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1. Fault rotation model
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In a previous publication [Olive and Behn, 2014], we proposed a conceptual
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framework for understanding the rapid rotation of normal faults from a steep 60º-
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initiation angle down to dip angles in the 30–45º range, over a few km of offset. We
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proposed that fault evolution proceeded in a manner that systematically minimized the
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mechanical work required to sustain fault slip, and showed that (1) this assumption
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results in rotation rates that scale roughly as the inverse of the faulted layer thickness, and
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(2) accounting for rotation is essential to correctly predict fault lifespan in a force balance
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model. Here we adopt a complementary approach, which is significantly less
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computationally challenging to implement, but produces very similar rotation rates (see
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below). Specifically, instead of assuming work minimization, we model fault rotation as
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the passive advection of the fault plane in the displacement field induced by flexural
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relaxation of the footwall and hanging wall blocks.
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If we denote
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fault rotation rate will scale roughly as
the average rotation rate of the near-fault displacement field, then the
(S1)
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To first order, the main source of rotation in the flexural displacement field is the lateral
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gradient of vertical motion (Figure 1a), which is maximal at the fault and equal to V tanθ,
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and becomes negligible a fraction of the flexural wavelength (γα, the lever arm, where γ
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is a scaling factor and α is the flexural wavelength) away from the fault. We therefore
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write
(S2)
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Plugging (S2) into (S1) with the constraint h = 2Vt yields
¶q
tanq
=¶h
4ga
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(S3)
Which can be integrated into
æ
æ 1
q = sin ç sinq 0 exp ç è 4g
è
-1
dh ö ö
ò0 a (h) ÷ø ÷ø
h
(S4)
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In addition, we taper the rotation rate at heaves greater than 80% of the faulted layer
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thickness so that rotation ceases at large offsets, when the fault has become curved and
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the idealized fault geometry depicted in Figure 1a no longer applies.
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Fault rotation enters the force balance model in several ways. As extension
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proceeds, the elastic-plastic iterations predict the fault-induced topography [Buck, 1988;
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1993]. At every extension step (h), we fit this topography with the elastic solution of
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Weissel and Karner [1989] and determine an equivalent elastic thickness for the entire
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plate. We then use it to calculate an equivalent flexural wavelength, α(h), which enters
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into the calculation of the next fault dip at step (h+Δh) following Equation (S4). We find
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that a scaling factor γ = 0.25 provides a good fit to the numerical models, regardless of
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layer thickness and other parameters. In other words, the lever arm associated with fault
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rotation corresponds to about a fourth of the effective flexural wavelength of the faulted
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layer.
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Once fault dip has been updated, it enters the calculation of the frictional force
FFRIC, following [Behn and Ito, 2008]
FFRIC =
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mr gH / 2 + HC
m sin 2 q + sinq cosq
(S5)
For reference, the threshold for breaking a new fault is given by
FBREAK =
mr gH / 2 + HC0
m sin 2 q 0 + sinq 0 cosq0
(S6)
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where θ0 corresponds to the Andersonian dip for a new fault breaking intact lithosphere.
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(All notations are summarized in Figure 1a).
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Fault dip also enters the calculation of the bending and topographic forces. A fault
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that rotates as it accumulates offset generates less throw than if it were to retain its initial
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angle. We therefore slightly modify the method of Weissel and Karner [1989] by using
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an effective dip θEFF in the loading term of the flexure equation (w*(x) in Olive and Behn,
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2014). For a given amount of heave (h) this effective dip yields the amount of throw that
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would have been reached by a rotating fault, and can be written:
æ1h
ö
ò tanq (h) dh÷ø
èh
q EFF = tan -1 ç
0
(S7)
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θEFF then enters the calculation of the fault-induced topography and the associated work
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terms. Specifically, the bending work WBEND is obtained by integrating the bending stress
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times the bending strain over the entire layer [Olive and Behn, 2014], while the
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topographic work WTOPO corresponds to the change in gravitational potential energy
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associated with offsetting the air-rock density contrast, Δρ, from its initial flat state, and is
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written
+¥
WTOPO =
ò Dr g y ( x ) dx
2
-¥
(S8)
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2. Comparison with the energy minimization model of Olive and Behn [2014]
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In [Olive and Behn, 2014], we suggested that fault rotation proceeds in a manner
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that minimizes the work needed to sustain fault slip, but did not explicitly consider the
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physical mechanism that causes rotation. The present approach suggests that passive
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advection of the fault plane in a flexural displacement field provides a good candidate to
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explain the kinetics of rotation observed in numerical experiments. Indeed, this new
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model predicts the rotation kinematics observed in our simulations (Figure 2b). Much
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like the energy minimization model, it accounts for rapid rotation of normal faults down
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to dip angles of 30–45º over fault heaves shorter than half of the faulted layer thickness.
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Further, the inverse scaling of rotation rate with layer thickness is evident from Equation
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(S3) and is derived from the fact that thicker layers bend on a larger wavelength, making
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faults rotate about a longer lever arm.
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Finally, we find that the present kinematic model for fault rotation predicts dips
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that remain close to the lowest energy configuration throughout fault evolution, even
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though energy minimization was not assumed. Supplementary Figure fs01 compares the
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kinematic elastic-plastic solution for a 15 km thick layer (red line in Figure 2b) to an
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energy minimization model where plasticity is accounted for by assuming an effective
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elastic thickness of 7.5 km. Colors indicate the total amount of work required to sustain
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slip on the fault. The kinematic rotation model predicts slightly slower rotation then the
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energy minimization model for small offsets, which provides a better fit to numerical
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experiments (Figure 2b).
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Supplementary captions
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Supplementary Figure fs01 – Total external work (WEXT = WFRIC + WBEND + WTOPO)
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required to keep a normal fault active as a function of fault dip and heave for increasing
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amounts of extension in a 15 km thick elastic pseudo-plastic layer with an effective
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elastic thickness of 7.5 km. The white dashed line indicates the lowest energy path
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computed following the model of Olive and Behn [2014] taking into account all energy
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terms: WFRIC, WBEND, and WTOPO, the last of which was previously ignored. The thick
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white line shows the kinematic rotation model used in this study.
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Supplementary Table ts01 – Summary of parameters and results of our numerical
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experiments.
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