1 Topographic controls on dike injection in volcanic rift zones 2 3 Mark D. Behn1*, W. Roger Buck2, and I. Selwyn Sacks3 4 5 1Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods 6 Hole, MA 02543 USA 7 2Lamont Doherty Earth Observatory, Columbia University, Palisades, NY 10964 USA 8 3Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, 9 DC 20015 USA 10 11 12 Abstract:

13 Dike emplacement in volcanic rift zones is often associated with the injection of “blade- 14 like” dikes, which propagate long distances parallel to the rift, but frequently remain 15 trapped at depth and do not erupt. Over geologic time, this style of dike injection implies 16 that a greater percentage of extension is accommodated by magma accretion at depth than 17 near the surface. In this study, we investigate the evolution of faulting, topography, and 18 stress state in volcanic rift zones using a kinematic model for dike injection in an 19 extending 2-D elastic-viscoplastic layer. We show that the intrusion of blade-like dikes 20 focuses deformation at the rift axis, leading to the formation of an axial rift valley. 21 However, flexure associated with the development of the rift topography generates 22 compression at the base of the plate. If the magnitude of these compressive stresses 23 exceeds the tensile stress associated with far-field extension, further dike injection will be 24 inhibited. In general, this transition from tensile to compressive stresses occurs when the 25 rate of accretion in the lower crust is greater than 50-60% of the far-field extension rate. 26 These results indicate that over geologic time-scales the injection of blade-like dikes is a 27 self-limiting process in which dike-generated faulting and topography result in an 28 efficient feedback mechanism that controls the time-averaged distribution of magma 29 accretion within the crust.

30 Key Words: Dike intrusion, faulting, rifting, mid-ocean ridge, topographic stress 31

*Corresponding Author, Dept. of Geology and Geophysics, Woods Hole Oceanographic Institution, 360 Woods Hole Road MS #22, Woods Hole, MA 02543, email: [email protected], phone: 508-289-3637, fax: 508-457-2187. -2-

31 1. Introduction

32 Extension in volcanic rift zones is accommodated through a combination of normal 33 faulting and local magma intrusion. Dike emplacement in these environments is often 34 characterized by “blade-like” dikes (identified by their large length-to-height ratios), 35 which propagate outward from a central magma reservoir parallel to the strike of the rift 36 (Figure 1). Blade-like dikes are typically associated with only minor surface eruptions, 37 implying that they form in a stress regime that favors lateral propagation rather than 38 upward growth and eruption. Ryan [1] hypothesized that such dikes propagate at the 39 level of effective neutral buoyancy. In regions where magma supply is continuous, this 40 level is determined by the vertical density structure of the crust. However, where the rate 41 of magma injection is less than the rate of extension, the level of effective neutral 42 buoyancy is controlled by a combination of rock and magma density, local stress state 43 and magma pressure, and broadly corresponds to the depth of the brittle ductile transition 44 [2]. Excellent examples of magma limited volcanic rift zones are slow-spreading mid- 45 ocean ridges, where seismic moment studies [3] and measurements of cumulative fault 46 throw [4] suggest that ~80% of seafloor spreading is accommodated by magmatic 47 accretion, while the remaining 20% occurs via extensional faulting.

48 Dike intrusion has a strong influence on faulting and topographic relief during rifting. 49 Regions overlying zones of dike emplacement are characterized by normal faulting, 50 subsidence, and graben formation (e.g., [5-7]). For example, individual diking events in 51 Iceland have been observed to produce up to 2 m of slip on normal faults with little or no 52 lava eruption [8]. Moreover, the 1–2-km deep rift valley observed along many parts of 53 the slow-spreading Mid-Atlantic ridge (e.g., [9, 10]), illustrates the potential for continual 54 magma injection to produce significant rift topography. However, while the mechanical 55 interaction between magma injection and faulting has been studied on the timescale of 56 individual dikes, little work has been done to understand how these processes will evolve 57 over geologic time.

