Rotational Extension Promotes Coeval Upper Crustal Brittle Faulting and DeepSeated RiftAxis Parallel Flow: Dynamic Coupling Processes Inferred From Analog Model Experiments
Abstract
The lower parts of warm, thick continental crust can flow in a ductile fashion to accommodate thinning of the upper brittle crust during extension. Naturally occurring continental rifts with a riftaxis parallel deformation gradient imply an underlying rotational component. In such settings, riftparallel crustal flow transports material perpendicular to the direction of rifting. We use analog experiments to investigate rotational rifting and coeval crustal flow. To test the effect of riftaxis parallel flow on rift evolution, we use different gravitational loads resulting in a range of horizontal pressure gradient magnitudes which drive horizontal lowercrustal flow. The use of (three dimensional) 3D Digital Volume Correlation techniques on Xray CT data combined with 3D Digital Image Correlation techniques applied to topographic stereo images provides detailed insights on the contemporaneous evolution of ductile flow patterns and brittle rift structures, respectively. Our results depict a complex flow field in the ductile lower crust during rotational rifting with: (a) extensionparallel horizontal inward flow and vertical upward flow that compensates thinning of the brittle upper crustal layer; (b) riftaxis parallel lateral flow, that compensates greater amounts of thinning further away from the rotation axis; and (c) different degrees of mechanical coupling between the brittle and viscous layers that change during rift propagation. Our analog experiments provide insights into ductile lower crustal flow patterns during rift evolution. The results emphasize the three dimensionality of rifting, which is an important effect that should be considered when estimating the amount of crustal extension from two dimensional (2D) cross sections.
Key Points

We use volumetric and surface image correlation techniques to depict coeval brittle rifting and ductile flow patterns in analog models

Rotational rifting promotes riftaxis parallel lateral flow, causing outflow of lower crustal material away from the rotation axis

Partial coupling of riftaxis parallel displacements in the brittle and viscous domain is transient and dynamic over time
Plain Language Summary
Stretching of continental plates produces rifts, that are characterized by narrow fault zones in the brittle, upper parts of the Earth's crust and flow of rocks in the more ductile, lower parts of the crust. As rifts often propagate in one direction, the flow pattern in the lower crust can be complex and challenging to study and interpret. Here, we use laboratory experiments to investigate flow in propagating rifts. The application of Xray CTscanning allows us not only to visualize the model internal deformation, but also to quantify the deformation which provides detailed insights into the coeval evolution of ductile flow patterns and brittle rift structures. Our results document a complex 3D flow pattern within the lower crust with an important component along the rift axis, emphasizing the need to consider the evolution of rifts in 3D.
1 Introduction
During continental rifting, extension in the upper parts of the crust is accommodated by brittle faulting. If stretching is distributed evenly throughout the crust (McKenzie, 1978), faulting in the upper crust is balanced by ductile deformation in the lower crust (e.g., Buck, 1991; Kusznir & Karner, 2007). However, at passive margins, subsidence analyzed from geological cross sections differs from expected values when applying the McKenzie uniform stretching model. Such observations led to the development of depthdependent stretching models (Wernicke, 1985) that account for higher amounts of extension in lower parts of the crust, which permit larger subsidence values. More recent studies of passive margins have suggested ductile crustal flow as an alternative mechanism for removal of lower crustal material. Consequently, mass is not conserved in 2D planestrain cross sections, which can explain discrepancies between estimated amounts of extension in the upper and lower crust (P. D. Clift, 2015).
Ductile flow in the lower crust should also occur in intracontinental rift settings. Indeed, lower crustal flow is expected in extensional settings where the crustal thickness exceeds 20 km and the upper mantle is strong (McKenzie et al., 2000 and references therein). This phenomenon has been extensively demonstrated in the Basin and Range Province, where flow of the lower crust toward the rift center compensates lateral crustal thickness variations (e.g., Buck, 1991; Kruse et al., 1991; Zuber et al., 1986). However, after extension ceases, lateral ductile flow may halt and reverse its direction (i.e., away from the rift center) due to increasing sediment loading (e.g., Morley & Westaway, 2006; P. Clift et al., 2002; P. D. Clift, 2015; P. D. Clift et al., 2015; Westaway, 1994). While most of these studies focus on ductile flow in twodimensions, the outofplane component (i.e., riftaxis parallel) should not be neglected. Especially in rift settings with an underlying extension gradient, riftaxis parallel ductile flow may compensate for increased thinning along the rift axis (Amato et al., 2004; Gautier et al., 2008; MacCready et al., 1997).
