosity for diffusion creep (=2.3067 × 10 10 Pa⋅ s), E diff is the activation energy for diffusion creep (=300 kJ mol 1 ), R is the gas constant (=8.3145J K 1 mol 1 ), η disl is the shear viscosity for dislocation creep, A disl is a constant for calculating the viscosity for dislocation creep (=2.8968 × 10 4 Pa⋅ s 1/n ), E disl is the activation energy for dislocation creep (=540 kJ mol 1 ), n is the stress exponent (=3.5), εI I is the second invariant of the strain rate tensor, which is given by: η plas is the shear viscosity for PY, C 0 is the cohesion (=10 7 Pa), μ is the friction coefficient (=0.6), g is the gravitational acceleration (=9.8 m s 2 ), and z is the z-coordinate.The values of most parameters are taken from Morishige (2015), with the exception of those associated with PY, which are taken from Behn et al. (2007).The shear viscosity in Equation 2 is then calculated as either: or: for the cases with and without PY, respectively, where η max is the maximum shear viscosity (=10 24 Pa⋅s).
Equations 1-3 are solved iteratively using the SEPRAN finite element code (Cuvelier et al., 1986) until the temperature difference between the current and previous iteration is sufficiently small.Linear tetrahedral elements are used, with the side length of each element ranging from 2 km near the mid-ocean ridge to 7 km far from plate boundaries.Subiterations are used to handle the non-linearities associated with the viscosity in Equation 2and latent heat of melting in Equation 3.

Step 2: Maximum Degree of Melting
The degree of depletion is predicted at two vertical cross-sections, which are located at x = 100 and 200 + d/2 km in the model (Figure 1b).The former includes the fracture zone, whereas the latter bisects the transform fault.The target locations where the degree of depletion is predicted are unevenly distributed in y-z space along each crosssection (Figure 1c).The degree of depletion is considered below 74 km depth in the case that the parameterization of hydrous mantle melting is used (Katz et al., 2003).I assume a passive tracer at each target location, and calculate its trajectory backward in time using the steady-state velocity field that was obtained in the first step.The degree of depletion is then defined as the maximum degree of melting that each tracer has experienced along the trajectory, following Morgan and Forsyth (1988) and Shen and Forsyth (1992).This definition is only correct when all of the melt is removed from the rock and migrates toward the ridge axis.Therefore, the predicted degree of depletion in this study should be considered the upper bound, since some of the melt will remain and freeze in the mantle rock (e.g., Keller et al., 2017).