58 In this study, we investigate the feedbacks between dike emplacement, faulting, and 59 the growth of topography in volcanic rift zones. Dike intrusion is simulated using an 60 elastic-viscoplastic continuum approach that allows us to model strain localization

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61 without presupposing the initial location of faults. We show that the emplacement of 62 blade-like dikes rapidly localizes strain onto inward dipping faults, resulting in the 63 formation of an axial rift valley. However, as topography grows, compressional stresses 64 associated with plate flexure accumulate at the ridge axis, which will suppress the 65 injection of future dikes. The result is an efficient feedback mechanism in which dike- 66 induced faulting and topography act to control the time-averaged depth distribution of 67 magma accretion in the crust.

68 2. Numerical Models of Volcanic Rifting 69 Most previous studies that examined the mechanical interaction between magma 70 injection and faulting assumed that the Earth behaves as an elastic half-space and 71 simulated faulting through induced slip on presupposed dislocations in the crust (e.g., [5, 72 6, 11]). These assumptions are likely valid on short timescales, but breakdown in the 73 case of repeated diking events over geologic time. Here we study the long-term 74 evolution of topography and faulting at volcanic rifts in a 2-D elastic-viscoplastic layer 75 (Figure 2) using the Fast Lagrangian Analysis of Continua (FLAC) technique of Cundall 76 [12]. This numerical approach has been used to simulate localized deformation (i.e., 77 faulting) in a variety of extensional environments [13-15] and is described in detail 78 elsewhere [16, 17].

79 In our model, material behavior is a function of the thermal structure, stress, and 80 accumulated plastic strain throughout the model space. In regions where deformation is 81 visco-elastic, the material behaves as a Maxwell solid. Viscous deformation is 82 incompressible and follows a non-Newtonian temperature and strain-rate dependent 83 power-law [18, 19]. Material properties appropriate for a dry diabase rheology [20] are 84 assumed throughout the model space. Plastic yielding in the brittle layer is controlled by 85 Mohr-Coulomb theory. The weakening of the brittle layer after failure is simulated by

86 assuming a strain-dependent cohesion law in which the initial cohesion, Co, decreases

87 linearly with the total accumulated plastic strain until it reaches a minimum value, Cmin,

88 after a critical increment of plastic strain of εc [14, 15, 17]. For the model runs presented

89 in this study we assume Co = 24 MPa and Cmin = 4 MPa, however we note that these

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90 parameters have little influence on our numerical results relative to the effects of dike 91 injection.

92 The initial numerical domain is 100 km wide by 20 km deep, with a grid resolution of 93 0.25 km x 0.25 km. Deformation is driven by imposing a uniform far-field extension

94 along the edges of the model space with a velocity, utectonic = 2.5 cm/yr (Figure 2). A 95 hydrostatic boundary condition is assumed for the base of the model space and the top

96 boundary is stress free. The thickness of the brittle layer, hlith, corresponds to the depth of 97 the brittle ductile transition (~650ºC) and is adjusted through the imposed thermal 98 structure. Temperature increases linearly with depth from 0ºC at the surface to the 650ºC

99 at hlith, below which it increases rapidly to a maximum of 1200ºC. This temperature 100 structure results in a sharp brittle ductile transition, minimizing the effect of viscous 101 strengthening in the lower crust. To ensure that rift behavior is not affected by necking 102 associated with perturbations in lithospheric thickness all calculations assume a uniform 103 initial plate thickness of 6 km. We note that the temperature structure remains fixed and 104 does not evolve throughout our numerical calculations.