Threedimensional crustal flow during extension has been demonstrated in the D'Entrecasteaux Islands in the Woodlark Rift (Figure 1) by Little et al. (2007, 2011, 2013). The Woodlark Rift is a continental rift zone east of the Papuan Peninsula separating the Australian plate from the Woodlark Microplate. To the east, the Woodlark Rift is confined by the westpropagating oceanic Woodlark Basin. At the transition between continental and oceanic crust (i.e., near the Moresby Seamount), continental breakup occurs along a characteristic Vshaped rift zone, with respect to the rift margin boundary faults (Benes et al., 1994; Taylor et al., 1999). This specific type of continentocean boundary can be attributed to a counterclockwise rotation of the Woodlark Plate (e.g., Wallace et al., 2014 and references therein). Based on GPS measurements, maximum opening velocities at the eastern end of the Woodlark rift are around 20 mm/a (Wallace et al., 2004) but, based on magnetic anomalies in the Woodlark Basin, were faster between 3.6 and 0.5 Ma (Taylor et al., 1999). The Woodlark Rift is bounded by a southdipping boundary fault in the north and the northdipping Goodenough Fault near the ca. 3 km high Papuan Orogen in the south (Little et al., 2011). Several metamorphic core complexes within the Woodlark Rift, among it the D'Entrecasteaux Islands, are indicative of its continental origin (Wallace et al., 2014) and document lower crustal flow (Little et al., 2007, 2011, 2013). A flat Moho at a depth of 20–30 km (Abers et al., 2002) indicates that the lower crust must have had a low enough viscosity to allow ductile riftaxis normal flow to compensate lateral pressure gradients (Little et al., 2007; Westaway, 2005). In addition, riftaxis parallel flow may explain excess subsidence when comparing observed and modeled bathymetric transects across the Woodlark Rift (P. D. Clift, 2015).
The Woodlark Rift example highlights the important contribution of crustal flow during rifting. However, the quantification of crustal flow as well as its dynamic behavior during continental rifting are still poorly understood processes. In particular: how does ductile flow compensate thinning of the upper crust? What is the effect of gradual opening due to rotation (as seen in the Woodlark Rift) on ductile crustal flow and how does a riftaxis parallel flow component (if present) affect the style of rift propagation? And how are brittle nearsurface and deepseated ductile deformation coupled? To answer these questions, this study presents brittleductile 3D analog models of crustalscale, rotational continental rifting with a particular focus on the role of riftaxis parallel ductile flow. While the previous work of T. C. Schmid, Schreurs, and Adam (2022) analyzed rift propagation and nearsurface deformation in detail, this study provides a quantitative analysis of internal deformation. By combining 3D stereo Digital Image Correlation (3D stereo DIC) and Digital Volume Correlation (DVC), we present a novel approach to quantify surface and internal model deformation and gain a comprehensive understanding of crustal flow and its role during rotational continental rifting with an alongstrike divergence velocity gradient.
2 Methods
We use an experimental apparatus which simulates rotational extension around a pivoting point (Figure 2). An identical setup has been used and described in previous studies of T. C. Schmid, Schreurs, and Adam (2022), Zwaan et al. (2020), and Zwaan and Schreurs (2020). Two short, curved and two mobile long sidewalls confine the model domain (Figure 2a). The mobile sidewalls rotate around a vertical axis which separates the model box into 25 and 65 cm long contractional and extensional model domains, respectively (Figure 2b). The applied maximum divergence velocity is defined along a circular segment at distance 65 cm with respect to the rotation axis. Simultaneously with the opening motion in the extensional domain, shortening occurs in the contractional domain. The initial model surface is defined by a quasirectangular area of 90 cm length and 31 cm width. For our analyses, we leave out the curved model parts close to the short sidewalls, resulting in a rectangular area of interest of about 80 by 31 cm. The model overlies a 5 cm thick foam base that is initially compressed and homogenously expands during the experiment runs. This setup allows for a gradually decreasing basal velocity profile with maximum divergence velocity at the mobile sidewalls and zero velocity near the rotational axis (Figure 2c). In the contractional domain, divergence velocities switch polarity and the foam undergoes further contraction. Rigid plexiglass bars sandwiched between the basal foam provide additional support to prevent the overlying model from sinking in. This basal setup has been successfully tested and applied for rotational rift models in T. C. Schmid, Schreurs, and Adam (2022).
The mechanical twolayer model consists of a crustal scale, brittleviscous setup in which a brittle layer simulates the upper crust and the underlying viscous layer the lower crust (Figure 2c). In the extensional domain, rifting in the brittle layer is localized by applying a viscous seed on top of the viscous layer (Figure 2a). This seed (a semicylindrical polydimethylsiloxane [PDMS]/corundum sand mixture rod with radius of ca. 0.5 cm) acts as a preexisting weak zone along which deformation initiates. We deliberately place the seed along the entire model length to control surface deformation. Note that the absence of a seed results in deformation localization solely along the mobile sidewalls. The applied divergence velocity gradient causes rift propagation toward the rotation axis over time with most mature rift stages developing furthest away from the rotation axis. We place an additional sand layer on top of the contractional domain in some of our models before starting the experiments. This additional gravitational load (see Table 1 for layer thickness) induces a horizontal pressure gradient and thereby, horizontal flow in the viscous basal layer in the opposite direction of rift propagation (Figures 2 and 3). Prior to the model runs, we sieve a fine 4 by 4 cm grid of corundum sand on the model surface for visual aid and sprinkle coffee powder to create a “salt and pepper” pattern for DIC.
