4.1. Compensation Depth of Lithostatic and Magmastatic Pressures: Extrusion or Intrusion?
 Buck et al.  proposed that the LMCD is the depth of the AMC for fast spreading ridges. However, the AMC depth is more likely controlled by the heat balance between hydrothermal circulation and magma supply [Hooft et al., 1997; Morton and Sleep, 1985], and the LMCD gives the minimum depth of the AMC as the root of erupting magma (Figure 6a). When the AMC depth is equal to the LMCD, eruption is only possible if there is a new supply of magma to the AMC (Figure 6b). When the excess magmatic pressure (Pex) on top of the AMC exceeds the tensile strength (Ts) of the chamber roof, fracturing of the roof will take place and magma will begin to ascend through the crack. As magma-filled crack ascends through the upper crust, Pex at the magma head in the crack first increases but gradually decreases as the crack tip approaches the surface. If there is a sufficient amount of magma filling the crack, magma will erupt onto the seafloor. With extrusion of magma, the crack decreases in Pex and closes from the bottom. The eruption terminates when the magma head in the crack loses Pex and the bottom of the crack rises to the LMCD. As the remaining magma in the crack cools, drain-back of lava occurs with retreat of the magma head into the fissure vent. Finally, the magma in the crack solidifies as a dike. Without supply of new magma, a dike intrusion can also be triggered by the increase in the horizontal deviatoric stress due to plate movement that contributes the increase in Pex to overcome Ts (Figure 6c). If there is an insufficient volume of magma, most dikes will be trapped at a level between the surface and the LMCD, where the level of neutral buoyancy resides [Rubin, 1990, 1995].
Figure 6. (a) Schematic model for crust with the minimum horizontal stress (Shmin, green solid lines) and magmastatic (Pm, red broken line) pressures which cross at moderate depth (star, LMCD), where the AMC is situated. Red solid lines (Figures 6a–6c) represent excess pressure of magma in the AMC. Broken lines in Figures 6b and 6c show the excess pressure of magma in a crack which changes in numerical order (from 1 to 3) as the magma-filled crack ascends toward the seafloor. (b) Supply of a new batch of magma increases magmatic pressure of the AMC, resulting in rupturing of the AMC roof and magma ascent thorough a crack. Eruption takes place when the magma head in the crack (broken green line 2) reaches the surface. The eruption ceases as the excess pressure of magma decreases and magma retreats into the crack that closes from above and bottom to solidify as a dike (blue broken line 3). (c) Dike intrusion without an eruption is favored when it is triggered by the increase in horizontal deviatoric stress (from green solid to broken line) without magma supply. As plate movement steadily increases horizontal deviatoric stress in the brittle upper crust, the excess pressure of the AMC magma increases and eventually exceeds the strength of the AMC roof. As a magma-filled crack grows, magma flows into the crack from the AMC. If the supplied magma is not enough, the magma head in the crack cannot maintain sufficient excess pressure for the crack tip to propagate upward. The magma-filled crack will stop growing before reaching the surface, solidifying as a dike without eruption.
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 In contrast to the assumptions of the model above, the LMCD in Site 1256D occurs at a shallow level within the extrusive rocks, whereas the estimated top of the AMC is at least 690–880 m deeper below the thick sequence of lavas and thin sheeted dike complex. This situation is schematically shown in Figure 7. The AMC is over pressurized and only a small increase in magmatic pressure and/or horizontal deviatoric stress is required to fracture the magma chamber roof (Figure 7b). As magma rises through a crack, Pex of the magma head in the crack increases as the crack grows vertically because buoyancy of the magma is proportional to the height of the crack [Maaløe, 1987; Rubin, 1990, 1995]. As long as the magma head retains a sufficient Pex to overcome the rock fracture toughness, the magma-filled crack continues to grow upward and eventually reaches the surface to erupt. As eruption continues, the AMC decreases in Pex and ultimately shuts off the magma outflow. As magma in the crack continues to extrude, the crack closes from its lower end, leaving fractured host rocks formed by the intrusion. Quenched magma in the crack against the wall rocks en route will be left as a thin dike (Figure 8). The resulting crack system will provide a locus for fluid circulation with resulting intense alteration and mineralization, and such phenomena are commonly observed throughout the sheeted dikes of Hole 1256D [Teagle et al., 2006]. Reuse of such cracks by a subsequent dike intrusion is very likely, with the consequent underpressure in the reopening cracks ahead of the propagating dike, absorbing pore fluids infiltrating from the host rocks [Curewitz and Karson, 1998; Rubin, 1995]. Branches of the dike propagating into the fluid-filled cracks will be fragmented into hyaloclastite by contact with hydrothermal fluid. This is the first direct evidence of deep cracking and hydrothermal fluid circulation beneath the ridge axis as suggested by a seismic anisotropy study on the East Pacific Rise at 9°N [Tong et al., 2004]. In the Troodos ophiolite, fracturing caused by episodic diking in the granulite-facies contact aureole within the basal sheeted dikes introduced hydrothermal fluids circulation to deeper levels in the upper crust [Gillis and Roberts, 1999].
