The Subductability of Continental Lithosphere: The Before and After Story

Abstract

The temporal evolution of internal forces in a collision environment controls first-order characteristics such as convergence rate, slab dip, subduction stall, and slab breakoff, amongst others. Foremost among these forces are the positive buoyancy provided by the subduction of felsic continental material and the negative buoyancy associated with the slab. In this work we use fully dynamic thermomechanical models coupled with thermodynamic/petrological formalisms to study the evolution of these forces during a continent–arc/microcontinent collision and their influence on the large-scale dynamics of the system. Two distinctive features of our models that allow a self-consistent assessment of collision dynamics are: (1) the use of a new thermodynamic database valid up to ~25–30 GPa that includes most of the major phases relevant to continental subduction, and (2) a fully dynamic approach in which no velocities are imposed to either force or stop subduction. The former allows realistic computations of the buoyancy forces driving the system as a function of P-T-composition. The latter assures that computed velocities emerge self-consistently in our simulations in response to the balance between internal forces in our numerical domain.

The main results from our experiments can be summarized as follows. (1) The delamination of the lithospheric mantle after a short episode of continental subduction is a viable scenario to end continental subduction; the associated evolution of convergence is comparable to those proposed for real collision setting. (2) We corroborate previous results showing that the main control on the dynamics and final configuration (type of slab breakoff) of the collision is the rheology and composition of the continental crust; strong mafic crusts favor deep subduction and recycling of significant volumes of continental material, while soft felsic crusts preclude them. (3) Subducted continental crust remains buoyant with respect to the surrounding mantle down to depths of ~250–300 km, thus allowing exhumation of deeply subducted crust as long as a detachment from the slab occurs. (4) Realistic compositional stratifications in the continental lithospheric mantle exert only a modest influence on the overall evolution of the collision system. (5) Subducted continental crust to depths >250–300 km becomes significantly denser than the surrounding mantle due to the appearance in the solid assemblage of high-density phases such as hollandite and stishovite; this provides extra negative buoyancy to the slab and precludes the exhumation of crustal components. This supports the idea of the existence of a “depth of no return” for continental material at around 250 km depth.

Appendix A

1.1 A.1 Governing Equations

The dynamics of the problem analyzed in this paper is governed by the usual conservation equations of fluid dynamics (under the continuum hypothesis), namely the conservation of mass, momentum, and energy (cf. Batchelor 1967; Kundu 1990). The conservation of mass is the most obvious condition and requires

$$ \frac{{\partial \rho }}{{\partial t}} + \frac{{\partial (\rho {u_i})}}{{\partial {x_i}}} = 0 $$

(3.1)

where t is time, ρ bulk density, u _i velocity, and x _i refers to a Cartesian coordinate system. Equation (3.1) is also known as the continuity equation.

The conservation of momentum is the result of applying Newton´s law of motion to an infinitesimal fluid element. In its differential form it reads

$$ \rho \left( {\frac{{\partial {u_i}}}{{\partial t}} + {u_j}\frac{{\partial {u_i}}}{{\partial {x_j}}}} \right) = \rho g - \frac{{\partial P}}{{\partial {x_i}}} + \frac{{\partial {{\tau \prime }_{ij}}}}{{\partial {x_j}}} $$

(3.2)

where

$$ {\tau \prime _{ij}} = 2\mu \left[ {\frac{1}{2}\left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {x_i}}}} \right) - \frac{1}{3}\left( {\frac{{\partial {u_i}}}{{\partial {x_i}}}} \right){\delta_{ij}}} \right] = 2\mu \left( {{\epsilon_{ij}} - \frac{1}{3}{\epsilon_{ii}}{\delta_{ij}}} \right) $$

(3.3)

is the deviatoric part of the total stress tensor ${\sigma_{ij}} = - P{\delta_{ij}} + {\tau \prime _{ij}}$ for a Newtonian fluid, P is pressure (i.e., −1/3 σ _ii ), g the acceleration of gravity, μ the dynamic shear viscosity, δ _ij the Kronecker delta, and ε _ij the strain rate tensor (cf. Batchelor 1967). Equation (3.2) is the 3D general form of the Navier-Stokes equation of motion. Note that the so-called Stokes assumption, in which the bulk viscosity is neglected, has been adopted in (3.3) (Kundu 1990).