Model name  Brittle layer thickness (cm)  Ductile layer thickness (cm)  Brittleductile thickness ratio T_{BD}  Brittleductile strength ratio S_{BD}  Additional gravitational loading (cm)  Comment 

M20  4  2  2  117  0  – 
M23  4  2  2  117  3  Loading 
M10  3  3  1  46  0  – 
M13  3  3  1  46  3  Loading 
M10*  3  3  1  46  0  CT scanned 
M13*  3  3  1  46  3  CT scanned; loading 
M11*  3  3  1  46  1  CT scanned; loading 
M10E*  3  3  1  46  0  CT scanned; ext. only 
M0.50  2  4  0.5  17  0  – 
M0.53  2  4  0.5  17  3  Loading 
2.1 Analog Model Setup
We conducted a series of crustal scale brittleviscous analog experiments with variable additional gravitational loadinduced pressure gradients causing lateral flow in the viscous layer. For models with an identical brittleductile thickness ratio T_{BD} (or identical brittleductile strength ratio, S_{BD}), we compare the deformation evolution in models with an initially flat model surface (i.e., no additional gravitational load) with corresponding models with an additional gravitational load. This thickness ratio T_{BD} is defined as the brittle layer thickness divided by the viscous layer thickness. The corresponding brittleductile strength ratio S_{BD} is listed in Table 1. The overall thickness of the twolayer model was 6 cm for all experiments, representing a 30 km thick, continental crust (McKenzie et al., 2000). However, the thickness ratio T_{BD}, between the brittle and ductile layers varies and defines three different model types with T_{BD} = ½, 1, and 2 simulating different crustal configurations. All experiments were run with a maximum divergence velocity of 10 mm/hr (symmetrically 5 mm/hr each side), which linearly decreases toward the rotation axis. This results in a maximum extension of 40 mm or about 13% maximum extension after 4 hr of each model run. Similarly, maximum shortening of 15.4 mm or 5% is reached at the final stage at the distant circular segment in the contractional domain. Lastly, we induced a pressuregradient driven horizontal flow in the viscous layer by adding an additional quartz sand layer on top of the contractional domain in some models prior to the experiment runs. Thus, the viscous layer flows in the opposite direction to that of rift propagation. For our model naming convention, Mxy*, the “x” and “y” denotes the brittleductile thickness ratio T_{BD} and the height of the additional quartz sand layer on top of the contractional domain, respectively. Models which have been analyzed by XRCT are further denoted with an asterisk. As a reference setup, we use the models M1x and M1x* with a brittleductile thickness ratio of 1. The complete model series consists of 10 experiments listed in Table 1.
The viscous basal layer consists of a mixture of corundum sand and PDMS with a mixing ratio 1:1 to achieve a density of 1,600 kg/m^{3}. The mixture has a quasilinear viscosity of 1 × 10^{5} Pa and a stress exponent of 1.05 (Zwaan, Schreurs, Ritter, et al., 2018). For the upper brittle layer of the analog models, we use dry quartz sand with a bulk density of 1,560 kg/m^{3} to simulate the upper brittle crust. The desired density is achieved by sieving sand into the model box from a height of 30 cm. The brittleviscous models thus have a density gradient increasing with depth avoiding density instabilities and spontaneous upwelling of the viscous lower layer. Dilation and localized deformation in the quartz sand locally lowers the peak friction coefficient over time, resulting in strain softening of about 16% (i.e., the difference between the coefficient of peak friction and dynamic friction coefficient normalized by the coefficient of peak friction (Panien et al., 2006) from mutual twopoint regression analysispresented in Santimano et al., 2015).
For XRCT scanned experiments, we mix small quantities (weight ratio 1:50) of Zirshot ceramic microbeads with the quartz and corundum sands which enhances volumetric patterns in CT scans to facilitate DVC analysis (Adam et al., 2013). Additionally, we sieve a thin layer of corundum sand from 30 cm height on top of the viscous mixture before adding quartz sand to enhance the Xray absorption contrast between the viscous and brittle domains. The influence of the ceramic microbeads and the thin corundum sand layer on the overall mechanical properties is negligible (Klinkmüller, 2011; Panien et al., 2006; T. Schmid et al., 2020a, 2020b) 2020a. All material properties are listed in Table 2.
Granular materials  Quartz sand  Corundum sand  Zirshot  Viscous material  PDMS/corundum mixture 

Density (kg/m^{3})  1,560*  1,960*  2,300  Density (kg/m^{3})  1,600 
Grainsize (μm)  60–250  88–175  150–210  Viscosity (Pa s)  1 × 10^{5} 
Peak Friction coefficient μ and angle  0.72–36°  0.78–38°  Stress exponent n  1.05  
Strain softening (%)  16  18  
Cohesion (Pa)  48 ± 26  55 ± 42 
 Note. Densities denoted with * are achieved by sieving from 30 cm height.
2.2 Scaling
We apply standard scaling equations from Hubbert (1937) and Ramberg (1981) to ensure proper scaling of our models with respect to nature. For brittle MohrCoulomb type materials and viscous materials, dynamic similarity is given by the stress ratio , and strainrate ratio , respectively.
