A Common Diffusional Mechanism for Creep and Grain Growth in Polymineralic Rocks: Application to Lower Mantle Viscosity Estimates

In a previous study (Okamoto & Hiraga, 2022, https://doi.org/10.1029/2022jb024638), we concluded that diffusion creep and grain growth in polymineralic rocks proceed by a common diffusional mechanism. Here, we built on that finding and estimated lower mantle grain size and viscosity during a single mantle convection cycle dominated by diffusion creep. We approximated the lower mantle as a two‐phase material consisting of bridgmanite + ferropericlase and post‐perovskite + ferropericlase, depending on depth. We used previously reported self‐diffusivities for bridgmanite and post‐perovskite. We predict a bridgmanite grain‐size of tens to hundreds of microns shortly after the phase transition at ∼660 km depth. This size remains relatively constant until the mantle material enters the post‐perovskite zone, which is marked by significant grain growth up to ∼9 mm just prior to upwelling. This size is sufficient to prevent further grain growth until the mantle material reaches the top of the lower mantle. These grain sizes combined with the diffusivities yield viscosities that vary laterally and with depth. At a lateral temperature difference of up to 800 K in the lower mantle, fine‐grained cold downwelling mantle is almost as viscous as, or more likely to be softer than coarse‐grained hot upwelling mantle. The lateral viscosity variations cannot be more than 2 orders of magnitude, and we estimated viscosities of 1018–1020 Pa · s in the upper lower mantle, 5 × 1020–5 × 1022 Pa · s in the lower bridgmanite zone, and 1017–1019 Pa · s in the post‐perovskite zone, which compare well with the values estimated in previous geophysical modeling studies.


Introduction
The viscosity of the lower mantle, which makes up ∼65% of Earth's overall mantle, is an important factor controlling mantle convection.Understanding the deformation mechanism that governs mantle viscosity and other properties is a subject of ongoing debate.Anisotropic elasticity in the mantle is primarily due to crystallographic preferred orientation (CPO) of the minerals that constitute the mantle.Previous studies have attempted to identify regions dominated by dislocation or diffusion creep based on isotropic and anisotropic seismic wave velocities, assuming that CPO uniquely results from dislocation creep (e.g., Karato & Wu, 1993).Strong anisotropy has been observed in the upper part of the upper mantle and the lowermost mantle (D″ layer) (e.g., Montagner & Kennett, 1996;Panning & Romanowicz, 2006), leading to the conclusion that dislocation creep occurs in these regions (Karato, 1998), while diffusion creep dominates elsewhere (Karato et al., 1995).However, a recent theoretical study has proposed dislocation climb creep, a mechanism that does not produce CPO due to climb instead of slip accommodating strain in the lower mantle (Boioli et al., 2017).Moreover, studies by Maruyama and Hiraga (2017a) and Miyazaki et al. (2013) have shown that CPO develops during diffusion creep due to preferential grain-boundary sliding (GBS) on boundaries parallel to low-index crystallographic planes.Maruyama and Hiraga (2017b) explained changes in isotropic and anisotropic mantle properties with depth in terms of the development of such easy-sliding grain boundaries.These findings challenge the conventional view that isotropic and anisotropic mantle correspond to flow by diffusion and dislocation creep mechanisms, respectively.
Given the uncertainty surrounding the deformation mechanism of the lower mantle, here we estimate mantle viscosity by assuming that the entire mantle deforms by diffusion creep.We rely on the diffusion creep model with using previously measured self-diffusivities, which is known to reproduce material strength during diffusion creep (Okamoto & Hiraga, 2022;Wang, 2000;Yabe et al., 2020).Its success is somewhat unique compared to other deformation mechanisms, which is mainly due to the clearer atomic process of the diffusion creep mechanism.Further, during diffusion creep, viscosity remains stress-independent due to Newtonian rheology, allowing the viscosity estimation without knowing stress.Diffusion creep occurs in parallel with dislocation creep, and thus, a diffusion-creep-based viscosity provides an upper limit on the mantle viscosity.If the estimated viscosity is much higher than that derived from geophysical approaches such as studying post-glacial rebound (Mitrovica & Forte, 2004), then we must invoke other deformation mechanisms as the processes that control mantle flow.However, the grain size dependence of diffusion creep makes such estimations unreliable when grain size constraints are limited.Previous studies have assumed particular grain sizes in the lower mantle, although these assumptions lack a robust physical and observational basis (e.g., Ammann et al., 2010;Tackley et al., 2013;Yamazaki & Karato, 2001).Moreover, assuming a constant grain size across the mantle is likely inappropriate.Unlike dislocation processes where dynamic recrystallization maintains a set grain size in response to stress, there is no driving force to preserve a particular grain size during diffusion creep, and grain growth occurs to reduce the interfacial energy of the system (Fei et al., 2021;Solomatov & Reese, 2008;Yamazaki et al., 1996).Consequently, the viscosity can increase over time.
In this study, we approximate lower mantle material as a two-phase aggregate of either bridgmanite + ferropericlase or post-perovskite + ferropericlase in the upper to middle and the lowermost portions of the lower mantle (i.e., D″ layer), respectively, following previous mantle viscosity estimates (F. Xu et al., 2022;Yamazaki & Karato, 2001).These two minerals account for 90% of lower mantle material (Ita & Stixrude, 1993) and effectively represent the mantle as consisting of a mechanically strong primary phase and a weak secondary phase (Yamazaki & Karato, 2001).Some studies have investigated deformation of two-phase materials that are influenced by grain growth (Glišović et al., 2015;Schierjott et al., 2020;Solomatov, 2001).However, these works modeled creep and grain growth processes separately with few constraints, particularly in terms of grain growth parameters, leading to viscosity estimates with large uncertainties for both processes.
Recently, we measured rates of grain growth and diffusion creep in forsterite (Mg 2 SiO 4 ) + 20% enstatite (MgSiO 3 ) and forsterite + 10% periclase (MgO) (Nakakoji et al., 2018;Okamoto & Hiraga, 2022).We found significant differences in the rates of both processes between these two systems, likely due to differences in silica activity affecting the rate-controlling atomic diffusion step.However, both grain growth and creep rates in each system can be effectively explained by a common set of diffusivities.Below, we first discuss the possibility of the same outcome for creep and grain growth in lower mantle materials.Then, we estimate grain sizes based on the common rate-controlling diffusivity for grain growth and creep.Finally, we calculate mantle viscosity based on these estimated grain sizes and common diffusivities.

Grain Growth and Creep of the Lower Mantle Materials
In this section, we discuss feasible grain growth and diffusion creep processes in the lower mantle, drawing insights from our previous work, especially regarding grain growth and creep of forsterite + enstatite (Nakakoji & Hiraga, 2018;Nakakoji et al., 2018) and forsterite + periclase (Okamoto & Hiraga, 2022).

Microstructures
Among various deformation mechanisms, diffusion creep is known to be highly sensitive to microstructure, necessitating a thorough understanding of its state in the deep Earth.As mantle material descends beyond 660 km, eutectoid decomposition of ringwoodite leads to the formation of bridgmanite and ferropericlase, the primary and secondary major minerals constituting the lower mantle.Initially during the phase transition, the two minerals appear as intermixed thin lamellae with a spacing of 0.1-1 μm (Kubo et al., 2000).These lamellae quickly degenerate into nearly equiaxed grains (El-Khozondar et al., 2002;Solomatov et al., 2002;Yamazaki et al., 1996), resulting in a well-dispersed mixture of bridgmanite and ferropericlase at the grain scale (Figure 1) (Yamazaki et al., 1996).The secondary ferropericlase phase typically resides in the intergranular regions of the primary bridgmanite phase, creating a characteristic microstructure in which grains of the secondary phase effectively pin the grain boundaries of the primary phase (Hiraga, Miyazaki, et al., 2010).
A pyrolite composition throughout the entire mantle is dominated by bridgmanite and post-perovskite phases, which account for over 70 vol% in both bridgmanite and post-perovskite zones in the lower mantle.This distribution predicts well-connected grains of the primary phase with isolated grains of the secondary phase (Fei et al., 2023).During the phase transition from bridgmanite + ferropericlase to post-perovskite + ferropericlase at depths of 2,600-2,700 km, multiple grains of post-perovskite can form from a single grain of bridgmanite phase.Similarly, during the transformation from postperovskite + ferropericlase back to bridgmanite + ferropericlase during upwelling from the bottom of the lower mantle, multiple grains of bridgmanite can form from a single grain of post-perovskite.During both phase transitions, the newly formed grains are not subject to grain-boundary pinning and quickly coalesce into fewer grains until they are ultimately surrounded by grains of the secondary phase, as schematically illustrated in Figure 1b.Consequently, the grain size becomes similar to that of the original phase, allowing us to approximate the same grain size before and after the phase transitions (Solomatov & Reese, 2008).
When the secondary phase grains effectively pin grain boundaries, the primary phase can grow if the number of pinning grains is reduced through Ostwald ripening of the secondary phase.This process is expected to produce simultaneous grain growth of the primary and secondary phases with a constant grain size ratio, which is a function of the fraction of the secondary phase ( f ΙΙ ): here, d Ι and d ΙΙ are grain sizes of the primary and secondary phases, respectively, while β and z are Zener parameters.Values of z ≈ 0.5 and β ≈ 0.7 have been found to explain the microstructures of pyroxene-bearing olivine aggregates in experiments (Hiraga, Tachibana, et al., 2010;Tasaka & Hiraga, 2013) and in nature (Linckens et al., 2011;Tasaka et al., 2014;Yabe & Hiraga, 2020).Grain size data from studies of bridgmanite + ferropericlase aggregates (Yamazaki et al., 1996(Yamazaki et al., , 2009) ) have been compiled in Figure 2. Initially, these aggregates were monomineralic olivine aggregates, leading to a calculated f ΙΙ (Equation 1) value of 0.32 for the volume fraction of ferropericlase.The grain size ratio of bridgmanite and ferropericlase follows Equation 1 when used with the Zener parameters determined for olivine and pyroxene (Figure 2).This not only confirms that the Zener relationship is valid during grain growth of bridgmanite + ferropericlase, but we have also demonstrated that the Zener relationship holds during diffusion creep (Kim et al., 2022;Yabe & Hiraga, 2020).

