Journal of Earth Science  2017, Vol. 28 Issue (4): 563-577 PDF     0
On the Numerical Modeling of the Deep Mantle Water Cycle in Global-Scale Mantle Dynamics: The Effects of the Water Solubility Limit of Lower Mantle Minerals
Takashi Nakagawa
Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, Yokohama 236-0001, Japan
Abstract: Water is the most important component in Earth system evolution. Here, I review the current understanding of the fate of water in the mantle dynamics system based on high-pressure and temperature experiments, geochemical analyses, seismological and geomagnetic observations, and numerical modeling of both regional-and global-scale mantle dynamics. In addition, as a numerical example, I show that the water solubility of the deep mantle is strongly sensitive to global-scale water circulation in the mantle. In a numerical example shown here, water solubility maps as functions of temperature and pressure are extremely important for revealing the hydrous structures in both the mantle transition zone and the deep mantle. Particularly, the water solubility limit of lower mantle minerals should be not so large as ~100 ppm for the mantle transition zone to get the largest hydrous reservoir in the global-scale mantle dynamics system. This result is consistent with the current view of mantle water circulation provided by mineral physics, which is also found as a hydrous basaltic crust in the deep mantle and the water enhancement of the mantle transition zone simultaneously. In this paper, I also discuss some unresolved issues associated with mantle water circulation, its influence on the onset and stability of plate motion, and the requirements for developing Earth system evolution in mantle dynamics simulations.
Keywords: numerical modeling    mantle convection    water cycle    water solubility limit
0 INTRODUCTION-CURRENT UNDERSTANDING OF WATER CIRCULATION IN THE EARTH'S SYSTEM

Determining the water budget in the Earth's system is one of the most important topics in the interaction between fluid and solid Earth sciences in terms of the coevolution of surface climate change and the dynamics of the Earth's deep interior (Foley and Driscoll, 2016; Franck et al., 2002; Tajika and Matsui, 1992; McGovern and Schubert, 1989). The existence of water under the surface environment is the most important component in the formation of the first bio-related material (e.g., Nisbet and Sleep, 2001), and the conditions necessary for the existence of water under the surface environment are determined by the thermal state of the atmosphere and ocean evolution (Hamano et al., 2013; Abe and Matsui, 1988). The Earth is only a 'habitable planet' because a water ocean can exist on a stable surface environment. A stable surface ocean may be supported by plate tectonics, which acts as the heat engine and is driven by mantle convection and geomagnetic field generation by the geodynamo resulting from the thermal and chemical convection of the Earth's core (Nakagawa and Tackley, 2015; Korenaga, 2011; Franck et al., 2002; Tajika and Matsui, 1992). In conclusion, the existence of liquid water on the Earth's surface due to tectonic plates driven by vigorous mantle convection is a key to understanding why the Earth is a 'habitable planet'; however, this issue remains controversial (Zahnle et al., 2007; Gaidos et al., 2005).

Although the water cycle and budget of the surface environment of a habitable planet such as the Earth is now well established based on the combination of observational data, geological evidence, ocean-atmosphere energytransfer computations, numerical simulations of the global atmosphere and ocean circulation and climate data analyses (e.g., Trenberth et al., 2009; Timm et al., 2008), the fate of water across deep Earth is not comprehensively understood. A few schematic images of the fate of water in the Earth's deep interior have been produced by integrating various geochemical analyses and high-pressure and temperature mineral physics experiments (see the review articles by Karato (2011) and Ohtani (2005)). One important finding water in the Earth's deep interior is that the mantle transition zone stores a huge amount of water (e.g., Pearson et al., 2014), which seems to be filtered by a partially molten region produced by the reduction of the melting temperature and the density cross-over between partially molten material and solid material of hydrous mantle material at the uppermost mantle transition zone (400 km); this zone is known as the 'Transition Zone Water Filter' (Bercovici and Karato, 2003). In addition, other high-pressure and temperature mineral physics studies have shown the possibility of forming hydrous basaltic piles (Mashino et al., 2016; Nishi et al., 2014; Ohira et al., 2014; Ohtani et al., 2014; Ohtani and Maeda, 2001). However, controversial seismological images of hydrous conditions in the mantle transition zone have been obtained from receiver function approach and tomographic images (Houser, 2016; Schmandt et al., 2014). Geomagnetic observations performed to compute the electrical conductivity of the Earth's mantle have also detected water or some fluid in the mantle transition zone (e.g., Matsuno et al., 2017). Regardless of whether the condition in the mantle transition zone is hydrous or less hydrous, a key physical phenomenon of water circulation in the mantle based on these findings is that the plate tectonics driven by mantle convection (in particular, plate subductions) plays a significant role with transporting the water from the surface environment to the deep Earth's interior (Iwamori, 2007; Maruyama and Okamoto, 2007).

