Allken, V., R. S. Huismans, and C. Thieulot (2012

Article
Volume 13, Number 1
XX Month 2012
QXXXXX, doi:10.1029/2012GC004077
ISSN: 1525-2027
1 Factors controlling the mode of rift interaction
2 in brittle-ductile coupled systems: A 3D numerical study
3 Vaneeda Allken and Ritske S. Huismans
4 Department of Earth Science, University of Bergen, Allégaten 41, N-5007 Bergen, Norway
5 (vaneeda.allken@geo.uib.no)
6 Cedric Thieulot
7 Department of Earth Science, University of Bergen, Allégaten 41, N-5007 Bergen, Norway
8 Now at Institute of Earth Sciences, Utrecht University, Budapestlaan 4, NL-3584 CD Utrecht,
Netherlands
9 [1] The way individual faults and rift segments link up is a fundamental aspect of lithosphere extension and
10 continental break-up. Little is known however about the factors that control the selection of the different
11 modes of rift interaction observed in nature. Here we use state-of-the-art large deformation 3D numerical
12 models to examine the controls on the style and geometry of rift linkage between rift segments during exten-
13 sion of crustal brittle-ductile coupled systems. We focus on the effect of viscosity of the lower layer, the
14 offset between the rift basins and the amount of strain weakening on the efficiency of rift linkage and rift
15 propagation and the style of extension. The models predict three main modes of rift interaction: 1) oblique
16 to transform linking graben systems for small to moderate rift offset and low lower layer viscosity, 2) prop-
17 agating but non linking and overlapping primary grabens for larger offset and intermediate lower layer
18 viscosity, and 3) formation of multiple graben systems with inefficient rift propagation for high lower
19 layer viscosity. The transition between the linking (Mode 1) and non-linking mode (Mode 2) is controlled
20 by the trade-off between the rift offset, the strength of brittle-ductile coupling, and the amount of strain
21 weakening. The mode transition from overlapping non-connecting rift segments (Mode 2) to distributed
22 deformation (Mode 3) is mainly controlled by the viscosity of the lower layer and can be understood from
23 minimum energy dissipation analysis arguments.
24 Components: 9700 words, 11 figures, 3 tables.
25 Keywords: extensional continental tectonics; geodynamics; lithosphere; numerical modeling; rift interaction; transform fault.
26 Index Terms: 8020 Structural Geology: Mechanics, theory, and modeling; 8109 Tectonophysics: Continental tectonics:
27 extensional (0905); 8175 Tectonophysics: Tectonics and landscape evolution.
28 Received 26 January 2012; Revised 20 March 2012; Accepted 20 March 2012; Published XX Month 2012.
29 Allken, V., R. S. Huismans, and C. Thieulot (2012), Factors controlling the mode of rift interaction in brittle-ductile coupled
30 systems: A 3D numerical study, Geochem. Geophys. Geosyst., 13, QXXXXX, doi:10.1029/2012GC004077.
31
Copyright 2012 by the American Geophysical Union 1 of 19

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32 1. Introduction
33 [2] The way individual faults and rift segments link
34 up is a fundamental aspect of lithosphere extension
35 and continental break-up. There has consequently
36 been widespread interest in the step-over regions
37 between segments, also referred to as transfer zones
38 [Rosendahl, 1987;Morley et al., 1990], relay struc-
39 tures [Larsen, 1988; Peacock and Sanderson, 1994],
40 accommodation zones [Bosworth, 1985], or overlap
41 zones [Childs et al., 1995], as the nature of rift
42 interaction is determined by the way extension is
43 accommodated in these structurally complex regions.
44 [3] The inherent 3D nature of the problem has mostly
45 restricted our understanding of the underlying pro-
46 cesses to observational evidence, semi-analytical
47 techniques and analogue modeling studies. Most of
48 the existing insight on rift interaction derives from
49 studies that were directed toward understanding the
50 origin of the orthogonal transform fault that occurs
51 between mid-oceanic ridges [Gerya, 2012, and
52 references therein]. Although the methodology and
53 setup used varied significantly in the different
54 approaches used, two end-member modes have
55 generally been recognized: a “connecting” and an
56 “overlapping” mode [e.g., Choi et al., 2008]. In the
57 connecting mode, the ridge segments are connected
58 by a transform-like fault whereas in the overlapping
59 mode, often called overlapping spreading center
60 [e.g., Hieronymus, 2004; Tentler, 2007; Acocella,
61 2008], the ridge segments develop a hook-shaped
62 configuration as they overlap and bend toward
63 each other.
64 [4] Little is known about what controls those spe-
65 cific types of interaction. Even though many of the
66 existing analogue studies simulating offset exten-
67 sional segments are limited by mechanical aniso-
68 tropies or basement discontinuities [e.g., Courtillot
69 et al., 1974; Mauduit and Dauteuil, 1996; Acocella
70 et al., 1999] in the interaction zone that have no
71 direct equivalent in nature, it is apparent that the
72 geometry of interaction zones and linkage pattern of
73 ridge segments are largely dependent on their initial
74 configuration [Pollard and Aydin, 1984; Tentler
75 and Acocella, 2010]. In many analogue models,
76 however, the model behavior is dominated by ten-
77 sile strain [Acocella, 2008; Tentler, 2003a, 2003b,
78 2007; Tentler and Acocella, 2010] whereas rifting
79 behavior is dominated by shear deformation.
80 [5] Using an elastic damage rheology, Hieronymus
81 [2004] suggests that shear damage by strain weak-
82 ening is a controlling factor for contrasting styles of
mid oceanic ridge spreading geometries. However, 83
for modeling large finite strains, a frictional-plastic 84
strain weakening rheology is more suitable. Choi 85
et al. [2008] use 3D numerical models with an 86
elasto-visco-plastic rheology to show that the style 87
of ridge connectivity depends on the ratio of ther- 88
mal stress to spreading-induced stress and on the 89
rate of strain weakening. While the models devel- 90
oped in the mid-oceanic ridge context provide 91
important insights on the possible controls and 92
modes of rift interaction, the emphasis on thermal 93
stress [Oldenburg and Brune, 1975; Choi et al., 94
2008] makes them less appropriate for the study 95
of rifting in the continental lithosphere where the 96
temperature gradient is not very large.
97
[6] A number of three dimensional forward models 98
have only recently been developed [Gerya and 99
Yuen, 2007; Braun et al., 2008; Petrunin and 100
Sobolev, 2008; Thieulot, 2011] that can be used to 101
study the 3D evolution of rifts and passive margins. 102
Existing 3D numerical studies on continental rifting 103
have focused on margin plateau formation [Dunbar 104
and Sawyer, 1996], pull-apart basins [Katzman et 105
al., 1995] and rift propagation [e.g., van Wijk and 106
Blackman, 2005] but until now few 3D numerical 107
studies have investigated rift interaction in the con- 108
tinental crust and lithosphere.