Results
Three half-spreading rates (1, 3, and 8 cm yr 1 ) and three transform fault lengths (50, 100, and 150 km) are tested for each rheology (with and without PY) for a total of 18 modeled cases.The predicted degree of depletion is based on an anhydrous mantle and modal mineralogical assemblage with 15 wt% clinopyroxene (cpx).
The temperature and horizontal velocity distributions at 20 km depth are shown for four selected cases in Figure 2.
A case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km is defined as the reference case (Figure 2a) because the effects of the transform fault on the degree of depletion are the most clearly seen in this case, as will be shown in Figures 3 and 4. The temperature beneath the transform fault increases when the transform fault length decreases to 50 km (Figure 2b).A higher half-spreading rate of 8 cm yr 1 causes the entire temperature field to increase owing to thinning of the oceanic plates (Figure 2c).The temperature beneath the transform fault increases significantly when PY is incorporated because the reduced viscosity along the fault enhances upward flow beneath it, as previously demonstrated by Behn et al. (2007) (Figure 2d).
The spatial variations in the degree of depletion in the vertical cross-sections bisecting the transform fault and including the fracture zone are shown in Figures 3 and 4, respectively, for all of the considered cases.The reference case is first considered (Figures 3a and 4a).The obtained distribution is nearly symmetric in the crosssection including the transform fault, as expected from the rotational symmetry of the model setting with respect to a vertical line passing through the center of the transform fault (Figure 3a).However, small deviations arise owing to the slightly rotationally asymmetric finite element distribution with respect to that line.The degree of depletion generally decreases with depth, with the exception of a region beneath the transform fault where the degree of depletion is less than that of the surrounding mantle, as previously demonstrated by Shen and Forsyth (1992).These results are the same in the cross-section across the fracture zone (Figure 4a), with the exception of an asymmetric degree of depletion relative to the fracture zone; these results are in agreement with those of Morgan and Forsyth (1988).The degree of depletion is lower beneath the older plate (y > 150 km) than beneath the younger plate (y < 150 km).
The effects of using different half-spreading rates, transform fault lengths, and rheologies are the same along the analyzed cross-sections, which bisect the transform fault and include the fracture zone.The degree of depletion generally increases with increasing half-spreading rate.For example, increasing the half-spreading rate to 8 cm yr 1 (Figure 3c) yields a maximum degree of depletion of 0.248, compared with 0.197 in the reference case (Figure 3a).The region beneath the transform fault or fracture zone where the degree of depletion is low relative to the surrounding mantle shifts to shallower depths as the half-spreading rate increases and transform fault length decreases.For example, the approximate depths at which the effects of the transform fault are clearly seen are 60, 20, and 30 km for Figures 3a, 3c, and 3g, respectively.The degree of depletion beneath the transform fault and fracture zone increases when PY is incorporated.For example, the maximum degree of depletion beneath the transform fault increases from 0.026 (Figure 3a) to 0.109 (Figure 3j).In some cases, the degree of depletion is higher beneath the transform fault than beneath the region far from the transform fault at the same depth (e.g., near y = 150 km and z = 25 km in Figure 3p).Representative particle trajectories are shown in Figures 5-7, along with the thermal structure and degree of melting, to better understand the origin of spatial variations in the degree of depletion.Figure 5 shows the particle trajectories in a region that is distal to the transform fault and fracture zone (y ∼ 1 km) for half-spreading rates of 1 and 8 cm yr 1 .These cases are considered to be almost identical to a simple two-dimensional (2-D) mid-ocean ridge setting because the velocity component that is parallel to the ridge axis is close to zero.The degree of depletion generally decreases with depth (Figures 3 and 4), because the degree of melting decreases with depth, and because the particle trajectories are similar to those expected for simple corner flow.Furthermore, the degree of melting increases when the half-spreading rate increases from 1 to 8 cm yr 1 because the oceanic plate thins and the temperature immediately beneath the ridge increases.Therefore, a higher degree of depletion is generally observed when a higher half-spreading rate is assumed (Figures 3 and 4).
The particle trajectories for the locations beneath the transform fault are shown in Figure 6 for selected cases.The trajectories are confined to a relatively narrow range in the y-direction, which is close to the transform fault (Figures 6a, 6c, 6d, 6f, 6g, 6i, 6j, and 6l), where the temperature is lower (and therefore the degree of melting is also lower) than that expected beneath a simple 2-D mid-ocean ridge setting (Figures 6b, 6e, 6h, and 6k; see also Figure 2 for the thermal structure).This explains why the degree of depletion is generally lower beneath the transform fault than in the surrounding region (Figure 3).A high half-spreading rate and/or short transform fault length generally result in limited temperature variations associated with the transform fault because the oceanic plate has not cooled sufficiently (Figures 6b, 6e, and 6h), with the region where the degree of depletion varies horizontally being shifted to shallower depths (e.g., Figures 3a, 3c, and 3g).Plastic yielding warms the region beneath the transform fault and increases the degree of melting (Figures 6b and 6k; see also Figure 2), thereby increasing the degree of depletion compared with the corresponding case without PY (Figures 3a and 3j).
Figure 7 shows the particle trajectories for the locations beneath the fracture zone.The trajectories differ significantly for the mantle beneath the older and younger oceanic plates.The material beneath the younger oceanic plate passes closer to the center of the region of partial melting beneath the mid-ocean ridge compared with the region beneath the older oceanic plate (Figures 7b and 7c), thereby producing an asymmetry in the degree of depletion with respect to the fracture zone (Figure 4).
The effects of assuming a hydrous mantle and different modal mineralogical assemblages in the parameterization of Katz et al. (2003) are also tested for the case with a half-spreading rate of 8 cm yr 1 and transform fault length of 150 km, and the results are shown in Figure 8.The degree of depletion above ∼25 km depth decreases for a modal mineral assemblage with 10 wt% cpx (i.e., a more depleted composition compared with 15 wt% cpx), with the maximum degree of depletion decreasing from 0.255 (Figures 3c and 4c) to 0.208 (Figures 8a and 8b).This occurs because cpx is exhausted earlier than in the corresponding cases for a modal mineral assemblage with 15% cpx, with the increase in the degree of melting in the upwelling mantle being notably suppressed once the cpx supply is exhausted.A hydrous mantle with an assumed 0.02 wt% bulk H 2 O content increases the overall degree of depletion (Figures 8c and 8d) compared with the corresponding anhydrous case (Figures 3c and 4c) because of the decrease in the solidus temperature due to the addition of water.Melting also initiates ∼20 km deeper (dashed black lines) compared with the corresponding anhydrous case.
The degree of depletion has only been calculated for a limited number of vertical cross-sections in previous studies (Morgan & Forsyth, 1988;Shen & Forsyth, 1992).However, the structure and tectonic complexities of mid-ocean ridge-transform fault systems warrant the need for 3-D constraints to accurately capture any spatial variations in the degree of depletion.Here this deficiency is addressed by providing 3-D constraints for the case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km (Figure 9), whereby the degree of depletion at 10 km depth varies over a relatively short distance in the x-direction (∼30 km) between the transform fault and fracture zone (Figure 9a).This type of 3-D information is necessary to fully investigate the impact of spatial variations in the degree of depletion on the structure and dynamics of the oceanic plate, as discussed in the following section.