105 Assuming blade-like dikes represent the dominant style of intrusion in magma limited 106 volcanic rifts such as slow-spreading ridges, a greater fraction of magma will be intruded 107 at depth than near the surface (e.g., Figure 1, [2]). We simulate this process 108 kinematically, by widening a column of model elements extending from the base of the 109 brittle layer to a fixed depth, d, below the surface (Figure 2). As deformation progresses, 110 d, remains fixed with respect to the surface and thus may migrate relative to the fixed 111 thermal structure. The rate of injection is described by the parameter M, which is defined 112 as the ratio of the rate of dike opening to the rate of far-field extension applied on either 113 side of the model space [21]. Thus, M = 1 corresponds to a case where the rate of dike 114 opening is equal to the rate of far-field extension, and M = 0 represents an amagmatic rift. 115 We have benchmarked our approach against the 2-D boundary element solution for an 116 opening dike in an elastic half-space [22]. The predicted vertical and horizontal surface 117 displacements closely reproduce the boundary element solution (Figures 3a & 3b), with 118 slight differences resulting from the finite width of the injection zone and the finite 119 thickness of the elastic plate. The predicted stresses are illustrated in Figure 3c, and show

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120 the expected regions of compression adjacent to the dike and tension above and below the 121 dike.

122 3. Stress Evolution and the Growth of Rift Valley Topography 123 In all simulations with M > 0, deformation initiates on two inward-dipping normal 124 faults that form at the top of the dike zone and propagate upward to the surface (note that 125 for M = 0 initial deformation will be distributed throughout the brittle layer). The 126 location of these inward dipping faults corresponds to the region of maximum tensile 127 stress predicted by calculations of a dike in an elastic half-space [e.g., Figure 3c, 5]. For 128 M > ~0.5, the inward-dipping normal faults remain pinned at the top of the dike creating 129 a small “key-stone” block that does not deform as the rift valley deepens (Figure 4a,b). 130 The long-term depth of the rift valley is limited by the development of off-axis faults, 131 which alleviate flexural bending stresses in the uplifted rift flanks [23].

132 For cases with M < ~0.5, a conjugate set of faults forms dipping outward from the 133 axis and cutting the entire brittle layer (Figure 4c,d). These conjugate faults relieve 134 tensile stresses that accumulate in the lower portion of the plate adjacent to the dike. The 135 formation of the conjugate faults produces a wider and shallower rift valley than for more 136 magmatic cases in which only inward dipping faults are observed. Furthermore, as 137 deformation continues alternating sets of inward and outward dipping faults form, 138 resulting in a pattern of horst and graben topography that is not predicted from simple 139 elastic models of dike injection.

140 Dike injection requires that the pressure in the dike is greater than the horizontal 141 stress in the lithosphere (Figure 1). Because magma chamber over-pressures are typically 142 small, this requires the deviatoric stress in the lithosphere to be tensile. As described 143 above, our simulations assume a kinematic formulation for dike injection, in which there 144 is no feedback between the modeled stress state and the depth distribution of magma 145 injection at a particular time step. However, the calculated axial stresses can be used to 146 assess whether conditions are consistent with continued dike injection.

147 For cases with low magma supply (M < 0.5), we find that stresses in the plate remain 148 tensile throughout the axial lithosphere (Figure 4c). However, for higher rates of magma 149 injection, compressional stresses accumulate rapidly in the lower lithosphere adjacent to

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150 the dike injection zone (Figure 4b). Although compression is expected adjacent to an 151 opening dike in an elastic half-space [5], injection alone will not result in compression in 152 an extending plate due to the net tensile stresses associated with the far-field boundary 153 conditions. Rather, the compression at the base of the plate develops in response to the 154 formation of the axial rift valley. As rift topography grows flexural bending stresses 155 caused by the uplifted rift flanks and depressed axial valley produce compression at the 156 base of the lithosphere beneath the rift axis.

