SEARCH

SEARCH BY CITATION

Keywords:

  • mid-ocean ridges;
  • faulting;
  • magmatism;
  • numerical modeling

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

We use 2-D numerical models to explore the thermal and mechanical effects of magma intrusion on fault initiation and growth at slow and intermediate spreading ridges. Magma intrusion is simulated by widening a vertical column of model elements located within the lithosphere at a rate equal to a fraction, M, of the total spreading rate (i.e., M = 1 for fully magmatic spreading). Heat is added in proportion to the rate of intrusion to simulate the thermal effects of magma crystallization and the injection of hot magma into the crust. We examine a range of intrusion rates and axial thermal structures by varying M, spreading rate, and the efficiency of crustal cooling by conduction and hydrothermal circulation. Fault development proceeds in a sequential manner, with deformation focused on a single active normal fault whose location alternates between the two sides of the ridge axis. Fault spacing and heave are primarily sensitive to M and secondarily sensitive to axial lithosphere thickness and the rate that the lithosphere thickens with distance from the axis. Contrary to what is often cited in the literature, but consistent with prior results of mechanical modeling, we find that thicker axial lithosphere tends to reduce fault spacing and heave. In addition, fault spacing and heave are predicted to increase with decreasing rates of off-axis lithospheric thickening. The combination of low M, particularly when M approaches 0.5, as well as a reduced rate of off-axis lithospheric thickening produces long-lived, large-offset faults, similar to oceanic core complexes. Such long-lived faults produce a highly asymmetric axial thermal structure, with thinner lithosphere on the side with the active fault. This across-axis variation in thermal structure may tend to stabilize the active fault for longer periods of time and could concentrate hydrothermal circulation in the footwall of oceanic core complexes.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

The style of tectonic faulting at mid-ocean ridges is highly sensitive to spreading rate, with slow spreading ridges characterized by larger and more widely spaced faults than fast spreading ridges [Searle and Laughton, 1981; Macdonald, 1982; Goff, 1991; Goff et al., 1997; Small, 1998]. The dependence of fault style on spreading rate is typically attributed to changes in the mechanical strength of the lithosphere caused by differences in the axial thermal structure, with overall cooler crust at slow spreading compared to fast and intermediate spreading ridges [Shaw, 1992; Malinverno and Cowie, 1993; Shaw and Lin, 1993, 1996]. On a regional scale, magma supply also influences axial thermal structure [Sleep, 1975; Neumann and Forsyth, 1993; Phipps Morgan and Chen, 1993; Shaw and Lin, 1996] and this influence has been thought to cause rapid variations in fault characteristics and axial morphology along ridges with relatively constant spreading rates (e.g., Galápagos Spreading Center [Canales et al., 1997; Detrick et al., 2002; Sinton et al., 2003; Blacic et al., 2004], Juan de Fuca Ridge [Canales et al., 2005], Lau Spreading Center [Martinez et al., 2006], and Southeast Indian Ridge [Goff et al., 1997]).

A correlation between fault style and magma supply is also observed along individual slow spreading ridge segments. Residual gravity anomalies [Kuo and Forsyth, 1988; Lin et al., 1990; Detrick et al., 1995] and seismic refraction data [Tolstoy et al., 1993; Canales et al., 2000; Hooft et al., 2000; Hosford et al., 2001] indicate that many slow spreading segments are characterized by along-axis variations in crustal thickness, with thicker crust near the segment center and thinner crust toward the segment ends. These variations in crustal thickness correspond to along-axis changes in fault style, with small, closely spaced faults at the segment center and larger, more widely spaced faults near the segment ends [Shaw, 1992; Shaw and Lin, 1993; Escartín et al., 1999; Hooft et al., 2000]. Thermal models that simulate focused magma supply toward segment centers predict the crust to be warmer in these regions than at segment ends [Phipps Morgan and Forsyth, 1988; Shen and Forsyth, 1992; Behn et al., 2007], and this thermal structure has been invoked to cause the segment-scale variations in faulting [Shaw, 1992; Shaw and Lin, 1993, 1996].

The inferred relationship between fault style, magma supply, and axial thermal structure is based on the premise that colder, thicker axial lithosphere can support longer lived faults that accumulate greater throw and are transported farther off-axis before a new fault is created at the ridge axis. This positive scaling between fault throw and lithospheric thickness is derived from a flexural model for lithospheric deformation, in which the stress near the fault is calculated assuming the lithosphere behaves as an elastic beam [e.g., Forsyth, 1992]. However, studies that incorporate a more realistic elastic-plastic rheology for the oceanic lithosphere show the opposite relationship, with fault offset increasing with decreasing lithospheric thickness [Buck, 1993; Lavier et al., 2000]. These predictions therefore present an inconsistency when interpreting the observed variations in ridge fault style solely in terms of differences in lithospheric thickness.

A shortcoming of most previous models for fault evolution is that they focused on the effects of magma supply and spreading rate on ridge thermal structure, but did not integrate the mechanical effects of dike injection. Dikes are well known to strongly influence the local stress field [Mastin and Pollard, 1988; Rubin and Pollard, 1988; Curewitz and Karson, 1998; Behn et al., 2006] and can even generate normal faulting, subsidence, and graben formation in volcanic rift zones [e.g., Abdallah et al., 1979; Sigurdsson, 1980; Pollard et al., 1983; Rubin, 1992; Hofton and Foulger, 1996; Bull et al., 2003; Wright et al., 2006]. Thus, variations in the rate and depth distribution of dikes at the ridge axis will likely influence the style of faulting [Chadwick and Embley, 1998; Carbotte et al., 2006]. In particular, the longevity of slip on normal faults has been linked to the rate of magma accretion at the ridge axis [Buck et al., 2005; Tucholke et al., 2008].

The goal of this study is to explore the relationship between magmatic and tectonic extension, axial thermal structure, and faulting at mid-ocean ridges. To do so, we develop a 2-D elastic-viscoplastic model for dike injection that accounts for both the thermal and mechanical effects of dike intrusion. On the basis of our numerical results we quantify the relationship between axial thermal structure, magma accretion rate, fault throw, and fault spacing. We find that fault throw and spacing increase for decreasing lithospheric thickness, decreasing rates of off-axis lithospheric thickening, and most importantly, for decreasing rates of magma accretion. Our results imply that magma supply influences faulting style predominantly by modulating the amount of magmatic extension, while the effects of axial thermal structure are secondary and more complex than often cited.