Discussion
This study models the variations in the degree of depletion for different half-spreading rates, transform fault lengths, and rheologies.In reality, transform faults can be much longer than 150 km (e.g., Ren et al., 2022), which is the maximum length modeled in this study, and can produce a lower degree of depletion beneath the transform fault and fracture zone than that predicted in this study.Roland et al. (2010) investigated the effects of a wide range of factors on the thermal structure of a mid-ocean ridge-transform fault system, including parameters related to PY, shear heating, and hydrothermal circulation, and found that hydrothermal circulation has the greatest impact on the thermal structure.If cooling via hydrothermal circulation is significant, then it will effectively lower the temperature beneath the transform fault and fracture zone, possibly leading to an even lower degree of depletion in these regions compared with that obtained in this study (Figures 3 and 4).For example, deep hydration along the Romanche transform fault in the Atlantic Ocean is supported by the presence of a low seismic velocity anomaly (Wang et al., 2022).The degree of depletion that is predicted in this study may be compared with the observed global variations in abyssal peridotite compositions.This comparison was conducted by plotting the degree of depletion at locations (x, y, z) = (200 + d/2, 1, 10) and (200 + d/2, 148, 10) km for all of the cases shown in Figures 3 and 4 (Figure 10).The former location may represent 2-D mid-ocean ridge settings, and the latter location is proximal to a transform fault.Figure 10 of Warren (2016) showed that the Yb contents in cpx, modal cpx, and spinel Cr# (=100 × Cr/(Cr + Al)) varied as a function of the spreading rate; these are largely consistent with the findings of the present study, at least in the following two aspects.First, the peridotite samples from locations where the halfspreading rate is higher than ∼6.5 cm yr 1 have low Yb contents in cpx, low modal cpx, and high spinel Cr#, which indicates that there is a high degree of depletion for high spreading rates (Figure 10).Second, a large degree of scatter is present in the peridotite samples from locations where the halfspreading rate is lower than ∼3 cm yr 1 .Although Warren (2016) attributed this scatter to melt-rock interactions and/or heterogeneity in the source composition, the data may be explained by differences in the transform fault length.Figure 10 shows that the variations in the degree of depletion due to different transform fault lengths decrease with increasing half-spreading rate, regardless of whether PY is incorporated.Since Warren (2016) analyzed peridotite samples from both transform faults and ridge axes, it is possible that the large scatter that is observed at low spreading rates may at least partly reflect variations in the transform fault length.
These modeling results may also be consistent with observations of the oceanic crust, although the comparison is less straightforward compared with that with the abyssal peridotite compositions due to a limited understanding of melt migration processes (e.g., Bai & Montési, 2015;Sim et al., 2020).Niu (2016) revealed a significant correlation between the spreading rate and mid-ocean ridge basalt (MORB) major element compositions, which were corrected for the effects of crustal-level processes to Mg# (=100 × Mg/ (Mg + Fe)) ≥ 0.72, whereby the degree of melting was observed to increase with the spreading rate.This correlation agrees with the prediction in this study that a higher degree of depletion occurs where there are higher spreading rates (Figure 10).
The findings in this study also have several implications for geophysical observations of mid-ocean ridge-transform fault systems, which are discussed here.The density difference due to the lower degree of depletion beneath the transform fault and fracture zone compared with the surrounding mantle can be predicted following Lee et al. (2005).The maximum degree of melting F max , which is equivalent to the degree of depletion in this study, is first converted to Mg# using the dMg#/dF max = 7.94 relationship; this Mg# value is then employed to calculate the density anomaly Δρ chem using the dΔρ chem /dMg# = 14.4 kg m 3 relationship (Lee, 2003).The results yield a maximum difference in F max of ∼0.15 across the transform fault and fracture zone for the case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km (Figures 3a and 4a).This roughly corresponds to variations in Mg# and Δρ chem of 1.2 and 17 kg m 3 , respectively.The isostatic topography is then calculated to further delineate the effects of this chemical density difference.The isostatic topography w is calculated as follows (e.g., Turcotte & Schubert, 2014): where H is the model thickness (=120 km), ρ w is the water density (=1,000 kg m 3 ), α is the coefficient of thermal expansion (=3 × 10 5 K 1 ), and T m is the reference temperature (=1623 K).Note that the key assumptions in this topography calculation are a constant oceanic crustal thickness of 7 km and an oceanic crustal density that is the same as the mantle density to clearly see the effects of the mantle density variations.The isostatic topography relative to y = 1 km for two selected cases is shown in Figure 11.The case without PY produces distinct subsidence at both the transform fault (solid lines in Figure 11a) and older oceanic plate (dashed lines in Figure 11a) because of the lower temperatures in these regions (Figure 2a).However, the case with PY produces uplift at the transform fault (solid lines in Figure 11b) because the mantle beneath the transform fault is warmer than in surrounding areas due to PY (Figure 2d).Including the effects of the chemical density difference, subsidence is up to ∼0.2 km greater than in the case where the density difference is ignored.This effect is smaller for the case that includes PY because the Figure 7. Particle trajectories for the case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km, and the following (x, y, z) locations: (100, 132, 10), (100, 132, 22), (100, 132, 34), (100, 168, 10), (100, 168, 22), and (100, 168, 34) km.The locations where the degree of depletion is predicted are marked with a cross symbol.(a) x-y cross-section.(b) x-z cross-section with the particle trajectories ending at y = 168 km.(c) x-z cross-section with the particle trajectories ending at y = 132 km.(d) y-z cross-section.The solid and dashed black lines in (a) indicate the locations of plate boundaries and fracture zones, respectively, and arrows show the direction of plate motion.Colored lines in each figure denote the particle trajectories, with the colors representing the degree of depletion.Solid and black dashed lines in (b) and (c) are the isotherms (300°C interval) and contours of the degree of melting (0.02 contour interval), respectively, along the y = 170 km (for b) and 130 km (for c) cross-sections.
horizontal variations in the degree of depletion are limited (e.g., Figures 3a  and 3j).The maximum subsidence at the transform fault is around 0.4 km.Given that some transform faults can be as much as ∼3 km deeper than the surrounding seafloor (e.g., Maia, 2019), the contribution of isostatic topography due to the variations in mantle density may not be geophysically discernible, even if the density difference associated with chemical effects is taken into account.