Diffusional Processes
In a microstructure with isolated grains of the secondary phase (Figure 1b), bulk deformation can only proceed by diffusion of the primary phase components (Figure 3a).Grain growth, driven by Ostwald ripening of the secondary phase, requires diffusion of components from smaller to larger grains of the secondary phase, along with counterdiffusion of primary phase components to accommodate the spaces created by growth and shrinking of the secondary phase grains (Okamoto & Hiraga, 2022) (Figure 3b).Such counter-diffusion may control the rate of grain growth, which has been proposed for most Earth materials, including those that constitute the lower mantle.The counter-diffusion can be viewed as deformation of the primary phase driven by the interphase boundary energy between grains of the primary and secondary phases.Typically, the interphase boundary energy is around ∼1 J/m 2 (Okamoto & Hiraga, 2022;Tasaka & Hiraga, 2013), which corresponds to stresses of ∼0.5 MPa and ∼5 × 10 2 Pa at grain sizes of 1 μm and 1 mm, respectively, based on γ IPB /2d (γ IPB : interphase boundary energy; d: grain size).These low calculated stress values indicate that the primary phase grains likely deform by a mechanism that dominates at The sample was annealed at a temperature of 1,573 K and a pressure of 25 GPa for 10 hr after the phase transition took place (modified from Yamazaki et al., 1996).Bridgmanite and periclase grains appear as light and dark gray, respectively.(b) Schematic illustration of the microstructure.
low stress, such as diffusion creep; consequently, grain growth rates are limited by the same diffusivity that controls creep.
In Okamoto and Hiraga (2022), we described the aggregates of forsterite + enstatite and forsterite + periclase used in our experiments and explained their suitability as analogs of lower mantle materials in terms of the chemical relationships between the primary and secondary phases.An approximated two-phase aggregate with the composition (Mg,Fe) SiO 3 + x • {(Mg,Fe)O} (where x < 1), resembling lower mantle material, displays a chemical relationship between the primary and secondary phases similar to that of forsterite + periclase, where Si is present only in the primary phase.Various studies indicate that Si (Yamazaki et al., 2000) or Mg (Holzapfel et al., 2005) is the slowest diffusing species in the primary phase, bridgmanite.The former scenario is comparable to forsterite + periclase, where the slowest diffusing species (Si) is contained only in the primary phase.The latter situation corresponds to forsterite + enstatite, where slowdiffusing Si is present in both the primary and secondary phases.

Viscosity Based On Diffusivity and Grain Size
The viscosity of the aggregate, η during diffusion creep is determined by its rate-controlling diffusivity, D cr and grain size, d: where α cr is a multiple of a geometric factor and the material parameters (Equation A1) and δ is the grain boundary width.D Latt cr and D GB cr are the rate-controlling diffusivities for an aggregate deforming by lattice and grainboundary diffusion creep mechanisms, respectively (Table 1).A factor of 1.7 was used to convert the 2D grain size to the real 3D grain size in the creep model (Okamoto & Hiraga, 2022).
For grain size determined by grain growth, which is driven by interfacial energy reduction, the size is likely to follow a static grain growth law (Ardell, 1972a):  et al. (1996, 2009).The Zener relationship (Equation 1) based on values of f ΙΙ = 0.32, z ≈ 0.5, and β ≈ 0.7 is also shown in the figure.where d(t) is grain size at time t, d 0 is the initial grain size (at t = 0), α Latt gr and α GB gr are multiples of the geometric and material parameters for grain growth controlled by lattice and grain-boundary diffusion, respectively (Equations A2 and A3), and D GB gr and D Latt gr are the grain boundary and lattice diffusivities that determine grain growth rate, respectively.When this law has been established for growth of the secondary phase, a factor of 0.7 f 0.5 ΙΙ (Equation 1) can be used to convert the grain size of the secondary phase to that of the primary phase.

Self-Diffusivities and Rate-Controlling Diffusivities for Creep and Grain Growth
The values of D gr and D cr should be related to the self-diffusivities of the constituent elements that are involved in diffusional processes such as grain growth and creep: where D i is the self-diffusivity of element i and n i is the stoichiometric factor for element i in each mineral.This relationship accounts for charge neutrality during diffusion, known as the compound effect.In this study, we used this relationship with previously reported D i values for the primary phases of the lower mantle, that is, bridgmanite and post-perovskite, to calculate grain size and viscosity (Equations 2 and 3).
In many cases, the D i of a particular element can be much smaller than that of the other constituents, such as Si in olivine (Fei et al., 2016).In such cases, D gr and D cr can be approximated using the diffusivity of the slowest diffusion species, j (Equation 4): where n j is the stoichiometric coefficient for element j.For the example of Si (=j) in bridgmanite (MgSiO 3 ), ∑ i n i n j = 5 where ∑ i n i = 5 and n j = 1.We have observed that δD GB cr in forsterite + enstatite aggregates corresponds to its predicted values based on δD Si GB measured in the same aggregates (Fei et al., 2016), using a α GB cr value that is larger than that of the classical Coble creep model by a factor of 10 (Nakakoji & Hiraga, 2018;Okamoto & Hiraga, 2022;Yabe et al., 2020).This finding is consistent with previous observations that the Coble creep model underestimates diffusion creep rates by a factor of ∼10 in various materials (Wang, 2000).The larger α GB cr values that best fit the experimental results are attributed to significant contributions from GBS (i.e., Rachinger sliding) during diffusion creep (Ashby & Verrall, 1973;Wang, 2000).In this study, we used an α GB cr value (Equation 2) that is consistently larger than the model-derived values by a factor of 10 (Equation A1).
Interestingly, a similar deviation has been identified for grain growth.Using the classical grain growth model for a polyphase system (Ardell, 1972a), δD GB gr has been found to be systematically larger by a factor of 9 compared to the values predicted based on δD Si GB (Nakakoji & Hiraga, 2018;Okamoto & Hiraga, 2022).Our proposition that grain growth can be viewed as part of the primary phase creep process led us to infer that this deviation is also due to a contribution from GBS during grain growth, which is not considered in the classical grain growth model.Thus, we used α GB gr and α Latt gr values (Equations 2 and 3) that are consistently larger than the model-derived values by a factor of 9 (Equations A2 and A3).
With these corrected α values, δD GB gr = δD GB cr and D Latt gr = D Latt cr , which are here denoted δD comGB and D comLatt , respectively.They can be summarized together with Equation 4as where δD i GB and D i Latt are the self-diffusivity of element i at grain boundaries and in the lattice, respectively.
The grain growth law is derived for grain-growth rates controlled by grain-boundary diffusion and lattice diffusion at small and large grain sizes, respectively, similarly to creep (Equation 2).The successful application of the creep law (Equation 2) and the grain growth law (Equation 3) to experimental results (Nakakoji & Hiraga, 2018;Okamoto & Hiraga, 2022) indicates that both the diffusion mechanism and the diffusion length scale over which creep and grain growth act are essentially the same (i.e., about half the grain size).Thus, the transition from grain-boundary to lattice diffusion that mainly controls the grain growth rate should occur at essentially the same grain size as for creep.This critical grain size d c is represented in terms of δD comGB and D comLatt (Equation 6) as

