Mantle convection simulations
The numerical modeling of global-scale mantle dynamics with water migration has been described by Nakagawa et al. (2015) and Nakagawa and Spiegelman (2017). This process is briefly described here. We assume the thermo-chemical multiphase mantle convection of a compressible and truncated anelastic approximation in a 2D spherical annulus geometry (Hernlund and Tackley 2008). The modeled mantle can be decomposed into depleted harzburgite and enriched basaltic material composed of two-phase transition systems, i.e., olivine-spinel-bridgmanite-post-perovskite and pyroxene-garnet-bridgmanite-post-perovskite, which are associated with changes in the basaltic material. A reference state for each phase transition system is computed as in Tackley (1996). All phase transition parameters can be found in Nakagawa and Tackley (2011). A partial melting effect is included to create an oceanic crust and to allow its segregation. The viscosity of the modeled mantle is dependent on temperature, pressure, water content, and yield strength and is determined via the following equations:
$$ {\eta}_d={A}_d{\sum}_{i,j=1}^{\mathrm{nphase}=3,4}\Delta {\eta}_{ij}^{\Gamma_{ij}f}\exp \left[\frac{E_d+p{V}_d}{RT}\right] $$
(1)
$$ {\eta}_w={A}_w{\left(\frac{C_w}{C_{w,\mathrm{ref}}}\right)}^{-1}{\sum}_{i,j=1}^{\mathrm{nphase}=3,4}\Delta {\eta}_{ij}^{\Gamma_{ij}{f}_j}\exp \left[\frac{E_w+p{V}_w}{RT}\right] $$
(2)
$$ {\eta}_Y=\frac{\sigma_Y\left(p,{C}_w\right)}{2\dot{e}} $$
(3)
$$ \eta ={\left(\frac{1}{\eta_d}+\frac{1}{\eta_w}+\frac{1}{\eta_Y}\right)}^{-1} $$
(4)
where Ad,w is the prefactor determined by T = 1600 K and the ambient pressure at the surface (the 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 and is assumed to be 620 ppm (Arcey et al. 2005); the exponent of the prefactor, which is dependent on the water content, is based on the results of deformation experiments (Mei and Kohlstedt 2000) and is assumed to be 1; Γij is the phase function; f is the basaltic composition (varying from 0 to 1); R is the gas constant (8.314 J K mol−1); T is the temperature; p is the pressure; σY(p, Cw) is the yield strength of the oceanic lithosphere, which is a function of pressure and water content; \( \dot{e} \) is the second invariant of the strain rate tensor; and ∆ηij is the viscosity jump associated with the phase transition, which is assumed to increase by 30 times during the phase transition from ringwoodite or garnet to bridgmanite.
The yield strength of oceanic lithosphere is dependent on pressure and the mantle water content and is determined as follows:
$$ {\sigma}_Y\left(p,{C}_w\right)={C}_Y+\mu \left({C}_w\right)p;\mu \left({C}_w\right)=\min \left[1,{\left(\frac{C_w}{C_{w,\mathrm{ref}}}\right)}^{-1}\right]{\mu}_0 $$
(5)
where μ0 is the Byerlee-type friction coefficient. We include only the water-weakening effect caused by hydrated rocks (Gerya et al. 2008).
Another important influence of hydrated mantle rocks is the variation in density caused by hydrated mantle minerals, as noted by Nakagawa et al. (2015):
$$ \rho \left({T}_{ad},p,C,{C}_w\right)=\rho \left({T}_{ad},C,p\right)\left(1-\alpha \left({T}_{ad},C,p\right)\left(T-{T}_{ad}\right)\right)-{\Delta \rho}_w{C}_w $$
(6)
where ρ(Tad, C, z) is the combined reference density between harzburgite and mid-ocean ridge basalt (MORB) compositions, with a 2.7% density difference, as shown in Fig. 1 (and a 3.6% density difference between olivine- and pyroxene-related phases); Tad is the adiabatic temperature in the mantle; ∆ρw is the density variation due to the water content; and Cw is the water content. Generally, the densities of hydrous minerals are less than those of dry minerals, but the value of ∆ρw is less constrained by high-pressure/high-temperature (high P-T) experiments; the densities of hydrous minerals are generally 0.1 to 1.0% less than those of dry mantle minerals (Richard and Iwamori 2010).