Regarding numerical modeling of mantle water circulation, several models are available that focus on the dynamics of the wedge mantle (Wilson et al., 2014; van Keken et al., 2011; Iwamori, 2007; Rüpke et al., 2004). In particular, Iwamori (2007) included a water solubility limit of hydrous mantle minerals in the upper mantle (Figs. 1a and 1b) to compute the excess water transport (dehydration process) based on a mass balance approach and found that the existence of a 'choke point' is a key for understanding the global-scale mantle water circulation; this choke point is a critical pressure for low water solubility limit even in temperature ranges corresponding to subducting plates appeared at approximately 5 GPa (Kawamoto, 2006; Iwamori, 2004; Komabayashi et al., 2004). For an instance of including such effects caused by water solubility limit, tomographic images obtained beneath the Honshu region of Japan (Nakajima and Hasegawa, 2007) suggest that the excess water generation associated with the water solubility limit of hydrous mantle minerals seems to be crucial for elucidating the generation of volcanic chains beneath the island arc, which is consistent with simple dynamics model of wedge mantle (e.g., Iwamori, 2007). If the slab temperature is higher than the choke point, the water stored in the oceanic crust/lithosphere should be released to a shallower region, whereas if the slab temperature is lower than the choke point, this water can be transported by the hydrous oceanic crust/lithosphere into the deep mantle. This effect is also evidenced by the results of mantle wedge dynamics modeled as two-phase flow physics (Wilson et al., 2014). Based on the first numerical mantle convection simulations, which were performed in terms of a global-scale water transport process, Richard et al. (2002) indicated that the mantle transition zone could store a huge amount of water in a hydrous mantle convection system with iso-viscosity and a simplified water solubility limit (i.e., not as a function of temperature). Fujita and Ogawa (2013) used a coupled magmatism-mantle convection system with dehydration melting to elucidate the global-scale mantle water circulation. They found hydrous basaltic piles above the core-mantle boundary (CMB), which were caused by the segregation of hydrous oceanic crust. However, these models did not consider the effect of the choke point of the water solubility limit of the hydrous mantle mineral shown in Iwamori (2007). Nakagawa et al. (2015) attempted to simulate global-scale mantle water circulation in a numerical thermo-chemical mantle convection model with water solubility maps. In this model, the evolution of the mantle water content is strongly regulated by the effect of the choke point and is maintained in the range of 1.2 to 1.5 ocean masses of mantle water content (1 ocean mass=1.4×1021 kg (Genda, 2016)). This evolution profile is very different from those of other global-scale mantle water circulation computed by semi-analytical approaches, which indicate the gradual increase of the mantle water content caused by the regassing-degassing balance of mantle water content but not excess water transportation (Crowley et al., 2011; Sandu et al., 2011). Thus, again, the water solubility limit of hydrous mantle minerals is a key property for understanding the global-scale mantle water circulation. However, a water solubility is limited to the temperature and pressure ranges of upper mantle minerals. Indeed, a water solubility limit has been a greatly uncertain in lower mantle temperature and pressure ranges because of the difficulty of obtaining experimental measurements. In the following section, the uncertainty in the water solubility of the lower mantle minerals is discussed.

 Download: larger image Figure 1. Water solubility maps of upper mantle rocks taken from Iwamori (2007). (a) Ambient mantle; (b) oceanic crust; (c) adiabatic temperature; and (d) water solubility profiles across the whole mantle, which vary with the water solubility of the lower mantle.
1 UNCERTAINTY IN THE SOLUBILITY LIMIT OF THE LOWER MANTLE

As mentioned in the previous section, the solubility limit of hydrous mantle minerals is an important component for understanding the dynamics of the wedge mantle, which can generate volcanic chains in the island arc (Iwamori, 2007). The solubility limit of hydrous mantle minerals has only been established for upper mantle minerals, as shown in Figs. 1a and 1b (e.g., Iwamori, 2004; Komabayashi et al., 2004). Note that those water solubility maps are indicated as the 'maximum' water solubility of upper mantle minerals but, the water solubility limits are also evaluated from many laboratory experiments at high pressure and temperature, which seems to be a factor of five smaller solubility compared to those in Figs. 1a and 1b (e.g., Hirschmann and Kohlstedt, 2012; Jacobsen and Smyth, 2006; Inoue et al., 2004; Kohlstedt et al., 1996) but not significantly changed for the schematic image on mantle water cycle because those mineral physics experiments still claim that the mantle transition zone may have a large water solubility. In the deep lower mantle, the water solubility of lower mantle minerals is not well constrained because of huge discrepancies resulting from experimental difficulties. For instance, Murakami et al. (2002) suggested that the water solubility of bridgmanite might be approximately 0.2 wt.%, which means that the lower mantle could store several ocean masses of water, whereas a different experimental study reported that as little as tens of ppm could be stored in the lower mantle minerals (Bolfan-Casanova, 2005). Recent results suggest that the water solubility of lower mantle minerals could be as low as tens to hundreds of ppm and that this value could not be resolved further because of uncertainty (Panero et al., 2015; Karato, 2011). Figure 1d shows profiles of the water solubility limit across the entire mantle, assuming three characteristic values of water solubility for the lower mantle and an adiabatic temperature profile (Fig. 1c). Moreover, the recent discovery of a hydrous mantle mineral that may be stable under the lower mantle conditions (Phase-H) has increased the uncertainty surrounding the lower mantle water solubility (Nishi et al., 2014; Ohtani et al., 2014), as does the high value of water partitioning of the post-perovskite phase found in the lowermost mantle (Townsend et al., 2016).