157 To illustrate the importance of rift topography on the stress at the rift axis we 158 examined the evolution of stress throughout deformation for the M = 1 simulation shown 159 in Figures 4a & b. To isolate the component of stress associated with topography, we 160 calculated flexural stresses in a 6-km-thick elastic plate loaded from above by the rift 161 topography [24]. Although this approach ignores complexities associated with variations 162 in plate thickness and the evolving material properties of the lithosphere, it provides a 163 good first order approximation for the flexural stresses associated with rift topography. 164 Figure 5 illustrates the relationship between the growth of the rift valley, the topographic 165 stresses predicted from flexure, and the total deviatoric stress at the base of the plate 166 beneath the rift axis. We find that for all time steps, the influence of the far-field 167 boundary conditions is to reduce the total deviatoric stress relative to the topographic 168 stress. However, both the topographic and deviatoric stresses are directly correlated with 169 the depth of the axial valley, and reach their respective maximum values only when the 170 axial valley stops deepening (Figure 5).

171 The transition between compressional and tensile deviatoric stresses at the base of the 172 plate occurs for M ≈ 0.5 (Figure 6). When M > 0.5, magma injection relieves the tensile 173 stresses associated with far-field extension, causing the rift valley topography to 174 dominate the stress state in the lithosphere. The compressional stresses in the lower 175 lithosphere promote enhanced tensile faulting above the dike, which further deepens the 176 rift valley leading to a positive feedback and greater compression at the base of the plate. 177 Of course, it is unlikely that magma injection will continue in the presence of such large 178 compressive stresses, and thus we view model runs with M > 0.5 to be unphysical on 179 geologic time-scales. In contrast, when M < 0.5, far-field extension dominates the stress

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180 state in the lithosphere. The result is net tension at all depths beneath the rift axis, which 181 in turn promotes extensional faulting throughout the entire brittle layer. Conjugate 182 outward dipping normal faults (e.g., Figure 4c) develop in response to these tensile 183 stresses, producing a wider, shallower rift valley that further decreases the magnitude of 184 the topographic stresses.

185 One oversimplification of the models presented above is that we assume all dikes 186 propagate upward until they reach a specified depth. In reality, there will be a 187 distribution in dike height through time, with some dikes reaching the surface and others 188 stopping at various depths below the surface. To investigate the influence of these effects 189 we examined two additional parameterizations for the depth distribution of magma. First, 190 we simulated cases in which M increases linearly from 0 at the surface to values ranging 191 from 0 to 1 at the bottom of the plate. Figure 6c illustrates axial stresses at the base of the 192 lithosphere after 5 km of total extension. Similar to the earlier calculations in which M 193 was constant within the dike zone, we find that cases in which M varies by more than 194 ~0.6 results in compressional stresses below the axial valley. We further explored 195 situations in which there was a linear variation in M throughout the plate, but with M > 0 196 at the surface, implying a certain fraction of dikes erupt. These calculations again show 197 that the transition between compressional and tensile stress at the base of the plate, which 198 occurs when the total variation in M exceeds 0.6–0.7 (Figure 6).

199 Discussion and Conclusions 200 Temporal and spatial variations in the rate of magma accretion are frequently invoked 201 to explain differences in faulting and axial morphology at slow-spreading ridges [21, 25, 202 26]. Regions with deep rift valleys are interpreted to reflect lower rates of magma 203 injection, while shallower and more distributed rift morphology is taken to imply 204 enhanced magmatism [9, 27]. In this study we have shown that the depth-distribution of 205 magma injection throughout the crust will also strongly influence rift morphology. 206 Specifically, larger gradients in the depth distribution of magma accretion will promote 207 enhanced normal faulting in the shallow crust and the development of deeper rift valleys 208 (Figure 6). Thus future interpretations of crustal magmatic processes from observed rift 209 morphology, must involve a better understanding of how accretion occurs at depth.

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210 There are strong feedbacks between dike injection, faulting, and the growth of 211 topography and our calculations illustrate the limitations of inferring lithospheric stress 212 and dike propagation in an evolving rift zone from a simple yield strength envelope. As 213 has been shown previously (e.g., [5]), the injection of blade-like dikes will promote 214 normal faulting and graben subsidence. However, magma injection simultaneously 215 relieves tensile stresses in the extending plate, leading to a situation in which rift 216 topography dominates the overall stress state in the lithosphere (e.g., Figure 6). The 217 result is an efficient negative feedback in which the deepening rift valley drives the base 218 of the plate toward more compressive stresses, inhibiting further dike injection.