109
[7] In a recent study of 3D extension in a single 110
layer brittle system we investigated the effect of 111
spacing between rift segments and the amount and 112
onset of strain weakening on the mode of rift 113
interaction [Allken et al., 2011]. In this earlier work 114
we demonstrated that for different combinations of 115
these parameters, three modes of interaction are 116
obtained: 1) grabens with a single relay zone, 2) 117
grabens with one or more secondary step-over 118
graben segments, and 3) large offset grabens with 119
no significant segment interaction.
120
[8] These models were, however, limited by the 121
absence of brittle-ductile coupling, which is known 122
to provide a first order control on the structural 123
style of rifting, where the localization of deforma- 124
tion is strongly linked to the viscosity of the lower 125
layer [Buck, 1991; Huismans et al., 2005; Buiter 126
et al., 2008]. The control of brittle-ductile cou- 127
pling on deformation mode has been confirmed 128
using analogue [Brun, 1999] and numerical experi- 129
ments [Buck et al., 1999; Huismans et al., 2005]. 130
Analytical mode transition criteria [Huismans et al., 131
2005; Buiter et al., 2008] based on a minimum 132
dissipation analysis, predicted the following modes 133
of deformation: 1) pure shear, 2) multiple conjugate 134
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135 or parallel shear zones, 3) two shear zones, 4) a
136 single shear zone forming an asymmetric basin.
137 According to these studies, the viscosity of the
138 lower layer, and the rate and degree of plastic strain
139 weakening control the selection of the mode of
140 deformation.
141 [9] Here we explore how the initial offset between
142 rift segments, frictional-plastic strain weakening of
143 the upper crust and the viscosity of a ductile lower
144 layer influence the symmetry, evolution, and
145 structural style of rifting as well as the mode of rift
146 interaction in a brittle-ductile coupled system. In
147 the following we first present the numerical model,
148 the model parameters, and the model setup. The
149 model results are then presented documenting the
150 effect and interaction of brittle-ductile coupling, rift
151 offset, strain weakening parameters. Three different
152 modes are distinguished: a connecting mode, an
153 overlapping non-connecting mode and a distributed
154 pure shear mode. Model behavior is subsequently
155 analyzed and compared to the predictions of mini-
156 mum dissipation analysis in the discussion.
157 2. Model Description
158 2.1. Governing Equations
159 [10] On geological timescales, the Earth’s litho-
160 sphere deforms at a sufficiently low rate that inertial
161 forces can be neglected (i.e. Reynolds number of
162 the flow is zero). The momentum equation, ignor-
163 ing inertial effects is given by:
D
s þ rg ¼ 0 ð1Þ
164 where s is the stress tensor, r is the mass density,
165 and g is the acceleration due to gravity. The flow is
166 assumed to be incompressible, which implies zero
167 divergence of the velocity field:
D
v ¼ 0 ð2Þ
168 The stress tensor s can be split into a spherical part
169 p1 and a deviatoric part s:
170 where the pressure p is given by
s ¼ p1 þ s ð3Þ
p ¼ 1
Tr½sŠ ð4Þ
3
For a Newtonian fluid, the deviatoric stress tensor 171
is related to the velocity gradient through the 172
dynamic viscosity m as follows:
173
s ¼ 2m_ ð5Þ
where _ is the strain rate tensor given by:
_ ¼ 1
2
D
þðD v
vÞ T
ð6Þ
174
In what follows the effects of temperature are not 175
considered.
176
[11] Equations (1)–(3), (5) and (6) form a closed set 177
of equations, which allow us to compute the 178
velocity and pressure:
179
D
ðm D
D
vÞ
D
p ¼ rg ð7Þ
v ¼ 0 ð8Þ
[12] The continuity equation (8) is replaced by 180
another equation, based on a relaxation of the 181
incompressibility constraint which is expressed as: 182
v þ p
¼ 0 ð9Þ
l
where l is the so-called penalty parameter [Hughes,
2000] that can be interpreted (and has the same
dimension) as a bulk viscosity. It is equivalent to
say that the material is weakly compressible. It can
be shown that if l is chosen to be a relatively large
number, the continuity equation D
v = 0 will be
approximately satisfied in the finite element context.
Equation (9) is used to eliminate the pressure
in equation (7) so that the mass and momentum
equation become:
D
ðm D
D
vÞþl D
ð D
vÞ ¼rg: ð10Þ
183
184
185
186
187
188
189
190
191
192
A new three dimensional Arbitrary Lagrangian 193
Eulerian (ALE) fully parallel Finite Element code, 194
FANTOM [Thieulot, 2011] is used to solve the 195
above equations. Earth materials are highly non- 196
linear so that the value of the viscosity m in 197
equation (10) depends on the state variables v and 198
p, and an iterative procedure is implemented to find 199
the solution of this equation.
200
2.2. Rheology
201
[13] The rheological behavior of the upper crust 202
is approximated with a pressure dependent 203
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Figure 1. (a) Model setup showing box of dimension 210 km 210 km 30 km representing the brittle upper crust
overlying a viscous lower crust. Extensional boundary conditions of 0.5 cm/yr are applied on 2 opposite sides of the
box. In the models, two weak regions each dimensioned 2.63 km 210 km 1.30 km are placed at two ends of the
box. In models 12–16, the offset D between the 2 weak seeds, is increased by a multiple of h, from 2h to 6h. (b) Rheological
profile implemented: Mohr-Coulomb plasticity in the upper crust and a fixed viscosity, mv in the lower crust.
(c) Frictional plastic strain weakening behavior of the upper crust. The yield stress value is constant until the strain
reaches the threshold value 1. For < 1, the cohesion and angle of friction are constant at C 0 and f 0 . When 1 < < 2,
the material strain weakens, i.e. C and f decrease linearly from C 0 and f 0 to C sw and f sw . Beyond 2, cohesion and
angle of friction remain constant at C sw , f sw .
204 Mohr-Coulomb yield criterion. The strength sp is
205 given by:
1
sp ¼
cosq þ 1ffiffi
½Psinf þ CcosfŠ
ð11Þ
p sinfsinq
3
206 where C is the cohesion, f is the angle of internal
207 friction, P is the dynamic pressure (mean stress)
208 and q is the Lode angle [see Thieulot, 2011,
209 Appendix B]. Brittle failure is approximated by
210 adapting the viscosity to limit the stress that is
211 generated during deformation, using the viscosity
212 rescaling method, implemented in the plastic model
213 during the finite element matrix building process
214 [Fullsack, 1995; Willett, 1999]. The lower layer
215 follows a linear Newtonian viscous flow law with a
216 constant viscosity, mv (Figure 1b).