By convention, σ^{∗} is defined as: σ^{∗} = σ_{m}/σ_{n}, where subscripts m and n indicate values for the model and nature, respectively. Further, ρ^{∗}, g^{∗}, h^{∗}, and η^{∗} are the density, gravity, length, and viscosity scale ratios, respectively. In our models, 1 cm represents 5 km in nature which yields a length scaling factor of h^{∗} = 2 × 10^{−6}. The stress and density scaling ratios are approximately σ^{∗} = 1.2 × 10^{−6} and ρ^{∗} = 0.6, respectively. The cohesion of quartz sand is about 60 Pa and upper crustal rocks are about 50 MPa (Byerlee, 1978), giving a cohesion scaling ratio c^{∗} = 1.2 × 10^{−6}.
Assuming a lower crustal viscosity of η = 5 × 10^{20} Pa s (e.g., Kruse et al., 1991) yields a viscosity scaling ratio and a strain rate scaling ratio . Further, the velocity and time scaling ratios are calculated from , yielding a velocity scaling ratio and time scaling ratio . Hence, 1 hr in our analog experiments translates to about 0.66 Ma in nature and the applied extension velocity (10 mm hr^{−1}) refers to a velocity of 8 mm a^{−1} and strainrate values of 8 × 10^{−15} s^{−1} in nature.
We ensure dynamic and kinematic similarity between nature and the experiments by calculating the Smoluchowski number S_{m}, and the Ramberg number R_{m} for the brittle and viscous layers, respectively. S_{m} is defined as the ratio between gravitational stress and cohesive strength (Ramberg, 1981) S_{m} = ρgh/(C + μρgh), where ρ, h, C, and μ are the density, thickness, cohesion and friction coefficient, respectively. Since upper crustal rocks have a cohesion values of 50 MPa and internal friction coefficients of ∼0.6 (Byerlee, 1978), this yields S_{m} ∼1 for both our experiments and nature. For our reference model setup with a brittleductile thickness ratio T_{BD} = 1, the Ramberg number R_{m} = ρgh^{2}/ηv yields values of 51 and 53 for our experiments and nature, respectively. For models with T_{BD} = 2, R_{m} values are 23 and 24 for the experiments and nature, respectively. For models with T_{BD} = 0.5, R_{m} ∼90 and ∼95 for the experiments and nature, respectively. The Reynolds number R_{e} = ρvh/η is defined as the ratio between inertial forces and viscous forces and is <<1 for all of our experiments and nature.
Based on the above scaling laws, the material properties and similar nondimensional numbers for the models and nature, we consider our analog experiments to be properly scaled. Model parameters and nondimensional numbers are given in Table 3.
General parameters  Brittle upper crust  Ductile lower crust  Dimensionless numbers  

Gravity (m/s^{2})  Crustal thickness (m)  Extension velocity (m/s)  Density (kg/m^{3})  Cohesion (Pa)  Density (kg/m^{3})  Viscosity (Pa s)  Smoluchowski S_{m}  Ramberg R_{m} ^{a}  Reynolds R_{e}  
Model  9.81  6 × 10^{−2}  2.8 × 10^{−6}  1,560  50  1,600  1 × 10^{5}  1  34  <<1 
Nature  9.81  3 × 10^{4}  1.4 × 10^{−10}  2,700  5 × 10^{7}  2,900  5 × 10^{20}  1  36  <<1 
Scaling ratios x* = x^{m}/x^{n} (dimensionless)  
σ*  ρ*  g*  h*  c*  η*  v*  t*  
1 × 10^{−6}  0.55  1  2 × 10^{−6}  1 × 10^{−6}  6 × 10^{9}  2 × 10^{−16}  1 × 10^{4}  2 × 10^{−10} 
 ^{a} Note that for calculating R_{m} the maximum velocity is used.
2.3 Contractional Boundary Condition and Induced Pressure Gradient
During opening of the extensional domain of the apparatus, shortening in the contractional domain induces a lateral pressure gradient, resulting in riftaxis parallel flow of the lower viscous layer from the contractional to the extensional model domain (Figure 3a). With subsequent deformation localization in the contractional domain, differential topography increases the lateral pressure gradient, enhancing riftaxis parallel flow (Figure 3b). Additionally, we simulate different additional gravitational loadings by using sand layers of 1 and 3 cm on top of the contractional model domain. To analyze the net effect of such additional gravitational load on the riftaxis parallel flow in the lower viscous layer, the flow component induced by contracting the sidewalls must be subtracted from the total flow. However, in our models the contribution of the contracting sidewalls is negligible.
2.4 Deformation Monitoring
We use an automated light and camera setup to monitor surface deformation by means of top view images for qualitative description as well as sets of stereo images for further 3D displacement analysis using 3D stereoscopic DIC (Adam et al., 2005). For the qualitative surface evolution description, we use a Nikon D200 (10.2 Mpx) DSLR camera mounted above the experiment apparatus. Two Nikon D810 (36 Mpx) DSLR cameras are aligned on a horizontal bar above the model surface with an angle of ca. 30° with respect to each other providing a setup for stereoscopic images and subsequent 3D DIC analysis.