2. Model for Magma Intrusion at Mid-Ocean Ridges

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

2.1. Conceptual Model

Seafloor spreading at mid-ocean ridges is accommodated through a combination of dike intrusion and tectonic faulting. Diking is typically focused within a 5–10 km neovolcanic zone at the ridge axis, with active faulting localized around the neovolcanic zone [Ballard and van Andel, 1977; Macdonald, 1986; Macdonald and Fox, 1988; Smith and Cann, 1993; Hooft et al., 1996]. Because dikes advect mass and heat vertically and laterally through the crust, they have the potential to influence near ridge faulting by altering both the axial lithospheric structure and the local stress field. Here we simulate the effects of magma intrusion at an idealized 2-D spreading ridge assuming a constant spreading rate and uniform depth distribution of intrusion (Figure 1). Although diking is a stochastic process, we approximate its long-term influence by assuming a constant injection rate with time. In our companion paper [Ito and Behn, 2008] we relax this condition and evaluate how temporal variations in the rate of magma intrusion affect faulting and the evolution of axial morphology. The rate of magma intrusion is described by the parameter M, which is defined as the fraction of the total spreading rate accommodated by magma accretion [e.g., Buck et al., 2005; Behn et al., 2006]. Thus, M = 1 corresponds to the case where the rate of dike opening is equal to the full spreading rate and M = 0 represents purely amagmatic spreading. Heat associated with the injection temperature of the magma and the latent heat of crystallization [Sleep, 1975; Phipps Morgan and Chen, 1993] is added to the ridge axis in proportion to the rate of intrusion.

image

Figure 1. Model setup used for numerical simulations of magmatic spreading at a mid-ocean ridge. The dike injection zone is illustrated with the thick black line. Gray region represents the portion of the model space that experiences brittle deformation, while the white region behaves viscously. Deformation is driven by applying a uniform half-spreading rate, us, to either side of the model space. The rate of magma injection is controlled by the parameter M, which is defined as the fraction of the total spreading rate accommodated by magma accretion; σn and τ denote normal and shear stress boundary conditions, respectively, and T is temperature.

Download figure to PowerPoint

2.2. Mathematical and Numerical Formulation

We use the Fast Lagrangian Analysis of Continua (FLAC) technique [Cundall, 1989; Poliakov et al., 1993] to solve the equations of conservation of mass and momentum in a visco-elastic-plastic continuum (see Appendix A1). This explicit hybrid finite element-finite difference approach has been used to simulate localized deformation (i.e., faulting) in a variety of extensional environments [e.g., Hassani and Chéry, 1996; Poliakov and Buck, 1998; Lavier et al., 2000; Behn et al., 2006] and is described in detail elsewhere [Poliakov et al., 1993; Lavier and Buck, 2002]. Dike intrusion is simulated kinematically by uniformly widening a vertical column of elements at the center of the model space at a rate, udike (Figure 1). Dike opening is achieved numerically by relating the stresses in the dike elements, σijdike, to the horizontal strain, ɛxxdike, by

  • equation image
  • equation image
  • equation image
  • equation image

where ν is Poisson's ratio, ɛxxdike = udikedt/xinj, udike is the widening rate of the dike, xinj is the width of the injection zone, and dt is the length of the time step. This approach allows the location of the dike zone to evolve with the deforming mesh and eliminates the need to impose nodal velocities explicitly on the dike elements.

Temperature evolution is modeled using a Lagrangian formulation, so heat advection occurs with the deforming grid. At each time step we use explicit finite differences to solve for the diffusion of heat with sources (see Appendix A2 for derivation),

  • equation image

where T is temperature and Cp is heat capacity. The three temperature intervals are needed to simulate the heat carried by the injected magma as well as the latent heat (L) with crystallization in which crystal fraction is assumed to increase linearly with decreasing T between the magma liquidus (Tliq) and solidus (Tsol). No magmatic or latent heat (per kg) is added in the first (top) interval, magmatic and latent heat are added and crystallinity changes with T in the second interval, and all of the magmatic and latent heat are added in the third interval. The effects of hydrothermal circulation are simulated by increasing effective diffusivity, κ, by a factor, Nu, in regions of the model space above a threshold depth of 7 km and where T < 600°C [e.g., Phipps Morgan et al., 1987].

Material behavior is a function of the thermal structure, stress, and accumulated plastic strain (Appendix A3). In regions where deformation is visco-elastic, the material behaves as a Maxwell solid. Viscous deformation is incompressible and follows a non-Newtonian temperature and strain rate dependent power law [Kirby, 1983; Chen and Morgan, 1990; Kohlstedt et al., 1995]. Material properties appropriate for a dry diabase rheology [Mackwell et al., 1998] are assumed throughout the model space. Plastic yielding is controlled by Mohr-Coulomb theory, in which cohesion is a function of the total accumulated plastic strain [Poliakov and Buck, 1998]. Cohesion, C, decreases linearly with the total accumulated plastic strain from its initial value, Co, to a minimum, Cmin, over a fault offset Δsc [Lavier et al., 2000]. Unless noted in the text we assume values of Co, Cmin, and Δsc equal to 44 MPa, 4 MPa, and 500 m, respectively (see Table 1 for a list of all model parameters).

Table 1. Summary of Model Parameters
VariableMeaningValueUnits
Ccohesion MPa
Coinitial cohesion44MPa
Cminminimum cohesion4MPa
Cpspecific heat of magma1000J kg−1 °C−1
EYoung's modulus30 × 109Pa
Fccrystal fraction = (TliqT)/(TliqTsol)0–1
hinjinjection zone height6.0km
Hlithospheric thickness km
Hoaxial lithosphere thickness, i.e., depth to 600°C3–9km
Llatent heat of crystallization5 × 105J kg−1
Mfraction of spreading accommodated by accretion0.5–1
Nuefficiency of hydrothermal circulation4–8
Taasthenosphere temperature1300°C
Tsolsolidus temperature of basalt1000°C
Tliqliquidus temperature of basalt1300°C
uAF, uIFvelocity of active and inactive faults cm a−1
ushalf spreading rate1–2.5cm a−1
xAFdistance an active fault migrates away from the axis km
xinjinjection zone width0.25km
μfriction coefficient0.6
Δsccritical slip for total cohesion loss0.5km
ΔSfault spacing km
Δtlifetime of active fault years
Δxtotal fault heave km
ϕlithoff-axis slope of lithosphere10–30°
θfault dip °
νPoisson's ratio0.25
κbase thermal conductivity2.5W m−1 K−1
ρdensity3300kg m−3

2.3. Model Domain and Boundary Conditions

The numerical domain is 60 km wide by 20 km deep, with a grid spacing that is smallest (0.25 km × 0.25 km) at the ridge axis and coarsens with distance from the ridge (0.75 km × 1 km at the lower left/right corners of the box). Deformation is driven by imposing a uniform far-field extension rate along the edges of the model space with a half-spreading velocity, us (Figure 1). The injection zone is situated at the ridge axis and extends from the surface down to a depth hinj (Figure 1). The rate of dike opening, udike, is controlled by the parameter M such that udike = 2·M·us. A hydrostatic boundary condition is assumed for the base of the model space and the top boundary is stress free. The top of the model space is set to 0°C, while the bottom temperature is set equal to the steady state solution for a cooling half-space at the depths coinciding to the model base and with a maximum asthenosphere temperature (Ta) of 1300°C. All models are run for a sufficient model time that all lithosphere was created at the axis during the model run (e.g., 30 km/us or 3 Ma for us = 1 cm/a).