To solve the equations of thermo-chemical mantle convection and to model the chemical composition, we use the numerical code of a finite-volume multigrid flow solver with tracer particles (StagYY; Tackley 2008). To determine the water migration processes, the tracer particle approach is used for the water advection process and degassing process via volcanic eruptions, and a numerical scheme involving a discrete migration velocity approximation is used for the dehydration process (Iwamori and Nakakuki 2013; Nakagawa et al. 2015; Nakao et al. 2016; Nakagawa and Spiegelman 2017). The dehydration process is modeled as the upward migration of excess water, which is defined as the difference between the actual water content and the water solubility at a certain temperature and pressure in a grid (see Fig. 2 in Nakagawa et al. 2015); this upward migration may have a velocity comparable to that of fluid movement in fully two-phase flow modeling (Wilson et al. 2014), which ranges from 0.5 to 50 m/year. Although this may affect the numerical resolution of the model, it should not have a significant influence on the results based on an assessment of the Appendix of Nakagawa and Iwamori (2017). Note that our model of water migration allows for migration only in the vertical direction, whereas in Wilson et al. (2014), the fluid component may migrate appreciably in the horizontal direction over several tens of kilometers near the corner of a mantle wedge. This conventional scheme seems to be valid for global-scale water circulation in a convecting mantle. In addition, we also assume the water partitioning of partially molten material, as in Nakagawa et al. (2015) and Nakagawa and Spiegelman (2017). The partition coefficient of water between solid mantle material and melt is set to 0.01 (Aubaud et al. 2008). In the numerical scheme of material transportation in Nakagawa and Iwamori (2017), two distinct compositional types of tracers are assumed, which can track the water migration for each composition separately; however, here, the chemical composition assigned to each tracer continuously varies with arbitrary melting (see Rozel et al. 2017; Lourenço et al. 2018). Hence, the water migration should be tracked using a bulk composition, which is given as:
$$ S\left(T,P,C\right)={S}_{\mathrm{Peridotite}}\left(T,P\right)\left(1-C\right)+{S}_{\mathrm{Basalt}}\left(T,P\right)C $$
(7)
where Speridotite and Sbasalt are the water solubilities of mantle peridotite and oceanic crust as functions of temperature and pressure, respectively, and C is the basaltic fraction. This represents a major difference between this study and Nakagawa and Iwamori (2017). The difference between the two different melting approaches is discussed in Appendix.
The numerical setup used in this study is described as follows: 1024 (azimuthal) × 128 (radial) grid points with four million tracers are used to track the chemical composition, melt fraction, and mantle water content. The boundary conditions for temperature are fixed temperatures at the surface (300 K) and the core-mantle boundary (CMB; 4000 K). The initial conditions include an adiabatic temperature of 2000 K at the surface plus a thin thermal boundary layer (to explain the thermal boundary conditions at the surface and CMB), a basaltic composition of 20%, and a dry mantle (zero water content). The composition of the mantle is assumed to be uniform so that partial melting can create heterogeneous features in the mantle. The mantle can become hydrated up to the boundary conditions of the mantle water content (described in the “Computing the water ocean mass”) via surface plate motion.
Water solubility maps
Figure 1 shows the maximum H2O content of the mantle peridotite system based on the work of Iwamori (2004, 2007) and Nakagawa et al. (2015) (Fig. 1a) and includes the stability fields of hydrous phases that are stable at lower mantle conditions (Fig. 1b). These data are used to compute the excess water migration in the convecting mantle. At the lower mantle condition, without DHMS solubility, the water solubility of the lower mantle minerals is set as 100 wt. ppm. With DHMS solubility, in addition to the existing DHMS (phases A to D), because a new hydrous mineral phase has recently been discovered to exist at lower mantle pressures, i.e., “phase H” (Komabayashi and Omori 2006; Nishi et al. 2014; Ohira et al. 2014; Walter et al. 2015; Ohtani 2015), we have added the stability field of DHMS including phase H to the water solubility map of the mantle peridotite system (Fig. 1c). Compared to the water solubility map for pressures of less than 28 GPa (Iwamori 2004, 2007), few experimental results are available for pressures of greater than 28 GPa to constrain the exact phase boundaries and the maximum amount of H2O in the peridotite system. The effect of aluminum on the stability of DHMS including phase H depends on the partitioning of Al among mantle minerals, including DHMS, phase H, and δ-AlOOH, which is currently poorly constrained. For instance, we assume that the stability of phase H in the natural peridotite system is similar to that of the pure MgSiO4H2 phase H (Ohira et al. 2016), which can be used to establish a minimum P-T stability range. Considering the bulk peridotite composition and the maximum modal amount of phase H in the peridotite system, we estimate that the maximum H2O content in the phase H-bearing P-T range is 8 wt.%. At pressures and temperatures higher than the stability field of DHMS with phase H, we set a maximum H2O content of 100 ppm below the solidus (e.g., Panero et al. 2015) and 0 ppm above the solidus, as in Nakagawa et al. (2015).
Computing the water ocean mass
In a previous study (Nakagawa and Iwamori 2017), we assumed a fixed boundary condition for mantle water migration in terms of the water ocean mass, such as a fixed value of 1.4 × 1021 kg (one ocean mass). This is not very realistic for understanding geologic records of the evolution of the water ocean associated with surface plate motion (e.g., Maruyama et al. 2013). To formulate a finite reservoir of surface seawater with a box model assumption, the mass of the water reservoir can be computed as:
$$ {X}_{w,s}={X}_{w,\mathrm{total}}-{X}_{w,m}\left({F}_R,{F}_H,{F}_{\mathrm{G}}\right) $$
(8)
where Xw, s is the mass of the water ocean, Xw, total is the mass of the total amount of water in the system, and Xw, m is the mass of mantle water as a function of regassing (FR), dehydration (FH), and degassing (FG). The boundary condition of mantle water migration at the surface is described as follows:
$$ {C}_w\left(\mathrm{surface}\right)=\left\{\begin{array}{c}{C}_{w,\mathrm{sol}}\left(300\ K,\mathrm{surface}\right)\ if\ {X}_{w,s}>0\\ {}0\ if\ {X}_{w,s}=0\end{array}\right. $$
(9)
where Cw, sol(300K, surface) is the water solubility of mantle rocks at the surface.