219 As the stress state beneath the ridge axis approaches lithostatic one of several 220 scenarios will occur. The first is that dike injection will simply cease until there has been 221 sufficient far-field extension to reduce the lithospheric stress. Alternatively, as the 222 gradient in the tectonic stress approaches lithostatic, dikes will propagate to shallower 223 depths, redistributing magma injection throughout the crust (Figure 7). Both of these 224 scenarios outline mechanisms in which the time-averaged gradient in magma accretion 225 evolves in such a way that the overall stress state beneath the rift axis remains tensile (i.e. 226 ∆M < 0.5–0.7 in Figure 6c). Finally, if magma is available over a sufficiently large area, 227 dike injection could migrate either along- or off-axis to regions with more favorable 228 stress conditions. Future studies in which dike injection is linked to the evolving stress 229 state in the lithosphere, as well to the flux and migration of melt beneath the rift axis, are 230 critical to distinguish between these different scenarios and gain a better understanding of 231 how volcanic rift zones evolve on geologic time-scales.

232

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232 Figure Captions

233 Figure 1: Schematic illustration of stress conditions leading to intrusion of blade-like

234 dikes. σtectonic represents the horizontal stress in the lithosphere and σlith is the lithostatic

235 stress. Dikes will propagate laterally if the driving pressure (difference between Pmagma

236 and σtectonic) at the dike center exceeds the driving pressure at the top and bottom of the 237 dike. This condition will occur in magma limited rifting environments such as slow 238 spreading ridges. Over geologic time a large proportion of dikes will not reach the 239 surface, implying that a greater percentage of the total extension will be accommodated 240 by magma accretion at the base of the plate than near the surface. Figure adapted from 241 [2, 28].

242 Figure 2: Model setup for numerical simulations of dike intrusion in an extending visco- 243 elastic layer. The dike injection zone is illustrated with the thick black line. Grey region 244 represents the portion of the model space that undergoes brittle deformation, while the 245 white region behaves viscously. Deformation is driven by applying a uniform horizontal

246 velocity, utectonic, to either side of the model space. Dike injection is simulated by 247 widening a vertical column of elements in the center of the model space at a fixed rate

248 udike. The rate of magma injection is defined by M = udike/ utectonic.

249 Figure 3: (a) Vertical and (b) horizontal displacements calculated at the surface for a 250 dike opening in a purely elastic layer (triangles). Displacements are normalized to the 251 amount of dike opening. The dike is embedded in a 20 km thick elastic layer and extends 252 from a depth of 3 to 6 km. Thick black lines show the results of a 2-D boundary element 253 solution [22]. (c) Horizontal stress normalized by the dike opening.

254 Figure 4: Numerical simulations of magmatic rifting showing the calculated strain-rate 255 and horizontal deviatoric stress after 5 km (200 kyr) of total extension for (a,b) M = 1 256 and (c,d) M = 0.4. The depth to the top of the dike injection zone is 2 km and the initial 257 lithospheric thickness is 6 km. For M = 1 two inward-dipping faults remain pinned at the 258 top of the injection zone throughout rifting resulting in the formation of deep rift valley. 259 Flexural stresses act to place the base of the plate in compression, which would act to 260 inhibit further magma injection. In contrast for M = 0.4, alternating inward and outward

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261 dipping faults form as extension continues, resulting in a shallow rift valley and tensile 262 stresses throughout the axial lithosphere.

263 Figure 5: Rift valley depth (filled circles) and topographic (grey squares) and total 264 deviatoric stress (open squares) at the base of the plate versus total extension for M = 1. 265 Topographic stresses are calculated by imposing model topography on a 6-km thick 266 elastic plate with a Young’s modulus of 30 GPa. Note that magnitude of the compressive 267 stresses at the base of the plate are directly correlated to the depth of the rift valley.