217 2.3. Strain Weakening
218 [14] Finite strain, which is computed and stored on
219 a three dimensional cloud of points of self-adapting
220 density, is used to include the effects of strain
221 weakening. The cohesion C and the angle of fric-
222 tion, f are both functions of the accumulated strain
223 (Figure 1c). For ≤ 1, C and f are set to C 0 and
224 f 0 respectively. As extension proceeds, the cloud
225 of points accumulates strain so that when the strain
226 in a cell reaches 1, the material starts to strain
227 weaken, i.e. the cohesion C and the angle of friction
228 f decreases linearly with strain until a final strain
229 weakened value f sw is reached at = 2. The yield
strength in the final strain weakened state is given
by:
s sw
p ¼
1
½ Š ð12Þ
cosq þ 1ffiffi p sinf
3
sw sinq P sinfsw þ C sw cosf sw
3. Model Setup
3.1. Material Layout and Boundary
Conditions
230
231
232
233
[15] Our experimental set-up aims at modeling 234
crustal extension. The model domain is a rectan- 235
gular cuboid of size Lx Ly Lz (Figure 1a),
containing a frictional-plastic material of density r
236
237
(the upper crust) overlying a viscous layer (the 238
lower crust) of the same density. The upper crust is
characterized by its plasticity parameters C
239
240
0 , C sw ,
f 241
0 , f sw along with the strain softening thresholds 1
and 2, while the lower crust is characterized by its
viscosity mv (Figure 1b). The numerical model is
non-dimensional and can therefore be scaled to
242
243
244
represent a variety of situations [Allken et al., 245
2011].
246
[16] Orthogonal extension is applied to the system, 247
through the following boundary conditions: free 248
slip on faces y =0,y = Ly and at the bottom of the 249
model domain (z = 0), imposed extensional veloc- 250
ities, vext, on faces x = 0 and x = Lx, and a free 251
surface at the top of the domain.
252
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253 [17] To initiate deformation, one or two weak seeds
254 are included. Strain in the weak seeds is set to
255 strain weakened values at the beginning of the
256 experiments. The weak seeds, of size one quarter
257 the length of the domain, are placed at the base of
258 the frictional-plastic upper crust. The geometry of
259 the weak seeds in the numerical models was pur-
260 posely simple, in order to study the basic first order
261 features of propagating rift segment interaction. The
262 seeds may be taken to represent inherited weak-
263 nesses in the crust. Natural inheritance may exhibit
264 more complex and unstructured characteristics that
265 will require further investigation.
266 3.2. Strain-Weakening Considerations
267 [18] In all experiments the initial cohesion and the
268 angle of friction of the upper crust are fixed. We
269 choose C sw and f sw so that R ≥ 1 is a uniquely
270 defined constant given by:
R ¼ C0 f0
¼
Csw f sw
ð13Þ
271 Assuming the angle of friction to be small enough,
272 a first-order Taylor expansion of the sine and cosine
273 terms can be carried out in equations (11) and (12).
274 This leads to:
sp
ssw ≈ R ð14Þ
p
275 Note that the strain weakening ratio R is to first-
276 order independent of the pressure (and by extension
277 of depth). In what follows, we use the ratio R to
278 characterize the amount of strain-weakening which
279 is present in the upper crust:
R ¼ 2: C ¼ 20MPa → 10MPa; f ¼ 15 → 7
R ¼ 3: C ¼ 20MPa → 6MPa; f ¼ 15 → 5
R ¼ 4: C ¼ 20MPa → 5MPa; f ¼ 15 → 4
R ¼ 5: C ¼ 20MPa → 4MPa; f ¼ 15 → 3
280 3.3. Model Nomenclature
281 [19] Six sets of models were run to test the sensi-
282 tivity to brittle-ductile coupling, to variable rift
283 offset, and to strain-weakening parameters.
284 [20] 1. In the first set of models (1–3), we test the
285 influence of brittle-ductile coupling in models with
286 a single discontinuous seed, for lower layer vis-
287 cosities, mv =1 10 19 ;1 10 20 and 1 10 21 Pa.s.
288 [21] In all the other models, we used two discon-
289 tinuous weak seeds, offset by a distance D, where
D is defined as a multiple of the brittle layer 290
thickness h (Figure 1a).
291
[22] 2. Models 4–11 test the effect of viscosity of 292
the lower crust for a fixed offset D = 5 h on mode 293
interaction.
294
[23] 3. Models 12–15 with a moderately weak 295
lower crust test the sensitivity to varying rift offset 296
D, which is systematically increased by h. 297
[24] 4. The next set of models tests the sensitivity to 298
the amount of strain weakening R = 2, 3, 5, for rift 299
offset D = 2 h, 3 h, 4 h, 5 h, 6 h.
300
[25] 5. Models 17–19 demonstrate the sensitivity to 301
strong lower crust for variable offset D (2 h, 4 h, 6 h). 302
3.4. Numerical Considerations
303
[26] The computational grid is composed of 160 304
160 23 = 588,800 elements resulting in a mesh 305
resolution of about 1.3 km. Trilinear velocity - 306
constant pressure elements are used, along with a 307
penalized formulation. The symmetric sparse matrix 308
is solved using the massively parallel IBM sparse 309
matrix solver WSMP [Gupta et al, 1997] on 64 310
cores on a Cray XT4. Typical time step size dt = 311
20 kyr were used, and each simulation took about 312
1.5 days to run.
313
4. Results
4.1. Influence of Brittle-Ductile Coupling
4.1.1. Models With a Single Seed
314
315
316
[27] We first test the effect of varying the viscosity 317
of the lower layer of the numerical model. In 318
models 1–3 (Figure 2), a single weak seed has been 319
placed at the base of the brittle layer. Plastic strain 320
weakening parameters are fixed (R = 5) and the
viscosity of the lower layer is set to 1 10
321
322
19 Pa.s,
mv =1 10 323
20 Pa.s, and mv =1 10 21 Pa.s for
models 1, 2 and 3 respectively.
324
[28] In models 1–3, deformation initially localizes 325
in two conjugate shear zones rooted in the weak 326
seed region forming an angle of 45 with the hori- 327
zontal. In model 1 at t = 1 Ma (Figure 2a) a sym- 328
metric graben has formed which has propagated 329
efficiently through the model domain. In model 2 330
(Figure 2b), at t = 3 Ma, an asymmetric half graben 331
has developed, which has propagated more than 332
halfway across the domain. The width of the graben 333
gradually decreases as the rift advances into the 334
unextended domain because extension of shear 335
zones starts later in this area. In model 3 336
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Figure 2. Models 1–3: Exploring model behavior on a simple case with a single seed (R = 5). Elevation, z (km) and
cross-section of strain rate (s 1 ) when brittle-ductile coupling is (a) very weak (mv =10 19 Pa.s), (b) weak (mv =10 20 Pa.s)
and (c) strong (mv =10 21 Pa.s). (d) Graph showing the variation of the rate of rift propagation, vy (cm/yr) with viscosity
of the lower layer, mv (Pa.s), where vy is the rate at which the material, at a given point along strike, start to strain weaken
( = 1).
337 (Figure 2c), deformation initially localizes in the
338 weak region. Further extension favors the forma-
339 tion of additional grabens over the propagation of
340 the initial rift graben along strike. At t = 10 Ma, the
341 rift has propagated slightly and deformation is dis-
342 tributed over three grabens with a characteristic
343 spacing. At the other end of the model domain, the
344 deformation is distributed approximating pure
345 shear, forming several small grabens. Symmetry is
346 conserved throughout the model evolution. The
347 degree of localization and the rate of rift propaga-
348 tion in these models are a direct function of the
349 viscosity of the lower layer. Both decrease with
350 increasing viscosity, mv (Figure 2d).