Comparison of Diffusivities in Lower Mantle Materials
Grain growth rates have previously been measured in bridgmanite + ferropericlase aggregates, as have selfdiffusivities in the bridgmanite lattice and grain boundaries, D i BrLatt and D i BrGB , respectively.Various values for D i BrGB (D O BrGB : Dobson et al., 2008;D Si BrGB : Yamazaki et al., 2000;Dobson et al., 2008; Dobson et al., 2008; D Mg BrLatt : J. Xu et al., 2011;Holzapfel et al., 2005; D Si BrLatt : Yamazaki et al., 2000;Dobson et al., 2008;J. Xu et al., 2011) are summarized in Arrhenius space in Figure 4. We also show D MgO BrGB values estimated from periclase (MgO) and quartz (SiO 2 ) reaction experiments (Nishi et al., 2013) in Figure 4a.Overall, we observe that D Si BrGB ≪ D O BrGB ≈ D MgO BrGB , and the comparison further indicates that Although there are some differences between studies, the lattice diffusivities can be summarized as 4b).
Grain growth rates controlled by grain-boundary diffusion in bridgmanite + ferropericlase aggregates (Fei et al., 2021(Fei et al., , 2023) ) were converted to a grain growth factor, k (Equation 1 in Fei et al. ( 2021)), which equates to α GB gr γ IPB δD GB gr in Equation 3.With α GB gr = 1.2 × 10 4 /T (m 3 /J) (see Appendix A), we were able to calculate the value of δD GB gr .To account for the compound effect (Equation 5), we evaluate the estimated δD GB gr values by comparing them to 5δD Si BrGB .The calculated δD GB gr values are about 200 times larger than expected from the Si diffusivity and they are also much smaller than expected from Mg and O diffusivities (Figure 4a).This result does not support our prediction of δD GB gr , which should be approximately equal to the diffusivity of the slowest species, that is, 5δD Si BrGB (Equation 6).At present, there is no good explanation for the fact that the diffusivities predicted from the grain growth rates (Fei et al., 2021(Fei et al., , 2023) ) are larger than the measured self-diffusivities (Dobson et al., 2008;Yamazaki et al., 2000).Parameters for estimating δD comGB and D comLatt over a wide range of pressures and temperatures in the lower mantle are known in more detail from the diffusion studies.Therefore, the present study relies on the previously measured self-diffusivities rather than grain growth rates.Later on, we will discuss how much using different D comLatt values changes our calculation results.
Creep rates of bridgmanite have been measured only during dislocation creep (Tsujino et al., 2022), which precludes a comparison of δD GB cr or D Latt cr with self-diffusivities (Equation 6).In the post-perovskite phase, only the theoretical D i Latt has been studied (Ammann et al., 2010).Dobson et al. (2019) used this D i Latt value to estimate strain rates at the base of the lower mantle with a grain size of 1 mm and a stress of 5 MPa.The chemical relationship between the primary and secondary phases, that is, post-perovskite and ferropericlase, respectively, and their microstructure is expected to be comparable to those of bridgmanite + ferropericlase.Although we are not able to confirm that lower mantle minerals have a common rate-controlling diffusivity from the previous experimental results, we adhere to our previous conclusion of the applicability of δD comGB and D comLatt (Equation 6) to the lower mantle in this study (Okamoto & Hiraga, 2022).

Diffusivity as a Function of Temperature and Pressure
To obtain D comGB and D comLatt values throughout the lower mantle, we need to know how D i changes with temperature and pressure.Diffusivity commonly takes the form where D 0 is a preexponential factor, H is the activation enthalpy for diffusion, R is the gas constant, and T is temperature.H can be expressed as: where Q is the activation energy for diffusion, P is pressure, and Ω is activation volume.The value of Ω as a function of P at very high pressure is often approximated by (Tackley et al., 2013) where V 0 is a reference volume and P 0 is a reference pressure.D i is related to the diffusivity of i vacancies, D v by where X v is the vacancy concentration, D 0v is a preexponential factor for vacancy diffusivity, H f is the enthalpy of vacancy formation, and H m is the migration enthalpy.We provide specific values for D 0v , H f , H m , V 0 , and P 0 based on previously reported theoretical and experimental results to obtain D comGB and D comLatt at various T and P conditions in the lower mantle.

Spatial and Time Scales
We adopted the convection model utilized in Solomatov (1996) and Solomatov et al. (2002)which consists of: (a) a 1-dimensional conceptual convecting fluid, (b) mantle material descending from the top of the lower mantle (660 km depth) to just above the core-mantle boundary (CMB) (2,890 km), (c) horizontal movement of mantle material at the same depth along a distance of 3,540 km, and (d) material finally returning to the upper lower mantle (Figure 5a).The velocity of material movement, v is assumed to be constant over the entire period, and we  15).(c) Temperature-depth profiles (Equation 17).Three different temperature profiles are shown: T 0/+0 K (black), T 300/+100 K (green), and T 600/+200 K (red).The dashed line represents the phase boundary between bridgmanite and post-perovskite (Tsuchiya et al., 2004).
consider cases with v = 0.1, 1, and 10 cm/year in this study.The total travel distance is 8,000 km; thus, a convective velocity of 1 cm/year involves a travel time of 800 million years, including 354 million years for horizontal flow at the base of the mantle.These assumptions are described by the relationships between the depth, z with v and t as

Viscosity Law
We estimated the lower mantle viscosity along the convection line described by Equation 12.We assume a viscosity determined by diffusion creep and static grain growth laws (Equations 2 and 3) using D comGB and D comLatt (Equation 6).Equation 2 can be rewritten to express viscosity at a particular time, t as where the coefficient of 1 3 is used for the conversion to shear viscosity.We used α cr values determined by Equation A1 (see Appendix A) for both bridgmanite and post-perovskite.The value of d(t) is determined as Equation 14 assumes that d(t) is much larger than the initial grain size, that is, ) of 1.57 is calculated from β = 0.7, z = 0.5 and f ΙΙ = 0.2, which should apply to bridgmanite (Figure 2) as well as to post-perovskite.We used an α GB gr of 2.1 × 10 4 /T (m 3 /J) and an α Latt gr of 8.7 × 10 5 /T (m 3 /J), which were determined by Equations A2 and A3 (see Appendix A) for both bridgmanite and postperovskite.We used a γ IPB value of 0.85 J/m 2 .This value was estimated from dihedral angles that formed in ferropericlase versus bridgmanite/bridgmanite triple junctions (Yamazaki et al., 2009) by assuming the same grain boundary energy for bridgmanite and olivine (=1 J/m 2 ) (Cooper & Kohlstedt, 1986).Using Equation 14to elucidate grain size in the lower mantle is essentially the same approach as used by Solomatov et al. (2002), although they used a constant D Latt gr value of 10 18 m 2 /s (Yamazaki et al., 2000), which corresponds to the Si diffusivity at the conditions of the uppermost lower mantle.

Temperature and Pressure
Pressure and temperature change during mantle convection, and hence D comGB and D comLatt change (Equations 8 and 9).Based on the PREM model (Dziewonski & Anderson, 1981), the relationship between z and P is (Figure 5b; Figure S1 in Supporting Information S1) P(GPa) = 3.12 × 10 6 z 2 + 3.90 × 10 2 z 3.62(km). (15) Temperature with respect to the depth in the lower mantle is expected to follow an adiabatic path, T ad (Katsura, 2022), which is closely approximated by (Figure S2 in Supporting Information S1) where ∆T represents the deviation from the mantle reference temperature (i.e., ∆T = 0) during downwelling and upwelling.Equation 16indicates T = 1,960 K at the uppermost lower mantle (i.e., z = 660 km) with ∆T = 0 and an adiabatic temperature gradient of 0.23-0.41K/km (Katsura, 2022).At the lowermost mantle, heat conduction from the core forms a thermal boundary layer, which can be modeled as instantaneous heating of a semi-infinite half-space.With this model, the geotherm from the CMB to the uppermost lower mantle is expressed as where T CMB is the temperature at the CMB, κ is the thermal diffusivity, and t e is the elapsed time.The term 2 ̅̅̅̅̅ ̅ κt e √ represents the thickness of the thermal boundary layer.Kawai and Tsuchiya (2009) calculated shear velocity values for various possible temperature profiles around the D″ region and compared them with seismic observations (e.g., Kawai, Geller, & Fuji, 2007;Kawai, Takeuchi, et al., 2007;Kawai et al., 2009).They found that T CMB and 2 ̅̅̅̅̅ ̅ κt e √ values of 3,500-4,000 K and 200-300 km, respectively, can reasonably explain the observations.In our study, we used a fixed value of 3,800 K for T CMB , which is also consistent with the result obtained by extrapolating the melting point of iron at the inner and outer core boundaries to the CMB using the adiabatic temperature gradient (Alfè et al., 2004) and a value of 200 km for 2 ̅̅̅̅̅ ̅ κt e √ .
We refer to the temperature-depth profile that follows Equations 16 and 17 with ∆T = 0 at depths above the thermal boundary layer as the T 0/+0 K profile and use it as a reference temperature in this study.For cold downwelling mantle and hot upwelling mantle above the thermal boundary layer, we assume their temperatures to be 600 K lower and 200 K higher, respectively, than T 0/+0 K at the same depths (Figure 5c) (i.e., ∆T = 600 and +200 K in Equation 16), and refer to them as T 600/+200 K .This temperature difference in the mantle should be sufficient to account for temperature heterogeneity at the same depths in the mantle, as derived from theoretical models of mantle convection (e.g., Motoki & Ballmer, 2015) and the conversion of seismic velocity to temperature (Bao et al., 2022).To observe the effect of lower mantle temperature heterogeneity, we also consider the case with temperatures 300 K lower and 100 K higher than T 0/+0 K , referred to as T 300/+100 K (Figure 5c).Any lateral temperature variation would be promptly relaxed in the liquid iron at the uppermost outer core, supporting the use of a constant T CMB value, which is the same as T 0/+0 K , at the bottom of the lower mantle, even for the lowermost downwelling and upwelling mantle.In the T 0/+0 K -profile, the temperature gradient at the thermal boundary layer is calculated to be ∼6.5 K/km (Equation 17) (Figure 5c).
Considering the bridgmanite/post-perovskite phase transition at ∼2,750 K and ∼125 GPa with a Clapeyron slope of ∼7.5 MPa/K (Tsuchiya et al., 2004), the bridgmanite/post-perovskite phase transition is predicted to occur at z = 2,610 and 2,710 km in downwelling and upwelling mantle, respectively, in the T 600/+200 K -profile (Figure 5c).These depths do not contradict the depth range of the D″ discontinuities (i.e., 200-300 km above the CMB) found in the S-wave velocity structure (Wysession et al., 1998).
Mantle potential temperature in the past 2.5-3 Gyr is considered to have been a maximum of 1,500-1,600°C, hotter than the present mantle potential temperature of 1,350°C (Herzberg et al., 2010).To estimate the effect of high temperature on grain growth in the ancient Earth, we also consider that grain growth occurs in the temperature profile 200 K higher than T 0/+0 K (referred to as T +200/+200 K ) (Figure 5c, same as the upwelling profile of T 600/+200 K ), assuming the mantle material at a constant depth, that is, no convection.The current temperature profile, T 0/+0 K , was used for viscosity calculations for the current mantle after the grain growth at the high temperatures.

