2.2. Numerical Model
 Numerical experiments are conducted which model mantle flow in the plume-ridge system as convection of a viscous, Boussinesq, Newtonian fluid within a two-dimensional (2-D), regular Cartesian domain (Figure 1b). The relevant system of dimensionless model equations to be solved includes the conservation of mass (1), momentum (2) and energy (3), which can be expressed in vector notation as
Dimensionless variables u, τ, p, T and D are the velocity vector, the viscous stress tensor, pressure, potential temperature and melt depletion, respectively, and is the unit vector in the vertical direction. ΦL represents heat loss due to latent heating during melting and is described below. Buoyancy forces acting on the fluid are accounted for in the body force term of the momentum equation (2) through a combination of the thermal Rayleigh number,
and a dimensionless buoyancy parameter,
which accounts for buoyancy due to the extraction of Fe from the residue during melting. Here ΔT is the total dimensional potential temperature contrast between the top and bottom of the model domain (1350°C), and β is the chemical buoyancy coefficient (0.06, appropriate for melting in the presence of residual garnet [Sparks and Parmentier, 1993; Manglik and Christensen, 1997]). Thus the extraction of 25% melt yields a buoyancy force on the residue equivalent to an increase in temperature of 430°C. The parameters g, α, d, κ, η0 and ρ0 are gravitational acceleration, thermal expansivity, fluid thickness, thermal diffusivity, reference mantle kinematic viscosity and reference mantle density.
 A non-diffusive Lagrangian particle technique is employed for tracking fluid chemistries and implementing melting [Kincaid et al., 1996; Manglik and Christensen, 1997; Kincaid and Hall, 2003]. Individual particles are transported through the model domain by solving the advection equation,
for each particle using a Runge-Kutta step. Here x is the position vector for the particle. Melting is modeled on the scale of the individual particles. For a given particle, the energy available for melting is related to the difference between the temperature of the particle and the temperature of the solidus for that particle at the same depth. In order to determine the incremental degree of melting (or depletion), ΔD, this energy is partitioned between heat loss to latent heating and isobaric melt production [Langmuir et al., 1992]:
where TR is the particle temperature, including the mantle adiabatic gradient (dT/dz = 0.5°C km−1). Ts is the solidus temperature, ΔS is the entropy of fusion for peridotite, Cp is the heat capacity of peridotite, and (dT/dD)∣p is one over the isobaric melt productivity, which is related to the temperature difference between contours of constant depletion at a given depth. Once the incremental degree of melting is determined, the temperature change due to latent heating is calculated,
and the temperature of the particle is adjusted accordingly (Figure 2). This is accomplished by applying the temperature correction, weighted by the local particle density, to the grid nodes of the computational element in which the particle is located. The total amount of melting (or depletion) experienced by each particle throughout its history is tracked. We use a parameterization of the peridotite solidus and liquidus [McKenzie and Bickle, 1988], and assume the contours of constant depletion to be spaced uniformly between the solidus and liquidus at any given depth. Melting is only calculated relative to the dry solidus. Thus we ignore the small amount of melt produced between the wet and dry solidi (∼1% [Hirth and Kohlstedt, 1996]). We adopt a value of 400 J kg−1 °C−1 for the entropy of fusion and a heat capacity of 1250 J kg−1 °C−1 [Manglik and Christensen, 1997]. Melt is extracted from the individual particles at every time step in what amounts to an incremental fractional melting model. Melt depletion at the level of the grid is then determined at the grid nodes by averaging the depletion of the particles in the adjoining computational elements.
Figure 2. Cartoon illustration of melting as implemented in the model. 1) A mantle particle ascending adiabatically approaches the solidus. 2) The particle crosses the solidus following an adiabatic path. 3) At the end of the advective step, the amount of melting undergone by the particle is computed by comparing its temperature to that of the solidus, balancing the available thermal energy between latent heating and elevating the temperature of the remaining solid (see equation (7)). The temperature of the particle is then corrected for latent heating. 4) The particle continues its ascent adiabatically. 5) At the end of the advective step, the amount of melting undergone by the particle is computed as in step 3, with the particle temperature being compared to that of the adjusted solidus (i.e., taking into account previous melting) to determine the available thermal energy. The temperature of the particle is then corrected for latent heating, as in step 3. Steps 4 and 5 are repeated with each advective step until the particle leaves the model domain.