The variations in the degree of depletion also affect the shear viscosity through dehydration.It has been suggested that the most significant viscosity increase occurs shortly after the mantle temperature exceeds the dry solidus because the remaining water is rapidly extracted from the rock and incorporated into the melt (Braun et al., 2000;Hirth & Kohlstedt, 1996).The viscosity can increase by more than an order of magnitude over a narrow depth range (a few kilometers) in this case.If we assume that this type of viscosity increase occurs where the degree of depletion reaches 0.001 (dashed black lines in Figures 3 and 4), then the results indicate that the depth of this viscosity change is shallower beneath the fracture zone relative to the surrounding mantle, and that the depth variations show an increase with decreasing half-spreading rate, increasing transform fault length, and without PY.For example, the viscosity increase due to dehydration occurs ∼10 km shallower beneath the fracture zone than in the surrounding mantle for the case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km (dashed black line in Figure 4a).It is also possible that the viscosity changes as a result of variations in Fayalite (Fe 2 SiO 4 ) content in olivine due to melting.However, the flow law given by Zhao et al. (2009) indicates that the strain rate for the case with a Fayalite fraction X Fa = 0.91 is only 13% and 11% higher than that for the case with X Fa = 0.89 at 1273 and 1473 K, respectively.Therefore, this effect is small and can be ignored.
The above-mentioned density and viscosity variations beneath the transform fault and fracture zone should affect the temporal evolution of the oceanic plate.Small-scale convection beneath the moving oceanic plate has been proposed to explain a wide range of geophysical observations, including surface heat flow and seafloor subsidence as the plates age (e.g., Parsons & Sclater, 1977), gravity anomalies (e.g., Buck & Parmentier, 1986), and seismic-wave velocities and anisotropy (e.g., Kawano et al., 2023;van Hunen & Čadek, 2009).The onset of  small-scale convection largely depends on the viscosity and density structure (e.g., Huang et al., 2003;Korenaga & Jordan, 2003).For example, the scaling law proposed by Korenaga and Jordan (2003) indicates that a decrease of 10 or 100 in the absolute value of the viscosity will cause the onset time of small-scale convection to shift forward by ∼80 or 95%, respectively.Previous studies have suggested that small-scale convection first develops near the fracture zone (e.g., Huang et al., 2003), as small-scale convection preferentially occurs where there is a large temperature (and therefore density) variation.It can be inferred from the present results that smallscale convection initially develops near the fracture zone in the context of depletion because the density is higher and the viscosity is lower beneath this region compared with the regions that are far from the fracture zone at the same depth.This is especially true for the case with a lower spreading rate and longer transform fault where a lower degree of depletion is predicted (Figure 4).It has also been proposed that the fracture zone could evolve into a subduction zone (e.g., Hall et al., 2003;Stern & Gerya, 2018).If the mantle beneath the fracture zone is chemically less viscous, then it may allow subduction to initiate more easily by kinematically decoupling the movement of the older plate from that of the younger plate.Geodynamic modeling that considers both the thermal and chemical effects is necessary to discuss the impacts of the fracture zone on both small-scale convection and the likelihood of initiating subduction in a more quantitative way.
This study demonstrates that both the half-spreading rate and transform fault length have significant impacts on the degree of depletion.If these parameters change over time (and this is at least true for the half-spreading rate, as demonstrated by e.g., Malinverno et al. (2020)), then the degree of depletion can also change with time.Although the presence of a transform fault is a key parameter that is investigated in this study, there are many other possible factors that may affect the spatial variations in the degree of depletion.First, variations in the mantle potential  Case with a half-spreading rate of 1 cm yr 1 , transform fault length of 150 km, and plastic yielding.Solid and dashed lines are the calculated topographies for the vertical cross-section at x = 275 km (including the transform fault) and x = 100 km (including the fracture zone), respectively.Red and black lines are the calculated topographies with and without the density difference caused by chemical effects (i.e., Δρ chem in Equation 12), respectively.PY, plastic yielding.
Geochemistry, Geophysics, Geosystems 10.1029/2023GC011227 temperature have not been considered in this study.An analysis of marine seismic refraction data by van Avendonk et al. (2017) revealed that the oceanic crust that formed about 170 million years ago is ∼1.7 km thicker on average, which corresponds to the cooling rate of 0.15-0.2K Myr 1 in the upper mantle if the observed thickness variations are caused by temperature effects.This suggests that the mantle potential temperature was higher and partial melting occurred deeper in the past, thereby leading to a higher degree of depletion.Joint analyses of seismic-wave velocities, ocean ridge depths, and MORB compositions have revealed that large variations in the mantle potential temperature exist beneath mid-ocean ridges (Dalton et al., 2014).These variations reach 100-150 K, even when only the ridge segments that are far from hot spots are considered.It is therefore expected that the degree of depletion varies widely between ridge segments.Second, mantle flow and melt migration beneath mid-ocean ridges have been extensively investigated, including the flow asymmetry caused by ridge migration, buoyancy effects due to temperature, melt, and depletion, and mechanisms of melt focusing toward the ridge axis (e.g., Bai & Montési, 2015;Braun et al., 2000;Conder et al., 2002;Katz & Weatherley, 2012;Keller et al., 2017;Pusok et al., 2022;Sim et al., 2020).Each factor will produce different distributions of the degree of depletion.For example, inefficient melt focusing toward the ridge axis will cause a large amount of melt to be trapped in the oceanic plate (Keller et al., 2017), thereby leading to a lower degree of depletion in general.Furthermore, heterogeneous depletion is expected as a result of porosity waves (e.g., Keller et al., 2017;Sim et al., 2020).All of these possibilities strongly suggest that the chemical structure of the oceanic plate is highly heterogeneous due to the specific melting history of the plate.Therefore, the structure and dynamics of the oceanic plate cannot be fully understood based solely on its age, as has been done in many previous studies (e.g., Morishige, 2022;Stein & Stein, 1992).