Figure 7. Schematic model for the Hole 1256D crust with a shallow LMCD. Symbols and abbreviations are the same as in Figure 6. (a) AMC lies much deeper than the LMCD, with the excess magmatic pressure as shown by the red solid line. (b) A small increase in excess pressure (shown by red solid line 1) due to supply of a new batch of magma into the AMC causes rupturing of the AMC roof, which leads to ascent of magma through a crack. If there is a sufficient amount of magma, the magma-filled crack grows with an excess pressure as shown by the pink broken line 2, and eventually eruption takes place. As the AMC extrudes magma, it loses the excess magmatic pressure and ultimately shuts off the magma outflow (shown by the green broken line 3). The magma left in the crack continues to extrude, with the crack closing from both above and bottom, which leaves a small amount of magma in the closed crack solidifying as a thin dike (blue broken line 4). (C) In magma-deficient conditions, the AMC loses the excess pressure before the magma-filled crack tip reaches the surface (broken lines 2 and 3). The magma head in the crack cannot maintain sufficient excess pressure to propagate upward until it reaches the surface. As the magma-filled crack moves upward, it closes from bottom and finally stops above the LMCD as a thin dike (broken line 4).
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Figure 8. Because of the shallow LMCD, dike intrusion under a small deviatoric stress leads to an eruption which ultimately expels magma out of the crack. As the eruption approaches to the end, the magma-filled crack closes from its lower end, leaving fractured host rocks formed by the intrusion. Such a crack system provides a locus of hydrothermal fluid circulation and localized alteration, as is observed in the crack-dike zone of Hole 1256D. Reuse of such cracks by a subsequent dike intrusion yields an underpressure in opening cracks ahead of the dike, which absorbs pore fluids infiltrating from the host rocks [Curewitz and Karson, 1998; Rubin, 1995]. Dike propagating into the fluid-filled cracks results in hyaloclastic fragmentation of magma in contact with hydrothermal fluid.
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 Magma-deficient conditions will form a dike without extrusion of magma, exhausting all magma in the AMC (Figure 7c). In both magma-sufficient and deficient conditions, the uppermost extrusive section above the LMCD host thin dikes as shallow roots of erupted flows, but these dikes will not evolve to a sheeted dike complex. Because subsequent eruptions bury the dike-hosting section and shift the LMCD upward, the dike-hosting horizons will remain at shallow levels in the crust.
4.2. Conditions of Dike Intrusions and Formation of the Sheeted Dike Complex in Hole 1256D
 The lithostatic and magmastatic pressure calculations above suggest that the LMCD at Site 1256 is too shallow to provide an effective LNB to trap ascending dikes in the upper crust. Nevertheless, we have the sheeted dike complex, although the ratio of intrusive to extrusive thickness in Hole 1256D is only 346 m/811 m = 0.43 compared to 1056 m/780.5 m = 1.35 in Hole 504B. The shallow LMCDs and the absence of effective LNB show that density contrast alone cannot explain the formation of the sheeted dike complex. We will consider the genetic conditions of the sheeted dike complex incorporating cooling of magma in the sheeted dikes and dynamics of the stress field at mid-ocean ridges.