3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

To analyze the basic cause for the behavior of our numerical models we first examine a more simplified set of calculations. Faulting at mid-ocean ridges is often assumed to occur sequentially, with the majority of tectonic extension focused on a single fault at any given time [e.g., Shaw and Lin, 1993]. In such a model, faults initiate near the ridge axis and are rafted off-axis until the stress required to continue to slip on the initial fault exceeds the stress required to break a new fault (Figure 2). Numerical studies show that in most cases the new fault will be antithetic and form on the opposite side of the ridge axis from the original fault [e.g., Melosh and Williams, 1989; Poliakov and Buck, 1998]. This results in asymmetric rates of off-axis fault transport on either side of the ridge axis, with the active fault migrating away from the axis at a rate, uAF = 2us(M − 0.5), and the inactive fault migrating at a rate uIF = us [Buck et al., 2005]. This sequential style of fault initiation and asymmetric off-axis migration implies that total fault heave, Δx, is a simple function of M and the distance the active fault is transported off-axis before the formation of the new fault, xAF [Buck et al., 2005]

  • equation image

Assuming that a fault migrates a distance, xAF while it is active plus a distance us(xAF/uAF) until the next fault on the same plate forms (Figure 2), the fault spacing, ΔS, is

  • equation image

The distance, xAF, that a fault will remain active depends on a combination of factors, including lithospheric thickness (H), the off-axis slope of the lithosphere (ϕlith), fault dip (θ), and the frictional properties of the lithosphere. Thus, observations of fault spacing and fault heave reflect upon the rate of magma injection (M) as well as the material properties of the oceanic lithosphere.

image

Figure 2. Schematic model for sequential fault development at mid-ocean ridges. Active faulting (red) occurs on one side of the ridge axis until the stress required to continue slip on the active fault exceeds that required to break a new fault nearer the ridge axis. This model predicts that both fault spacing, ΔS, and total fault heave, Δx, can be predicted from the fraction of magmatic extension, M, and the distance, xAF, that the active fault migrates off-axis before breaking a new fault. ϕlith denotes the off-axis slope of the base of the lithosphere, which in the numerical models is defined at the 600°C isotherm. In runs with a fixed thermal structure, ϕlith is imposed as a linear gradient; in runs with thermal evolution, ϕlith is calculated from the resulting thermal structure.

Download figure to PowerPoint

To compare equations (3) and (4) with predictions from our numerical models below, we compute xAF by following the concepts of previously proposed force-balance models [e.g., Forsyth, 1992; Buck, 1993; Lavier et al., 2000]. These models assume that the depth-integrated tensile stress (here and in Appendix B we are referring to the nonlithostatic stress) required to keep an existing fault active equals or exceeds the depth-integrated stress to break a new fault near the ridge axis. We assume the net pulling forces needed to keep a fault slipping are those associated with friction and reduced cohesion (C) on the fault, FF, and those associated with bending an elasto-plastic lithosphere around the fault FB [Buck, 1993; Lavier et al., 2000]. The strength of the unfaulted, axial lithosphere FI is supplied by friction as well as the full cohesion (C0). Hence, the criteria for forming a new fault (which determines xAF) is

  • equation image

While FI remains constant, the other two change off-axis as lithospheric thickness H and fault heave Δx increase and fault dip θ decreases (Figure 3). Specifically, FF decreases quickly owing to the loss of cohesion with fault slip, and causes the fault to become weak and remain active for a period time. However, once cohesion is fully reduced, FF and FB increase slowly with distance x as the fault moves into increasingly thick lithosphere, until eventually (equation (5)) is achieved at x = xAF and a new fault forms on the opposite plate.

image

Figure 3. Calculation of the depth-integrated tensile strength due to friction on the active fault, FF (thin red line), depth-integrated tensile strength associated with bending an elasto-plastic layer due to slip on the fault, FB (thin blue line), and depth-integrated tensile strength of unfaulted lithosphere, FI (thick dashed line), as a function of fault heave, Δx′, for Ho = 6 km and M = 0.7. Note that total fault heave is related to the distance the fault has migrated off-axis (xAF) through equation (3). FF initially decreases owing to cohesion loss on the active fault surface and then increases owing to the off-axis thickening of the lithosphere. FB increases with heave Δx′ as elastic bending stresses accumulate in the lithosphere. The force required to continue extension on the active fault (FF + FB, thick solid line) exceeds FI when Δx′ ∼ 4.9 km, corresponding to xAF ∼ 3.3 km.

Download figure to PowerPoint

4. Numerical Models of Fault Development at Mid-Ocean Ridges

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

4.1. Fixed Ridge Thermal Structure

To investigate the influence of lithospheric structure on fault style, we conducted a series of model runs with a fixed axial thermal structure. At the ridge axis, temperature was imposed to increase linearly with depth from 0°C at the surface to 600°C at a depth Ho (representing the approximate depth of the brittle-ductile transition). Below Ho temperature increases rapidly to the asthenospheric temperature (1300°C) over a depth interval of 0.5 km. An important aspect of this set of calculations was that we imposed a linear increase in lithospheric thickening off-axis with a slope that was proportional to the on-axis thickness Ho, i.e., the lithosphere doubled in thickness over 20 km. The maximum depth of seismicity at the Mid-Atlantic Ridge ranges from 3 to 9 km [Toomey et al., 1988; Kong et al., 1992; Wolfe et al., 1995; Barclay et al., 2001], and seismic moment release studies [Solomon et al., 1988] and estimates of cumulative fault slip [Escartín et al., 1999] indicate that 70–90% of spreading is accommodated by magmatic intrusion. Thus, we first examined a series of runs for Ho = 3–9 km with M = 0.7 for a half spreading rate us = 2.5 cm/a. The height of the magma injection zone was set to be equal to the axial lithospheric thickness (hinj = Ho).

In all cases, faults initiate near the axis where the lithosphere is thinnest and are subsequently rafted off-axis by continued magma accretion (Figure 4). This style of fault initiation and migration is consistent with the sequential fault model shown in Figure 2, with a single fault actively accumulating slip at any time. Fault heave (Δx) and spacing (ΔS) are estimated from the average value of each quantity for all faults formed during a model run. Comparing cases with Ho = 3–9 km, we find that faults formed in thinner axial lithosphere accumulate greater slip and are more widely spaced than those formed in thicker lithosphere (Figures 5 and 6a6c). These results contrast with the predictions of a purely elastic lithosphere, which predicts Δx ∼ H1/4 [Forsyth, 1992], but are consistent with the scaling for an elastic-plastic layer [Buck, 1993; Lavier et al., 2000] (and hence our use of their scaling for FB).

image

Figure 4. Model results after 1 Ma of deformation for two cases with M = 0.7 and (a) Ho = 4 km and (b) Ho = 8 km. Top panels show horizontal surface velocity; bottom panels show second invariant of the strain rate tensor. High strain rates within the lithosphere are found along the active fault and within the dike injection zone. Surface velocity of the active fault follows kinematic prediction of Buck et al. [2005]. Note that faults have greater heave and are more widely spaced in the case with thinner lithosphere (Ho = 4 km).