For imaging the internal model deformation, we run models with a reference thickness ratio T_{BD} = 1, that is, a 3 cm thick sand layer overlying a 3 cm thick viscous layer, with initial flat topography (M10*), and models with additional quartz sand layers of 1 cm (M11*), and 3 cm (M13*) on top of the contractional domain in a medical XRCT scanner (64 slice Siemens Somatom Definition AS Xray CTscanner). The models are scanned with a 20min interval (corresponding to 3.3 mm divergence or 1% maximum extension). Additionally, we scanned an experiment with T_{BD} = 1 with no material (i.e., upper, and lower crust and additional sand layer) placed on the compressional part of the apparatus (M10E*). The resolution of the scans is 512 by 512 pixels for each slice with a size of 37 by 37 cm (∼0.72 by 0.72 mm/px). For every time step, models must be halted and scanned three times for quantitative DVC. Each individual scan has a duration of ca. 1 min with a time gap of ca. 30 s between the three scans. The CT scanner required such cooldown periods which were kept at a minimum by reducing the horizontal scan area (68 by 31 cm; Figure 2b) to avoid ongoing deformation in the timedependent viscous layer.
2.5 Data Analysis and Post Processing
Since E_{max} is invariant to the coordinate system, we use it to describe deformation in our experiments where rigid body rotation occurs around a fixed axis.
Unlike cutting wetted analog models at their final stage, CT scanning provides a nondestructive manner to gain insights into transient internal deformation during experiment runs. As in our surface strain analysis, we use commercial DaVis software (Version 10.2, LaVision) for quantifying internal 3D deformation captured in CT scan images. The DVC module of this software computes 3D displacement fields by cross correlating intensity patterns from subsequent volumetric (CT scanned) data sets collected over time (Adam et al., 2013; Poppe et al., 2019; Zwaan, Schreurs, & Adam, 2018). The data volume is divided into smaller voxel subvolumes to determine local displacement vectors by identifying similar intensity patterns in subsequent volume data sets. In contrast to 2D correlation, peaks are represented by a spherical blob in 3D and for each subvolume the best matching position is associated with displacement vector components dx, dy, and dz from the initial to the deformed volume. As each subvolume yields only one displacement vector, spatial vector resolution is increased by placing subvolumes next to each other at a distance d, smaller than the voxel edge length L to create a subvolume overlap. Making use of a multipass cross correlation, DaVis first applies a computationally efficient Fast Fourier Transform (FFT) algorithm for coarse subvolumes to predict displacement fields for subsequent iterations which are done using a direct correlation (DC) algorithm.
Using the smallest XRCT slice thickness of 0.6 mm/px the volumetric data consists of voxels with dimensions of 0.72 by 0.72 by 0.6 mm/px. Since DVC requires cubic voxels with the same edge length along all three axes, the images must be reprocessed before conducting quantitative analysis. In our anisotropic voxels the y axis (slice thickness of 0.6 mm/px) provides the best resolution in the volumetric data set, and we adjust pixel sizes (0.72 mm by 0.72 mm) in the xzplane to obtain isotropic 0.6 mm sided voxels. We apply a nearest neighbor resampling algorithm in ImageJ to increase the pixel number in the xzplane (Figure 4) without creating artificial pixel intensity values due to interpolation. As the models are scanned multiple times per time step, we further stack the voxel data to increase intensity values. In addition, this step reduces noise created by the high Xray reflectivity Zirshot ceramic beads and improves the image quality. For DVC analyses, we use subvolume sizes of 128 pixels with an overlap of 50% for the first step (FFT). Further iterations (DC) subsequently decrease subvolume sizes to 64, 32 and 16 pixels with overlaps of 50% and 75% for the final step. Where subpixel displacements are not captured by the discrete correlation peak, fitting of a Gaussian curve restores accuracy. The final subvolume displacement vectors are assembled to construct incremental (here dt = 20 min) 3D displacement fields. Incremental displacements are in the order of 1–2 mm for all three displacement components with an uncertainty of 0.03 and 0.06 mm in planes parallel and normal to the CT scan slices, respectively.
Processing of 3D DIC data follows the protocol of T. C. Schmid, Schreurs, and Adam (2022). Like DIC data, we process data from DVC analyses in MATLAB (Version 18b) and Python (Version 3.7) to quantify deformation. For volumetric visualization we use Tecplot360 (Version 2018) software. A more detailed overview of DIC and DVC parameters can be found in T. Schmid et al. (2021) and T. C. Schmid, Rudolf, et al. (2022), respectively.
3 Results
The complete model series consists of 10 experiments (see Table 1). Before quantifying surface deformation, we describe the firstorder evolution based on top view images and CT scans, to provide qualitative insights into the deformation evolution. Firstorder deformation features such as rift propagation show strong similarities in all models. Hence, we use a model with the reference brittleductile thickness ratio T_{BD} = 1 and an additional sand layer of 3 cm on top of the contractional domain, to describe the overall deformation evolution. Note, that an identical setup with an initially flat topography has been previously described in Zwaan et al. (2020) and T. C. Schmid, Schreurs, and Adam (2022).