Conclusions
This study investigated the effects of the half-spreading rate, transform fault length, and PY rheology on spatial variations in the degree of upper-mantle depletion in a mid-ocean ridge-transform fault system using 3-D thermomechanical modeling.The degree of depletion increased with increasing half-spreading rate.The lowest degree of depletion was observed beneath the transform fault and fracture zone, although this feature was only observed at shallower depths for the cases with higher half-spreading rates and shorter transform fault lengths.The degree of depletion was lower beneath the older plate compared with that beneath the younger plate in a vertical cross-section that included the fracture zone.Incorporating PY resulted in a higher degree of depletion beneath the transform fault and fracture zone compared with the corresponding case without PY.The predicted degree of depletion may at least partly explain the observed global variations in abyssal peridotite compositions.The spatial density variations that were expected from the obtained degree of depletion contributed to a subsidence of up to 0.2 km in the isostatic topography for the parameter range considered in this study.The density and viscosity variations due to the degree of depletion may also promote an earlier onset of small-scale convection beneath the fracture zone, and potentially facilitate the evolution of the fracture zone into a subduction zone.The oceanic plate is considered to be chemically highly heterogeneous because other factors could also contribute to the observed variations in the degree of depletion, such as temporal variations in the half-spreading rate and transform fault length, spatiotemporal variations in mantle potential temperature, and details of the mantle flow and melt migration beneath mid-ocean ridge-transform fault systems.It is therefore critical to constrain the spatial variations in the chemical composition of oceanic plates in mid-ocean ridge-transform fault systems to obtain a better understanding of their structure and dynamics.