Download figure to PowerPoint

image

Figure 5. Model topography after 1.5 Ma of spreading for axial lithospheric thicknesses (Ho) ranging from 3 to 9 km and M = 0.7. Small vertical bars and dots show locations of fault tops and bottoms, respectively. Solid dots denote the base of active faults, while open dots show the base of inactive faults. Note that fault heave and spacing increase for thinner axial lithosphere, although total rift relief is greater in thicker lithosphere. Remeshing of the model grid causes topography to smooth sharp corners with time, which introduces an insignificant uncertainty in measurements of fault properties.

Download figure to PowerPoint

image

Figure 6. Total fault heave (Δx), fault spacing (ΔS), and final dip angle (θ) versus (a–c) axial lithospheric thickness (Ho) and (d–f) the rate of magmatic accretion (M). Models shown in Figures 6a–6c are calculated for M = 0.7. Models in Figures 6d–6f assume Ho = 6 km. Filled symbols represent the average values of each parameter estimated from all faults formed during a single model run. Open symbols in Figures 6d and 6e indicate model runs with Δx and ΔS that are greater than the vertical axis. Error bars represent ±25% uncertainty on estimates of Δx and ΔS, which is typically sufficient to encompass the variability between individual faults in a model run. Error bars on θ are based on uncertainty in determining dip angle due to the finite width of the fault zone (∼3–4 grid elements) in the numerical experiments. Solid curves show predictions from the force-balance model (see Appendix B). Fault dip is calculated assuming an initial dip of 55°, a rotation rate of dθ/dΔx = 5°/km, and a minimum dip of 35°.

Download figure to PowerPoint

A prediction of our numerical models that is important to the force-balance calculations is that fault dip decreases with increasing fault heave (Figure 7). Initially, fault dip (θ) is controlled by the coefficient of friction as predicted by Mohr-Coulomb theory, and is ∼55° in our calculations where μ = 0.6. As slip accumulates on the fault, θ decreases to a minimum value of ∼30–35° over a heave of ∼4 km and then remains nearly constant thereafter. This behavior appears to be consistent over a range of Ho and M (Figure 7). To account for this effect in the force-balance model, we assume a linear decrease in θ with Δx. Figure 6c compares the final dip angle measured in the numerical runs with that predicted by the force-balance model. Comparing the predictions of fault heave and spacing from the force-balance model to our numerical results, we find that the force-balance model provides a good fit, slightly over-predicting fault spacing for thin axial lithosphere (Figures 6a and 6b).

image

Figure 7. Evolution of fault dip as a function of fault heave for a series of model runs with a fixed axial thermal structure and Ho = 4–8 km and M = 0.6–0.8. For all cases, faults initiate with dips of ∼55° consistent with Mohr-Coulomb theory and rotate to a minimum dip angle of 30–35° over ∼4 km of fault heave.

Download figure to PowerPoint

In the next set of model calculations we examine the effect of lithospheric slope (ϕlith) on fault characteristics. Over a range of slopes from 10°–30°, fault heave and spacing decrease with increasing slope (Figure 8). These numerical results are consistent with the force-balance model, which predicts that the net force on either side of the active fault (FF + FB) will increase with distance away from the axis in proportion to the rate of off-axis, lithosphere thickening. Because (FF + FB) increases with lithospheric slope, but FI remains constant, (FF + FB) = FI at smaller values of xAF and, thus, smaller values of Δx and ΔS.

image

Figure 8. Total (a) fault heave (Δx) and (b) fault spacing (ΔS) as a function of lithospheric slope for Ho = 6 km and M = 0.7. Filled circles represent the average values of each parameter estimated from all faults formed during a single model run with error bars calculated as in Figure 6. Solid curves show predictions from the force-balance model (see Appendix B).

Download figure to PowerPoint

We next investigated a series of calculations with M = 0.5–1 and Ho = 6 km to quantify the mechanical effects of magma intrusion rate on fault heave and spacing (Figures 6d6f). These model runs demonstrate that fault heave and spacing increase rapidly with values of M approaching 0.5. In particular, for M < 0.6, a single fault remains stable for the entire length of the model run. The force-balance model provides a good fit to these numerical results, predicting a sharp increase in fault heave and spacing for M < 0.6 due to the (M − 0.5)−1 dependence of equations (3) and (4). We next used the force-balance model to extend these results to a wide range of geologically reasonable values for M and Ho (Figure 9). Classifying large-offset faults as those faults for which Δx > Ho, we find that large-offset faults form where Ho is small and M approaches 0.5 (Figure 9a). Figure 9 also illustrates the sensitivity of fault heave and spacing to the initial cohesion of the brittle layer (Co) and the critical fault offset required for complete cohesion loss (Δsc). Specifically, we find that while heave and spacing are relatively insensitive to Δsc, they both decrease significantly with decreasing values of Co.

image

Figure 9. Total fault heave and spacing calculated from the force-balance model (Appendix B) as a function of axial lithospheric thickness and magma accretion rate, M. Results are shown for (a and b) Co = 44 MPa, Cmin = 4 MPa, and Δsc = 500 m; (c and d) Co = 30 MPa, Cmin = 4 MPa, and Δsc = 500 m; and (e and f) Co = 44 MPa, Cmin = 4 MPa, and Δsc = 5000 m. Dashed line denotes boundary between large and small offset faults, where large offset faults are characterized by Δx > Ho. In general, fault heave and spacing are more sensitive to M than to axial lithospheric thickness.

Download figure to PowerPoint

4.2. Thermal Evolution Incorporating Magmatic Heat Input

We next examine a series of simulations in which temperature was allowed to evolve with time. Models were run for half rates appropriate for slow (us = 1.0 cm/a) and intermediate (us = 2.5 cm/a) spreading ridges, Nusselt numbers of 4 and 8, and values of M between 0.5 and 1. As in the cases with fixed thermal structure, active faults alternate between sides of the ridge axis. The overall axial thermal structure is dominantly controlled by spreading rate, with the rate of magma injection and the efficiency of hydrothermal circulation influencing temperature to a lesser degree (Figure 10). In general, the depths of the 600°C and 1100°C isotherms beneath the ridge axis decrease with increasing M.

image

Figure 10. Depth of the (a) 600°C and (b) 1100°C isotherms as a function of magmatic accretion rate, M. (c) Slope of axial lithosphere calculated from the change in depth of the 600°C isotherm from the ridge axis to a point located 5 km off-axis. Filled and open symbols show results for slow (us = 1.0 cm/a) and intermediate (us = 2.5 cm/a) spreading rates, respectively. Circles and squares denote Nusselt numbers of 4 and 8, respectively. Dashed line in Figure 10b illustrates threshold depth at which an axial magma chamber (AMC) would be predicted to form within a 6-km thick crust on the basis of the depth of the 1100°C isotherm.