3.1 General Rift Evolution for Models With Gravitational Load
Differences in strain evolution among models with or without additional gravitational load, and with varying brittleductile ratios are inconspicuous and hence, difficult to assess. This highlights the necessity to further analyze deformation quantitatively. However, we briefly describe characteristic stages of rift evolution under the influence of an additional gravitational load. Figure 5 shows the rift evolution in model M13*. After 3.3 mm of maximum extension (i.e., 20 min), a set of conjugate dipslip faults forms, expressed by short segments at the surface 400–600 mm away from the rotation axis (Figures 5a, 5d and 5g). While this is difficult to distinguish in top views (Figure 5a), XRCT scans of the corresponding transect (Figure 5d) clearly depict a set of conjugate normal faults with a dip angle of about 70° (Figure 5g). With increasing bulk extension, these conjugate normal faults propagate toward the rotation axis, forming a continuous rift structure delimited by rift boundary faults. After 23.1 mm of maximum extension (i.e., 140 min; Figures 5b, 5e and 5h), the rift boundary faults have gently rotated about a horizontal axis to a dip angle of about 64° (Figure 5h). Simultaneously, a new set of conjugate normal faults forms in between the rift boundary faults (intrarift faults). Like the rift boundary faults, the intrarift faults initially form as isolated segments within the rift (Figures 5b and 5e). At the final stage (i.e., 40 mm maximum extension and 240 min; Figures 5c, 5f, 5i and 5j), the rift boundary faults have propagated as far as the rotation axis, where bulk extension is zero (Figures 5c and 5f). Formerly isolated intrarift fault segments have partially connected and consist of longer fault segments propagating toward the rotation axis.
The intrarift faults show a less advanced stage of propagation when compared to models with an identical crustal configuration but flat topography (e.g., M10*; T. C. Schmid, Schreurs, & Adam, 2022). Figure 6 compares rift morphologies of models M10* and M13* at the final stage. The rift morphology in model M10* shows three distinct pairs of conjugate normal faults with successively younger faults toward the rift center (Figures 6a, 6b and 6d). Each fault generation bounds a graben structure resulting in increasing subsidence toward the rift center. In contrast, rifting in model M13* shows a less advanced stage with only two generations of conjugate normal faults present (Figures 6e, 6f and 6h). Note that thrusting in the contractional domain is entirely suppressed in model M13*. XRCT scan imagery from the corresponding cross section reveals that the gravitational load entirely inhibits thrust development (Figures 6c and 6g).
3.1.1 Propagation of Rift Boundary Faults
We investigate lateral rift propagation by tracking the position of the rift boundary fault tips with respect to the rotation axis over time. For this, we set a threshold of 10% of cumulative surface strain identical to T. C. Schmid, Schreurs, and Adam (2022). Figure 7 shows the mean rift tip position (i.e., the mean position of the two conjugate rift boundary faults) for all 10 experiments. For all models, rift propagation follows a distinct trend with rapid rift lengthening in an early stage and slower propagation in a second stage. All models show onset of rift propagation at around 1.3% of maximum extension. However, rift onset in models with an initially flat topography tends to occur earlier compared to models with gravitational loading. For models with a brittleductile thickness ratio T_{BD} = 2, the mean rift propagation path is nearly identical but with a decreasing brittleductile ratio, the onset of rift propagation is successively retarded for models with a gravitational load.
3.1.2 Inward Migration of Fault Activity
The delayed rift propagation in experiments with gravitational loading is clearly visible in surface strainrate maps obtained from DIC analysis (Figure 8) which we use as a proxy for fault activity. T. C. Schmid, Schreurs, and Adam (2022) document three generations of conjugate normal faults for a brittleductile thickness ratio T_{BD} = 1. For an initial flat topography (M10*; Figures 8a–8c), fault activity progressively abandons rift boundary faults and migrates inwards onto intrarift faults. After about 23 mm of maximum extension, strainrate values decrease along boundary fault segments that are further away from the rotation axis. Simultaneously, fault activity migrates inward onto conjugate intrarift normal faults. At the final stage (i.e., after 40 mm of maximum extension), three different fault generations are active in different rift segments with increasing bulk strain along the rift axis. In model M13* (Figures 8d–8f) with an additional gravitational load, early stages (e.g., at 20 min; Figures 8a and 8d) show similar strainrate patterns along rift boundary faults compared to model M10*. However, the onset of inward migration is delayed for model M13*. After about 23 mm of maximum extension, inward migration initializes fault activity along intrarift faults (Figure 8e) whereas in model M10* intrarift faults consist of longer segments in the distal model part (Figure 8b). In contrast to model M10* only two generations of conjugate fault pairs are active at the final stage (Figure 8f) indicating a less advanced rift maturity for model M13*. Presumably, the rift structure in model M13* will become as mature as that in model M10* at a later time.