Diffusivities at Lower Mantle Conditions
After the temperature and pressure of the lower mantle have been established as a function of z and hence t, the temperature-and pressure-dependence of the self-diffusivities, D i in the bridgmanite and post-perovskite phases allow us to determine δD comGB and D comLatt (Equation 6) along the convection line (Figure 5a).

Bridgmanite
Ammann et al. ( 2010) conducted calculations of D 0v and H m (Equation 11) for Mg, Si, and O lattice diffusion in bridgmanite and post-perovskite using density functional theory at various pressure conditions.Their theoretically calculated H m value for bridgmanite was found to be nearly equivalent to the H value determined in diffusion experiments (Yamazaki et al., 2000), implying that D i ≈ X 0v D 0v exp H i m RT ) (Equation 11).This suggests that the vacancy concentration is solely determined by the concentration of extrinsic defects, and hence Journal of Geophysical Research: Solid Earth 10.1029/2023JB027803 that X v remains constant at any temperature and pressure conditions.Tackley et al. (2013) assumed Mg lattice diffusion to be the rate-controlling process for lower mantle flow.They fitted the theoretically calculated H Mg m at a particular P and T for lattice diffusion in bridgmanite (Ammann et al., 2010) by varying Q, V 0 , and P 0 to obtain H Mg m as a function of T and P (Equations 9 and 10).They used a reference viscosity (η 0 ) of 3 × 10 23 Pa • s at 1,600 K and 0 Pa, which results in a bridgmanite strength larger by a factor of 10 compared to the upper mantle at the T and P conditions at a depth of 660 km.Tackley et al. (2013) then estimated viscosities throughout the lower mantle using H Mg m (P,T) and Equations 2 and 8).
In this study, we determined H i m following the procedure of Tackley et al. (2013); however, to account for compound effects, H i m was determined for Si and, O as well as for Mg (Equation 4).Earlier diffusion experiments revealed that D O BrLatt is 2 orders of magnitude larger than D Si BrLatt and D Mg BrLatt (Figure 4b) (Dobson et al., 2008;J. Xu et al., 2011) 4).Similar to the approach for Mg in Tackley et al. (2013), we determined the Q, V 0 , and P 0 values for Si lattice diffusion that best explain the previously calculated values of H Si m in bridgmanite (Ammann et al., 2010) (Figure S3a in Supporting Information S1) (Equations 9 and 10).We then fit the results of the diffusion experiments of J. Xu et al. (2011) with Equation 11, using the H Mg m and H Si m values for the pressure at which the diffusion experiments were conducted, and obtained D 0v X 0v values for Mg and Si.
The resulting Arrhenius-type equation for Mg and Si diffusion is shown as solid lines in Figure 4b.
We also used grain-boundary diffusivities determined from diffusion and reaction experiments (Dobson et al., 2008;Nishi et al., 2013;Yamazaki et al., 2000) (Figure 4a).Because no theoretical calculations of grainboundary diffusivity are available, we simply used the Si grain-boundary diffusivity to obtain δD comGB for bridgmanite, which is supported by the previous finding that D Si BrGB ≪ D Mg BrGB and D O BrGB (Dobson et al., 2008;Nishi et al., 2013;Yamazaki et al., 2000).We used a pre-exponential factor D 0 of 7.1 × 10 9 m 2 /s and an activation energy Q of 311 kJ/mol for Si grain-boundary diffusion in bridgmanite, which were determined from diffusion experiments conducted at 25 GPa and 1,673-2,073 K (Yamazaki et al., 2000).The effect of pressure on D Si BrGB is unknown, so we assumed Ω for Si diffusion to be 3.8 cm 3 /mol (Equation 9), which corresponds to that for Mg grain-boundary diffusion estimated from reaction experiments between stishovite and periclase at 24-50 GPa and 2,000 K (Nishi et al., 2013).

Post-Perovskite
Due to the extreme conditions needed for stability of the post-perovskite phase, no diffusion experiments have been performed on the constituent elements of post-perovskite.Thus, we must rely on the results of previous theoretical studies on post-perovskite lattice diffusion to estimate grain growth and creep rates in the lowermost mantle (Ammann et al., 2010;Dobson et al., 2019).Theoretical calculations have shown that the diffusion of Si and Mg in post-perovskite is highly anisotropic, with diffusivities varying by about 8 orders of magnitude between the fastest <100> and slowest <010> directions (Ammann et al., 2010).Similarly to bridgmanite, we estimated the Q, V 0 , P 0 , and D 0 values (Equations 9 and 10) that best explain the pressure (P)-enthalpy (H) relationship for lattice diffusion of each constituent element (i.e., Mg, Si, and O) in each of the three principal directions in post-perovskite (Ammann et al., 2010) (Table 2; Figure S3b in Supporting Information S1).Using these values, and assuming that the X v values are equal to those in bridgmanite (Table 2), we obtained D i for each post-perovskite constituent.All the parameters Q, V 0 , P 0 , and D 0v X v used to determine D i PpvLatt are summarized in Table 2.We followed the method of Dobson et al. (2019) to determine a reasonable D comLatt from the highly anisotropic self-diffusivities in post-perovskite.They concluded from diffusion experiments on a material with a crystal structure similar to post-perovskite that the Voigt-Reuss-Hill (VRH) average can be used to represent the bulk diffusivity.We used the VRH average of D i PpvLatt in three crystallographic directions in post-perovskite (<100>, <010>, and <001>) for Mg, Si, and O (Figure S4 in Supporting Information S1).

Depth Profiles
Once D i as a function of T and P was determined (Equation 11), we converted it to D i as a function of depth.We calculated these D i (z) profiles for bridgmanite and post-perovskite within their stability fields, where T follows the T 0/+0 K and T 600/+200 K -profiles (Figure 6).However, we were unable to plot D i (z) for post-perovskite grain boundaries due to the lack of relevant diffusion data.In the T 0/+0 K -profile, all D i values for bridgmanite gradually decrease with depth, indicating that the effect of pressure to decrease D i prevails over the temperature effect, which increases D i (Figure 6a).Similarly, all the D i Br values, except for D Si BrLatt , decrease gradually with depth in downwelling mantle in the T 600/+200 K -profile (Figure 6b).This profile results in a temperature difference of 800°C between downwelling and upwelling mantle at a certain depth in the bridgmanite zone (Figure 5c), leading to D i that is 4-5 orders of magnitude larger in upwelling mantle compared to that in downwelling mantle (Figure 6b).The differences in D i between bridgmanite and post-perovskite at the same T and P result in a sudden increase in D i of 2-4 orders of magnitude at the depth of the boundary between the bridgmanite and the post-perovskite zones in the T 0/+0 K -profile (Figure 6a).This jump increases to about 4 orders of magnitude in downwelling mantle in the T 600/+200 K -profile, while it decreases by 1-2 orders of magnitude in upwelling mantle (Figure 6b).The D i values for post-perovskite all increase with depth.
All of the calculated D i values were used to obtain δD comGB and D comLatt (Figure 6).The resulting depth profiles essentially follow that of the smallest D i GB and D i Latt at a given depth.Except for in the deepest portion in the bridgmanite zone, D comGB is always larger than D comLatt at a given depth in either downwelling or upwelling  mantle; however, this does not indicate that grain boundary transport dominates over lattice transport.Such a determination is influenced not only by the ratio of D comGB to D comLatt , but also by the grain size (Equation 7), the latter of which is estimated within the scope of this study.

No Convection
First, we examined grain size and viscosity evolution in the case where mantle material remains at a constant depth, that is, no convection (Figure 7).We used the T 0/+0 K -profile (Figure 5c), which was fixed over a period of 45 million, 450 million, or 4.5 billion years.Grain size increases by a factor of 2.2 with each order of magnitude increase in the time period for grain growth (Equation 14), where lattice diffusion controls the growth rate.For a given time, the grain size is larger at shallower depths in the bridgmanite zone, while larger sizes occur at deeper depths in the post-perovskite zone.At the depth of the phase transition from bridgmanite to post-perovskite, the grain size jumps by about a factor of 10.Using the age of the Earth yields grain sizes of up to 5 and 20 mm at the depths of bridgmanite and post-perovskite stability, respectively.We also used T +200/+200 K -profile which was fixed over 4.5 billion years, corresponding to the extreme situation for the grain growth in the ancient hot mantle.Its final grain size is twice as large as the estimate in the T 0/+0 K profile (Figure 7a).The depth of the grain size jump in the T +200/+200 K profile is 25 km deeper than the depth of the grain size jump in the T 0/+0 K profile.This corresponds to the change in depth of the phase transition from bridgmanite to post-perovskite.Using these grain sizes and the calculated diffusivities of δD comGB and D comLatt for the T 0/+0 K -profile (Figure 6a), we calculated the lower mantle viscosity as a function of depth (Equation 13) (Figure 7b).The viscosity increases by a factor of ∼5 with each order of magnitude increase in the grain growth period at all depths.The viscosity increases weakly with depth in the bridgmanite zone, while it decreases sharply in the post-perovskite zone.Using the age of the Earth indicates a maximum viscosity of 3 × 10 20 Pa • s at the bottom of the bridgmanite zone and a minimum value of 10 19 Pa • s at the bottom of the post-perovskite zone.The calculated viscosity of the coarse-grained mantle as the product of the large grain growth in the ancient hot mantle is greater than the viscosity estimate in the  14) (a) and viscosity (Equation 13) (b) when all mantle material remains at a constant depth, that is, no convection.We used the T 0/+0 K -profile (Figure 5c), which was fixed over periods of 45 million, 450 million, and 4.5 billion years.We also used the T +200/+200 K -profile, which was fixed over periods of 4.5 billion years, to calculate the maximum grain size in the ancient hot mantle.
reference temperature profile (T 0/+0 K ) by a factor of 4. The negative and positive viscosity jumps at the depths 2,710 and 2,735 km are due to the phase transition and the grain size jump, respectively.