Because of these uncertainties, no numerical model of global-scale mantle dynamics is currently available that can fully describe water circulation in the Earth's deep interior. However, such information is inferred from subduction-scale numerical models (Nakao et al., 2016; Iwamori and Nakakuki, 2013; van Keken et al., 2011; Iwamori, 2007; Rüpke et al., 2006) and even a global-scale mantle convection model (Nakagawa and Spiegelman, 2017; Nakagawa et al., 2015). In Nakagawa et al. (2015) and Nakagawa and Spiegelman (2017), they used pressure and temperature dependence of water solubility of upper mantle hydrous minerals. For lower mantle minerals, they used a constant value (100 ppm (Karato, 2011)) that depends on temperature and pressure and the assumption of zero solubility above the solidus temperature. In the next section, I discuss the effect of the uncertainty in the water solubility limit of lower mantle minerals on global-scale water circulation in the mantle convection system.

2 AN EXAMPLE OF GLOBAL-SCALE MANTLE WATER CIRCULATION-EFFECTS OF THE WATER SOLUBILITY OF THE LOWER MANTLE 2.1 Brief Description of the Numerical Model of Hydrous Mantle Convection

Here, an example of a mantle watercycle in mantle convection simulations that affects the water solubility of the lower mantle is introduced. Again, the water solubility maps of upper mantle minerals are shown in Figs. 1a and 1b and of lower mantle minerals are shown in Fig. 1d. The details of the numerical model can be found in Nakagawa et al. (2015) and Nakagawa and Spiegelman (2017). Briefly, the mantle is modeled as a compressible and truncated an elastic fluid with the temperature-, pressure-, yield strength-and water content-dependent viscosity given as

 ${\eta _{\rm{d}}} = {A_{\rm{d}}}\sum\nolimits_{i,j = 1}^{n{\rm{phase = 3,4}}} {\Delta \eta _{ij}^{{\Gamma _{ij}}f}} {\rm{exp}}\left[ {\frac{{{E_{\rm{d}}} + p{V_{\rm{d}}}}}{{RT}}} \right]$ (1)
 ${\eta _{\rm{w}}} = {A_{\rm{w}}}{\left( {\frac{{{C_{\rm{w}}}}}{{{C_{{\rm{w,ref}}}}}}} \right)^{ - r}}\sum\nolimits_{i,j = 1}^{n{\rm{phase = 3,4}}} {\Delta \eta _{ij}^{{\Gamma _{ij}}f}} {\rm{exp}}\left[ {\frac{{{E_{\rm{w}}} + p{V_{\rm{w}}}}}{{RT}}} \right]$ (2)
 ${\eta _Y} = \frac{{{C_Y} + \mu p}}{{2\dot e}}$ (3)
 $\eta = {\left( {\frac{1}{{{\eta _{\rm{d}}}}} + \frac{1}{{{\eta _{\rm{w}}}}} + \frac{1}{{{\eta _{Y{\rm{d}}}}}}} \right)^{ - 1}}$ (4)

respectively, where Ad, w is the prefactor determined by T=1 600 K and the surface (subscripts d and w indicate dry and hydrous mantle, respectively); Ed, w is the activation energy; Vd, w is the activation volume; Cw is the water content in the mantle; Cw, ref is the reference water content, which is assumed to be 620 ppm (Arcay et al., 2005); r is the exponent of the prefactor dependence of the water content, which ranges from 0.3 to 2.0 (Fei et al., 2013; Korenaga and Karato, 2008; Mei and Kohlstedt, 2000) (in this study, I assume values of 0.3 and 1.0, where 0.3 indicates 'weakly water-dependent viscosity', and 1.0 indicates 'strongly water-dependent viscosity'); Γij is the phase function; f is the basaltic composition (varies from 0 to 1); R is the gas constant (=8.314 J·K·mol-1); T is the temperature; p is the pressure; CY is the surface-cohesion factor; μ is the friction coefficient; ${\dot e}$ is the second invariant of the strain-rate tensor; p is the pressure, and Δηij is the viscosity jump associated with the phase transition, which is assumed to increase 30 times for the bridgmanite transition. Note that subscripts i (up to 3) and j (up to 4) in Eqs. (1) and (2) indicate the index of the phase transition from deeper to shallower transitions (i=j=1: post-perovskite phase boundary; i=j=2: bridgmanite transition boundary; i=3: spinel phase boundary; j=3: garnet phase boundary; j=4: basalt-eclogite transition boundary), as described by Keller and Tackley (2009). Here the water dependent viscosity is formulated with actual water content in the mantle but, generally, water in the mantle minerals should be found in -OH atomic group. To convert actual water content to OH content, there is an empirical law derived from high P-T experiments (Li et al., 2008). However, this empirical law seems to be only applicable for the continental and oceanic crust not for the deep mantle materials. To avoid such a difficult, I use the reference water content (Cw, ref). The density is also changed with the mantle water content given as

 $\rho = \rho \left( {{T_{{\rm{ad}}}},C,p} \right)\left( {1 - a\left( {{T_{{\rm{ad}}}},C,p} \right)\left( {T - {T_{{\rm{ad}}}}} \right)} \right) - \Delta {\rho _{\rm{w}}}{C_{\rm{w}}}$ (5)

where ρ(Tad, C, p) is the combined reference density between ambient mantle and basaltic material (here assumed as MORB: mid-oceanic-ridge-basalt) compositions with 2.7% density difference, Tad is the adiabatic temperature, C is the chemical composition, and Δρw is the density change due to mantle water content. The density of hydrous mantle minerals is generally less dense than that of dry mantle minerals, which ranges 0.1% to 1% (Mao et al., 2008; Wang et al., 2006; Inoue et al., 1998). The density reduction of hydrous mantle minerals is assumed as 0.125 wt.% with Cw=1 (100 wt.% of mantle water content) but recent estimate seems to be reduced for ~1.4 wt.% with 1 wt.% of mantle water content (e.g., Ye et al., 2012). Here I assume very small contribution to the density reduction due to mantle water content.