The equation of conservation of energy relevant to our study includes five terms:

$$ \rho \frac{{DE}}{{Dt}} = {\sigma_{ij}}\frac{{\partial {u_i}}}{{\partial {x_j}}} + H + \rho Ls\frac{{D\chi }}{{Dt}} + \rho Lm\frac{{DF}}{{Dt}} - \frac{{\partial q}}{{\partial {x_i}}} $$

(3.4)

where E the is internal energy per unit mass, H the internal heat generation (volumetric) due to radioactive decay, Ls the latent heat of transformation arising from solid-solid phase changes, Lm the latent heat of melting, q the conductive heat flow, χ the mass fraction of the new phase across a divariant solid-state reaction, F the mass fraction of melt, and D/Dt refers to the material derivative. The first term in the right-hand side of (3.4) is a mechanical contribution due to compressions and deformations of a fluid element without change of its velocity (Batchelor 1967). Using the above definition of total stress tensor σ _ij , (3.4) can be rewritten as

$$ \rho \frac{{DE}}{{Dt}} = - P\frac{{\partial {u_i}}}{{\partial {x_i}}} + \Phi + H + \rho Ls\frac{{D\chi }}{{Dt}} + \rho Lm\frac{{DF}}{{Dt}} - \frac{{\partial q}}{{\partial {x_i}}} $$

(3.5)

where

$$ \Phi = 2\mu {\left[{\epsilon_{ij}} - 1/3\frac{{\partial {u_i}}}{{\partial {x_i}}}{\delta_{ij}}\right]^2} = \tau_{ij}^\prime {\epsilon_{ij}} $$

(3.6)

is the viscous dissipation through shear deformation [i.e., there are no volume changes associated with the tensor Equation (3.6)], sometimes referred to as “shear heating” in the literature. Using the following thermodynamic identities

$$ TdS = dE + PdV $$

(3.7)

$$ dV = - \frac{{d\rho }}{{{\rho^2}}} $$

(3.8)

which imply

$$ \frac{{TDS}}{{Dt}} = \frac{{DE}}{{Dt}} - \frac{P}{{{\rho^2}}}\frac{{D\rho }}{{Dt}} $$

(3.9)

we can rewrite (3.5) as

$$ \rho T\frac{{DS}}{{Dt}} = \Phi + H + \rho Ls\frac{{D\chi }}{{Dt}} + \rho Lm\frac{{DF}}{{Dt}} - \frac{{\partial q}}{{\partial {x_i}}} $$

(3.10)

where we made use of (3.1) to eliminate the terms affected by P. Equation (3.7) implicitly includes all the “non-conventional” entropy production terms (e.g., due to phase changes) in the E term. Recalling that for a homogeneous fluid the rate of change of entropy is

$$ \frac{{DS}}{{Dt}} = \frac{{Cp}}{T}\frac{{DT}}{{Dt}} - \frac{\alpha }{\rho }\frac{{DP}}{{Dt}} $$

(3.11)

we obtain the more familiar form

$$ \eqalign{ \rho Cp\left( {\frac{{DT}}{{Dt}} - \frac{{T\alpha }}{{\rho Cp}}\frac{{DP}}{{Dt}}} \right) \cr \quad= \Phi + H + \rho Ls\frac{{D\chi }}{{Dt}} + \rho Lm\frac{{DF}}{{Dt}} - \frac{{\partial q}}{{\partial {x_i}}}} $$

(3.12)

where α is the coefficient of thermal expansion and Cp the isobaric specific heat capacity. Since we introduce latent heat effects directly into the energy equation, α and Cp do not need to be replaced by their “effective” or “apparent” counterparts in the phase change temperature range (e.g., Reddy and Gartling 2000; Li and Gerya 2009). Note that the second term on the left-hand side of (3.12) can be written as

$$ \frac{{T\alpha }}{{\rho Cp}}\frac{{DP}}{{Dt}} = {\left( {\frac{{\partial T}}{{\partial P}}} \right)_{\hskip-3ptS}}\frac{{DP}}{{Dt}} = \frac{{T\alpha }}{{\rho Cp}}\left( {\frac{{\partial P}}{{\partial t}} + {u_i}\frac{{\partial P}}{{\partial {x_i}}}} \right) $$

(3.13)