Download figure to PowerPoint

Fault growth also influences axial temperatures, creating an asymmetric thermal structure across the ridge axis. Specifically, elevated temperatures are predicted near the base of the lithosphere on side of the ridge axis where the active fault is located. This asymmetry results from the advection of warm mantle material into the footwall of the active fault (Figure 11). After a new fault forms on the opposite side of the ridge axis the thermal perturbation is advected off-axis and gradually decays by diffusion (Figure 11g). The asymmetry is most extreme in the case of long-lived faults (e.g., M = 0.5 shown in Figures 11a11d) because the axial thermal structure has sufficient time to fully equilibrate with the steady fault geometry. Asymmetry is more subtle, but still present, at higher M values when fault size and spacing are small (e.g., M = 0.8 in Figures 11e and 11f).

image

Figure 11. Model runs showing deformation and thermal structure after 0.95 Ma of extension for (a–d) M = 0.5 and (e–h) M = 0.8 assuming us = 2.5 cm/a and Nu = 8. Illustrated are (a and e) strain rate, (b and f) temperature, (c and g) temperature difference, ΔT, between the two sides of the ridge axis at a given distance from the axis and depth beneath the seafloor, and (d and h) ΔT as a function of time at a point located 3 km off-axis and at a depth of 6 km. ΔT is greater in the case of M = 0.5 because the thermal structure equilibrates with the large-offset fault that forms on the left side of the ridge axis. In contrast, the small-offset faults formed in the M = 0.8 remain cooler and fluctuate as faulting alternates sides of the ridge axis.

Download figure to PowerPoint

Like the results of our runs with imposed thermal structure, these calculations also show that fault heave and spacing are primarily controlled by M, with values of M approaching 0.5 producing larger and more widely spaced faults (Figure 12). The importance of lithospheric structure is secondary, but still evident by comparing fault heave and spacing at the same M but at different spreading rates. Intriguingly, the slow spreading runs (us = 1 cm/a) with larger H0 (as inferred from depth of the 600°C isotherm) produce larger faults than do the intermediate spreading cases (us = 2.5 cm/a) with smaller H0 (Figure 12a). This trend is opposite to what was observed in the fixed thermal structure runs, where thicker axial lithosphere produces smaller, more closely spaced faults (e.g., Figures 6a and 6b).

image

Figure 12. (a) Total fault heave and (b) fault spacing as a function of M for models with thermal evolution. Gray and white symbols show results for slow (us = 1.0 cm/a) and intermediate (us = 2.5 cm/a) spreading rates, respectively. Squares and circles denote Nusselt numbers of 4 and 8, respectively. Error bars are calculated as in Figure 6. Solid and gray lines show predictions from the force-balance model (Appendix B) assuming Ho = 2 km, ϕlith = 30° and Ho = 6 km, ϕlith = 15°, respectively.

Download figure to PowerPoint

The more important factor, therefore, appears to be the rate of off-axis lithospheric thickening. Figure 10c shows the lithospheric slope near the ridge axis as calculated from the change in depth of the 600°C isotherm from the axis to a point 5 km off-axis. Cases with us = 1 cm/a display a significantly shallower slope than those for us = 2.5 cm/a (the smaller us creates a larger H0 and thus less subsequent lithospheric thickening off-axis). This effect is enhanced for values of M approaching 0.5 and in runs with efficient hydrothermal cooling (e.g., Nu = 8). The importance of lithospheric slope is illustrated by comparing a force-balance calculation for Ho = 2 km and ϕlith = 30° to a calculation with Ho = 6 km and ϕlith = 15° (Figure 12). Specifically, the later case with a shallower slope predicts larger values of fault heave and spacing for M = 0.5–1, even though the axial lithospheric thickness is greater.

In light of this result, the previously noted asymmetry in lithospheric slope associated with faulting could be yet another important factor controlling fault behavior. Our numerical calculations suggest that the relative difference in lithospheric thickness across the ridge axis becomes greater for lower values of M and for slower spreading rates. For the cases in which M ≤ 0.6 and us = 1 cm/a, this asymmetry becomes sufficiently large that it tends to promote faulting on only one side of the axis, i.e., the side with thinner lithosphere. This asymmetry has two effects on the predicted fault characteristics. The first is to complicate the scaling between fault heave and spacing, which is largely controlled by the behavior, at larger M and us, of sequential faults forming on opposite sides of the lithosphere. Specifically, for M ≤ 0.6 and us = 1 cm/a, ΔS is smaller for a given Δx than is predicted for cases with faulting on alternating plates. The second effect is to further enhance the lithosphere asymmetry, thereby further stabilizing long-lived normal faults.

5. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

Our numerical models display a style of sequential faulting (e.g., Figure 2) that is consistent with observed seismicity at the Mid-Atlantic Ridge, which shows that active deformation effectively ceases once a fault migrates off-axis beyond the crest of the rift valley [Smith et al., 2003]. We find that total fault heave (Δx) and spacing (ΔS) tend to increase for decreasing values of axial lithospheric thickness (Ho) as predicted by Buck [1993], but opposite to the positive relationship inferred from geologic and geophysical observations at mid-ocean ridges [e.g., Shaw, 1992; Malinverno and Cowie, 1993; Shaw and Lin, 1993, 1996]. It thus appears that the more important influence of lithospheric structure on faulting, is the increase in Δx and ΔS with decreasing lithospheric slope. However, the primary control on fault development is the fraction of plate motion accommodated by magmatic accretion (M). Both fault heave and spacing depend nonlinearly on M, because the rate that an active fault migrates into thicker and stronger lithosphere off-axis is proportional to (M − 0.5). Thus, as M approaches 0.5 faults can be nearly stationary and stable for vary long times [Buck et al., 2005].

If correct, this interpretation suggests that intrasegment variations in fault heave and spacing at slow spreading ridge are primarily caused by the mechanical, and secondarily by the thermal, effects of reduced magmatic accretion near segment ends. Reduced magma supply at segment ends is consistent with concepts of a high flux of melt being delivered from the mantle to the segment center, redistributed at shallow depths by along-axis dike propagation, and approaching the distal segment ends with reduced frequency [e.g., Fialko and Rubin, 1998].