The mantle water circulation can be computed as follows. The transport equation of water in the mantle is given as

 $\frac{{\partial {C_{\rm{w}}}}}{{\partial t}} + u \cdot \nabla C = \frac{{D{C_{\rm{w}}}}}{{Dt}} = {S_{\rm{w}}}\left( {{F_R},{F_G},{F_E}} \right)$ (6)

where u is the convective velocity, and Sw is the source-sink term related to regassing (water entrance caused by plate subductions), dehydration (i.e., non-magmatic excess water migration that is computed by the subtraction of actual water content from water solubility limit), and degassing (including dehydration melting; the water can be removed from the mantle convection system due to the volcanic eruptions). A schematic diagram of the numerical implementation of non-magmatic water migration is shown in Fig. 2. To compute source-sink water fluxes, the conventional equations are

 $\left\{ {\begin{array}{*{20}{l}} {{\rho _{\rm{m}}}{u_z}{C_{\rm{w}}}}&{\left( {{\rm{if}}\;{u_z} < 0} \right)}\\ 0&{\left( {{\rm{if}}\;{u_z} \ge 0} \right)} \end{array}} \right.$ (7)
 ${F_G} = \frac{d}{{dt}}\left( {{\rho _{\rm{m}}}{C_{{\rm{w,erupt}}}}} \right)$ (8)
 ${F_E} = {\rho _m}{C_{{\rm{w,ex}}}}{u_{\rm{f}}}$ (9)
 Download: larger image Figure 2. Schematic diagram of the numerical implementation of the non-magmatic excess water migration process. Time goes from left to right. If the water content in a certain grid cell exceeds the water solubility corresponding to the temperature and pressure in the grid, excess water (actual water content in excess of the grid-cell water solubility) should move upward, and the water content after the excess water moves upward is equal to the water solubility because the density of an aqueous fluid is generally lower than that of the surrounding mantle rock.

where ρm is the mantle density ${{\dot M}_{{\rm{erupt}}}}$; Cw is the water content; uz is the radial velocity; Cw, erupt is the water content of the erupted material, which is equivalent to the water content of the molten material (Eq. (9)); Cw, ex is the excess water content relative to the water solubility; and uf is the conventional form of fluid migration velocity expressed numerically, which is computed as the ratio of the radial grid spacing to the time step size (Nakagawa and Spiegelman, 2017; Nakao et al., 2016). I also assume that the partitioning of water into partially molten material is given as

 ${C_{{\rm{w,erupt}}}} = \frac{{{C_{{\rm{w,0}}}}}}{{{D_{\rm{w}}} + f\left( {1 - {D_{\rm{w}}}} \right)}}$ (10)
 ${C_{{\rm{w,solid}}}} = {D_{\rm{w}}}{C_{{\rm{w,melt}}}}$ (11)

where Cw, melt is the water content of the molten material; Cw, solid is the water content of the solid mantle; Dw is the partition coefficient of water into the solid, which is assumed to be 0.01 here (Aubaud et al., 2008; Kohn and Grant, 2006); f is the melt fraction; and Cw, 0 is the bulk water content of the solid plus the melt. The solidus temperature of the mantle material is assumed to be reduced by the mantle water content according to the scaling law provided by Katz et al. (2003).

By combining the water migration process described above with the numerical code of thermo-chemical mantle convection 'StagYY' (Tackley, 2008) (some details can be found in Nakagawa et al. (2015) and Nakagawa and Spiegelman (2017)), I can simulate a thermal-chemical-viscous-hydrous structure in the global-scale mantle dynamics. To solve the governing equations of mantle convection, I use the numerical code 'StagYY' (Tackley, 2008). I assume a 2-D spherical annulus geometry (Hernlund and Tackley, 2008) and the reference state formulated as described by Tackley (1996) with parameters from Nakagawa and Tackley (2011). The numerical resolution used is 1 024×128, with 4 million tracers tracking the chemical composition, melt fraction and water content via the tracerratio method. The melt-induced material differentiation allows the creation of the oceanic crust as in Christensen and Hofmann (1994) and Xie and Tackley (2004). The average spatial resolution is ~15 km. I integrate up to 4.6 billion years from the initial conditions, which are assumed to be adiabatic temperature (2 000 K at the surface) plus thin thermal boundary layers with small random perturbations, 20% basaltic materials and zero water content in the mantle. Fixed temperatures are applied for the surface and the CMB: 300 and 4 000 K, respectively. For the boundary condition of the water content, I assume the maximum water content in rocks under surface pressure and temperature conditions (6 wt.% water content in the basaltic material, which is constant over time because the surface temperature is assumed as a fixed temperature). Because of the viscosity formulation with the boundary condition of mantle water transport, the viscosity at the surface is automatically reduced by a factor of up to 102 because of the surface water content, which is equivalent to imposing weak hydrous oceanic crust at the surface. This boundary condition represents the surface water ocean that could be found on the early Earth with an inventory of 1 ocean mass after the solidification of the surface magma ocean (e.g., Hamano et al., 2013). The initial condition of the mantle water content may not influence the long-term mantle water content evolution, as discussed in Nakagawa and Spiegelman (2017). All physical parameters used in this study are listed in Table 1.