351 4.1.2. Variation of Viscosity for a Fixed Offset,
352 D = 5 h (Models 4–11)
353 [29] In a previous study [Allken et al., 2011], we
354 demonstrated that for single layer systems, the
355 transition between modes of interaction (connect-
356 ing vs non connecting grabens) occurs at a mod-
357 erate offset of D = 5 h. For this critical offset, we
358 now explore how the strength of the coupling to a
359 viscous layer affects the mode of rift interaction.
360 The plastic strain weakening parameters are fixed
361 (R = 5), and the viscosity mv of the lower crust is
362 varied in small increments between 10 19 and 10 21 Pa.
363 s in models 4–11 (Figure 3).
364 [30] In model 4 (Figure 3a) (mv =10 19 Pa.s), the
365 grabens propagate and connect after 2.5 Ma
366 through an oblique transform fault forming an
367 angle of 61 with the primary grabens. In model 5
368 (Figure 3b) the viscosity of the lower layer is
369 increased to 2.5 10 19 Pa.s. The transform fault
linking the primary grabens forms at t = 3.2 Ma. At 370
this stage the grabens have propagated further than 371
in model 4. The transform shear zone is conse- 372
quently more oblique making an angle of 71 with 373
the original trend of the grabens. In model 6
(Figure 3c), at mv = 5.5 10
374
375
19 Pa.s, linkage of the
primary grabens is even further delayed and occurs 376
when the tips of the propagating grabens are 377
aligned, leading to full transform linkage shear zone 378
perpendicular to the trend of the primary grabens. 379
In model 7 (Figure 3d) when viscosity is increased
to 6 10
380
381
19 Pa.s, the grabens initially propagate,
curve around the central region, and do not connect.
Model 8 (Figure 3e) at mv =1 10
382
383
20 Pa.s shows
similar behavior to model 7. The grabens in this 384
case propagate even further before curving around 385
the central region. When the viscosity is increased
to 2.5 10
386
387
20 Pa.s in model 9 the primary grabens
do not seem to interact significantly and propagate 388
largely independently (Figure 3f). In model 10
(Figure 3g), at mv = 5.0 10
389
390
20 Pa.s, the deformation
is even more distributed than in model 9 with 391
the formation of additional shear bands adjacent 392
to the boundary of the model. In model 11 393
(Figure 3h), when viscosity is very high (mv =1
10
394
395
21 Pa.s), the propagation of the grabens is halted.
Deformation is now more distributed throughout 396
the model domain and new grabens are formed at a 397
characteristic distance from the original primary 398
grabens.
399
[31] As viscosity of the lower layer increases, the 400
mode of interaction between the rift segments 401
changes from connecting, to overlapping non-con- 402
necting and finally to distributed deformation for 403
high viscosities.
404
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4.2. Weak Lower Crust: Sensitivity to Rift
Offset and Strain-Weakening Ratio
405
406
[32] Next we test the sensitivity of the brittle-ductile 407
coupled system to varying offset and strain weak- 408
ening ratio with a constant lower layer viscosity of
mv =10
409
410
20 Pa.s. Models 12–15 explore the consequence
of varying rift offset for a constant strain- 411
weakening ratio, R = 5, whereas models 12-R2, 412
13-R2, 14-R2, 15-R2, 16-R2 and 12-R3, 13-R3, 413
14-R3, 15-R3, 16-R3 test the sensitivity to the 414
amount of strain-weakening.
415
4.2.1. Variation of Offset
4.2.1.1. Offset D = 2 h (Model 12)
416
417
[33] In model 12 (Figure 4), the two weak seeds 418
placed at the base of the plastic layer are offset by a 419
distance D = 2 h. At t = 0.5 Ma (Figure 4d), 420
deformation initially localizes in 2 conjugate shear 421
zones rooted in the weak seeds, forming an angle of 422
approximately 45 with the horizontal. The inner- 423
most plastic shears with opposing dip are almost 424
aligned at the surface. At t = 1.0 Ma (Figure 4e), the 425
faults propagate along strike into the area without a 426
seed, as the initial shear zones start to strain weaken 427
leading to the formation of grabens. In the central 428
region between the grabens deformation is initially 429
diffuse. At t = 1.5 Ma (Figure 4f), as strain weak- 430
ening sets in, strain localizes preferentially in the 431
innermost shear zones and the outermost plastic 432
shears are abandoned allowing for largely asym- 433
metric deformation in both graben segments. A 434
change of polarity between the normal faults can 435
thus be observed in the more advanced stages of rift 436
propagation. There is a slight flank uplift as the 437
viscous material from the lower crust layer flows 438
into the necking zone of the plastic layer. At t = 439
2.0 Ma (Figure 4g), the main active plastic shear 440
zones connect the two asymmetric half graben 441
segments in one continuous graben structure. The 442
central area where the primary plastic shears with 443
opposing dip link is characterized by distributed 444
accumulation of strain.
445
Figure 3. Models 4–11, with D = 5 h and R = 5, where
the viscosity mv of the lower crust ranges from 10 19 Pa.s
to 10 21 Pa.s. (left) Strain accumulated after a representative
time: the red regions are the parts of the system that
have reached strain weakened values. (right) Strain rate
at the same time (as strain) showing the part of the system
which is actively deforming at this given time. The
models 4–11 are shown at different times, as the rate of
propagation depends on mv.
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Figure 4. Evolution of model 12, where the weak seeds are offset by D = 2 h and the brittle-ductile coupling is weak.
3D view of deformed domain after 2 Ma of extension, showing (a) free surface elevation superimposed on the frictional-plastic
upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain. (d–g)
Top view of elevation, juxtaposed with side view of cross-section of strain rate after 0.5 Ma, 1 Ma, 1.5 Ma and 2 Ma.
446 4.2.1.2. Offset D = 3 h (Model 13)
447 [34] In model 13 (Figure 5), the two weak seeds
448 placed at the base of the plastic layer are offset by a
449 distance D = 3 h. At t = 0.6 Ma (Figure 5d),
450 deformation initially localizes to form two shear
451 zones rooted in the weak seeds. At t = 1 Ma
452 (Figure 5e), two symmetric rift grabens are formed.
453 At t = 1.6 Ma (Figure 5f), the rift grabens propagate
454 in unextended region. At this stage, strain preferen-
455 tially localizes in the innermost shear zones leading
456 to a loss of symmetry and a change of polarity
457 between the normal faults formed in the two primary
458 graben segments. At 2.0 Ma (Figure 5g), the rifts
459 curve toward one other and are almost connected in a
460 single continuous structure. As in model 12 defor-
461 mation in the linkage area is distributed.