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 Viscosity is modeled using an Arrhenius law relationship modified by several pre-exponential coefficients [Kincaid et al., 1996],
where E is a dimensionless parameter related directly to the activation energy, ε, and inversely to the total dimensional potential temperature contrast across the model, ΔT, and the ideal gas constant, R (i.e., E = ε/RΔT). The exponential term creates a highly viscous plate along the cold upper boundary of the model domain. For a given temperature profile, the thickness of this rheological boundary layer (RBL) is controlled by E (Figures 3a and 3b). The “max” function, which returns the maximum of either the exponential term or one, allows viscosity decreases due to high temperatures associated with the plume (T > 1) to be isolated in the pre-exponential terms and controlled explicitly. The pre-exponential coefficient A1 is used to model viscosity reduction due to the temperature anomaly associated with the plume. This additional temperature dependence of the viscosity law is handled explicitly so that a variety of plume viscosities could be investigated without altering the structure of the RBL. This relationship is given by
where γ is a parameter controlling the sharpness of the decrease in viscosity, ΔηT controls the magnitude of the viscosity reduction, and Tp is the dimensionless plume temperature [Kincaid et al., 1996]. For example, a value of 10 for ΔηT (the maximum used in this study) results in a minimum value of 0.1 for A1 when T = Tp, and therefore a maximum of a factor of 10 reduction of viscosity in (9). In the experiments presented here plumes are assumed to have a potential temperature 200°C above that of ambient mantle, while the ambient mantle temperature is taken to be 1350°C, giving Tp = 1.038. We simulate the effect of dehydration on viscosity with the second pre-exponential coefficient, A2. We assume an abrupt increase in viscosity by a factor of 50 during the first 1% melting at the dry solidus, so that A2 has a value of 1 for D = 0 and a value of 50 for D ≥ 0.01 (Figure 2c) [Ito et al., 1999]. A2 increases linearly between D = 0 and D = 0.01. A maximum cut-off viscosity of ηmax = 100η0 is used, and the minimum model viscosity, found in the hot plume, is ηmin = 0.1η0. This gives a maximum viscosity variation of 103 within the model.
Figure 3. Profiles of T (a) and η (b) versus depth in the model domain, demonstrating the relative thermal and rheological boundary layers for various values of the dimensionless activation energy, E. For a given temperature profile, larger values of E lead to a thicker, more strongly sloping rheological boundary layer. c) The effect of latent heat loss and melting on viscosity. The profile follows a single plume particle from the base of the model domain to the surface, with and without consideration of dehydration effects. Viscosity is initially diminished relative to the reference mantle viscosity because of the higher plume temperature. As the particle ascends, it begins to melt at a depth of ∼125 km. If dehydration is considered (dashed path), the viscosity of the particle increases by a factor of 50 over the first 1% of melting. Further upwelling brings additional melting and a decrease in particle temperature due to loss of heat to latent heating. This decrease in temperature leads to increases in viscosity.
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 These numerical experiments are conducted in a 2-D model domain representing a vertical slice through the upper mantle oriented perpendicular to the ridge axis and passing through the center of the plume (Figure 1b). The model domain has an aspect ratio of 3:1, corresponding to horizontal and vertical dimensions of 1200 km and 400 km, respectively. The computational grid is regular and Cartesian, with 288 elements in the horizontal and 96 in the vertical. The top boundary is impermeable and has a fixed temperature (T = 0). The bottom boundary is permeable and stress-free with a fixed temperature (T = Tm). Sidewalls are permeable, stress-free and insulating. Material is free to enter or leave the domain through the bottom or either of the side boundaries.
 Spreading plates are represented by a divergent horizontal velocity imposed along the upper boundary. A horizontal velocity ±U0, corresponding to the plate velocity, is assigned along the top boundary. This prescribed plate speed is zero at the ridge axis and increases linearly to a constant value within three grid points on either side of the ridge axis. The velocities on each side of the ridge axis are equal in magnitude, but opposite in sign, so as to induce a symmetric corner flow. Plate scale flow within the model domain is therefore assumed to be a passive response to the motion of plates which are driven by body forces concentrated far outside the model boundaries.
 In order to best utilize the model to investigate flow between the plume and the ridge axis, the ridge and plume are both positioned off-center within the model domain. The ridge is located to the left of center, while the plume is located to the right. While previous studies have demonstrated that the flux of plume material from an off-axis plume to a ridge decreases with increasing plume-ridge separation [Kincaid et al., 1996], the distance between the plume and the ridge is held constant in these experiments (Lpr = 450 km).
 Separate clouds of Lagrangian particles, designated “plume mantle” and “ambient mantle” particles, are initially distributed within the model domain as in Figure 1b. Additional particles are added to the domain along the bottom boundary as particles are advected away with the flow so as to maintain a constant density of particles. Particles leaving the model domain through the bottom or either of the side boundaries are no longer tracked. Thermal buoyancy is imparted to the plume particles by assigning temperatures that are elevated relative to the ambient mantle rising through the bottom boundary to the nodes in the plume source region. We assume a plume temperature anomaly, or excess temperature, of 200°C in this region.
 The system of model equations are solved subject to the stated boundary and initial conditions using a penalty function, finite element technique which utilizes bilinear splines [Hughes, 1987]. These methods have been employed previously for studies of mantle convection in plume-ridge systems [Kincaid et al., 1996] as well as at subduction zones [Kincaid and Sacks, 1997; Kincaid and Hall, 2003], and have been thoroughly benchmarked [Travis et al., 1991]. Experiments are first run without plumes to produce a steady state thermal structure characteristic of spreading ridges, in which the cold RBL thickens with distance away from the ridge axis as a function of spreading rate. The temperature and viscosity fields from these runs serve as initial conditions for the plume-ridge experiments.