Table 1 Mantle model physical parameters
2.2 Numerical Results 2.2.1 Sensitivity of the lower mantle solubility limit

Figures 3 and 4 show that the thermal-chemical-viscous-hydrous structure obtained in global-scale mantle dynamics simulations varies with the water solubility of the lower mantle minerals for two strengths of rheological properties of hydrous mantle minerals. Generally, as the water solubility of the lower mantle increases, the water content in the lower mantle increases. This behavior does not strongly affect the strength of the rheological properties of the hydrous minerals, but more heterogeneous features of the mantle water content can be found for weakly water-dependent viscosity than for strongly water-dependent viscosity. As shown in Nakagawa et al. (2015), since cases with strongly water-dependent viscosity may have higher numbers of subducting boundaries than those with weakly water-dependent viscosity, more water can be transported via plate-like behavior in the former. However, thermal and chemical structures in the deep mantle differ substantially between weakly and strongly water-dependent viscosity. For a weakly water-dependent viscosity, large-scale thermal and chemical structures can be found in the deep mantle, whereas for a strongly water-dependent viscosity, thick and strong cold downwelling flow is found from the surface to the deep mantle. As a result, the length scale of the chemically dense piles is not as large as that for weakly water-dependent viscosity cases; this result does not change when the water solubility of the lower mantle is varied.

 Download: larger image Figure 3. Viscous-chemical-hydrous structure for r=0.3 (weakly water-dependent viscosity) taken at 4.6 Ga after the initial state.
 Download: larger image Figure 4. Viscous-chemical-hydrous structure for r=1.0 (strongly water-dependent viscosity) taken at 4.6 Ga after the initial state.

Figure 5 shows the time-series plots of mantle temperature and mantle water content. As noted by Nakagawa et al. (2015), the rheological feedback to temperature, which is enhanced for the heat transfer caused by the viscosity reduction due to water-dependent viscosity and, as a result, the mantle temperature is more effectively cooled down compared to the dry mantle situation, is much stronger for cases with strongly water-dependent viscosity than for weakly water-dependent viscosity, but the profiles do not differ substantially for different water solubility limits of the lower mantle. Thus, the heat transfer may be controlled by the surface plate motion rather than the convective dynamics in the lower mantle. The largest mantle water content corresponds to the largest water solubility limit of the lower mantle. However, for strongly water-dependent viscosity, this difference is not very clear because the mantle temperature is much lower than that for weakly water-dependent viscosity. Thus, as shown in the solubility maps (Fig. 1), the cold downwelling slabs in the hydrous oceanic crust can transport water into the mantle transition zone without non-magmatic excess water generation caused by the choke point. Therefore, the choice of the water solubility of the lower mantle exerts limited effects on the mass-averaged diagnostics. However, as described below, a relatively small value of the water solubility limit of the lower mantle is preferable.

 Download: larger image Figure 5. Mass-averaged mantle temperature and water content as functions of time, (a) weakly water-dependent viscosity and (b) strongly water-dependent viscosity.

To determine the optimal value of the water solubility limit of the lower mantle, the hydrous structure as a function of depth and its time variations are very useful. Figure 6 shows the time dependence of 1-D horizontally averaged profiles of mantle water content taken from 0.5 to 4.6 Ga. For the first 1 to 2 Ga, the mantle transition zone cannot store the water transported from cold subducting slabs because the temperature of the cold subducting slabs is not lower than the choke point at a certain pressure; thus, the water stored in the subducted oceanic crust is mostly released at a depth corresponding to the choke point. As the mantle temperature decreases, the temperature of the cold subducting slabs is likely to be lower than the choke point, and thus, the hydrous oceanic crust can transport the water into the mantle transition zone. As a result, the water enhancement of the mantle transition zone can occur except for a case with weakly water-dependent viscosity and the larger water solubility limit of lower mantle (a plot at the top in Fig. 6b) because the hydrous state of lower mantle seems to be still under-saturated situation and the lower mantle mineral can still absorb the water. This seems to be caused by the activity of plate subduction that may transport the water in deep mantle and smaller amount of water transport found in weakly water-dependent viscosity due to smaller number of plate boundaries compared to strongly water-dependent viscosity (Nakagawa et al., 2015). Therefore, to generate the water enhancement of the mantle transition zone, a smaller water solubility of the lower mantle is slightly preferable. However, for revealing the Transition Zone Water Filter proposed from simple theoretical model and high pressure experiments, the density cross-over caused by hydrous silicate melt is required to prove such a hypothesis (Bercovici and Karato, 2003) but not included in this study. Thus, the smaller water solubility of lower mantle minerals is required for the water enhancement in the mantle transition zone but not sufficient for proving the Transition Zone Water Filter hypothesis. Again, to prove a hypothesis, we should include the phase relationship of hydrous silicate melt.