462 4.2.1.3. Offset D = 4 h (Model 14)
463 [35] In model 14 (Figure 6), the two weak seeds
464 placed at the base of the plastic layer are offset by a
465 distance D = 4 h. At t = 1 Ma, rift grabens are
formed in the weak seed regions (Figure 6d). In 466
this case, the rifts initially propagate independently 467
into the central area. As in the previous models 468
the feedback effect owing to strain weakening 469
causes strain to accumulate in the innermost shear 470
zones, forming asymmetric grabens. At t = 3 Ma 471
(Figure 6f), the rifts have propagated almost half- 472
way across the domain and as deformation accu- 473
mulates in the central region, the tip of the rifts 474
segments curve around the central region. At t = 475
4 Ma (Figure 6g) as the central region strain weak- 476
ens a dextral transfer fault links the primary grabens. 477
4.2.1.4. Offset D = 5 h (Model 15)
478
[36] In model 15 (Figure 7), the two weak seeds 479
placed at the base of the plastic layer are offset by a 480
distance D = 5 h. As in the previous models, strain 481
accumulates on the innermost shear zones, forming 482
asymmetric grabens. At t = 2 Ma (Figure 7e), the 483
asymmetric rift segments start to propagate largely 484
independently of each other into the non seed 485
region. At t = 3 Ma (Figure 7f), the rifts propagate 486
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Figure 5. Evolution of model 13, where the weak seeds are offset by D = 3 h and the brittle-ductile coupling is weak.
3D view of deformed domain after 2 Ma of extension, showing (a) free surface elevation superimposed on the
frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain.
(d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 0.6 Ma, 1 Ma, 1.6 Ma
and 2 Ma.
487 along each other, curving slightly around the cen-
488 tral region. At t = 4 Ma (Figure 7g), the primary
489 grabens propagate further, forming two hook-
490 shaped faults around the central region which is left
491 intact.
492 4.2.2. Effect of Strain Weakening
493 [37] In all previous models, a strain weakening ratio
494 R = 5 was used. To investigate the sensitivity of the
495 model behavior to the amount of strain weakening,
496 models 12–16 were run with R = 2 and R =3
497 (Figure 8). As described above for R = 5, the rifts
498 propagate and connect efficiently for offsets D =
499 2 h, 3 h and 4 h, while a strike-slip dominated
500 feature is promoted for D = 4 h, and rift linkage is
501 inefficient for D > 4 h. Decreasing the strain
502 weakening ratio to R = 3 only affects the style of rift
503 linkage for intermediate offset D = 4 h. In this case
504 the transform linkage structure is suppressed and
505 more diffuse strain accumulation takes place in the
506 linkage area. Decreasing the strain weakening ratio
507 even further to R = 2 has a significant impact on rift
linkage. The most striking difference is observed 508
for an offset D = 4 h. In these conditions the rifts do 509
not connect and at t = 3.6 Ma, start to curve around 510
the central region, exhibiting similar behavior to 511
model 15 (R =5,D = 5 h).
512
4.3. Strong Lower Crust: Variation of Offset
513
[38] In the last set of models 17–19, we examine the
effect of a strong lower crust (mv =10
514
515
21 Pa.s) on rift
mode for various rift offsets (D = 2 h, 4 h, 6 h) with 516
strain weakening ratio, R = 5 (Figure 9).
517
[39] Model 17–19 show all similar behavior with 518
little dependence on the offset of the weak seeds. 519
Deformation at t = 4 Ma initially localizes in the 520
weak seeds to form asymmetric primary grabens 521
above the weak seeds. At t = 6 Ma deformation is 522
distributed, with the coeval deformation of the pri- 523
mary grabens and the formation of secondary shear 524
bands. The secondary shears form initially sym- 525
metric graben structures at a characteristic distance 526
of about 4h from the primary grabens. Strain 527
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Figure 6. Evolution of model 14, where the weak seeds are offset by D = 4 h and the brittle-ductile coupling is weak.
3D view of deformed domain after 4 Ma of extension, showing (a) free surface elevation superimposed on the
frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain.
(d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 1 Ma, 2 Ma, 3 Ma and 4 Ma.
528 weakening of secondary plastic shear zones leads to
529 asymmetry in the secondary grabens. For t ≥ 8Ma
530 lateral propagation of both primary and secondary
531 graben structures leads to complex interaction
532 structures in the central region. The detailed rift
533 linkage structure in the central domain depends on
534 the offset between the regularly spaced graben
535 structures on either side of the model domain.
536 When the offset of the initial weak seeds is a mul-
537 tiple of the evolving characteristic spacing between
538 the grabens, rift propagation leads to continuous
539 linking structures as in model 18 and 19 for
540 t > 10 Ma. Complex linkage structure result when
541 the primary graben offset is a fraction of the char-
542 acteristic spacing (model 17, t > 10 Ma).
543 [40] Brittle-ductile coupling provides a strong con-
544 trol on the localization of deformation. Spectral
545 analysis of the free surface elevation (Figure 10) for
546 models 17–19 with offsets 2 h, 4 h and 6 h, indicate
547 a characteristic spacing of about 70–75 km. For
548 each of these three rift offsets, the dominant topo-
549 graphic wavelength for the evolving rift system is
550 around these values. This indicates that the wave-
551 length between the graben structures is independent
of the position and offset of the initial weak seeds 552
and mostly controlled by the viscosity of the lower 553
layer.
554
5. Discussion
555
[41] Our results indicate three distinct modes of 556
interaction between two interacting rift segments 557
for brittle-ductile coupled crustal systems in three 558
dimensions. The modes, summarized in Figure 11, 559
are: 1) oblique to transform linking graben systems 560
for small to moderate rift offset and low lower layer 561
viscosity, 2) propagating but non linking and 562
overlapping primary grabens for larger offset and 563
intermediate lower layer viscosity, and 3) formation 564
of multiple graben systems with inefficient rift 565
propagation for high lower layer viscosity. 566
[42] Strain weakening, rift offset, and brittle ductile 567
coupling provide the main controls on these modes. 568
The degree to which the primary grabens localize 569
deformation, propagate and link depends on: 1) the 570
efficiency of strain accumulation and resulting 571
strain weakening above the weak seeds, which 572
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Figure 7. Evolution of model 15 where the weak seeds are offset by D = 5 h and the brittle-ductile coupling is weak.
3D view of deformed domain after 4 Ma of extension, showing (a) free surface elevation superimposed on the
frictional-plastic upper crust (red) and viscous lower crust (green), (b) strain and (c) cross-sections of the strain.
(d–g) Top view of elevation, juxtaposed with side view of cross-section of strain rate after 1 Ma, 2 Ma, 3 Ma and 4 Ma.
573 leads to the formation of localized plastic shears, 2)
574 the offset of the primary grabens, which affects the
575 efficiency of strain accumulation in the linkage
576 zone versus along strike propagation, and 3) the
577 viscosity of the lower viscous layer that controls the
578 efficiency of localized versus distributed strain
579 accumulation in the brittle-ductile coupled system.
580 The trade-off between these factors determines the
581 relative efficiency of localized versus distributed
582 strain accumulation.