5.1. Comparison to Observations at Slow and Intermediate Spreading Ridges

To test the hypothesis that fault style is controlled by the rate of magma injection at mid-ocean ridges, we used bathymetric data from a range of slow and intermediate spreading ridges to estimate fault heave, fault spacing, and M. Fault heave and spacing were calculated from the average value of each quantity measured from individual across-axis bathymetry profiles (Figure 13). M was estimated by plotting cumulative fault heave as a function of distance off-axis and finding the best fit linear slope to this distribution [Escartín et al., 1999]. For most of the ridges we studied, M ranges from ∼0.7–0.85, consistent with previous estimates from cumulative fault throw [Escartín et al., 1999] and seismic moment release [Solomon et al., 1988] at slow spreading ridges. However, several profiles from the Mid-Atlantic Ridge displayed M values of 0.4–0.6. These locations typically correspond to the end of spreading segments and coincide with previously identified oceanic core complexes (e.g., Kane megamullion at 23.5°N MAR [Tucholke et al., 1998] shown in Figure 13d).

image

Figure 13. (a) Example bathymetry profiles across the Galapagos Spreading Center (GSC), Southeast Indian Ridge (SEIR), and Mid-Atlantic Ridge (MAR). Oblique Mercator projections of multibeam bathymetry show the locations of the above profiles across the (b) GSC, (c) SEIR, and (d) MAR. Long vertical (red) dashed line marks our picks of the approximate center of the axes. Small vertical bars and dots mark our picks of fault tops and bottoms (not shown in Figures 13b–13c), respectively. West of the MAR, bars mark breakaway locations, and the solid circles mark terminations of oceanic core-complex identified by Tucholke et al. [1998].

Download figure to PowerPoint

In general, these data show that fault heave and spacing increase as M decreases toward 0.5 in agreement with the trends predicted by the force-balance model and our numerical simulations (Figure 14). We find that the data can be best fit with a maximum cohesion of 30 MPa and lithospheric slopes ranging from 10° to 30°, consistent with the slopes predicted by our thermal models (Figure 10c). The observed values of fault heave are slightly smaller than predicted by the force-balance model at values of M approaching 0.5. Estimates of heave and spacing in regions of low M are sensitive to additional processes not included in the force-balance model. For example, the model does not account for flexural faulting in the footwall of an oceanic core complex. Small faults associated with unbending of the footwall will reduce average fault heave, spacing, as well as estimates of M from bathymetric data (e.g., see MAR profiles in Figure 13d), thus biasing the direct comparison of the mid-ocean ridge data with the force-balance model. However, the general inverse relationship between fault heave and spacing and values of M between 0.5 and 1 (Figure 14) supports our assertion that fault style is strongly controlled by the rate of magmatic accretion at mid-ocean ridges.

image

Figure 14. (a) Average fault spacing and (b) heave versus fraction nontectonic strain, M, as estimated from our picks of fault tops and bottoms along the GSC, SEIR, MAR, Lau Basin (Lau), and Juan de Fuca Ridge (JDF). M is estimated from the best fitting slope of cumulative fault heave versus distance on both sides of the axis. Errors in M are based on an estimate of 20% error in this slope. Errors in spacing and heave are ± one standard deviation of the population for each profile. Theoretical predictions are calculated from the force-balance model (Appendix B) assuming lithospheric slopes of 10° and 30°. Calculations assume an axial lithospheric thickness Ho = 6 km, a fault rotation rate of dθ/dΔx = 5°/km, a minimum fault dip of 35°, and an initial and minimum cohesion of 30 and 4 MPa, respectively. Kinks in curves at M ∼ 0.55 correspond to situations in which large offset faults rotate to the minimum dip angle before a new fault is formed. In these cases, θ no longer evolves with fault heave in equation (B2), resulting in the change in curvature of the theoretical predictions.

Download figure to PowerPoint

5.2. Role of Magmatism in the Evolution of Oceanic Core Complexes

Large-offset normal faults are an important class of faults at mid-ocean ridges because of their potential to exhume lower crustal and upper mantle rocks at the seafloor in oceanic core complexes [Karson and Dick, 1983; Cannat, 1993; Tucholke and Lin, 1994; Cann et al., 1997; Tucholke et al., 1998]. Several competing models have been proposed regarding the role of magmatism in the formation of oceanic core complexes. Tucholke and Lin [1994] proposed that core complexes form during periods of amagmatic spreading and are terminated by later magmatic periods. In this model, the core complex (or megmamullion) is created by a long-lived detachment fault that cuts the entire lithosphere and soles near the brittle-ductile transition [Tucholke et al., 1998, 2001]. Dredging [Karson and Dick, 1983; Cannat, 1993; Cannat et al., 1995, 1997; Blackman et al., 1998; Escartín et al., 2003] and residual gravity anomalies [Tucholke and Lin, 1994; Tucholke et al., 1997; Blackman et al., 1998] indicate that gabbros and serpentinized peridotites are exposed in the fault footwall, leading to the suggestion that deformation on the detachment is localized by the presence of serpentine [Cann et al., 1997; Escartín et al., 1997; Tucholke et al., 1998].

An alternative model has been proposed by Dick et al. [1991, 2000], who argue that core complexes form during magmatic periods and that the detachment soles into a melt-rich region in the shallow lithosphere located near the dike-gabbro transition. This model is supported by evidence for melt-assisted deformation in fault rocks sampled from Hole 735B on Atlantis Bank (57°17′E Southwest Indian Ridge) [Dick et al., 2000; Natland and Dick, 2001]. However, in contrast most footwall rocks exposed in core complexes on the Mid-Atlantic Ridge show no evidence for high-temperature deformation [Escartín et al., 2003], suggesting a much cooler thermal regime during fault formation than proposed by Dick et al. [2000]. Ildefonse et al. [2007] proposed a hybrid model for core complex formation, in which the detachment is initiated by the rheologic contrast between gabbroic intrusions and surrounding peridotite. In this scenario, core complex formation is associated with mafic intrusions, but does not require active faulting in the presence of melt.

More recently, Buck et al. [2005] proposed that oceanic core complexes form at intermediate rates of magmatism, when the accretion accommodates ∼50% of the total plate separation rate. This model is supported by geologic and geophysical observations that indicate megamullion frequency along mid-ocean ridges correlates with intermediate rates of magmatism. Tucholke et al. [2008] used residual gravity anomalies to show that some detachment terminations correspond to periods of crustal thickening, while others correspond to crustal thinning. On the basis of these observations, Tucholke et al. [2008] concluded that oceanic core complexes form under restricted conditions when the rate of magmatism is neither too high nor too low (the “Goldilocks hypothesis”).

Our results showing increased fault throw and heave as M approaches 0.5 support the “Goldilocks hypothesis” for the formation of core complexes. In addition, our numerical results suggest that depending on the spreading rate and efficiency of hydrothermal circulation, core complexes can form in either the presence (e.g., us = 2.5 cm/a, Nu = 4) or absence (us = 1.0 cm/a, Nu = 4) of a crustal magma chamber (Figure 10). In our models, the key for generating large offset faults is the rate of accretion in the brittle portion of the lithosphere. Thus, so long as ∼50% of the extension above the brittle-ductile transition is supported by magmatism a large-offset fault is predicted to form. This result may help to explain the apparently discrepant temperature conditions recorded in fault rocks from different detachments, and implies that a one-size-fits-all thermal model is not necessary for all oceanic core complexes.