3.2 Vertical Surface Motions
The cumulative vertical displacement, Dz (Figure 9), is used to compare the surface topography in the final stage of all experiments with an additional gravitational load (Mx3 and Mx1) with the corresponding experiments with an initially flat topography (Mx0). Note that these motions only indicate net uplift and subsidence and do not necessarily represent the absolute topography. The cumulative horizontal displacement vector field is also indicated in Figure 9 by gray arrows (not to scale) as a visual aid. Regardless of the brittleductile thickness ratio, T_{BD}, the final topography of models with an initially flat topography (Figures 9a–9e) are nearly identical, with maximum subsidence values in the rift structure occurring at distances far away from the rotation axis. For T_{BD} = 2, 1, and 0.5, maximum subsidence values are about −20, −15, and −10 mm, respectively. Maximum uplift values occur in the contractional domain (except model M10E*; Figure 9d) and are in the order of about 5 mm for all models with an initially flat topography. The zeroelevation line (black line) separates zones of uplift and subsidence which, in the case of initially flat topography models, coincides with domains separating contraction from extension. Note that the extensional domain shows patterns of diffuse subsidence outside the rift. This regional subsidence is due to the homogeneous thinning of the lower viscous layer, which is placed on top of rigid basal setup that does not allow isostatic compensation. The maximum rift and popup structure width depends on the absolute thickness of the brittle layer. Maximum subsidence values decrease with lower T_{BD} ratio.
In models with an additional gravitational load (Figures 9f–9j), parts of the extensional domain, that are located farther away from the rotation axis show identical subsidence patterns when compared to models with identical T_{BD} but an initially flat topography. However, experiments with an additional gravitational load show enhanced vertical motions in model parts closer to the rotation axis in both the contractional and extensional domain. For models with T_{BD} = 2, the zone of uplift in model M23 (Figure 9f) shrinks (in contrast to model M20; Figure 9a), as the zeroelevation contour migrates into the compressional domain. Absolute maximum uplift values are of the same order of magnitude, however, the rift tip in the extensional domain is flanked by zones of minor net uplift in model M23 (Figure 9f) compared to model M20 (Figure 9a). Models with T_{DB} = 1 and an additional gravitational load of 3 cm (M13 and M13*; Figures 9g and 9h) show a clear distinction between a zone of maximum subsidence of −2 mm in the contractional domain (i.e., location of the additional gravitational load) and a bulgelike uplift zone with maximum values of 2 mm at the transition from contraction to extension. The rift is not influenced by this uplift as it propagates toward the rotation axis and through the bulge, resulting in uplifted flanks on either side of the propagating rift. In model M11 (Figure 9i) with T_{BD} = 1, the additional gravitational load is reduced to a sand layer with a thickness of 1 cm. Identical to models M13 and M13*, a zone of subsidence develops in the contractional domain. However, maximum subsidence values do not exceed 1 mm and do not overprint net uplift in the contractional domain farther away from the rotation axis where thrusting persists. The adjacent uplifted bulge in the extensional domain is only minor with values <1 mm. The effect of the gravitational load is most pronounced in model M0.53 with T_{BD} = 0.5. Here, subsidence values within the contractional domain are largest and reach −7 mm. The surface of the contractional domain subsides homogeneously and has a sharp boundary to the adjacent uplift bulge, where maximum uplift values are around 5 mm. The uplift bulge continues far into the extensional domain and eventually uplift is large enough to suppress net subsidence in the rift near the rotation axis (Figure 9j). Note that the horizontal vector field in models with an additional gravitational load deviates increasingly from the reference state (i.e., models with initial flat topography) as T_{BD} decreases.
3.3 Internal Deformation Analysis
We now focus on quantification of the internal deformation in XRCT scanned models with an intermediate brittleductile thickness ratio T_{BD} = 1 (M10E*, M10*, M11*, and M13*). Specifically, we analyze the flow field in the viscous domain by means of cumulative displacement vector fields and focus on the horizontal component of riftparallel flow (i.e., along the model yaxis). Figure 4 gives an overview of the data slices extracted from DVC analyses of all four XRCT scanned models. For each model, we extract individual 3D displacement components on five xslices (i.e., yzplanes; x_{1} − x_{5}) and five yslices (i.e., xzplanes; y_{1} − y_{5}). Additionally, we extract displacement vectors from horizontal zslices in both the brittle and viscous domains (i.e., xyplanes; z_{1} and z_{2}). For all slices, the displacement vectors represent the total 3D displacement, projected into the plane of the corresponding CT slice. The entire slice compilation is available in an additional data publication (T. C. Schmid, Rudolf, et al., 2022). Here, we present a representative selection of displacement data to document quantitative 3D deformation features in our XRCTscanned models.
3.3.1 Displacement Components in Vertical Rift Perpendicular Transects y_{1} and y_{5}
Apart from differences in the absolute values of displacements, similar observations are made in the yslices of all XRCT scanned models. Therefore, we only show the y_{1} and y_{5} slices of model M10* in Figure 10, which plots the horizontal displacement components Dx (a and d) and Dy (b and e), and the vertical displacement Dz (c and f) at the final stage after 4 hr of the model run (i.e., 40 mm max. extension). In addition, white arrows indicate the 3D displacement projected into the plane. The left column shows displacement components in position y_{1}, in the contractional domain, whereas the right column shows displacement components in position y_{5} at the far end of the extensional domain. Values of the cumulative horizontal displacement components Dx, Dy, and Dz range from −15 to 15, −5 to 5, and −8 to 8 mm, respectively.