For the Earth’s mantle it is appropriate to neglect the term ∂P/∂t and assume that the vertical gradient in P is dominant and equal to ρg (i.e., hydrostatic profile; Schubert et al. 2001). Accordingly, (3.13) becomes

$$ \rho Cp\left( {\frac{{DT}}{{Dt}} - \frac{{{H_{ad}}}}{{\rho Cp}}} \right) = \Phi + H + \rho Ls\frac{{D\chi }}{{Dt}} + \rho Lm\frac{{DF}}{{Dt}} - \frac{{\partial q}}{{\partial {x_i}}} $$

(3.14)

where H _ad  ≡ Tαρu _z g is the energy gained/lost by adiabatic compression/decompression and u _z is the vertical component of velocity. An alternative useful form of this latter relation is

$$ \eqalign{ \rho Cp\left( {\frac{{\partial T}}{{\partial t}} + {u_i}\frac{{\partial T}}{{\partial {x_i}}}} \right) \cr \quad= \frac{\partial }{{\partial {x_i}}}\left( { - {k_{ij}}\frac{{\partial T}}{{\partial {x_i}}}} \right) + \Phi + H + {H_{ad}} + \rho Ls\frac{{D\chi }}{{Dt}} + \rho Lm\frac{{DF}}{{Dt}}} $$

(3.15)

where k _ij is the thermal conductivity tensor. Equations (3.1), (3.2), and (3.15) represent the fundamental set of equations governing the heat transfer and flow of a compressible Newtonian fluid. The insertion of an equation of state describing density as a function of other intensive variables (e.g., P, T, F, χ) completes the definition of the mathematical problem (except for the non-Newtonian case). In what follows, we describe a number of approximations and modifications to the fundamental set of equations.

1.2 A.2 Approximations

We adopt a modified version of the extended Boussinesq approximation (e.g., Christensen and Yuen 1985) in solving the system (3.1), (3.2), and (3.15). The Prandlt Pr and Match M numbers of a fluid determine the effects of inertia and compressibility on the flow (cf. Jarvis and McKenzie 1980; Schubert et al. 2001). For the Earth\prime s mantle, the product Pr M ² is $\ll 1$ (Jarvis and McKenzie 1980), and therefore the material can be considered incompressible except in regions affected by phase transitions (see below). Equation (3.1) thus becomes

$$ \frac{{\partial {u_i}}}{{\partial {x_i}}} = 0 $$

(3.16)

Likewise, given the effectively infinite Prandlt number of the mantle, inertia terms in the momentum (3.2) are neglected. Using (3.16), we rewrite (3.2) as

$$ 0 = \rho g - \frac{{\partial P}}{{\partial {x_i}}} + \frac{\partial }{{\partial {x_j}}}\left[ {\mu \left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} + \frac{{\partial {u_j}}}{{\partial {u_i}}}} \right)} \right] $$

(3.17)

Besides the inertia terms, the only difference between the (3.2) (compressible fluid) and (3.17) (incompressible fluid) is a term that includes the divergence ▽ = ∂/∂x _i of the velocity field

$$ - \frac{\partial }{{\partial {x_j}}}\left[ {\frac{2}{3}\mu \left( {\frac{{\partial {u_i}}}{{\partial {x_i}}}} \right){\delta_{ij}}} \right] $$

The energy equation is made compatible with (3.16) by removing the term affected by the divergence of velocity in the viscous dissipation term (3.6), which becomes

$$ \Phi = 2\mu {\epsilon_{ij}}^2 $$

(3.18)

The latent heats L in the energy equation can be written as ΔS T, where ΔS is the entropy change of the phase transformation (solid-solid or solid-melt). Since entropy is an extensive quantity, the magnitude of ΔS in a rock experiencing a phase transition will depend on the actual amount of matter being transformed. For instance, although the ΔS associated with the spinel-garnet transformation is ~25 J Kg⁻¹ K⁻¹ in the system MgAl₂O₄ (spinel) + 4 MgSiO₃ (orthopyroxene) $\rightleftarrows$ Mg₃Al₂Si₃O₁₂ (pyrope) + Mg₂SiO₄ (forsterite), it becomes only ~4 J Kg⁻¹ K⁻¹ in a natural lherzolite with 4.5 wt% Al₂O₃ (see also Iwamori et al. 1995). Similarly, our thermodynamic calculations (see below) indicate that most metamorphic reactions typically involve ΔS < 8 J Kg⁻¹ K⁻¹. Such ΔS values result in local variations of the order of 5–10°C in the thermal structure and thus they can be ignored without losing any generality in large-scale models. On the other hand, reactions involving large mass fractions, such as the olivine-wadsleyite phase transitions in lherzolites (~60 vol% of the rock), can result in ΔS of the order of 30–45 J Kg⁻¹ K⁻¹. In this case, the local thermal perturbation close to the phase transition can be as high as 60–70°C for fast vertical flow, and therefore the contribution from latent heat release should not be neglected in the energy equation. We introduce latent heat effects by approximating the material derivative of χ by (no time effects)