 Assuming the wall rock temperature of ∼200–300°C in the sheeted dike complex as indicated by the hydrothermal mineralogy, rising magma in a crack will solidify to form chilled margins <5 cm thick in 1.5 h [Carslaw and Jaeger, 1959], which is comparable to the observed duration of the 1998 eruption at Juan de Fuca Ridge [Fox et al., 2001] and the estimated period of the 1991 April eruption at the northern EPR [Gregg et al., 1996]. Most lava flows observed at the EPR axis are lobate sheets with a typical volume of 0.01–0.1 km3 with the median value of 0.09 km3 [Sinton et al., 2002]. Length of fissure vent range from <1 to >18 km and larger eruptions tend to have been fed through longer fissures [Sinton et al., 2002]. Assuming lava viscosity of 100 Pa·s and fissure lengths and erupted lava volumes of 2–10 km and 0.01–0.1 km3, respectively, minimum extrusion rates for lobate sheet flows can be estimated as 300–1440 m3/s using the model of Gregg and Fink . This gives the maximum duration of eruptions of 580–1160 min, during which dike margins will solidify only 10–15 cm in thickness if the wall rock temperature is 300°C [Carslaw and Jaeger, 1959]. Consequently, typical eruptions will form dikes with thicknesses less than 30 cm at the end of eruptions. If the extension of the lower part of the upper crust is provided by the emplacement of these dikes, the average eruptive frequency will be every 1.4 years. This is rather higher than the estimate of every 6.9 years for the fast spreading southern EPR [Sinton et al., 2002], even if we take into account the difference in spreading rate between Site 1256 and the present southern EPR. Although it is virtually impossible to know the statistically meaningful frequency distribution of dike thicknesses in a single hole that drilled through only a limited number of dikes, the massive basalt-dolerite structures in the sheeted dike complex indicate the presence of dikes more than a few tens centimeters in thickness [Teagle et al., 2006; Tominaga et al., 2007]. Likewise, the sheeted dike complex in the Oman Ophiolite, considered to have been formed at a fast spreading ridge system, shows a thickness distribution of individual dikes strongly skewed toward thinner dikes, where 36% and 50% of the sheeted dikes are thinner than 20 cm and 40 cm, respectively [Umino et al., 2003]. Conversely, the rest 50% of the sheeted dikes is thicker than 40 cm, ranging in thickness up to 15 m. Thus, the presence of thicker (>30 cm) dikes may also be common in Hole 1256D sheeted dike complex, that urges us to consider the necessary conditions of their formation.
 The lithosphere at spreading centers is under a large horizontal tectonic stress due to plate motions that alter the stress gradient through the upper crust to yield an apparent level of neutral buoyancy (apparent LNB) [Rubin, 1990, 1995; Takada, 1989]. The driving force of dike propagation is the magmatic excess pressure Pex, which depends on both the density difference between the magma and the host rock and the gradient of the minimum principal stress [Takada, 1989; Watanabe et al., 1999]:
where Dr, Dm, g and Shmax − Shmin are the host rock and the magma density, acceleration due to gravity and the horizontal deviatoric stress. Z is the depth and Zb and Zt denote the bottom and top depths of the dike. Emplacement of a dike is favored at the apparent LNB because Pex is largest there.
 The upper limit of the horizontal deviatoric stress is given by brittle rock strength, which is dependent on temperature, strain rate and rock types. In the upper crust below a spreading axis, the horizontal compressive stress is most likely smaller than the vertical stress, because tensile cracks and normal faults are common. Rock deformation is focused on faults, so the rock strength is given by the frictional strength of fault planes rather than by the brittle rock strength [Sibson, 2002];
where Fs is frictional strength, n is coefficient of internal friction and given as 0.4 for normal faults [Byerlee, 1978], Sv is vertical stress and given as DrgZ, and b is a ratio of fluid pore pressure to vertical stress. As hydrostatic fluid pressure prevails in shallow fluid-saturated crust, we may assume b = 0.4 [Sibson, 2002]. Then, the frictional strengths at the top and bottom of the upper crust are 4.7–9.9 MPa, which gives the lower limit of the minimum horizontal stress under elastic strain due to the plate spreading (Figure 5).
 Because of a large thermal gradient ∼50°C/100 m, from greenschist through amphibolite and granulite facies to hypersolidus temperatures in the lowermost sheeted dikes and the roof zone of the AMC [Gillis and Roberts, 1999; Jupp and Schultz, 2000; Teagle et al., 2006], the rheology of the host rocks largely changes from brittle basalt dikes through viscoelastic dolerite and gabbros to plastic hypersolidus roof zone mushes. As a result, the applied stress tends to concentrate in the brittle upper crust and the reduction in the horizontal stress parallel to the spreading direction becomes larger with depth, attaining the maximum at the bottom of the sheeted dike complex (Figure 9a).
Figure 9. Model of dike emplacement at the apparent level of neutral buoyancy (ALNB) in the upper crust generated under a horizontal deviatoric stress. (a) Green solid curve shows the reduction of horizontal stress (Shmin) normal to the rise axis from “Sh = Sv” to “Shmin,” caused by elastic and viscous deformation of the brittle upper crust and ductile lower crust in response to plate movement. This generates an ALNB through the upper crust, which can accommodate a thick dike. Reduction in Shmin is largest at the bottom of the sheeted dike complex, and Shmin rapidly increases to the lithostatic pressure through the ductile gabbro-mush zone surrounding the AMC. The increase in Pex is mainly caused by the increase in the horizontal deviatoric stress (Sv − Shmin). A dike intrusion begins when the excess pressure (Pex) overcomes the tensile strength (Ts) of the AMC roof. (b) If sufficient magma is available, the head of magma filling the crack reaches the surface to extrude with a large excess pressure (shown by the red line 1). As eruption continues, cooling of magma in the crack and the decrease in Pex to a certain level will lead to a closure of the vent of the AMC roof (light green line 2). The magma-filled crack begins to close from the bottom, extruding magma exceeding the amount emplaced as a dike, which relaxes the deviatoric stress in the upper crust. At the end of the eruption, a thick dike is emplaced through the entire upper crust (blue line 3; the horizontally hatched area denotes the amount of stress relaxed by the dike intrusion). (c) In magma-rich conditions, the increase in Pex is mainly caused by the increase in the magmatic pressure of the AMC. (d) Eruption occurs with a large absolute magmatic pressure (red line 1) and under a small deviatoric stress. Because the dike intrudes under a small deviatoric stress, the emplaced dike (blue horizontally hatched area) should be thinner than in Figure 9b.
