Download Scaling models into their natural prototypes ensures

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1. Model scaling
Scaling models into their natural prototypes ensures fulfillment of similarity in geometry,
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kinematics, dynamics and rheology [Ramberg, 1981].
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It is widely accepted that a continental lithosphere with a normal crustal thickness (30-35km) and
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stable geotherm, extending at strain-rate magnitude of about 10-15 s-1, can be mechanically
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described by a four-layer strength profile with alternating high- and low-strength (Figure1b) [Brace
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and Khoelsted, 1980; Davy and Cobbold, 1991; Brun and Beslier, 1996].
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Stresses scale down according to:
𝜎 ∗ = 𝜌∗ 𝑔∗ 𝐿∗
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(1)
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where the starred symbols are the dimensionless ratios between model and natural prototype for the
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following parameters: 𝜎* is stress, ρ* is density, g* is gravity, and 𝐿* is the length scale-factor (L*
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= Lmodel / Lnature = 6.66 x 10-7, that is 15km in nature correspond to 1cm in the model). Time scaling
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for the viscous material is achieved by
𝑡 ∗ = 𝑡𝑚 ⁄𝑡𝑛 = 𝜂 ∗ ⁄𝜎 ∗
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(2)
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where t* is the dimensionless ratio of time, tn and tm the time in nature and in model respectively,
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and 𝜂* is the dimensionless ratio for viscosities.
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The values of dimensionless ratios used to scale the lower crust in the models are provided in Table
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1, while in Table 2 the resultant values for parameters in the models and in nature are listed.
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Viscous materials have a Newtonian rheology and their strength is provided by [e.g. Ranalli,
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1995]:
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τ = 𝛾̇ 𝜂
(3)
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where 𝛾̇ is the shear-rate (V/h, Velocity/layer thickness).
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The brittle rheology is simulated by pure quartz sand [Davy and Cobbold, 1991] with average grain-
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size of 260µm and negligible cohesion. The ductile rheology is simulated by using silicone putty
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SGM36. In particular, pure silicone putty was mixed with the proper weight of barite powder in
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order to tune its density and rheology (Table 2). Silicone putty has an almost Newtonian rheology
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(parameter ‘n’ is 1,053 and 1,067 for the silicone simulating the crust and the mantle, respectively)
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and is widely used to reproduce the mechanical behavior of ductile layers in normal gravity
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experiments [e.g. Davy and Cobbold, 1991; Brun and Beslier, 1996; Brun, 1999; Autin et al., 2010].
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The experimental lithosphere of our models consists of a four-layer multilayer that, from bottom to
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top, includes a 12mm thick silicone layer simulating ductile mantle material, a 10mm thick sand
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layer simulating brittle mantle material, a 7mm thick silicone layer simulating the ductile lower
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crust, and a 13mm thick sand layer simulating the brittle upper crust. This multilayer floats on
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glucose syrup (𝜌𝐴 =1450 kg/m3) simulating the asthenosphere [i.e. Davy and Cobbold, 1991]. Model
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velocity (1cm/h) is representative of prototype velocities ranging between 0,2 - 20cm/y according to
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the lower crust viscosity range (1020 - 1022 Pa.s). To simulate a natural extensional velocity of
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≈4mm/y we assume a viscosity of 5 x 1021 Pa.s, and a strain-rate of 5,87x10-14 for the prototype
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lower crust (Table 1). The calculated model to nature ratios of both the “Ramberg number” (Rm* =
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1) and the “Smoluchowsky number” (Sm* = 1,03), for the ductile and the brittle domains,
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respectively (Table 1), ensure dynamic scaling of the model (e.g. [Corti et al., 2003]).
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2. Thinning factor calculation
Top and bottom laser-scanned surface pairs underwent a post-acquisition processing to tie them up
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to the same x-y space matrix. The thickness of the undeformed lithosphere (T0) is calculated as the
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difference between top (Zt0) and basal (Zb0) surfaces at each point, then averaged over the entire
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undeformed model surfaces:
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𝑇0 = 𝑁 ∑𝑛 [𝑍𝑡0 − 𝑍𝑏0 ](𝑛)
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where N is the number of nodes into the matrix space (N = i x j).
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The whole lithosphere-stretching factor (β) [McKenzie, 1978] is calculated as the ratio between the
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mean initial lithosphere thickness (T0) and the lithosphere thickness at the end of experiment (Tf):
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[𝑇𝑓 ]
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𝛽(𝑛) =
1
(𝑛)
= [𝑍𝑡𝑓 − 𝑍𝑏𝑓 ]
𝑇0
𝑇𝑓 (𝑛)
(𝑛)
(4)
(5)
(6)
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Rather than β, it is more convenient to use the thinning factor (TF) [eg. Reston, 2007], which varies
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between 0 and 1:
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𝑇𝐹(𝑛) = 1 − 𝛽
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TF is calculated for the crust (Figure 4a), the lithospheric mantle (Figure 4b), and the whole
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lithosphere (Figure 4c).
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1
(𝑛)
(7)