 Download: larger image Figure 6. 1-D horizontally averaged mantle water content and its time dependence, (a) 0.01 wt.% water solubility of the lower mantle, (b) 0.2 wt.%, and (c) 0.001 wt.%. Top: Weakly water-dependent viscosity; bottom: strongly water-dependent viscosity.
2.2.2 Hydrous basalt in the deep mantle

As suggested by high-pressure experiments involving hydrous mantle minerals (Mashino et al., 2016; Ohira et al., 2014; Ohtani et al., 2014; Ohtani and Maeda, 2001), the hydrous basaltic crust seems to be transported into the deep mantle. Fujita and Ogawa (2013) suggested that hydrous basalt could be found in the deep mantle, but that the numerical model was not fully accurate regarding the water solubility limit of the deep mantle minerals because of the large uncertainty. To examine this issue, Fig. 7 shows the distributions of mantle water content for both weakly and strongly water-dependent viscosity (total water content, water in the ambient mantle and water in the basaltic material). In basaltic material, relatively high water content can clearly be observed the deep mantle for weakly water-dependent viscosity but not as clearly for strongly water-dependent viscosity. However, the basaltic water content is obscured by the water stored in the ambient mantle material. Thus, whether hydrous basaltic piles exist in the total water content is difficult to determine.

 Download: larger image Figure 7. Correlation between the mantle compositional structure and water content. The mantle water content is decomposed into the ambient mantle (third column) and basaltic material (right column). Top: Weakly water-dependent viscosity; bottom: strongly water-dependent viscosity.
2.2.3 With or without non-magmatic water migration

For the global-scale water circulation in the mantle in the parameterized mantle model, the mantle water content is described as a balance between regassing and degassing, but non-magmatic water migration (i.e., the effects of water solubility maps that are functions of temperature and pressure) is ignored (Crowley et al., 2011; Sandu et al., 2011). Figure 8 shows the mantle water distribution and 1-D horizontally averaged mantle water content with and without the non-magmatic water migration process. Apparently, the mantle water content without non-magmatic water migration remains nearly constant at 1.0 wt.%, except between 100 and 150 km, which corresponds to the partially molten zone (Nakagawa and Spigelman, 2017). No water enhancement can be found in the mantle transition zone, as indicated in Fig. 6. However, when the effects of non-magmatic water migration are included, the mantle water content is strongly regulated by the water solubility map and can develop a structure. To understand global water circulation in the mantle and its hydrous structure, the water solubility map of mantle minerals is crucial for identifying the water-enhancement zones in both the mantle transition zone and the deep mantle simultaneously. Figures 9 and 10 show the source-sink flux (determined by computing the surface integrals of Eqs. (7) and (9) for regassing and non-magmatic excess water migration and the volume integral of Eq. (8) for degassing) and mass-averaged mantle water content evolution with and without non-magmatic excess water migration. Clearly, non-magmatic excess water migration strongly regulates the water circulation system across the mantle (Figs. 9a, 9c, 10a, 10c). With non-magmatic excess water migration (Figs. 9a and 10a), the mantle water content appears to be explainable by a balance between regassing and non-magmatic excess water transportation, and degassing flux plays a minor role in this system. Without non-magmatic excess water migration (Figs. 9b and 10b), degassing seems to have a major influence on the mantle water content. However, such degassing events are likely to occur within a depth of 150 km, and the water content in the deep mantle is not strongly influenced by these degassing events (Nakagawa and Spiegelman, 2017). Therefore, the mantle water content appears increase monotonically with time (Figs. 9c and 10c). Again, the regulation of the mantle water content caused by non-magmatic excess water migration is a critical process for revealing the dynamics and evolution of hydrous mantle convection.

 Download: larger image Figure 8. Comparison of the water content with and without non-magmatic excess water transport. (a) 1-D horizontally averaged mantle water content for weakly water-dependent viscosity; (b) same as (a) but assuming strongly water-dependent viscosity; (c) mantle water content with non-magmatic excess water transport (dehydration process; left: weakly water-dependent viscosity; right: strongly water-dependent viscosity); (d) same as (c) but without a dehydration process.
 Download: larger image Figure 9. Source-sink water flux profiles and mass-averaged mantle water content as functions of time for weakly water-dependent viscosity. (a) Source-sink fluxes with non-magmatic excess water migration (described as dehydration here), (b) source-sink fluxes without non-magmatic excess water migration, and (c) mass-averaged mantle water content.
 Download: larger image Figure 10. Source-sink water flux profiles and mass-averaged mantle water content as functions of time for strongly water-dependent viscosity. (a) Source-sink fluxes with non-magmatic excess water migration (described as dehydration here), (b) source-sink fluxes without non-magmatic excess water migration, and (c) mass-averaged mantle water content.
3 SUMMARY AND DISCUSSION 3.1 Global Mantle Water Cycle with the Effects of the Water Solubility Limit of the Lower Mantle