$$ \frac{{D\chi }}{{Dt}} \equiv {u_i}\frac{{\partial \chi }}{{\partial {x_i}}} $$

(3.19)

where ∂χ/∂x _i is assumed to be a step function (i.e., sharp transition) which is zero everywhere except at the transition, where it takes a value of 1/u _i Δt. Appropriate values of ΔS are calculated as a function of bulk rock composition using a Gibbs free energy minimization formalism (see below).

The latent heat of melting is addressed in a similar manner, except that in this case the material derivative DF/Dt is a continuous function based on a parameterization of partial melting as a function of P, T, water content, and depletion (see below).

As stated above, the incompressibility assumption (3.16) is appropriate for modelling flow within the Earth as long as the Boussineq assumptions are fulfilled. This is not longer the case when density changes produced by mineral phase transitions are included in the model. Neglecting volume changes associated with phase transitions in the conservation equations has the net effect of “creating/destroying” mass when the material experiences a phase change (i.e., mass conservation is not ensured). This carries the obvious consequence of over/underestimating the buoyancy forces of the model, which in the case of a subducting plate has a major long-term effect on its buoyancy and thus on the dynamics of the model. We tackle this problem by modifying (3.16) and (3.17) to account for compressibility (i.e., non-zero divergence of the velocity field) in the regions affected by phase transitions. Equation (3.16) thus becomes

$$ \frac{{\partial {u_i}}}{{\partial {x_i}}} = - \frac{1}{\rho }\frac{{\partial \rho }}{{\partial {x_i}}}{u_i} \approx - \frac{1}{\rho }\frac{{\Delta \rho }}{{\Delta t}} $$

(3.20)

where the material change in density $\tfrac{{\displaystyle\Delta \rho }}{{\displaystyle\Delta t}}$ is computed in each Lagrangian particle as the difference between the density in the current and previous time step (we only account for density changes due to mineral phase transitions, density variations related to melting are not included). The application of this non-zero divergence generates a localized velocity field that effectively compresses/expands the material at the time of the transition, therefore enforcing local mass conservation. In practice, since our numerical box is closed, we apply the mass correction only to lithospheric materials. This plays the role of simulating an “open” domain (i.e., sublithospheric convecting mantle can flow through the boundaries of the box to compensate for lithospheric volume changes). We note that this correction is not equivalent to modelling a fully compressible material, but rather numerically impose the changes in volume needed to conserve the mass in specific regions of the simulation domain, i.e., lithospheric materials. Similar strategies have been used by (e.g., Warren et al. 2008; Burov and Yamato 2008).

Equations (3.16) through (3.20) are solved using the finite-element platform Underworld (Moresi et al. 2003), which has been modified accordingly to account for all the processes described in this paper. The numerical accuracy of the mass correction term included in (3.20) is evaluated in each simulation by keeping track of the mass of the subducting plate trough time. However, this calculation is not trivial due to the difficulty in estimating the volumes of all bodies (i.e., the geometry of a body is defined by a set of moving particles that tend to mix with the particles of surrounding materials during the evolution of the model). We circumvent this problem by constructing Voronoi diagrams based on the particles and computing the mass of each particle as the product of its density times the area of the corresponding Voronoi cell. Therefore, the mass of a body is simply the sum of the mass of its constituents particles. We find that typical accumulated errors associated with mass conservation are <1%.

1.3 A.3 Rheological Relationship

Our numerical models have both plastic (brittle) and viscous (Newtonian and non-Newtonian) rheologies. The brittle behaviour of rocks is assumed to follow a modified version of the von Mises criterion, in which the material yields (plastically) when the following condition is met

$$ \sqrt {{{{\tau \prime }_{II}}}} \ge {c_0} + \mu (\rho gz) $$

(3.21)

where ${\tau \prime _{II}}$ is the second invariant of the deviatoric stress tensor, c ₀ is the “cohesive strength” and μ the coefficient of internal friction. Parameters c ₀, and μ are specific for each material and vary among simulations. High-pressure failure of rocks (e.g., Shimada 1993; Renshaw and Schulson 2004) is modelled assuming a critical (constant) yield stress τ _c between 250 and 350 MPa (See Fig. 3.2).