583 [43] In all models, strain initially accumulates above
584 the weak seed, leading to the formation of the pri-
585 mary grabens. Distributed deformation occurs in the
586 region ahead of the primary grabens. Rift propaga-
587 tion is achieved when the accumulated strain in this
588 area reaches strain weakening values. At the same
589 time, distributed strain accumulates in the rift link-
590 age area and in the areas offset from the primary
591 grabens. Whether rift linkage (mode 1), rift propa-
592 gation (mode 2), or off axis distributed deformation
593 is preferred depends on the relative efficiency of
594 strain accumulation and resulting strain weakening
595 in each of these areas.
5.1. Factors Controlling Mode of Rift
Interaction
5.1.1. Effect of Brittle-Ductile Coupling
596
597
598
[44] The first two sets of models (Figures 2 and 3) 599
demonstrate that viscosity of the lower layer plays a 600
key role in the evolution and mode of deformation 601
of extensional grabens and the nature of their 602
interaction with other graben segments.
603
[45] At low viscosities when the brittle-ductile 604
coupling is very weak, strain localizes efficiently in 605
the shear zones above the weak seed. As viscosity 606
of the lower layer (and strength of coupling) 607
increases, deformation becomes increasingly dis- 608
tributed throughout the system, demonstrating an 609
inverse relationship between strain accumulation 610
and strength of brittle-ductile coupling. The weaker 611
the coupling, the faster the strain weakening 612
threshold is reached in the initial shear zones. This 613
leads to fast and efficient rift propagation. This is 614
apparent in Models 1–3, where in the time that is 615
required for a rift to propagate throughout a very 616
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Figure 8. Influence of strain weakening on final structure when the lower crust is weak (mv =10 20 Pa.s). A different
amount of strain weakening (R = 2, 3 and 5) has been applied in each row, which shows the structure formed for different
offsets (D =2h–6 h). Different times are shown for the different offsets because the larger the offset, the more
time it takes for the rifts to connect, if at all.
617 weakly coupled system (e.g. Model 1), only a
618 fraction of that distance is covered by rifts with a
619 more strongly coupled system (e.g., Model 2 and 3)
620 (Figure 2). The different rates of rift propagation
621 explain why the variation in the width of the rift
622 along strike becomes larger as mv increases. At any
623 given time, the shear zones formed above the weak
624 seed have accommodated more extension and have
625 been advected sidewards by extension, than the
626 shear zones formed subsequently by rift propaga-
627 tion in the previously homogeneous regions. This
628 lateral variation in the initiation and formation of
629 the rift results in the formation of a V-shaped rift,
630 which becomes more pronounced for higher
631 viscosities.
632 [46] The influence of the viscosity of the lower
633 layer on rift segment interaction is studied with
634 models 4–11 (Figure 3). Varying the viscosity of
635 the lower layer, m v, in small increments allows three
636 contrasting modes of rift interaction. At low vis-
637 cosities, each of the rift segments localizes strain
638 and propagates efficiently. Distributed strain accu-
639 mulating in the rift linkage area results in efficient
640 connectivity between the rift segments. At these low
641 viscosities all the extension is accommodated within
the innermost shear zone, which is favored as the 642
rifts exert an influence over each other leading to 643
asymmetry.
644
[47] As viscosity increases, the rift becomes more 645
symmetric as the outermost shear zone also accu- 646
mulate strain and strain weaken. Beyond a certain 647
viscosity, strain accumulation in the linkage area is 648
less efficient and no longer sufficient to cause the 649
rifts to link. The competition between strain accu- 650
mulation in the central region and along strike 651
causes the rifts to curve around the central region. 652
[48] At high viscosities, deformation is distributed 653
and the rifts no longer interact with each other. 654
Localized grabens form with a characteristic wave- 655
length of about 5 times the brittle layer thickness. 656
While the distributed mode is only obtained for high 657
viscosities, the selection of a connecting mode or an 658
overlapping non-connecting mode also depends on 659
offset between rift segments and the amount of 660
strain weakening.
661
5.1.2. Offset D Between Rift Segments
662
[49] When the offset, D, is small, the linkage area 663
coincides with the propagation area facilitating 664
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Figure 9. Model 17–19: Overview of the evolution of model behavior for different rift offset (offset D = 2 h, 4 h, 6 h)
of the initial rift segments when coupling between the brittle and the ductile layer is very strong (mv =10 21 Pa.s).
665 efficient connectivity. When the offset is large,
666 distributed strain accumulates in each of these areas
667 leading to competing domains of strain localization.
668 [50] Offset between the rift segments is the second
669 major control on the mode of rift interaction. When
670 the offset D, is small, the zones of distributed
671 deformation in front of the propagating rift seg-
672 ments are close enough to overlap. When the brit-
673 tle-ductile coupling is weak, strain accumulates
674 more efficiently in the rift linkage area than along
675 strike promoting linkage. Beyond an offset of
676 around D = 4 h (depending on mv and R), the strain
677 accumulated in the central region is no longer suf-
678 ficient to induce linkage and an overlapping hook-
679 shaped mode of rift interaction is obtained.
680 5.1.3. Amount of Strain Weakening R
681 [51] For intermediate offsets, a decrease in the
682 amount of strain weakening causes a change in the
mode of rift interaction from transform linking 683
mode (M1) to overlapping non linking mode (M2), 684
as a result of the decrease in the efficiency of 685
strain weakening in the linkage area. Using a dif- 686
ferent approach in which weakening was achieved 687
through energy-dependent damage, Hieronymus 688
[2004] showed similarly that weakening favors a 689
shift in mode interaction from transform fault to 690
overlapping spreading center.
691
[52] For low strain weakening ratios (R = 3–4) 692
the transition from linking to non-linking graben 693
systems occurs at viscosities in the range [1–5]
10
694
695
19 Pas, whereas for higher strain weakening
ratios (R = 5) this mode transition occurs at
viscosities of [0.5–1] 10
696
697
20 Pas. This can be
explained by the trade-off between strain weaken- 698
ing in the linkage area and the resistance to locali- 699
zation in this area provided by brittle ductile 700
coupling.
701
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Figure 10. Y-averaged spectra of the (x, z) free surface elevation for models 17, 18 and 19, showing dominant wavelength
at 70–75 km.
702 5.2. Interplay Between Mode Transition
703 Controls
704 [53] Figure 11 illustrates the primary controls on
705 rift interaction and their trade-off effects. The
706 mode of interaction between rift segments in the
707 system depends on the combined effects of brittle-
708 ductile coupling, the onset and magnitude of strain-
709 weakening, and the offset between the rift segments.
710 While increasing the offset, D, tends to cause a
711 shift from linking mode 1 to non-linking mode 2,
712 decreasing strain weakening has the same effect for
713 intermediate offsets (Figure 8). Increasing viscosity
714 on the other hand, for a given offset and strain
715 weakening ratio R, can cause the system to switch
716 from localized mode (M1) to distributed mode (M2
717 and M3) of interaction (Figure 3). Oblique to
718 transform linking mode (M1) is favored by small
719 offset, a large amount of strain weakening, and low
720 viscosity. Localized but non-linking grabens mode
721 (M2) is obtained for intermediate to large offsets
722 and intermediate viscosities, and is favored by a
723 small amount of strain weakening. The distributed
724 deformation mode (M3) results when the brittle-
725 ductile coupling with the lower layer is strong.