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 A dike intrusion begins when the excess pressure (Pex) overcomes the tensile strength (Ts) of the AMC roof. Increase in both horizontal deviatoric stress (Sv − Shmin) and magmatic pressure of the AMC contribute the increase in Pex. When the increase in the deviatoric stress surpasses that of magmatic pressure, the dike intrusion is promoted by a large deviatoric stress but with a small absolute magmatic pressure (Figures 9a and 9b). If sufficient magma is available, the head of magma filling the crack reaches the surface to extrude with a large excess pressure. As eruption continues, cooling of magma in the crack and the decrease in Pex to a certain level will lead to a closure of the vent of the AMC roof. For a two-dimensional dike with a thickness 2w and a height of 2l, elastic pressure for dike widening ΔP is given by
where M is elastic stiffness and is approximately several GPa in nature [Rubin, 1995]. Given M to be 5 GPa and 2h of 1000 m, the elastic pressure is 1.5–2 MPa if the dike thickness is 30–40 cm. Considering the dike margins solidifying during typical eruptions to be ∼15 cm in thickness, Pex ∼ 1 MPa on the AMC roof would be the critical excess pressure to maintain the magma conduit. In addition to the solidification of dike margins, relaxation of the deviatoric stress by the dike intrusion and the outflow of magma from the AMC reduce Pex < 1 MPa and ultimately choke the conduit. The magma-filled crack is detached from the AMC and begins to close from the bottom. Magma in the crack continues to extrude, leaving a thick dike in the upper crust (Figure 9b).
 The amount of magma to be emplaced as a dike depends on the accumulated deviatoric stress, the excess magmatic pressure and the volume of available magma. A large deviatoric stress field enables a thick dike intrusion associated with or without an eruption. If sufficient magma is supplied, eruptions will follow dike intrusions (Figure 9b). To the contrary, magma-starved conditions lead only to a dike intrusion without any eruption. All magma in the AMC may be exhausted for a dike emplaced in some part of the upper crust. A magma-filled crack grows as the AMC expels magma into the crack. The crack head continues to rise as far as it retains sufficient Pex to overcome the surrounding rock fracture toughness, while the crack bottom closes as the magma-filled crack moves upward. Finally the dike relaxes the deviatoric stress mainly of the lower upper crust where the dike is emplaced. This situation is consistent with the hydrothermal circulation model proposed for the magma-starved ridge segments, where deep magma intrusions and crustal permeability control the vent locations and hydrothermal activity [Haymon, 1996].
 On the other hand, magmatically robust ridge segments would have high supply rates of magma to the AMC, which gives rise to a rapid increase in magmatic pressure (Figure 9c). This raises the contribution of the AMC pressure to the increase in the excess pressure compared to that of the accumulation of deviatoric stress due to plate movement, which results in a dike intrusion under a small deviatoric stress and a large absolute magmatic pressure (Figure 9d). This condition is relatively compressive compared to that under a large deviatoric stress and allows only a thin dike emplacement in the upper crust. Thus, the supplied magma to the crust is more likely to extrude onto the seafloor. Common presence of AMC reflectors beneath the fast spreading EPR [e.g., Hooft et al., 1997] strongly suggests magma-rich conditions for faster spreading ridge segments, which allows more magma to erupt rather than only to remain in the upper crust as a dike. This explains the high ratio of the extrusive to intrusive rocks in the superfast spread oceanic crust at Site 1256. Because accumulation rate of deviatoric stress depends on the plate movement, it is considered to be more stable than supply rate of magma to AMCs, which may vary on the order of 1,000 to 100,000 years [Reynolds et al., 1992; Scheirer and Macdonald, 1993; Shah et al., 2003]. A subtle fluctuation in magma supply rate may change the contribution of the absolute magmatic pressure to the increase in excess pressure, which results in different ratios of magma partitioning into the extrusive and intrusive rocks in the upper crust.