The assumed general flow law for both diffusion and dislocation mechanisms has the following form (e.g. Hirth and Kohlstedt 2003)

$$ \dot{\epsilon } = A{(\sigma )^n}{d^{ - m}}{\Psi_{{H_2}O}}\;\exp \left( { - \frac{{{E^*} + P{V^*}}}{{R\;T}}} \right) $$

(3.22)

where d is the average grain-size, σ the differential stress, A the pre-exponential factor, n the stress exponent, m the grain-size exponent, E ^* the activation energy, V ^* the activation volume, R the gas constant, and ${\Psi_{{H_2}O}}$ a parameter dependent on the water content (see below). Note that the activation energy E ^* and volume V ^* already include the T and P dependence of OH⁻ dissolution in olivine (cf. Hirth and Kohlstedt 2003; Karato 2008). Applying the Levy-von Mises formalism to purely viscous fluids (Ranalli 1995; Karato 2008) allows defining the effective viscosity as

$$ \eta = \frac{1}{2}{({\tau \prime _E})^{1 - n}}{A^{ - 1}} = \frac{1}{2}{({\dot{\epsilon }_E})^{\tfrac{{1 - n}}{n}}}{A^{ - \tfrac{1}{n}}} $$

(3.23)

where τ′ _E and ${\dot{\epsilon }_E}$ are the “effective” second invariants of the deviatoric stress tensor $[ = {(\tfrac{1}{2}{\tau \prime _{ij}}{\tau \prime _{ij}})^{1/2}}]$ and of the strain rate tensor $[ = {(\tfrac{1}{2}{\dot{\epsilon }_{ij}}{\dot{\epsilon }_{ij}})^{1/2}}]$, respectively. Parameter A is a factor derived from the empirical Equation (3.22) and the type of experiment [for details see Chap. 4 of Ranalli (1995) and Chap. 3 of Karato (2008)]. All relevant parameters used to solve (3.21)–(3.23) are listed in Table 3.3.

Since diffusion and dislocation creep are thought to act simultaneously in the mantle, two different viscosities η _diff and η _disl are computed separately and then combined into an effective viscosity η _eff. The latter is computed as the harmonic mean of η _diff and η _disl:

$$ \frac{1}{{{\eta_{\rm{eff}}}}} = \left( {\frac{1}{{{\eta_{\rm{diff}}}}} + \frac{1}{{{\eta_{\rm{disl}}}}}} \right) $$

(3.24)

This expression is truncated if the resulting viscosity is either greater or lower than two imposed cutoff values (10¹⁸ to 10²⁴ Pa s). Although grain-size may change due to grain growth and dynamic recrystallization processes, its dependence on mantle flow conditions is poorly known and therefore we consider only constant grain sizes. For average grain sizes d ≳3.5 mm, dislocation creep represents an important component of the effective viscosity at depths < 200 km. For smaller values of d, diffusion creep becomes dominant. In this context, we note that synthetic seismological models of oceanic mantle suggest d ≳ 3.5 mm (Afonso et al. 2008a, b). All simulations shown in this work are carried out with a constant d = 5 mm.

1.4 A.4 Melting Model

We adopt a simple melting model based on the parameterizations of Katz et al. (2003). Melting of crustal components are not considered in the present work. The melt fraction of mantle rocks as a function of P, T, water content, and depletion ξ is expressed as

$$ {F_{(T,P,{H_2}O,\xi )}} = {\left( {\frac{{T - {T_s}}}{{{T_l} - {T_s}}}} \right)^{\beta 1}} $$

(3.25)

where

$$ {T_s} = {A_1} + {A_2}P + {A_3}{P^2} - \Delta {T_w} + \Delta {T_\xi } $$

(3.26)

$$ {T_l} = {B_1} + {B_2}P + {B_3}{P^2} - \Delta {T_w} + \Delta {T_\xi } $$

(3.27)