Figure 1 .
Figure 1.Model setting.(a) Model domain and boundary conditions.(b) Locations of the vertical and horizontal crosssections shown in other figures.(c) Locations along the y-z cross-sections where the degree of depletion is predicted.The degree of depletion is not predicted above 10 km depth because this region corresponds to the oceanic crust.The locations below 74 km depth are only considered when the parameterization of hydrous mantle melting is used(Katz et al., 2003).

Figure 2 .
Figure 2. Temperature and horizontal velocity distributions at 20 km depth for selected cases.The temperature distribution is color-coded.Black lines are isotherms (50°C interval) and black arrows indicate horizontal velocities.Solid and dashed white lines are locations of the plate boundaries (mid-ocean ridges and transform faults) and fracture zones, respectively.PY: plastic yielding.

Figure 3 .
Figure 3. Degree of depletion along the vertical cross-section bisecting the transform fault (x = 200 + d/2 km), with the transform fault located at the center (y = 150 km).The mantle is color-coded based on the degree of depletion.Only the region spanning 50-250 km in the y-direction along the cross-section is shown because the horizontal variations in the degree of depletion are very limited outside this region.Solid black lines are depletion contours (0.02 contour interval), with bold lines indicating a value of 0.2.Dashed black lines indicate a depletion of 0.001.PY, plastic yielding.

Figure 4 .
Figure 4. Degree of depletion along the vertical cross-section at x = 100 km, with the fracture zone located at the center (y = 150 km).The right and left sides of each panel correspond to the older and younger oceanic plates, respectively.The mantle is color-coded based on the degree of depletion.The region spanning 50-250 km in the y-direction along the cross-section is shown because the horizontal variations in the degree of depletion are very limited outside of this region.Solid black lines are depletion contours (0.02 contour interval), with bold lines indicating a value of 0.2.Dashed black lines indicate a depletion of 0.001.PY: plastic yielding.