With Convection
The evolution of grain size and viscosity during mantle convection were calculated at velocities of 0.1, 1, and 10 cm/year in the T 600/+200 K -profile (Equations 13 and 14) (Figure 8).After the phase transition to form bridgmanite at 660 km depth, the grain size soon reaches a certain value that remains almost constant until the downwelling material reaches the depth of the phase boundary between bridgmanite and post-perovskite (Figure 8a).Slower convection speeds increase the size at a given depth, but reducing the convection speed by 1 order of magnitude only increases grain size by a factor of ∼2.As a result, the bridgmanite grain size reaches a range of 20-100 μm at rates of 0.1-10 cm/year in the downwelling mantle.Grain growth rates are determined by grain-boundary diffusion at depths shallower than ∼800 km, and by lattice diffusion at deeper depths.Immediately after the phase transition from bridgmanite to post-perovskite at 2,610 km, large grain growth occurs with an increase in size of about twofold.Grain growth continues during downwelling to the bottom of the lower mantle, allowing growth to sizes of 1-4 mm at rates of 0.1-10 cm/year.Further grain growth occurs during horizontal flow at the base of the lower mantle, eventually reaching a grain size of 4-20 mm.This grain size does not change during subsequent upwelling until the material reaches the top of the lower mantle.
Using these grain sizes and calculated diffusivities for the T 600/+200 K -profile (Figure 6b), we calculated lower mantle viscosities as a function of depth (Equation 13) (Figure 8b).After the phase transition to bridgmanite at 660 km, the viscosity is ∼10 19 Pa • s at a velocity of 1 cm/year in the downwelling mantle, with viscosity increasing/ decreasing by a factor of ∼5 as the convection velocity is reduced/increased by 1 order of magnitude.The viscosity increases rapidly with depth during downwelling, reaching ∼10 20 Pa • s at 1,000 km and increasing slightly to 10 21 Pa • s at greater depth until the bridgmanite-post-perovskite phase transition occurs.Immediately after the downwelling material experiences the phase transition, the viscosity decreases rapidly to ∼10 18 Pa • s, with viscosity again increasing/decreasing by a factor of ∼5 as the convection velocity is reduced/increased by 1 order of magnitude.The viscosity increases slightly with depth and then decreases, eventually reaching 2 × 10 16 -  14) (a) and viscosity (Equation 13) (b) during mantle convection at velocities of 0.1, 1, and 10 cm/year in the T 600/+200 K -profile (Figure 5c).Arrows indicate the directions of material migration in the lower mantle.Profiles with dotted and solid curves correspond to the regions where grain growth and creep are each controlled by grain-boundary diffusion and lattice diffusion, respectively.
5 × 10 17 Pa • s at the bottom of the lower mantle at rates of 0.1-10 cm/year.The viscosity then increases continuously during horizontal flow along the base of the lower mantle and can be as much as 20 times higher by the time horizontal flow ceases.During ascent from the bottom of the lower mantle with a rate of 1 cm/year, the viscosity further increases to ∼3 × 10 19 Pa • s, the maximum viscosity in the post-perovskite zone.The phase transition from post-perovskite to bridgmanite causes a rapid increase in viscosity of up to 2 orders of magnitude.A small rise in the viscosity during upwelling occurs for ∼100 km after the phase transition depth, which produces the maximum viscosity, estimated as ∼6 × 10 21 Pa • s at a rate of 1 cm/year in upwelling mantle.Further upwelling decreases the viscosity monotonically, reaching ∼2 × 10 19 Pa • s at the top of the lower mantle with viscosity increasing/decreasing by a factor of ∼5 as the convection velocity is reduced/increased by 1 order of magnitude.
To see the effect of temperature differences between downwelling and upwelling mantle, we also calculated grain size (Figure 9a) and viscosity (Figure 9b) for T 300/+100 K and T 0/+0 K (Figure 5c) at a rate of 1 cm/year.The different temperature profiles essentially only affect grain size in downwelling mantle (Figure 9a), where the grain size can increase by up to 1 mm at T 0/+0 K , which is 20 times larger than in the case of T 600/+200 K in the bridgmanite zone.In upwelling mantle, the grain size is constant at 9 mm, irrespective of the temperature profile.At T 0/+0 K , the downwelling mantle is weaker than the upwelling mantle (Figure 9b).The viscosity difference between the softer downwelling mantle and the stiffer upwelling mantle lessens as the temperature difference increases.At T 600/+200 K , the viscosity of downwelling mantle becomes almost equal to that of upwelling mantle.

Diffusion Mechanism
With decreasing grain size, grain-boundary diffusion increasingly dominates over lattice diffusion (Equation 7).This situation is only seen in the early stage of downwelling in the bridgmanite zone, where grain size is smaller than ∼60 μm (Figure 8a), while grain growth produces larger sizes elsewhere.The point where the controlling process switches from grain-boundary diffusion to lattice diffusion occurs at depths of 700-900 km, leading to a change in the depth dependence of viscosity.In the post-perovskite zone, grain growth and creep rates have only  14) and viscosity (b) (Equation 13) during mantle convection in different temperature profiles: T 0/+0 K , T 300/+100 K , and T 600/+200 K .The convection rate is fixed at 1 cm/year.Arrows indicate the direction of material migration in the lower mantle.Profiles with dotted and solid curves correspond to the regions where grain growth and creep are each controlled by grain-boundary diffusion and lattice diffusion, respectively.
been estimated from lattice diffusivity, leaving the possibility that grain-boundary diffusion might dominate.In that case, since both grain-boundary and lattice diffusion mechanisms are coupled in parallel, the predicted viscosity would be lower than that shown in Figure 8b in the post-perovskite zone.

Grain Size
We assumed the value of d 0 to be 0 μm for the grain growth calculation (Equation 14).The grain size immediately reached ∼10 μm, even in the T 600/+200 K -profile (Figure 8a), which is consistent with the grain size at the topmost lower mantle estimated based on the grain growth rate determined in Fei et al. (2021).This rapid grain growth of bridgmanite at very small grain sizes supports our assumption of the initial grain size.The differences in grain growth results in this study are mainly due to differences in temperature and the grain-growth period.Higher temperatures at a given depth (Figure 9a) and slower convection velocities, that is, a longer grain-growth period (Figure 8a), both contribute to increasing grain size.
The diffusivity governing bridgmanite grain growth decreases slightly with depth, whereas the diffusivity of postperovskite is much greater than that of bridgmanite and increases rapidly with depth (Figure 6).Thus, during mantle convection, grain growth is practically restricted to a limited depth range (660 km-1,000 km) in the bridgmanite zone and to depths where downwelling and horizontal flow occur in the post-perovskite zone (Figures 8a and 9a).As a result, grain size in the bridgmanite zone does not vary strongly with depth and is estimated to be around 50 μm for the downwelling and around 10 mm for the upwelling mantle in the T 600/ +200 K -profile, for example, (Figure 8a).
Different temperature profiles result in significantly different grain sizes in the downwelling mantle (Figure 9a).However, since T CMB is constant across the different temperature profiles (Equation 17), grain growth in the postperovskite zone results in the same grain sizes for different temperature profiles, for instance, 2 mm at the bottom of the lower mantle at a convection rate of 1 cm/year (Figure 9a).Grain growth in the post-perovskite zone is so rapid that no additional increase in grain size occurs during subsequent upwelling in any temperature profile.
The calculated grain size in the upwelling mantle is even larger than that after the grain growth in the static mantle over the age of the Earth in the T +200/+200 K -profile (Figures 7a and 9a); this finding indicates that the movement of mantle by convection limited in the bridgmanite zone does not significantly increase grain size.The grain size in the post-perovskite zone increases with residence time in the zone, such that the mantle material remaining in that zone over the age of the Earth could have grain sizes of up to ∼20 mm (Figure 7a).Solomatov and Reese (2008) estimated the grain size to be 1-10 cm immediately after the crystallization in the magma ocean which probably existed before the sub-solidus mantle was formed.Even in the T +200/+200 K -profile for 4.5 billion years, our estimated grain size in the static mantle is less than 1 cm at most depths (Figure 7a), indicating that grain growth after the crystallization in the magma ocean is negligible.The mantle, which has not experienced the phase transition from the upper mantle material to bridgmanite + ferropericlase throughout Earth's history, may keep the ancient grain sizes that record the crystallization in the magma.