268 Figure 6: (a) Model topography and (b) horizontal deviatoric stress versus depth at the 269 rift axis after 5 km (200 kyr) of total extension. (c) Horizontal deviatoric stress at the 270 base of the plate versus the difference in M between the surface and base of the plate. 271 Filled circles illustrate cases where the depth to top of the dike is 2 km and M is constant 272 within the injection zone. Triangles illustrate runs where M varies linearly with depth

273 from the surface to the base of the plate with Msurface = 0 (blue) and Msurface = 0.25 (red). 274 Note that for ΔM > 0.5–0.7 continuous magma injection is inhibited by compression at 275 the base of the plate.

276 Figure 7: Comparison of stress conditions for rifts with and without the influence of rift 277 topography. As the rift topography reduces the tensile stress at the base of the plate,

278 dikes with the same driving pressure, Pdrive, will propagate upward to shallower levels in 279 the crust or become trapped beneath the brittle layer. The net result is to suppress the 280 formation of blade-like dikes and reduce the total gradient in magma accretion 281 throughout the crust.

282

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282 References

283 [1] M.P. Ryan, Neutral buoyancy and the mechanical evolution of magmatic systems, 284 in: B.O. Mysen, (Ed), Magmatic Processes: Physicochemical Principles Spec. Pub. 285 1, Geochem. Soc., 1987, pp. 259-287. 286 [2] A.M. Rubin, D.D. Pollard, Origin of blade-like dikes in volcanic rift zones, in: 287 R.W. Decker, T.L. Wright, P.H. Stauffer, (Eds), Volcanism in Hawaii Professional 288 Paper 1350, U.S. Geol. Surv., 1987, pp. 1449-1470. 289 [3] S.C. Solomon, P.Y. Huang, L. Meinke, The seismic moment budget of slowly 290 spreading ridges, Nature 334(1988) 58-60. 291 [4] J. Escartín, P.A. Cowie, R.C. Searle, S. Allerton, N.C. Mitchell, C.J. Macleod, A.P. 292 Slootweg, Quantifying tectonic strain and magmatic accretion at a slow spreading 293 ridge segment, Mid-Atlantic Ridge, 29ºN, J. Geophys. Res. 104(1999) 10,421- 294 410,437. 295 [5] A.M. Rubin, D.D. Pollard, Dike-induced faulting in rift zones of Iceland and Afar, 296 Geology 16(1988) 413-417. 297 [6] D.D. Pollard, P.T. Delaney, W.A. Duffield, E.T. Endo, A.T. Okamura, Surface 298 deformation in volcanic rift zones, Tectonophys. 94(1983) 541-584. 299 [7] J.M. Bull, T.A. Minshull, N.C. Mitchell, K. Thors, J.K. Dix, A.I. Best, Fault and 300 magmatic interaction within Iceland's western rift over the last 9 kyr, Geophys. J. 301 Int. 154(2003) F1-F8. 302 [8] A. Björnsson, K. Saemundsson, P. Einarsson, E. Tryggvason, K. Gronvald, Current 303 rifting episode in north Iceland, Nature 266(1977) 318-323. 304 [9] J.-C. Sempéré, J. Lin, H.S. Brown, H. Schouten, G.M. Purdy, Segmentation and 305 morphotectonic variations along a slow-spreading center: The Mid-Atlantic ridge 306 (24º00'N - 30º40'N), Mar. Geophys. Res. 15(1993) 153-200. 307 [10] K.C. Macdonald, The crest of the Mid-Atlantic Ridge: Models for crustal 308 generation processes and tectonics, in: P.R. Vogt, B.E. Tucholke, (Eds), The 309 Geology of North America The Western North Atlantic Region, v. M, Geological 310 Society of America, 1986, pp. 51-68. 311 [11] A.M. Rubin, Dike-induced faulting and graben subsidence in volcanic rift zones, J. 312 Geophys. Res. 97(1992) 1839-1858. 313 [12] P.A. Cundall, Numerical experiments on localization in frictional materials, Ing. 314 Arch. 58(1989) 148-159. 315 [13] R. Hassani, J. Chéry, Anelasticity explains topography associated with Basin and 316 Range normal faulting, Geology 24(1996) 1095-1098. 317 [14] A.N.B. Poliakov, W.R. Buck, Mechanics of strectching elastic-plastic-viscous 318 layers: Applications to slow-spreading mid-ocean ridges, in: W.R. Buck, P.T. 319 Delaney, J.A. Karson, Y. Lagabrielle, (Eds), Faulting and Magmatism at Mid- 320 Ocean Ridges Geophys. Mono. 106, AGU, Washington, D.C., 1998, pp. 305-323. 321 [15] L.L. Lavier, W.R. Buck, A.N.B. Poliakov, Factors controlling normal fault offset in 322 an ideal brittle layer, J. Geophys. Res. 105(2000) 23,431-423,442. 323 [16] A.N.B. Poliakov, An explicit inertial method for the simulation of viscoelastic 324 flow: An evaluation of elastic effects on diapiric flow in two- and three-layers 325 models, in: D.B. Stone, S.K. Runcorn, (Eds), Flow and Creep in the Solar System: 326 Observations, Modeling and Theory, Kluwer Acad. Pub., 1993, pp. 175-195.