Parameters A _n and B _n are fitting parameters obtained from regressions through a large number of experimental data (Katz et al. 2003). ΔT _w is a parameter that takes into account the temperature decrease in the solidus caused by the presence of water and has the simple form (Katz et al. 2003)

$$ \Delta {T_w} = KX_{{H_2}O}^\gamma $$

(3.28)

where K = 43°C wt%^−γ, γ = 0.75, and ${X_{{H_2}O}}$ is the weight percent of water in the melt. The latter is obtained assuming batch melting and a representative bulk partition coefficient D _w (Katz et al. 2003). Although it is expected that in natural systems D _w will vary with P,T, and bulk composition, here we adopt a constant D _w  = 0.008 as a representative average for our simulations; this value is in agreement with various experimental estimations (cf. Hirschmann 2006). The concomitant extraction of bound water during melting of hydrous assemblages is tracked using the same bulk partition coefficient D _w as above.

In our simulations we assume “instantaneous” batch melting during each time step. Thus, all melt generated during a time step is extracted in the next time step, except for a small amount that remains in the solid assemblage as residual porosity (see below). The ΔT _ξ parameter is introduced here to include the effect of incremental depletion on the solidus temperature. This effect, not considered in the original parameterization of Katz et al. (2003), arises due to the progressive extraction of fusible elements from the solid rock by partial melting, which in turn increases the solidus temperature of the solid residue. We adopt the following linear form

$$ \Delta {T_\xi } = 250F $$

(3.29)

where the result is in °C and F is in weight fraction (e.g., 0 < F < 1). The numerical value 250 is identical to that estimated by Phipps Morgan (2001) and used by Afonso et al. (2008a). The effect of melt depletion on bulk density is accounted for by tracking the total amount of melt extraction F _T experienced by mantle particles and applying the following correction: ρ _r  = ρ _s  − ζ F _T , where ρ _r is the bulk density of the residual peridotite, ρ _s is the bulk density of the unmelted (fertile) mantle, and ζ is a correction factor = 2 kg m⁻³ %F ⁻¹. For instance, if a mantle parcel has experienced a total of 10% melt extraction at a certain time of the simulation, the density of the residual solid assemblage will be 20 kg m⁻³ less than its fertile counterpart at identical P-T conditions.

The density of the partially melted rock is computed as

$$ {\rho_{eff}} = {\rho_s}(1 - F) + {\rho_m}F $$

(3.30)

where ρ _s is the density of the solid aggregate and ρ _m is the density of the melt. The former is retrieved from the energy minimization scheme; the latter is estimated as a linear function of P and T according to the relation

$$ {\rho_m} \;\quad=\quad\; {\rho_0} + {\left( {\frac{{\partial {\rho_0}}}{{\partial T}}} \right)_P}(T - {T_0}) + {\left( {\frac{{\partial {\rho_0}}}{{\partial P}}} \right)_T}(P - {P_0}) $$

(3.31)

where ρ ₀ is a reference value at P ₀ = 1 atm and T ₀ = 800 K. The derivatives are estimated using the method of Lange and Charmichael (1987) for a melt with an average MgO content of 10 wt%. The results are (∂ρ ₀/∂T) _P ~ −10⁻⁴ kg m⁻³ K⁻¹ and (∂ρ ₀/∂P) _T ~0.065 kg m⁻³ Gpa⁻¹. We assume the existence of a critical residual porosity F _c below which melts are immobile (Wark et al. 2003). Here we use a conservative value of 1% for F _c . When the calculated F at a certain P-T-H₂O condition becomes greater than F _c , the “excess” melt (i.e., F − F _c ) is extracted from the solid and transported instantaneously to the surface. This assumption is entirely justified by the high velocities of melts in the mantle wedge (e.g., Turner et al. 2001) in comparison to the duration of a typical time step in our simulations (50,000–200,000 years).

1.5 A.5 Darcy Flow of Water

Aqueous fluids released by dehydration reactions are assumed to migrate through the solid matrix as porus flow along the solid grain boundaries (Darcy\prime s flow). The effect of adding water into nominally-anhydrous minerals by diffusion is neglected here due to its minute contribution to the water balance compared to hydration/dehydration reactions.