Viscosity at a Constant Grain Size
First, we discuss the characteristics of the lower mantle viscosity at a constant grain size, that is, no grain growth, to observe the sensitivity of mantle viscosity to diffusivity alone (Figure 6b).Our analysis predicts minimal grain growth throughout the entire mantle upwelling stage (Figure 8a).At depths where post-perovskite is stable, the mantle viscosity increases rapidly during upwelling (Figure 8b) because the decrease in diffusivity with decreasing temperature outweighs the increase in diffusivity from decreasing pressure (Equations 8 and 9) (Figure 6b) in a large temperature gradient.At depths where bridgmanite is stable, the mantle viscosity gradually decreases during upwelling (Figure 8b).This is because under a small temperature gradient, the pressure effect outweighs the temperature effect and increases the diffusivity.The viscosity change in downwelling mantle varies with depth similarly to upwelling mantle (Figure 8b).
The sensitivity of diffusivity to temperature is apparent in the viscosity differences in upwelling mantle between different temperature profiles at the same depth (Equation 8).For example, the viscosity of upwelling mantle in the bridgmanite zone in the T 600/+200 K -profile is an order of magnitude lower than in the T 0/+0 K -profile, where temperature is 200 K lower, at the same grain size (Figure 9b).The pressure effect on the diffusivity is demonstrated by the differences in upwelling mantle viscosity with the same temperatures in the T 0/+0 K -profile, which appear at a depth of 700 km greater than in the T 600/+200 K -profile, for example, (Figure 5c).This increase in depth produces a viscosity increase of ∼2 orders of magnitude, solely due to the effect of pressure on the diffusivity (Equations 8 and 9) (Figure 9b).
Given that the initial grain sizes for grain growth are given by the sizes of the crystallization in the magma, we estimated essentially zero grain growth throughout the Earth's history (Equation 3).The initial grain size of 1 cm at all depths gives the viscosity profile in the bridgmanite zone as essentially the same as that of the upwelling mantle in the T 0/+0 K -profile (Figure 9b).

Viscosity in Conjunction With Grain Growth
A larger diffusivity alone would increase creep rates, but would also lead to increased grain growth, resulting in a larger grain size, which in turn reduces the creep rate.Solomatov (1996Solomatov ( , 2001) ) specifically considered the effect of temperature on both creep and grain growth and proposed the possibility of hot mantle being more viscous than cold mantle.This combined effect is represented by the apparent enthalpy of creep, H app , in which a negative value indicates material strengthening due to enhanced diffusion in both creep and grain growth.Our assessment that a common rate-controlling diffusivity exists for both grain growth and creep gives H app = H/m (where H is the activation enthalpy common to grain growth and creep and m = 3 and 4 for lattice-diffusion control and grainboundary-diffusion control, respectively) (Okamoto & Hiraga, 2022).H app is small but always positive, which consequently results in generally lower viscosities for larger diffusivities (Figure 9b).Strictly speaking, the temperature-and pressure-dependence of viscosity, represented by H app , is based on the assumption that the ratecontrolling diffusivity is constant over time, which is only valid for the non-convecting scenario (Figure 7).Indeed, viscosity is lower at depths with larger grain sizes.
This trend continues into the convective mantle (Figure 8); however, it is limited during downwelling in the postperovskite zone, where grain growth is enhanced due to the increment of the diffusivity with depth.We found that the viscosities of the cold downwelling mantle and the hot upwelling mantle are almost the same or that the downwelling mantle is likely to be less viscous than the upwelling mantle (Figure 9b).The increment of the viscosity due to significant grain growth during the period of horizontal flow at the bottom of the lower mantle (Figure 9a) overwhelms the viscosity reduction due to temperature increases in the upwelling mantle.

Comparisons With Geophysical Estimates of Lower Mantle Viscosity
We compared our calculated viscosity profile with various previous geophysical estimates of lower mantle viscosity (Figure 10).Mitrovica and Forte (2004) proposed a viscosity of ∼10 21 Pa • s at shallow depths that increases monotonically with depth to ∼10 23 Pa • s at a depth of ∼2,000 km.Their predicted values then decrease with depth up to the bridgmanite-post-perovskite phase transition, below which the post-perovskite zone has a viscosity of ∼10 20 Pa • s, much lower than that of the bridgmanite zone.Newer studies presented viscosity profiles with lower values by 1-2 orders of magnitude particularly in the 1,000-2,000 km depth range (Argus et al., 2021;Lau et al., 2016).The result of Lau et al. (2016) shows almost monotonic increase of the viscosity to ∼10 22 23 Pa • s toward the post-perovskite zone (Figure 10).These estimates do not consider lateral viscosity heterogeneity.Čížková et al. (2012) showed that a slab sinking rate of 1.2 ± 0.3 cm/year estimated from global seismic tomography (van der Meer et al., 2010) is best explained by the surrounding mantle having a viscosity of ∼10 22 Pa • s.
Based on plume shapes identified by seismic tomography, Korenaga (2005) estimated that mantle plumes are either similar or more viscous than their surroundings, which differs from the general view of hot, and hence less viscous, plumes.His estimated viscosity ranges from 10 21 23 to 10 20 22 Pa • s, in the bridgmanite and post-perovskite zones, respectively.Nakada et al. (2012) estimated very low viscosity at the bottom of the lower mantle, around 10 16 17 Pa • s, based on the decay time of the Chandler wobble and tidal deformation of the Earth (Figure 10).
The range of viscosity that dynamic mantle deforming by diffusion-creep mechanism can have is colored gray (Figure 10).The dynamic mantle corresponds to the mantle with the convection velocity of 1 cm/year (Figure 9b), which is comparable to the estimated sinking rate of a slab (Domeier et al., 2016;van der Meer et al., 2010).The T 0/+0 K -and T 600/+200 K -profiles as the constraint on the temperature range of the lower mantle give the viscosity range consisting of stronger upwelling mantle and weaker downwelling mantle.
A mantle plume viscosity of ∼10 21 Pa • s (Korenaga, 2005) is consistent with the properties of upwelling mantle in our model.A plume that is stiffer or similar in strength to its surrounding (Korenaga, 2005) can also Journal of Geophysical Research: Solid Earth 10.1029/2023JB027803 be explained by our results of the relative viscosity of downwelling and upwelling mantle.These small lateral changes in mantle viscosity and an even weaker downwelling mantle than the surroundings are characteristic results of this study.Korenaga's and our proposals differ from the orthodox view of a hot and hence weak upwelling mantle; however, a recent geophysical modeling study (Yang & Gurnis, 2016) shows that the slab is weaker than its surroundings, which can be interpreted as the same conclusion as ours.A rather large range of viscosities have been estimated for the base of the lower mantle.The lowest estimate, 10 16 17 Pa • s (Nakada et al., 2012), is reasonably well explained by our calculated mantle viscosity in the early stage of horizontal flow, where grain size is small (Figure 8a).Significant grain growth during horizontal flow and/or a jump in viscosity from post-perovskite to bridgmanite where D″ layer is absent (Wysession et al., 1998) would bring the viscosity closer to the larger estimated values of around 10 20 22 Pa • s (Lau et al., 2016;Mitrovica & Forte, 2004).With the exception of Mitrovica & Forte's (2004) estimate of high viscosity at 1,000-2,000 km depth, the viscosity estimates based on geophysical modeling are within our possible viscosity range for dynamic mantle (Figure 10).Within the range, the geophysical viscosity estimates roughly follow the viscosity in the T 600/+200 K -profile that gives almost similar viscosities of downwelling and upwelling mantle.Small deviations that are often larger viscosity in the geophysical estimates than the viscosity in the T 600/+200 K -profile require our estimations for the stronger upwelling mantle.
It is also possible that the geophysical viscosity estimates correspond to the viscosity of the mantle that is not involved in mantle convection or that has not experienced a bridgmanite↔post-perovskite phase transition during mantle convection.We estimate the viscosity in these cases from the viscosity of the mantle that has been stationary for 4.5 billion years in the T 0/+0 K -and T +200/+200 K -profiles.The viscosity is relatively low compared to the dynamic mantle and deviates from what would be expected to explain the larger viscosity estimated from geophysical modeling as in Mitrovica and Forte (2004) compared to our predictions (Figure 10).Our calculation shows that grain growth under the static conditions is too slow to reach sizes (i.e., ∼3 cm) that give high viscosity particularly at the shallow depths (i.e., 1,000-2,000 km) in Mitrovica and Forte (2004).In a recent study (Fei et al., 2023), portions of the lower mantle with high viscosity were considered to have a small fraction of ferropericlase (i.e., f II < 0.05-0.1)that promotes grain growth via reduced grain boundary pinning, which in turn increases mantle viscosity.Based on the relationship d ∝ f II 0.5 (Equation 3), at T +200/+200 K for 4.5 billion years and at the shallow depths of 1,000-2,000 km, the final grain size is 6-20 mm, even for a very small ferropericlase fraction of 2 vol%.The effect of ferropericlase fraction is too small to explain the high viscosity of Mitrovica and Forte (2004).Only grain sizes that have recorded crystallization in the magma ocean appear to correspond to the large grain sizes, while it is only applicable to static mantle (Solomatov & Reese, 2008).
Obviously, our dynamic and static models of the mantle are extreme situations, and the mantle can be any in between, including parts of the dynamic mantle that do not reach the CMB and experience significant grain growth.We have shown that the viscosity of the static mantle essentially ranges within the lateral viscosity range of the dynamic mantle (Figure 10).This fact does not contradict with the dynamic mantle viscosity providing reasonable explanations for geophysical estimates of mantle viscosity except for the high viscosity of Mitrovica and Forte (2004).Among different viscosity estimates by geophysical modeling, our study prefers the estimates of low viscous lower mantle and does not support the proposal of Mitrovica and Forte (2004).modeling with our estimates based on mantle convection at a velocity of 1 cm/year for the T 0/+0 K -and T 600/+200 K -profiles.Our calculated viscosity range for dynamic mantle based on diffusion creep is shown in a gray hatch.We also show the viscosity of static mantle (i.e., no migration) for a T 0/+0 K -and T +200/+200 K -profile for 4.5 billion years.Red arrows indicate the direction of material movement in the lower mantle.The solid black line (Mitrovica & Forte, 2004) and dashed black line (Lau et al., 2016) denote the mantle viscosity estimated from geophysical observations and modeling; open circles denote the plume viscosities estimated by Korenaga (2005) (The depth of each plume, which is taken from Montelli et al. (2004), is the deepest depth at which the tomographic image of each plume is observed.);the black double arrow denotes the estimated viscosity near the base of the lower mantle based on the decay time of the Chandler wobble and Earth's tidal deformation (Nakada et al., 2012).