Figure 5 .
Figure 5. Particle trajectories for the following (x, y, z) locations in the model: (100, 1, 10), (100, 1, 22), and (100, 1, 34) km.The locations where the degree of depletion is predicted are marked with a cross symbol.Only the x-z cross-sections are shown because the change in the trajectories in the y-direction is less than ±0.06 km.(a) Case with a half-spreading rate of 1 cm yr 1 and transform fault length of 50 km.(b) Case with a half-spreading rate of 8 cm yr 1 and transform fault length of 50 km.Colored lines in each figure denote the particle trajectories, with the colors representing the degree of depletion.Solid black lines are the isotherms (300°C interval), and dashed black lines are contours of the degree of melting (0.02 contour interval) along the y = 1 km cross-section.

Figure 6 .
Figure 6.Particle trajectories for the following (x, y, z) locations for four modeled cases:(275, 140, 10),(275, 140, 22), and (275, 140, 34)  km in the case of a transform fault length of150 km, and (225, 140, 10), (225, 140, 22), and (225, 140, 34)  km in the case of a transform fault length of 50 km.The(a, d, g, i)  x-y, (b, e, h, k) x-z, and (c, f, i, l) y-z cross-sections are shown for each case.The locations where the degree of depletion is predicted are marked with a cross symbol.(a-c) Case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km.(d-f) Case with a half-spreading rate of 1 cm yr 1 and transform fault length of 50 km.(g-i) Case with a half-spreading rate of 8 cm yr 1 and transform fault length of 150 km.(j-l) Case with a half-spreading rate of 1 cm yr 1 , transform fault length of 150 km, and plastic yielding.Colored lines in each figure denote the particle trajectories, with the colors representing the degree of depletion.Solid and dashed black lines in (a), (d), (g), and (j) are the locations of the plate boundaries and fracture zones, respectively, and the arrows indicate the direction of plate motion.Solid and dashed black lines in (b), (e), (h), and (k) are the isotherms (300°C interval) and contours of the degree of melting (0.02 contour interval) along the y = 140 km cross-section, respectively.PY: plastic yielding.

Figure 8 .
Figure 8. Degree of depletion along the vertical cross-section at (a and c) x = 275 km and (b and d) 100 km, for the case with a half-spreading rate of 8 cm yr 1 and transform fault length of 150 km.(a and b) A modal mineral assemblage with 10 wt% cpx is assumed.(c and d) A hydrous mantle with a 0.02 wt% bulk water content is assumed.The mantle is color-coded based on the degree of depletion.Only the region spanning 50-250 km in the ydirection along the cross-section is shown because the horizontal variations in the degree of depletion are very limited outside of this region.Solid black lines are depletion contours (0.02 contour interval), with bold lines indicating a value of 0.2.Dashed black lines indicate a depletion of 0.001.

Figure 9 .
Figure 9. 3-D variations in (a) the degree of depletion and (b) temperature for the case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km.The solid and dashed gray lines at the top of each panel denote the plate boundaries (mid-ocean ridges and transform fault) and fracture zones, respectively.The degree of depletion is only calculated in half of the model domain, where x (km) ranges from 0 to 275, because of the rotational symmetry of the model setting.

Figure 10 .
Figure10.Relationship between the half-spreading rate and degree of depletion at 10 km depth for the cases (a) without and (b) with plastic yielding.Circle, triangle, and cross symbols correspond to transform fault lengths of 50, 100, and 150 km, respectively.Black and red symbols represent the results at y = 1 and 148 km, which are 149 and 2 km from the transform fault, respectively.

Figure 11 .
Figure11.Isostatic topography relative to that at y = 1 km.(a) Case with a half-spreading rate of 1 cm yr 1 and transform fault length of 150 km.(b) Case with a half-spreading rate of 1 cm yr 1 , transform fault length of 150 km, and plastic yielding.Solid and dashed lines are the calculated topographies for the vertical cross-section at x = 275 km (including the transform fault) and x = 100 km (including the fracture zone), respectively.Red and black lines are the calculated topographies with and without the density difference caused by chemical effects (i.e., Δρ chem in Equation12), respectively.PY, plastic yielding.

Table 1
Definitions of the Variables Used in This Study