Lithospheric structure may also play a secondary role on the longevity of oceanic detachment faults. The reduced rate of off-axis lithospheric thickening beneath the active fault will tend to stabilize the active fault by making it more difficult to initiate a fault in the thicker lithosphere on the opposite side of the ridge axis. In situations where the asymmetry is extreme (e.g., cases with low us and M approaching 0.5), it may also result in the breakdown of the sequential fault model, focusing deformation onto one side of the ridge axis for multiple fault cycles.

5.3. Fault Controls on Hydrothermal Venting

The two primary mechanisms proposed for promoting high-temperature hydrothermal venting are the heat supplied by crustal magma chambers [Cann et al., 1985] and cracking fronts into hot, but unmelted rock [Lister, 1974; Wilcock and Delaney, 1996]. The coincidence of high-temperature vent sites above seismically identified crustal magma chambers [e.g., Detrick et al., 1987; Singh et al., 1998, 2006; Martinez et al., 2006; Van Ark et al., 2007] supports cooling and crystallization of basaltic melts as the heat source for volcanically hosted hydrothermal systems at magmatically robust ridges [e.g., Cannat et al., 2004]. In contrast, tectonically hosted vent systems typically form at slow spreading, magma poor ridges, and sometimes in the absence of a crustal magma chamber. Modeling suggests that if cracking fronts extend deep enough into hot (but unmelted) lithosphere, they can supply sufficient heat to drive these hydrothermal [Cannat et al., 2004; German and Lin, 2004]. These theoretical studies are supported by recent seismic data from the TAG segment of the Mid-Atlantic ridge, which shows no evidence for crustal melt beneath the TAG hydrothermal mound [Canales et al., 2007].

The TAG mound sits directly above a low-angle normal fault, which has been interpret to represent the early stages of formation of an oceanic core complex [Canales et al., 2007; deMartin et al., 2007]. Thus, its location corresponds to the region of elevated temperatures in the footwalls of the large-offset faults produced in our M = 0.5–0.6 models (e.g., Figures 11b and 11c). Moreover, tectonically hosted, high-temperature vent sites and deposits associated with ultramafic lithologies (suggesting significant unroofing on normal faults) have been identified at Rainbow and Logachev vent fields on the Mid-Atlantic Ridge [Cherkashev et al., 2000; Charlou et al., 2002] and along the Knipovich [Connelly et al., 2007] and Southwest Indian Ridge [German et al., 1998; Bach et al., 2002]. Taken together these observations suggest that the upward advection of heat due to exhumation of the footwall on long-lived normal faults combined with increased permeability in the active fault zone may play an important role in focusing tectonically hosted hydrothermal systems. Future modeling studies that fully couple crustal deformation with fluid circulation are necessary to better constrain the feedbacks between hydrothermal circulation, alteration, and detachment faulting.

6. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

Using numerical models we have shown that faulting at mid-ocean ridges typically follows a sequential pattern of initiation, growth, and termination. Under all conditions examined in this study a single active fault initiates near the ridge axis and is subsequently rafted off-axis by continued magmatic accretion, until it becomes mechanically easier to break a new fault than continue deforming on the original fault. This new fault forms on the opposite side of the ridge axis from the existing fault, where tensile stresses are highest. As this process continues through time it generates seafloor morphology similar to the abyssal hill topography seen at many slow and intermediate spreading ridges.

Faulting is influenced by a number of competing factors including lithospheric structure, rheology, and the rate of magma accretion at the ridge axis. In agreement with previous studies of the mechanics of faulting and contrary to qualitative inferences, we find that fault heave and spacing decrease with increasing lithospheric thickness. We also find that fault heave and spacing increase with decreasing rate of off-axis lithosphere thickening. However, what appears to be the dominant factor controlling fault growth is the ratio of magmatic accretion to the rate far field extension (M), with the largest, most widely spaced faults occurring for values of M decreasing toward 0.5. Observations from slow and intermediate spreading ridges support the inverse relation between fault heave and spacing with M predicted by our force-balance and numerical calculations.

Our results are consistent with a model in which oceanic core complexes form under intermediate rates of magma accretion [Buck et al., 2005; Tucholke et al., 2008]. Furthermore, fault growth generates a strongly asymmetric axial thermal structure, with elevated temperatures at the base of the lithosphere below the active fault. This asymmetry results from the advection of warm mantle material into the footwall of the active fault and is most pronounced for long-lived, large-offset normal faults. These across-axis variations in thermal structure will further stabilize slip on large-offset normal faults and offer a plausible mechanism for localizing hydrothermal circulation on the footwall of oceanic core complexes.

Appendix A:: Model Description

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References
A1. Conservation of Mass and Momentum

The FLAC method employs an implicit time-marching scheme to solve for conservation of mass [Cundall, 1989]

  • equation image

and momentum

  • equation image

where vi is the nodal velocity in the xi direction, ρ is density, g is the gravitational acceleration, σij is the stress tensor, and equation image denotes the material time derivative. The nodal accelerations (left-hand side of equation (A2)) are integrated in time to give updated nodal velocities, strains and strain rates. Using the constitutive laws the strains and strain rates are used to calculate new elastic and viscous stresses, respectively (right-hand side of equation (A2)). Finally, these stresses are used to determine the accelerations for the next time step.

The length of the time step is the minimum of the Maxwell relaxation time (2η/E) and the time required for an elastic P wave to propagate across the local grid spacing. Because of the high grid resolution (∼0.25 km) used in our calculations, the elastic propagation time would result in extremely short time steps and large computational times. To circumvent this problem we employ the adaptive density scaling method of Cundall [1982]. This approach assumes that when the inertial term in equation (A2) is small, the density on the left-hand side of equation (A2) (i.e., the inertial density) and hence the time step can be increased. Following Lavier et al. [2000] we chose a ratio for the imposed boundary velocity to the P wave velocity (reduced by increasing inertial density) of 5 × 10−5, resulting in a time step of 1–5 years.

The computation mesh consists of quadrilateral elements that are subdivided into 4 triangular subelements. The final solution in each quadrilateral element is the average of the solution in the 4 triangular subelements. Because the FLAC method is Lagrangian, large deformations result in highly a distorted numerical grid, which decreases numerical accuracy. We account for this problem by remeshing the model space when the minimum angle in any triangular element decreases below 5°. During remeshing strains are interpolated from the old (deformed) to the new (undeformed) grid by linear interpolation [Lavier and Buck, 2002]. This interpolation results in artificial accelerations that decay over several hundred time steps. To reduce the model time during which these artificial accelerations influence solutions (thus minimizing their total influence in a time-integrated sense), we decrease the time step by an order of magnitude immediately following a remeshing event and then increase it linearly to its original value over 1000 time steps.