Quantifying and Overcoming Uncertainties
Because our study relies heavily on the diffusivities of lower mantle minerals, the uncertainty in our conclusions is largely a product of uncertainties in the diffusivity values used, especially those of post-perovskite.These values have been theoretically derived with the aid of several assumptions and lack experimental support.Our conclusions are based on the best available knowledge of the transport properties of the mineral at the present time and should be refined and improved by future studies.It should be noted that the combined effect of grain growth and creep will result in viscosity being proportional to D Latt 1/ 3 (Equations 2 and 3); thus, the viscosity values will change less than D PpvLatt .
Given the uncertainties in the diffusivities for post-perovskite, an alternative approach is to consider the grain sizes observed in natural peridotite as the likely size at the bottom of the lower mantle.Grain growth at the base of the lower mantle may be rapid enough that grain size will remain unchanged until the material reaches the top of the upper mantle.This scenario is partially supported by our grain growth calculation that indicates essentially zero growth during upwelling from the bottom of the lower mantle (Figure 8a).Solomatov and Reese (2008) considered the evolution of grain size during the phase transition that occurs during upwelling from the top of the lower mantle to the upper mantle.This transition is characterized by MgSiO 3 + MgO → Mg 2 SiO 4 and should begin at interphase boundaries between MgSiO 3 and MgO grains.In this reaction, Mg 2 SiO 4 grains will grow rapidly until MgO is consumed, and the Mg 2 SiO 4 grains will eventually be surrounded by the transformed but unreacted low-pressure MgSiO 3 phase and/or other secondary phases.The final microstructure will correspond to that controlled by Zener pinning (Figure 1b), with a final grain size equal to or twice the size of bridgmanite.This consideration led Solomatov and Reese (2008) to determine that the 1-10 mm grain sizes observed in natural peridotite correspond to the size produced by grain growth at the bottom of the lower mantle.Here, we also consider two natural grain size observations of 1-3 mm (Harigane et al., 2011;Liu et al., 2019) and 7 mm (Ave Lallemant et al., 1980).The former is the olivine grain size identified in coarse-grained mantle xenoliths that likely constituted oceanic upper mantle.As olivine and pyroxene grain sizes have been shown to follow the Zener law (Equation 1) (Yabe & Hiraga, 2020), this size is consistent with our assumptions for the microstructure resulting from grain growth in a polymineralic system.The latter coarse grain size represents olivine grains in mantle xenoliths from the African craton that would have constituted continental upper mantle.Olivine grain size is depth-independent and nearly constant from the top to the bottom of thick (i.e., >100 km) lithosphere (Ave Lallemant et al., 1980).Chu and Korenaga (2012) noted that this size was too small to be controlled by the stress loaded throughout the tectonic history of the craton and attributed it to Zener pinning by secondary phases.For a 1 mm grain size at all depths in the bridgmanite zone, the viscosity becomes 9 2 times lower than that at a 9 mm grain size (Figure 8), which is too low to be consistent with the geophysical estimates (Figure 10).In fact, for a 1 mm grain size in mantle rising from the post-perovskite zone at a rate of 1 cm/year in the T 600/+200 K -profile, grain growth occurs in the bridgmanite zone to bring the size to ∼2 mm at the top of the lower mantle (Figure S5 in Supporting Information S1).Thus, a grain size smaller than 1 mm is required to produce a 1 mm grain size at the top of the lower mantle, resulting in an even more unrealistically low viscosity for the upwelling mantle.In contrast, a 7 mm grain size rising from the post-perovskite layer is large enough to prevent further grain growth in the bridgmanite zone.The viscosity associated with this size is less than ∼2 times lower than the estimated viscosity based on grain growth modeling in the post-perovskite layer (Figure 10), which may support the estimate of the lower mantle viscosity from this approach.
Previously, we obtained the viscosity at the bottom of the lower mantle by assuming a 1 mm grain size at the bottom of the lower mantle (Okamoto & Hiraga, 2022).The temperature and pressure in the horizontal direction in the lowermost mantle are constant, so that the rate-controlling diffusivity for grain growth and creep is likewise constant, yielding a viscosity simply determined by grain size and the time it takes to reach that size (i.e., a grain size-time viscometer) (Okamoto & Hiraga, 2022).If we give the viscometer a grain size of 9 mm instead of 1 mm and a time of 350 million years (=3,540 km ÷ 1 cm/year) to reach this size, which corresponds to the period of horizontal flow at the bottom of the lower mantle (Figure 5a), we obtain ∼5 × 10 18 Pa • s.This viscosity corresponds to that of mantle material just before it rises.Here, we obtained a value of 2 × 10 18 Pa • s at a velocity of 1 cm/year with the T 600/+200 K -profile (Figure 8b), confirming the validity of the viscometer presented previously.

Roles of Other Deformation Mechanisms
In this study, we have imposed deformation by a diffusion creep mechanism on the entire mantle to estimate the lower mantle viscosity.We have recently shown that power-law creep appears at lower stresses and smaller grain size conditions than for diffusion creep in olivine (Nakakoji et al., 2018;Yabe et al., 2020).This creep corresponds to interface-controlled diffusion creep, where the reaction rate of vacancy creation and absorption at grain boundaries determines the creep rate.Interface-controlled diffusion creep can contribute to increasing viscosity because it proceeds sequentially with respect to normal diffusion creep.At present, it is difficult to predict its operation in lower mantle minerals, and if present, it is likely to be in fine-grained (e.g., 10 μm) downwelling mantle (Figure 8a).Tsujino et al. (2022) performed uniaxial deformation experiments on polycrystalline bridgmanite at temperatures of 1,473 to 1,673 K and pressures of 23-27 GPa.They constructed a deformation mechanism map for a temperature of 1,900 K and a pressure of 25 GPa and proposed that the uppermost lower mantle flows by dislocation creep when stresses are larger than 2 × 10 4 -3 × 10 5 Pa and grain sizes are larger than 3-8 mm.They used a similar lattice diffusivity for bridgmanite to that in this study to estimate lower mantle viscosity (Figure 4b); however, they did not consider the compound effect (Equation 5).Further, they used an α cr value (Equation 2) that is 10 times smaller than the value used in this study.As a result,