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Behn et al: Controls on Magmatic Rifting, submitted to EPSL December 2005 Segment End % Magmatic Extension Stress 0 100

Segment Time-averaged top of dikes Center σ σtectonic P σ tectonic magma lith Pmagma Depth Base of brittle layer

Region of Dike Injection

Figure 1 Rift

Axis σn = 0; τ = 0

udike d Brittle M = hlith utectonic udike Depth u u tectonic Ductile tectonic

Across-Axis Distance σn = −ρgz; τ = 0

Figure 2 0.15 a) 0.1 0.05 0 Vert. Disp. / Dike Opening −0.05

0.2 b) 0.1 0 −0.1 Horiz. Disp. / Dike Opening −0.2

0 c) −2 −4 −6

Depth (km) −8 −10 −20 −10 0 10 20 X−Distance (km)

−5 0 5 compression σ (MPa/m) tension xx

Figure 3 Depth (km) Depth (km) − − − − 10 10 − 0 0 5 5 30 a) c) M =0.4 M =1.0 − − 14.5 20 Across strain − − 14 10 − − Axis Distance(km) rate (log − 13.5 0 10 − 13 10 s − 1 ) − 12.5 20 30 esl Compressiv Tensile − 30 b) d) − 40 − 20 Across − 20 − 10 − Axis Distance(km) σ xx (MPa) 0 0 10 20 Figure 4 20 40 30 e Time (kyr) 0 50 100 150 200 250 300 0 200

150 σ

−1 Topographic Stress Total Stress xx 100 (MPa)

50 −2 Rift Valley Depth Depth (km) 0

−3 −50 0 1 2 3 4 5 6 7 Total Extension (km)

Figure 5 a) b) Depth Below Seafloor (km) c) 2 0 σ

const M (M = 0) xx surf lin M (M = 0) 100 @ base of plate (MPa) 1 surf −2 lin M (M = 0.25) surf 0 50 −4 Injection −1 Inhibited M = 1 0 M = 0.8 −6 Injection

Topography (km) −2 M = 0.6 Allowed M = 0.4 −3 −8 −50 −30 −20 −10 0 10 20 30 −50 0 50 100 0 0.25 0.5 0.75 1 Across−Axis Distance (km) σ (MPa) ∆ M xx

Figure 6 Total Stress Field Total Stress Field without Topographic Stresses with Topographic Stresses Stress Stress

P magma σ σ P tectonic If stress at base of plate tectonic magma σ lith become compressive, dike Depth Depth injection will cease. Pdrive Pdrive

Figure 7