A2. Solution of the Energy Equation for Magma Intrusion

To calculate temperature we solve the energy equation accounting for the heat added via magma emplacement and crystallization within the injection zone

  • equation image

Here I is the specific internal heat (i.e., I = CpT), Cp is the specific heat, mL is the mass of the added magma, mC is the crystal mass, q is the diffusive heat flow, and ρ is density. The first term on the right-hand side of equation (A3) represents the heat carried by the injected liquid magma,

  • equation image

where Tliq is the liquidus temperature, udike is the injection rate, and xinj is the width of the injection zone. The second term on the right hand side of equation (A3) accounts for the heat released by generating crystals. The first part is

  • equation image

where L is the latent heat released per kg of melt crystallized and m represents the mass of the infinitesimal control volume. The change in crystal mass has two parts,

  • equation image

The first part represents the change in crystallinity with changes in the local temperature T and the second part represents the new crystal mass forming out of the injected mass as it (instantaneously) equilibrates to the local temperature.

  • equation image

where Fc is the fraction of crystallinity approximated to be a linear function of temperature between the liquidus (Tliq) and solidus (Tsol). Combining equations (A3), (A4), and (A7) results in the following heat equation

  • equation image

Note equation (A8) is the heat diffusion equation with a source (equation (2)). The first term represents diffusion with a modified thermal diffusivity. Diffusivity is further modified to incorporate the effects of hydrothermal circulation by increasing κ by a factor, Nu, in regions of the model space above a depth of 7 km and where T < 600°C [e.g., Phipps Morgan et al., 1987]. The threshold depth is an approximation for the maximum pressure at which cracks are predicted to remain open for hydrothermal circulation, and the cut-off temperature is consistent with the maximum alteration temperature observed in rocks from the lower oceanic crust [Gillis et al., 1993].

Numerically we implement the source term in equation (A8) by adjusting the temperature in the injection zone at each time step, k, by

  • equation image

where Tk-1 is the temperature at the previous time step, and ɛxxdike is the increment of horizontal strain across the injection zone imposed at the time step.

A3. Rheology

In our model, material behavior is a function of the thermal structure, stress, and accumulated plastic strain throughout the model space [Poliakov and Buck, 1998; Lavier et al., 2000]. In regions where deformation is visco-elastic, the material behaves as a Maxwell solid in which the deviatoric stresses, σdev, are related to the deviatoric strains, ɛdev, by

  • equation image

where E is Young's modulus, η is the effective viscosity, and dotted quantities represent derivatives with respect to time. The isotropic part of the deformation field is assumed to be purely elastic, with the isotropic stress and strain related by

  • equation image

where K is the bulk modulus. Viscous deformation is incompressible and follows a non-Newtonian temperature and strain rate dependent power law of the form [Kirby, 1983; Chen and Morgan, 1990; Kohlstedt et al., 1995]

  • equation image

in which A is a material constant, n is the power law exponent, equation imageII is the second invariant of the strain rate tensor, Q is the activation energy, and R is the universal gas constant.

Plastic yielding in the brittle layer is controlled by Mohr-Coulomb theory [Poliakov and Buck, 1998]. The failure criteria is given by [Jaeger and Cook, 1979]

  • equation image

where τ is the shear stress required for faulting, μ is the coefficient of friction, σn is the normal stress, and C is the cohesion. Because σn is dominated by the lithostatic stress, τ increases linearly with depth. Following Poliakov and Buck [1998], we simulate the weakening of brittle materials after failure by using a strain-dependent cohesion law. The initial cohesion, Co, decreases linearly with the total accumulated plastic strain until it reaches a minimum value, Cmin, after a critical increment of plastic strain of ɛc. For a characteristic fault zone width, Δd, the strain required for complete cohesion loss can be interpreted in terms of a critical fault offset, Δsc = ɛcΔd [Lavier et al., 2000]. In general, Δd is 2–4 times the grid resolution, therefore for a grid resolution of 250 m and ɛc = 0.5 we find Δsc = 250–500 m. To reduce the broadening of fault zones due to numerical diffusion caused by regridding (see below) we also include an annealing time in our calculations in which plastic strain decays over 1012 s.

Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

Fault evolution is modeled following a force-balance approach [Forsyth, 1992; Buck, 1993; Lavier et al., 2000], such that a new fault will form when the force required to continue slip on the active fault exceeds the force required to initiate a new fault at the axis:

  • equation image

Strictly, FF + FB represent the depth-integrated, nonlithostatic, tensile stress on the lithosphere on either side of the fault and FI is the depth-integrated yield strength of the unfaulted (i.e., with full cohesive strength) lithosphere. Following Forsyth [1992],

  • equation image

and

  • equation image

Force FF represents the force needed to cause further slip on the fault plane; it is identical in form to FI, but FI depends on the initial fault dip (θo), cohesion, and axial lithospheric thickness (Ho), whereas FF depends on the values of these properties as they evolve on the fault as it moves off-axis. Immediately after the initiation of a new fault, FF decreases as the cohesion on the fault drops with the accumulated plastic strain. However, after fault slip exceeds the critical offset (Δsc) required for complete cohesion loss, FF gradually increases owing to the off-axis thickening of the lithosphere (Figure 3).

Following the scaling law for a elastic-plastic layer of Lavier et al. [2000],

  • equation image

where A and B are constants related to the elastic-plastic properties of the lithosphere. Equation (B4) describes the force needed to bend the lithosphere as the fault heave (Δx′) increases. Equation (B4) includes plastic weakening of the plate with continued bending and therefore predicts a peak in FB with a maximum value of AH2 (Figure 3).

By solving equation (B1) for Δx′ we can determine xAF, and then predict total fault heave (Δx) and fault spacing (ΔS) using equations (3) and (4), respectively. Figure 3 illustrates an example calculation of FF, FB, and FI for Ho = 6 km and M = 0.7. A new fault is predicted to form when Δx′ ∼ 5 km, corresponding to the point at which the active fault has migrated xAF ∼ 3.4 km off-axis. Calculating fault spacing from equation (3) implies ΔS ∼ 12 km, which is consistent with the numerical results shown in Figure 6b.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References

Funding for this research was provided by NSF grants OCE-0327018 (M.D.B.), OCE-0548672 (M.D.B.), OCE-0327051 (G.I.), and OCE-03-51234 (G.I.). This work has benefited from many fruitful discussions with R. Buck, B. Tucholke, R. Qin, S. Sacks, J. P. Canales, L. Lavier, G. Hirth, J. Lin, and E. Mittelstaedt. We are indebted to A. Poliakov, L. Lavier, and R. Buck for their extensive work in adapting the FLAC code for mid-ocean ridge applications and for generously making this code available to us. Finally, we thank R. Buck and an anonymous reviewer for careful reviews that improved this manuscript. Several figures were produced with GMT 3.4 [Wessel and Smith, 1995].

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model for Magma Intrusion at Mid-Ocean Ridges
  5. 3. A Simple Model for Sequential Faulting at Mid-Ocean Ridges
  6. 4. Numerical Models of Fault Development at Mid-Ocean Ridges
  7. 5. Discussion
  8. 6. Conclusions
  9. Appendix A:: Model Description
  10. Appendix B:: Force-Balance Model for Faulting at Mid-Ocean Ridges
  11. Acknowledgments
  12. References