726 [54] Other factors may also affect the selection of
727 these modes. Strain rate or velocity of rifting pro-
728 vide a similar control on brittle-ductile coupling as
729 lower layer viscosity [e.g., Huismans et al., 2005;
Buiter et al., 2008; Choi et al., 2008], with 730
increased rate of extension leading to distributed 731
deformation modes. Second, the onset of strain 732
weakening has been demonstrated to affect the 733
efficiency of rift linkage [e.g., Allken et al., 2011]. 734
This suggests that different strain weakening 735
mechanisms characterized by contrasting onset, 736
magnitude, and rate of strain weakening may lead 737
to varying efficiency of rift linkage. Last, the exis- 738
tence of the ductile lower layer also impacts on the 739
modes of interaction. Comparison of the models 740
presented here with our earlier work on 3D exten- 741
sional systems that included a plastic layer on a 742
fixed base [Allken et al., 2011] only demonstrates 743
this. The inclusion of a fluid viscous layer allows for 744
isostatic compensation, resulting in shallower com- 745
pensated basins and subdued rift flank topography. 746
Furthermore brittle-ductile coupling accounts for 747
the new features observed in the models presented 748
here, i.e. the orthogonal transform fault and the 749
distributed mode of rifting.
750
5.2.1. Emergence of Orthogonal Transform
Faults
751
752
[55] Orthogonal transform faults only occur when 753
coupling is weak, for intermediate offsets (4 h or 754
5 h) at the transition between mode 1 and 2. This 755
feature is observed when the conditions are such 756
that the zones of distributed deformation from each 757
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rift segment overlap and reach strain weakening 758
values when the rifts have each propagated halfway 759
throughout the model. Orthogonal transform faults 760
only occur within a narrow parameter space, for a 761
given combination of rate of rift propagation, 762
largely controlled by the viscosity of the lower layer 763
and strain weakening parameters. Using a purely 764
elastic damage rheology, Hieronymus [2004] sug- 765
gests that the formation of transform faults depends 766
on the rate of shear damage formation relative to 767
that of ridge propagation. This is consistent with the 768
results presented here which indicate that orthogo- 769
nal transform faults can form by gradual focusing of 770
diffuse damage.
771
5.2.2. Distributed Pure Shear Mode
772
[56] The mode transition between overlapping non- 773
connecting rift segments to distributed deformation 774
can be understood using minimum energy dissipa- 775
tion arguments. Mode transition can be predicted if 776
it is assumed that the preferred mode minimizes the 777
internal rate of energy dissipation. Earlier analytical 778
work [Huismans et al., 2005; Buiter et al., 2008] 779
predicts the transition viscosity mt beyond which
2
the distributed mode is favored over the strain
780
781
weakened symmetric graben mode. Adapting this 782
formulation (see Appendix A) to the rheology used 783
in our models gives us the following transition 784
viscosity mt2 :
785
mt2 ¼ 6hphbLx
V L2 x 8h2 b
Ccosf C sw cosf sw þ rghp
2 ðsinf sinfsw Þ
ð15Þ
For reasonable parameter values (Appendix A) the 786
mode transition viscosity can be calculated for 787
varying amount of strain weakening ratio R 788
(Table 1). The values of the predicted transition 789
viscosity mt are consistent with the model results
2
presented here that indicate the mode transition for
viscosities ranging from mv = 2.5 10
790
791
792
20 to
5 10 793
20 Pa.s (Figures 3 and 11).
Figure 11. Summary of modes predicted (including
some models not all shown in the paper), illustrating
how the 3 modes of rift interaction are influenced by
the viscosity, mv, the strain weakening ratio, R and the
offset D and the trade-off between those parameters.
The connecting mode (M1: blue), is obtained for small
D, low m v and is favored by low R. The overlapping
non-connecting mode (M2: green) can be subdivided
into a hook-shaped mode (M2a: dark green) and a propagating
mode (M2b: light green) modes. Mode 2 is
obtained for larger D and intermediate mv and is favored
by low R. The distributed pure shear mode (M3: yellow)
is obtained for high m v.
15 of 19

t1:1 Table 1. Model Parameters
t1:3 Symbol Meaning Value
t1:4
t1:5
t1:6
t1:7
t1:8
t1:9
Lx
Ly Lz hp
hv
r
Length of model along x-axis
Length of model along y-axis
Length of model along z-axis
Thickness of plastic layer
Thickness of viscous layer
Density
210 km
210 km
30 km
15 km
15 km
2800 kg.m 3
t1:10 g Acceleration due to gravity 9.81 m.s 2
t1:11 C 0
t1:12 f
Initial cohesion 20 MPa
0
Initial angle of friction 15
t1:13
t1:14
1
2
Strain softening threshold
Onset of full strain weakened
0.25
1.25
state
t1:15 v ext Extensional velocity 0.5 cm/yr
t1:16 l Penalty factor 10 32 Pa.s
794 5.3. Model Limitations
795 [57] 1. Temperature dependent rheologies, although
796 available as an option, have not been included.
797 Transient variations in the depth of the brittle-
798 ductile are expected to affect rift evolution. Given
799 that no conductive cooling and associated deepen-
800 ing of the brittle-ductile transition is included, the
t2:1 Table 2. Model Parameters Used: Seed Arrangement,
t2:2 Viscosity of Lower Layer, mv, and Strain Weakening, R
t2:4 Figure
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Model
Number
Seed
Arrangement mv (Pa.s) R
t2:5 Figure 2 1 Single seed 1.0 10 19
t2:6 Figure 2 2 Single seed 1.0 10 20
t2:7 Figure 2 3 Single seed 1.0 10 21
t2:8 Figure 3 4 D = 5 h 1.0 10 19
t2:9 Figure 3 5 D = 5 h 2.5 10 19
t2:10 Figure 3 6 D = 5 h 1.0 10 19
t2:11 Figure 3 7 D = 5 h 6.0 10 19
t2:12 Figure 3 8 D = 5 h 1.0 10 20
t2:13 Figure 3 9 D = 5 h 2.5 10 20
t2:14 Figure 3 10 D = 5 h 5.0 10 20
t2:15 Figure 3 11 D = 5 h 1.0 10 21
t2:16 Figure 4 12 D = 2 h 1.0 10 20
t2:17 Figure 5 13 D = 3 h 1.0 10 20
t2:18 Figure 6 14 D = 4 h 1.0 10 20
t2:19 Figure 7 15 D = 5 h 1.0 10 20
t2:20 Figure 8 16 D = 6 h 1.0 10 20
t2:21 Figure 8 12-R2 D = 2 h 1.0 10 20
t2:22 Figure 8 12-R3 D = 2 h 1.0 10 20
t2:23 Figure 8 13-R2 D = 3 h 1.0 10 20
t2:24 Figure 8 13-R3 D = 3 h 1.0 10 20
t2:25 Figure 8 14-R2 D = 4 h 1.0 10 20
t2:26 Figure 8 14-R3 D = 4 h 1.0 10 20
t2:27 Figure 8 15-R2 D = 5 h 1.0 10 20
t2:28 Figure 8 15-R3 D = 5 h 1.0 10 20
t2:29 Figure 8 16-R2 D = 6 h 1.0 10 20
t2:30 Figure 8 16-R3 D = 6 h 1.0 10 20
t2:31 Figure 9 17 D = 2 h 1.0 10 21
t2:32 Figure 9 18 D = 4 h 1.0 10 21
t2:33 Figure 9 19 D = 6 h 1.0 10 21
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
2
3
2
3
2
3
2
3
2
3
5
5
5
Table 3. Transition Viscosity mt2 as a Function of t3:1
Strain Weakening Ratio R
t3:2
R m t2 (Pa.s)
2 2.67 10 t3:5
20
3 3.46 10 t3:6
20
4 3.79 10 t3:7
20
5 4.11 10 t3:8
20
models presented here should be interpreted in the 801
context of moderate to fast extended crustal sys- 802
tems at the advective limit.