Neglecting compaction effects and assuming that pressure variations from the lithostatic state are small (Burov and Yamato 2008), the fluid (melt or aqueous) velocity relative to the matrix velocity obeys the relation

$$ {V_f} - {V_s} = \frac{{\Delta \rho g{k_\phi }}}{{{\mu_f}\phi }} $$

(3.32)

V _f and V _s are the velocities of the fluid and solid matrix, respectively, φ the fluid fraction, μ _f the dynamic viscosity of the fluid, and Δρ the density difference between the solid aggregate and the fluid. The permeability k _φ is given by (Wark et al. 2003)

$$ {k_\phi } = \frac{{{d^2}{\phi^3}}}{{270}} $$

(3.33)

where d is the mean grain diameter (5 mm). As stated in (3.32), the densities and viscosities of the aqueous fluids are required to compute their velocities. Here we estimate the density with the Pitzer-Sterner equation of state for pure water (Pitzer and Sterner 1994). This assumption will underestimate the real density of fluids as the solubility of silica-rich components in water increases significantly at high pressures (Audetat and Keppler 2004), thus increasing the density of the fluid. The viscosities are somewhat more problematic due to their order-of magnitude variations over conditions pertaining to the mantle wedge (Audetat and Keppler 2004). In order to avoid overcomplicating the model with poorly constrained parameters, here we assume the viscosities of fluids to be constants and equal to 1 Pa s.

1.6 A.6 Thermodynamic Databases and Free Energy Minimization Strategy

We compute stable assemblages and all their relevant bulk properties (ρ, C _p , α, bulk modulus, etc.) by free energy minimization using the software Perple_X (Connolly 2009). All calculations were performed in the system K₂O–Na₂O–TiO₂–FeO–CaO–MgO–Al₂O₃–SiO₂ + H₂O (KNTFCMAS + H₂O), which accounts for ≳99 wt% of the Earth´s crust and mantle. All stable assemblages and relevant physical properties were computed a priori as function of P, T, bulk composition, and water content and saved in “look-up” tables with a spacing of 75 MPa for P, 5.6 K for T, and 1 wt% for water content. Given the size of our numerical domain and the large range of compositions, we used two different thermodynamic databases and formalisms. For ultramafic upper mantle lithologies and for all crustal materials, we used an augmented-modified version of the Holland and Powell (1998) database (revised in 2002). The original data set has been augmented with the following phases: CAS* (CaAl₄Si₂O₁₁), K-hollandite* (KAlSi₃O₈), Si-wadeite* (K₂Si₄O₉), K-Cymrite* (KAlSi₃O₈⋅H₂O), Mg-majorite (Mg₄Si₄O₁₂), Na-majorite (Na₂Al₂Si₄O₁₂), Ca-perovskite* (CaSiO₃), Mg-wadsleyite (Mg₂SiO₄), Fe-wadsleyite (Fe₂SiO₄), high-pressure (C2/c) clinoferrosilite (Fe₄Si₄O₁₂), and high-pressure (C2/c) clinoenstatite (Mg₄Si₄O₁₂); * denotes stoichiometric phases (i.e., not part of a solid-solutions). The so-called NAL and Egg phases as well as Ca-ferrite are stable at very high-pressures along subduction P-T paths (e.g., Ono 1999; Miyajima et al. 2001). However, they are not included in the present version of the database due to their small volume fraction in comparison with other dominant phases.

In order to obtain a reliable internally consistent data set within the system KNTFCMAS + H₂O for the wide range 0 < P < 30 GPa and 273 < T < 2,000 K, the following phases in the original Holland and Powell (1998) database had to be slightly modified (always within experimental uncertainty): forsterite, fayalite, ferrosilite, enstatite, stishovite, tschermakite, Ca-tschermakite, Mg-tschermakite, grossular, pyrope, diopside, and A-phase. The modification involved mainly a refitting of the polynomials describing C _p and α to make them compatible with high-pressure experiments, and in some cases a slight modification of the enthalpies of formation and molar volumes. However, we stress that the predictions of the original database at low and moderate (crustal and shallow upper mantle) T-P conditions were not affected to any significant extent by these modifications, since they mostly apply at high-pressure conditions. A full description of all solid-solution models and end-members will be published elsewhere (Afonso and Zlotnik, in preparation).