Conclusions
Our finding of a common rate-controlling diffusivity for grain growth and diffusion creep in polymineralic systems has allowed for a more reliable estimate of lower mantle rheology under the assumption of mantle deformation by a diffusion creep mechanism.Using the best-known diffusivities and diffusion parameters of bridgmanite and post-perovskite phases, we estimated grain sizes in the lower mantle through grain growth modeling.These grain sizes and diffusivities provided the viscosity of the convective mantle.Our best estimates for possible range of the grain size in the lower mantle, where lateral temperatures can be 600 and +200 K from the reference temperature profile (i.e., mantle geotherm), are summarized as follows.Grain size in downwelling mantle in the bridgmanite zone is 20-400 μm.Grain size in the post-perovskite zone varies widely from 0.1 to 1 mm upon entering the post-perovskite zone, to 9 mm just before mantle ascent from the bottom of the lower mantle.Grain size in the upwelling mantle of the bridgmanite zone remains constant at 9 mm, which is comparable to the size observed in coarse-grained mantle peridotite.Based on these grain sizes, the best estimate of possible range of the lower mantle viscosity is summarized as follows.Downwelling mantle experiences a monotonic increase in viscosity from 4 × 10 17 -10 19 Pa • s to 5 × 10 20 -4 × 10 21 Pa • s with increasing depth from 700 to 2,600 km in the bridgmanite zone, an abrupt viscosity reduction to ∼10 18 Pa • s just beneath the depth of the bridgmanite/post-perovskite phase transition, followed by a nearly monotonic decrease in viscosity to 10 17 Pa • s at the base of the lower mantle, which is the lowest value for the entire lower mantle.There is roughly an order of magnitude increase in viscosity up to 2 × 10 18 Pa • s during horizontal flow at the bottom of the lower mantle.Mantle ascent from the bottom of the lower mantle is accompanied by increasing viscosity, which brings the mantle to ∼(3-7) × 10 19 Pa • s just beneath the depth of the bridgmanite/post-perovskite phase transition.
Immediately after bridgmanite formation, the viscosity is expected to increase rapidly to 6 × 10 21 -4 × 10 22 Pa • s, corresponding to the most viscous portion of the upwelling mantle.A monotonic decrease in viscosity then occurs with decreasing depth in upwelling mantle, reaching ∼2 × 10 19 -10 20 Pa • s at the top of the lower mantle.These viscosities indicate that fine-grained cold downwelling mantle is often softer than coarse-grained hot upwelling mantle.Furthermore, lateral viscosity variations cannot exceed 2 orders of magnitude and become smaller as the lateral temperature difference increases.For a temperature difference of 800 K, downwelling mantle is almost as viscous as upwelling mantle at any given depth.These estimates are in agreement with the vertical and lateral changes in viscosity as well as the absolute viscosity values estimated by geophysical modeling.We conclude that this mineral-physics approach to lower mantle viscosity can reveal fine-scale viscosity structures that have been difficult to resolve using geophysical modeling.
Indeed, this study is based on the simplifying assumption that the lower mantle deforms primarily by diffusion creep.While other deformation mechanisms may be involved in mantle flow, they should mainly act to reduce the viscosity, based on our estimates.We have shown that this is unnecessary.We proposed that most of the Earth deforms by diffusion creep, which we termed the superplastic (diffusion creep) Earth hypothesis (Maruyama & Hiraga, 2017b).This hypothesis has been challenged by the general understanding that diffusion creep cannot generate CPO, and thus cannot produce seismic anisotropy.We have demonstrated the mechanism by which CPO develops during diffusion creep, which involves grain rotation due to preferential GBS on grain boundaries parallel to low-index crystallographic planes (Maruyama & Hiraga, 2017b).Grain rotation occurs for grains with both crystallographically controlled (anisotropic) and isotropic shapes, with the former and the latter developing CPO and uniform (random) fabrics, respectively.We have discussed elsewhere how bridgmanite and postperovskite are likely to exhibit isotropic (Yamazaki et al., 2009;Yoshino & Yamazaki, 2007) and anisotropic shapes (McCormack et al., 2011;Yoshino & Yamazaki, 2007), respectively.Anisotropic grains can form from isotropic grains through oriented growth of grains that become two times larger than the original size (Kim et al., 2022).We have shown that such grain growth is easily established in the post-perovskite zone, which explains the observed transition from a seismically isotropic bridgmanite zone to a seismically anisotropic postperovskite zone.Bridgmanite grains that back-transform from post-perovskite during upwelling can inherit such CPO; however, their shapes are predicted to be isotropic, and their CPO can be easily removed by grain rotation during subsequent diffusion creep (Maruyama & Hiraga, 2017b).
The superplastic Earth hypothesis has also been challenged by the fact that diffusion creep is a grain-sizedependent mechanism, and therefore its operation is limited to rocks with smaller grain sizes.The very high temperatures of up to 3,500 K and very long residence times of several hundred million years in the lower mantle and/or the crystallization in the ancient magma ocean are favorable for grain growth, which promotes larger grain sizes.This fact implies that the mantle is not likely to deform primarily by a diffusion creep mechanism.Our studies have demonstrated that the grain size effect is minimized by the phase transition and the polymineralic nature of deep Earth materials, and that the resulting grain sizes are small enough to suitably explain the lower mantle viscosity.The diffusion of elements in both primary and secondary phases over a distance equal to the primary phase grain size (Okamoto & Hiraga, 2022) suppresses grain growth more effectively than in monomineralic systems as grain size increases.Additionally, the common rate-controlling diffusivity for grain growth and creep in polymineralic rocks minimizes the sensitivity of the mantle to the extreme conditions of the Earth's interior.The wide-ranging temperature and pressure values in the lower mantle produce rate-controlling diffusivities spanning 5 orders of magnitude in the bridgmanite zone.Higher diffusivity alone would increase the creep rate, but it would also increase grain growth rate, resulting in larger grain size, which in turn reduces the creep rate.Grain growth at the bottom of the lower mantle exaggerates this trade-off, reducing the effect of the large diffusivity change in the bridgmanite zone on mantle viscosity, and even produces hot but stronger upwelling mantle relative to cold downwelling mantle.We propose that the lower mantle viscosity is a unique consequence of superplastic (diffusion creep) polymineralic Earth.

Figure 1 .
Figure 1.Microstructure of a synthetic bridgmanite or post-perovskite (Br or Ppv) + (ferro-)periclase (Pc or Fp) aggregate.(a) Back-scattered electron image of the aggregate after the phase transition from synthetic forsterite.The sample was annealed at a temperature of 1,573 K and a pressure of 25 GPa for 10 hr after the phase transition took place (modified fromYamazaki et al., 1996).Bridgmanite and periclase grains appear as light and dark gray, respectively.(b) Schematic illustration of the microstructure.

Figure 3 .
Figure 3. Schematic illustration of diffusion processes that are required for diffusion creep and for grain growth (ripening) of the secondary phase (modified after Okamoto & Hiraga, 2022).Gray and white grains represent primary and secondary phases, respectively.(a) Diffusion creep proceeds by diffusion of components contained in both primary and secondary phases.The direction of compression is indicated by large arrows.(b) Grain growth proceeds by diffusion of components in the secondary phase (solid arrows) and counter-diffusion of components in the primary phase (dashed arrows).Small arrows indicate either growth or shrinkage of the secondary phase.

Figure 4 .
Figure 4. (a) Plots of Si, Mg, and O grain-boundary diffusivity in bridgmanite against inverse temperature.(b) Plots of Si, Mg, and O lattice diffusivity in bridgmanite against inverse temperature.The solid line represents a least-squares fit to the data using the Arrhenius equation obtained from theoretical calculations(Ammann et al., 2009).

Figure 5 .
Figure 5. Parameters for modeling the evolution of grain size and viscosity during mantle convection in this study.(a) Travel distances with a schematic of possible microstructure changes in the lower mantle.(b) Pressure-depth profile (Equation15).(c) Temperature-depth profiles (Equation17).Three different temperature profiles are shown: T 0/+0 K (black), T 300/+100 K (green), and T 600/+200 K (red).The dashed line represents the phase boundary between bridgmanite and post-perovskite(Tsuchiya et al., 2004).

Figure 6 .
Figure 6.Depth profiles of D i , D comGB , and D comLatt for bridgmanite and post-perovskite along the T 0/+0 K -profile (a) and T 600/+200 K -profile (b).A grain boundary width of 1 nm was assumed in order to determine D i GB .

Figure 7 .
Figure 7. Depth profiles of grain size (Equation14) (a) and viscosity (Equation13) (b) when all mantle material remains at a constant depth, that is, no convection.We used the T 0/+0 K -profile (Figure5c), which was fixed over periods of 45 million, 450 million, and 4.5 billion years.We also used the T +200/+200 K -profile, which was fixed over periods of 4.5 billion years, to calculate the maximum grain size in the ancient hot mantle.

Figure 8 .
Figure 8. Depth profiles of grain size (Equation14) (a) and viscosity (Equation13) (b) during mantle convection at velocities of 0.1, 1, and 10 cm/year in the T 600/+200 K -profile (Figure5c).Arrows indicate the directions of material migration in the lower mantle.Profiles with dotted and solid curves correspond to the regions where grain growth and creep are each controlled by grain-boundary diffusion and lattice diffusion, respectively.

Figure 9 .
Figure 9. Depth profiles of grain size (a) (Equation14) and viscosity (b) (Equation13) during mantle convection in different temperature profiles: T 0/+0 K , T 300/+100 K , and T 600/+200 K .The convection rate is fixed at 1 cm/year.Arrows indicate the direction of material migration in the lower mantle.Profiles with dotted and solid curves correspond to the regions where grain growth and creep are each controlled by grain-boundary diffusion and lattice diffusion, respectively.

Figure 10 .
Figure10.Comparison of viscosities previously estimated by geophysical modeling with our estimates based on mantle convection at a velocity of 1 cm/year for the T 0/+0 K -and T 600/+200 K -profiles.Our calculated viscosity range for dynamic mantle based on diffusion creep is shown in a gray hatch.We also show the viscosity of static mantle (i.e., no migration) for a T 0/+0 K -and T +200/+200 K -profile for 4.5 billion years.Red arrows indicate the direction of material movement in the lower mantle.The solid black line(Mitrovica & Forte, 2004) and dashed black line(Lau et al., 2016) denote the mantle viscosity estimated from geophysical observations and modeling; open circles denote the plume viscosities estimated byKorenaga (2005) (The depth of each plume, which is taken fromMontelli et al. (2004), is the deepest depth at which the tomographic image of each plume is observed.);the black double arrow denotes the estimated viscosity near the base of the lower mantle based on the decay time of the Chandler wobble and Earth's tidal deformation(Nakada et al., 2012).

Table 1
Notation and Material Properties (Ammann et al., 2010)determined temperature dependence of D i BrLatt indicates that H O Latt is larger than H Si Latt and H Mg Latt .These results indicate that D O BrLatt is larger compared to D Si BrLatt and D Mg BrLatt , even at temperatures higher than those used in the experiments.D O Latt is theoretically predicted to be much larger than D Si Latt and D Mg Latt under pressures of up to 135 GPa(Ammann et al., 2010).For these reasons, we decided to disregard D O BrLatt and used only D Si BrLatt and D Mg BrLatt to determine the value of D comLatt for bridgmanite (Equation
Tsujino et al. (2022)calculated a diffusion creep rate 50 times slower than used in our study at the same conditions.The transition from diffusion creep to dislocation creep should occur at a grain size ∼7 times larger than the size estimated byTsujino et al. (2022), that is, 20-50 mm.Dislocation processes operate in parallel with diffusion creep.Using a ∼10 mm grain size as our maximum size estimate, we conclude that diffusion creep is dominant over dislocation creep in most of the lower mantle.This does not contradict the existence of a seismically isotropic bridgmanite zone.