803
[58] 2. The boundary conditions applied in these 804
models, orthogonal extension, are kept simple. 805
Natural systems may extend by combinations of 806
both oblique extension and varying amounts of 807
extension along strike. We do not expect that vary- 808
ing amounts of extension along strike would sig- 809
nificantly change the modes observed in our simple 810
models. This, however, needs further investigation. 811
Oblique extension with respect to inherited hetero- 812
geneity (weak seeds) should result in oblique rift 813
modes but is beyond the scope of this paper. 814
[59] 3. Surface processes (erosion and deposition) 815
are not included. These are expected to have an 816
effect on the structural style of sedimentary basin 817
formation but are beyond the scope of the present 818
paper.
819
6. Conclusions
820
[60] We use state-of-the-art large deformation 3D 821
numerical models to study the structural style of 822
crustal extension. We examine the controls on the 823
style and geometry of rift linkage between rift 824
segments during extension of crustal brittle-ductile 825
coupled systems, and test the sensitivity of varying 826
the viscosity of the lower layer, the offset between 827
the rift basins, and the amount of strain-weakening 828
on the efficiency of rift linkage and rift propagation, 829
and the style of extension. We conclude that: 830
[61] 1. The major controls on the mode of rift 831
interaction in brittle-ductile coupled crustal systems 832
are, in decreasing order of importance: 1) strength 833
of brittle-ductile coupling, 2) rift offset, and 3) the 834
amount of strain weakening.
835
[62] 2. For small to moderate rift offset and low 836
lower layer viscosity, Mode 1, oblique to transform 837
linking graben is preferred. For larger offset and 838
intermediate lower layer viscosity, Mode 2 with 839
propagating but non linking and overlapping pri- 840
mary grabens is promoted, and for high lower layer 841
16 of 19
t3:4

842 viscosity, Mode 3 with the formation of multiple
843 graben systems and inefficient rift propagation is
844 obtained.
845 [63] 3. The mode transition between the linking
846 Mode 1 and non-linking Mode 2 is controlled by
847 the trade-off between the rift offset, the strength of
848 brittle-ductile coupling and the amount of strain
849 weakening.
850 [64] 4. The transition from Mode 2 overlapping
851 non-connecting rift segments to Mode 3 distributed
852 deformation is mainly controlled by the viscosity of
853 the lower layer and can be understood using mini-
854 mum energy dissipation arguments.
855 Appendix A: Transition Between Modes
856 Predicted From Dissipation Analysis
857 [65] Dissipation analysis in 2D [Huismans et al.,
858 2005] predicted that as viscosity increases, the sys-
859 tem will deform in asymmetric graben (AP) mode,
860 symmetric graben (SP) mode and pure shear (PS)
861 mode respectively, the criteria for mode selection
862 being based on the minimization of energy dissipa-
863 tion. At a transition between two modes, the total
864 dissipation in these modes are equal:
_W mode1
I
þ _W mode1
G
¼ _W mode2
I þ _W mode2
G
ðA1Þ
865 where _W I is half the rate of internal viscous-plastic
866 dissipation per unit volume and _W G is the total rate
867 of work against gravity. Modes 1, 2, and 3 represent
868 AP, SP and PS modes. It was considered in this
869 study that mode selection could be estimated with
870 sufficient accuracy from the internal dissipation _W I
871 alone. The internal dissipation is the sum of the
mode V
872 viscous dissipation _W I and the plastic dissimode
P
873 pation _W I , which depends on the strength of
874 the frictional-plastic layer, s. Since the strength of
875 the plastic layer is pressure-dependent, we take the
876 mean strength, sm:
sðz ¼ 0Þþsðz ¼ hpÞ
sm ¼
2
¼ 1 ccosq
2
cosq þ 1 þ
pffiffi sinfsinq
3
rghp sinf þ ccosq
cosq þ 1 2
3
6
7
6
7
4
5
pffiffi sinfsinq
3
ðA2Þ
¼
Geochemistry
Geophysics
Geosystems G 3
G 3 ALLKEN ET AL.: 3D MODELING—MODES OF RIFT INTERACTION 10.1029/2012GC004077
1
cosq þ 1 pffiffi sinfsinq
3
þ ccosq þ rghp sinf
2
At transition between modes:
mode V _W
mode P
mode V
I þ _W I ¼ _W I þ _W
Internal dissipation for:
Pure shear mode:
_W PS
I
Symmetric graben mode:
_W SP
I ¼ 4mv2 hv hb
Lx
hv
¼ 4mv2 þ 2vhpsm
ðA4Þ
L
þ 4 hb
mv2
3 Lx
þ 1
3
mode P
I
ðA3Þ
Lx
mv2 þ 2vhpsm ðA5Þ
hb
where hb is the boundary layer and v is the exten- 877
sional velocity.
878
[66] At the transition between the symmetric graben 879
mode and the pure shear mode:
880
SP V _W
SP P
_W SP
I ðSW Þ¼ _W PS
I
PS V
I þ _W I ðSW Þ¼ _W I þ _W
SP V _W I
PS V _W
I ¼ _W
mv 2 L2 x 8h 2 b
3hbLx
PS P
I
¼ 2vhp sm s sw
m
[67] Let a be cosq þ 1ffiffi p sinf sinq
3
sm s sw
m
1
¼
a C cosf þ rghp sinf
2
1
asw Csw cosq þ rghp sinf sw
2
¼ C cosf þ rghp sinf sw
2
a
asw Csw cosf þ rghp sinf sw
2
PS P
I
ðA6Þ
ðA7Þ
SP P _W I ðSW Þ ðA8Þ
ðA9Þ
ðA10Þ
881
Given that the Lode angle q lies between p/ 882
6

887 Acknowledgments
888 [68] This work was funded through Norwegian Research
889 Council Grant 177489/V30 to Huismans, who also acknowl-
890 edges support through an EU International Reintegration Grant
891 and from the Bergen Center for Computational Science. We
892 thank Suzon Jammes and Philippe Steer for valuable discus-
893 sions and constructive comments that helped significantly
894 improve the manuscript. Taras Gerya and an anonymous
895 reviewer are thanked for their quick reviews and helpful
896 suggestions.
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