In this paper, I introduce relevant information from mineral physics and geochemistry for the numerical modeling of water circulation in the Earth's mantle. In summary, to construct the global-scale mantle water cycle in numerical mantle convection simulations, the following information (at a minimum) is required: 1. water solubility maps, 2. the rheological properties of hydrous mantle minerals, and 3. the partition coefficients of water into partially molten material, which may include three components of the water migration process from the surface to the deep interior (regassing) or from the deep interior to the surface (degassing and non-magmatic excess water migration). By using the developed numerical model of hydrous mantle convection for simulations, the effects of the water solubility limit of the lower mantle minerals are introduced through a numerical example. For a large amount of water to be stored in the mantle transition zone, the water solubility limit of the lower mantle should not be very large; a value of approximately 100 ppm is preferable (Karato, 2011) over the 0.2 wt.% value suggested by earlier experiments under high temperature and pressure (Murakami et al., 2002). In addition, high water solubility limit of the lower mantle minerals seems to be unfavorable because, in such a case, the mass-averaged mantle water content is too large to maintain the mass of the surface water ocean over the geological time scale (~1 billion years) and seems to be difficult to generate the hydrous mantle transition zone. From this perspective, the preferable water solubility limit of the lower mantle minerals is, again, on the order of 100 ppm or less. Moreover, non-magmatic water migration is crucial for understanding the hydrous structure of the deep mantle because the mantle water content seems to be nearly uniform when the non-magmatic water migration of water solubility maps is not included. Based on the various results from mineral physics (e.g., Karato, 2011 for review), the hydrous structure in the Earth's mantle is, at least, heterogeneous. In addition, hydrous basaltic crust can be found in the deep mantle, which is a key for the water migration in the global-scale mantle dynamics system and also appears to be consistent with the findings from mineral physics experiments (Ohtani, 2005 for review). However, the water solubility limit of the lower mantle minerals remains uncertain because, according to the recent discovery of a new hydrous phase that seems to be stable under the lower mantle pressure with coldslab geotherms (Nishi et al., 2014; Ohtani et al., 2014), the current picture of mantle water circulation should not change substantially. Thus, the main scenario of mantle water circulation should not be substantially altered.

3.2 Future Perspectives-Plate Tectonics and Earth System Evolution 3.2.1 Plate tectonics

As mentioned in Section 0, plate tectonics is one of the key processes that makes the Earth a 'habitable planet'. However, the plate tectonics in the mantle dynamics model remains an unresolved issue in the numerical modeling community.

First, the rheological properties of mantle rocks are typically implemented with 'pseudo-plastic' yielding to generate plate-like behavior in global-scale mantle convection simulations (Foley and Becker, 2009; van Heck and Tackley, 2008; Tackley, 2000a, b; Moresi and Solomatov, 1998; Trampert and Hansen, 1998). For regional-scale models, such as mantle wedge scale or spreading center models, more complicated rheological properties of mantle rocks are used to reproduce more realistic plate subduction (Gerya et al., 2008) and transform faults into mid-oceanic ridges (Gerya, 2012). However, these complicated rheological properties are not used in global-scale mantle dynamics modeling because of the huge computational costs (i.e., requirement of huge numbers of numerical grids). If both numerical techniques and the computational infrastructure were to be greatly improved, more complicated and realistic rheological properties of the mantle material would become applicable. To obtain a more realistic plate boundary, more realistic boundary conditions are also important, such as those used in the 'free surface' approach. Gerya et al. (2008) and Crameri et al. (2012) applied the 'sticky air' approach, in which a thin and less viscous layer is set in a few grids at the surface. As a result, 'single-sided' subduction can be reproduced in some parameter ranges for both regional-and global-scale mantle dynamics modeling. In the future, 'free surface' boundary conditions will likely be a key for understanding more realistic plate motion with mantle water circulation. In addition, the grain-size dependence of rheological properties and historical dependence of mantle rock (Bercovici and Ricard, 2016, 2014) are essential for achieving more realistic surface plate motions.

Second, the onset time and stability of plate tectonics have been deeply debated in the past several years. The onset time of plate tectonics ranges from 2.5 to 4.3 Ga according to geological records (Condie, 2016; Hopkins et al., 2008). The stability of plate tectonics suggests two time scales (long: ~1 billion years; short: several tens to hundreds of Ma) of episodic behavior (van Hunen and Moyen, 2012; O'Neill et al., 2007). In global-scale mantle convection simulations, the plate-like behavior seems to begin approximately 60 Ma after the solidification of the magma ocean, and the short time scale episodic behavior is essential for understanding some constraints on the thermal history of the core and mantle (Nakagawa and Tackley, 2015). When the effects of water are included, the onset time is much earlier, but short time scale episodic behavior is still exhibited (Nakagawa and Spiegelman, 2017). As discussed in relation to regional-scale modeling (Gerya et al., 2015), complicated and realistic rheological properties of mantle rocks are required for the onset of plate motion in mantle convection model, in addition to the surface boundary condition. In current global-scale mantle dynamics modeling, the yield strength of the oceanic lithosphere is assumed to be between several tens of MPa and hundreds of MPa at the surface, which is much weaker than the value measured in laboratory experiments (up to 1 GPa) (e.g., Kohlstedt et al., 1995). To obtain a more robust value for the onset time of plate motion, more complicated and realistic rheological properties must be included in global-scale mantle dynamics modeling.

Third, the continental lithosphere is needed to reveal the onset and stability of plate tectonics (Rey et al., 2014; Coltice et al., 2012; Rolf et al., 2012), but the formation mechanism of the continental crust is not included in the global-scale model because it requires a complicated thermodynamic formulation (de Smet et al., 1998) and remains under debate in the geochemical community (e.g., Kelemen and Behn, 2016).