Our database has been calibrated based on experimental results in simplified systems. The thermodynamic parameters of the mentioned phases were refined in successive steps in a hierarchical manner. First, a consistent database was obtained for the MS, FS, MAS, and FAS systems. Succeeding refining stages considered the ternary systems KAS, CAS, FMS, CMS, and quaternary systems KAS + H₂O, CMAS, FMAS. A subsequent comparison against experimental results in complex system such as NKCMFAS, KNTFCMAS + H₂O, and KCMAS + H₂O showed good agreement, and permitted a final refining of some phases and/or solutions (e.g., garnets and clinopyroxenes). Figure 3.15 shows a comparison of calculated phase diagrams for a peridotite using three different databases. It can be seen that at low P-T conditions our database (Fig. 3.15b) predicts similar equilibrium assemblages than the original database of Holland and Powell (1998) (Fig. 3.15a). At higher pressures, however, the original database fails to predict representative assemblages due to the lack of solid solution models for the solubility of pyroxenes into garnet, the transformation of olivine into wadsleyite, and the high-pressure monoclinic polymorph of orthopyroxene with C2/c symmetry. At high P-T conditions our phase diagram is therefore more similar to that predicted from the high-pressure database of Xu et al. (2008) (Fig. 3.15c).

Fig. 3.15

Predicted phase diagrams of dry peridotite using the database of Holland and Powell (1998) (a), our augmented/modified version (b), and the database of Xu et al. (2008) (c). (d), (e) and (f) are the respective modal proportions along a 1330 °C adiabat.

Figure 3.16 shows the high T-P experimental results of Irifune and Isshiki (1998) together with predictions from our database for the same P-T and compositional conditions. Although the agreement with the experimental data is good, there are some differences at high pressures. Firstly, the experiments of Irifune and Isshiki (1998) did not constrain the amount of the monoclinic (clinopyroxene-structured) polymorph of orthopyroxene (e.g., Woodland and Angel 1997; Akashi et al. 2009), and therefore we cannot constrain well the effect that the presence (and disappearance) of the C2/c phase will have on the transfer of atomic species between garnet and clinopyroxene (see also Fig. 3.15). Although the behaviour of the C2/c phase in our calculations is in perfect agreement with experiments in the simple system FMS, it is still uncertain how this phase behaves in more complex systems such as the one in Fig. 3.16. Secondly, our database predicts high pressure garnets (P > 10 GPa) with relatively low contents in Na and high in Fe with respect to experiments on continental crust compositions (Irifune et al. 1994; Wu et al. 2009). Clinopyroxene, on the other hand, tend to be deficient in Fe and Mg. Although we have modified some end-member properties to make the database more reliable at high pressures, the above observations suggest that the solution models adopted in this study from experiments at low-moderate pressures require further refinements. Unfortunately, there are only a few high-P-T experiments in complex systems that could be used for this purpose, and not all of them are consistent (e.g., Irifune et al. 1994; Wu et al. 2009). Despite this limitation, Figs. 3.16 and 3.17 shows that our database reproduce high-P-T experimental observations satisfactorily for a wide range of compositions.

Fig. 3.16

Comparison between experimental and theoretical phase proportions in dry pyrolite. Black circles with error bars are from Irifune and Ito (1998) based on high-T-P experiments. Red (grey) circles are predictions from our dataset. Stars are predictions from the dataset of Xu et al. (2008).

Fig. 3.17

Comparison between experimental results (left column) and predictions from our dataset (right column). Experimental studies are (a) Irifune et al. (1994), (c) Ono (1998), (e) Wu et al. (2009), (g) Ono (1998).

The above discussion indicates that our database will predict reliable phase diagrams up to ≲30 GPa only for felsic, intermediate, and mafic compositions. For ultramafic compositions, however, predictions are only reliable up to ~15 GPa (i.e., above the wadsleyite-ringwoodite transition). Therefore, for mantle materials below this transition we use the thermodynamic database and formalism of Xu et al. (2008). This database is suitable for ultramafic compositions (but not for felsic, intermediate, or hydrated compositions) at transition zone and lower mantle conditions in the simplified system NCFMAS. For similar NCFMAS compositions, the differences in bulk ρ, α, and seismic velocities between the two databases are < 1%, ≲ 1.5, < 0.8%, respectively. Therefore, the discontinuity of such properties when passing from our database to that of Xu et al. (2008) is unlikely to have any significant effect on the dynamics of the system. In practice, we combine both databases at the Ol-Wads transition, where the properties of the system have a natural and significant discontinuity (i.e., the so-called 410d). We apply an interpolation function to smooth out the sharp contrasts in the physical properties.