3.2.2 Earth system evolution

Interactions between the solid and fluid Earth systems might be addressed by varying the surface boundary condition of the mantle dynamics model. The surface temperature can be determined through radiative-convective equilibrium modeling of the atmosphere (Nakajima et al., 1992), and the surface water ocean must be obviously finite in size. Several simple semi-analytical models have been presented for the interaction between the exosphere (including either surface temperature evolution or the evolution of the amount of surface water ocean) and deep Earth's interior. However, this interaction remains poorly understood because the models are too simplified to determine the effects of mantle dynamics (Korenaga, 2011; Rüpke et al., 2004; Franck et al., 2002; Tajika and Matsui, 1992; McGovern and Schubert, 1989) or because results are only available for Venus rather than the Earth (Gillmann et al., 2016; Gillmann and Tackley, 2014). Here, only the water cycle has been discussed, but other volatile elements, such as carbon, sulfur, and nitrogen, are also relevant. For example, carbon is expected to be a key volatile element for reconciling the interaction between fluid and solid Earth systems because carbon dioxide is a major element in greenhouse gases. However, addressing the effects of carbon on the evolution of the solid Earth system is difficult because the solubility model of C-H-O (carbon-hydrogen-oxygen) fluid in the mantle minerals must be extended to the pressure range in the CMB region and the partitioning of carbon between solid and molten silicate. To improve our current understanding of the 'deep carbon cycle' suggested by various geochemical and high P-T experiments, some of researchers are organizing a research consortium known as the 'Deep Carbon Observatory' (https://deepcarbon.net) (e.g., Dasgupta and Hirschmann, 2010). Understanding the volatile cycle in the Earth's mantle would also be beneficial for reconciling the distribution of energy resources found in the Earth's surface (e.g., Arthur and Cole, 2014) and the amount and species of light elements in the Earth's core (e.g., Poirier, 1994). Additionally, information about the presence of hydrogen rather than carbon is crucial for explaining the density and elastic structure of the Earth's core (Nakajima et al., 2015; Umemoto and Hirose, 2015).

4 CONCLUDING REMARKS

In this paper, I describe the current understanding of the numerical modeling of global-scale water circulation in mantle dynamics and provide a numerical example of global-scale mantle dynamics with water migration processes (regassing, degassing, and non-magmatic excess water transport). In the numerical modeling community, regional models that include water migration are applicable for various geologic phenomena, such as island arc volcanism, subduction initiation and the formation of a spreading center, with complicated rheological and material properties of mantle minerals and quite high numerical resolution (less than ~1 km) (e.g., Gerya et al., 2008). However, for global-scale modeling, because of the high computational cost of evolving the early Earth to the present Earth, the numerical resolution is limited to ~5 to 15 km and simplified rheological and material properties. Despite these issues, such models are still useful for discussing the behavior of water in mantle dynamics systems. In the numerical example shown here, water solubility maps as functions of temperature and pressure were required to understand the hydrous structure of the mantle suggested by mineral physics experiments and theory (Karato, 2011; Ohtani, 2005). The next steps for the numerical modeling of hydrous mantle convection are as follows: 1. developing the Earth system evolution (i.e., the evolution of a coupled system of core-mantle-plate-ocean); 2. including the effects of the mantle carbon cycle that are crucial for understanding the interaction between deep-mantle dynamics and climate change; 3. applying more realistic rheological properties, such as grain-size-dependent viscosity, complicated deformation mechanisms (Bercovici and Ricard, 2016, 2014) and more complicated phase relationships between hydrous mantle minerals, to reveal the effects of global-scale seismic structures, as done for some global-scale mantle dynamics models with dry mantle minerals (Nakagawa et al., 2010); and 4. modeling at much higher resolutions to observe plate tectonics generation in the mantle convection system more robustly, if the computational costs are sufficiently low. Note that these future directions are just suggestions. For further development and progress of numerical mantle dynamics modeling including water and other volatiles migration, more inputs from experimental and observational studies should be required.

ACKNOWLEDGMENTS

The author thanks Hikaru Iwamori, Tomoeki Nakakuki, Atsushi Nakao and Marc Spiegelman for constructive discussions; Paul Tackley for providing his numerical mantle convection code (StagYY); and Prof. Timothy M. Kusky for inviting this paper. The author also thanks Masanori Kameyama and two anonymous reviewers for improving the original manuscript greatly. Financial support was obtained from JSPS KAKENHI (Nos. JP16K05547, JSPS/MEXT), and the Grant-In-Aid for Scientific Research on Innovative Area (Interaction and Coevolution of the Core and Mantle-Towards Integrated Deep Earth Science, No. JP15H05834), and MEXT as "Exploratory Challenge on Post-K Computer" (Frontiers of Basics Science: Challenging the Limits-Subproject C: Structure and Properties of Materials in Deep Earth and Planets allocated at Computational Astrophysics Laboratory, RIKEN). Numerical computations were performed at SCI ICE-X/UV in the JAMSTEC, YETI HPC cluster in Columbia University in the City of New York and at the K-Computer in AICS, RIKEN. The final publication is available at Springer via http://dx.doi.org/10.1007/s12583-017-0755-3.

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