The magnitudes of the maximum horizontal displacement component Dx are smaller than expected based on the maximum divergence velocity (20 mm/hr on each side). This results partly from the fact that the applied maximum divergence velocity is defined along a circular segment. However, this deviation is on the order of 1% and therefore negligible. More importantly, the DVC analysis does not capture the entire model length (see Figure 2b) and is also limited by a fixed data mask width along the y axis and can therefore not account for the increasing differences in model width along the y axis over time. Therefore, the DVC analysis cannot capture maximum displacements near the mobile sidewalls (i.e., model widths, −150 and 150 mm) which results in smaller maximum values of the horizontal displacement component Dx.
In the contractional domain (Figures 10a–10c), Dx values range between −5 and 5 mm (Figure 10a). The left part (i.e., from x = −150 to 0 mm) moves to the right (indicated by light red colors) while the right part (i.e., from x = 0–150 mm) moves to the left (indicated by light blue colors). This results in a sharp displacement gradient in the brittle domain representing a thrust fault (Figure 10a). The horizontal out of plane displacement Dy is distinctly different between the brittle and viscous domains. Dy values in the viscous domain range between −5 and 5 mm with positive values indicating flow from the contractional to the extensional domain and vice versa (Figure 10b). In the contractional domain, the Dy displacement component is characterized by a wide zone (ca. −100 to 100 mm) of enhanced displacement with values ≤5 mm (Figure 10b). In the brittle domain, Dy values are mostly negative indicating a cumulative displacement away from the extensional domain toward the contractional part. Only a central area confined by thrust faults shows positive values (Figure 10b). Vertical displacement values Dz are small and generally positive in the contractional domain (Figure 10c), where thrusting in the brittle domain leads to a minor uplift of about 5 mm near the model center (from −50 to 50 mm).
Generally, the horizontal displacement components Dx and Dy are more pronounced in the extensional domain (Figures 10d–10f). The Dx component shows diffuse stretching in the viscous layer, whereas deformation in the brittle layer localizes above the viscous seed, resulting in a graben system (Figure 10d). Note that the displacement polarity changes at the brittleductile interface which results in pockets of reversed viscous flow below the rift axis (from ca −50 to 50 mm). The change in the horizontal displacement direction at the brittleductile interface is coupled with the vertical displacement Dz (Figure 10f). In the viscous layer below the rift zone, the viscous seed material rises, while the surrounding viscous and brittle material subsides due to gravitational unloading in the rift center. The combination of horizontal Dx, and vertical Dz displacement causes a convectionlike cell indicated by the displacement vectors. The horizontal displacement component Dy (Figure 10e) has maximum values in the viscous layer below the rift. In contrast to the contractional domain, the zone of pronounced displacement is restricted to the model center.
3.3.2 RiftAxis Parallel Horizontal Displacement Dy in the x_{3} Longitudinal Transect
Longitudinal transects of all XRCT scanned models show that the riftaxis parallel horizontal displacement (Dy) is focused in the viscous domain (Figure 11). D_{y} displacement values range between −5 and 5 mm for models M10E* (Figure 11a) and M10* (Figure 11b), and between −15 to 15 and −30 to 30 mm for models M11* (Figure 11c) and M13* (Figure 11d), respectively.
For model M10E*, maximum values of Dy occur in the viscous layer at ∼100 mm from the rotation axis and continue evenly to the extensional domain at the far end at ∼550 mm (Figure 11a). Near the rotation axis (0–100 mm), riftaxis parallel displacement values are negative. Similarly, riftaxis parallel displacement in model M10* (Figure 11b) indicates a continuous channel flow with Dy values around 3 mm in the center of the viscous layer. However, two distinct zones of maximum values with 5 mm displacement occur in the contractional domain (at −100 mm distance to the rotation axis) and at the far end of the extensional domain (500 mm). In experiments with an additional gravitational load, a continuous channel of riftaxis parallel displacement in the viscous domain is subsequently localized below the topographic step (Figures 11c and 11d). In Model M11*, maximum displacement values of 15 mm are located below the topographic step at the rotation axis and gradually decrease to 0 at ∼200 mm distance from the rotation axis (Figure 11c). This is also valid for model M13* (Figure 11d). However, in this experiment maximum values of riftaxis parallel displacement double and reach up to 30 mm.
3.3.3 Vertical Displacement Dz in the x_{3} Longitudinal Transect
The increased riftaxis parallel displacement in models with an additional gravitational load also affects the vertical displacement component Dz (Figures 12c and 12d). In the brittle layer, negative values as large as −8 mm reflect subsidence within the rift. For models without an additional gravitational load (M10E*, M10*; Figures 12a and 12b), subsidence of the brittle layer occurs across the entire extensional domain and reaches maximum values near the far end at ∼500 mm from the rotation axis. In the viscous layer, maximum values of Dz = 8 mm occur near the far end at ∼500 mm and indicate upward flow simultaneously with the thinning of the brittle layer (Figures 12a and 12b; see also Figure 10f). In model M10* (Figure 12b), subsidence in the brittle layer decreases toward the rotation axis and changes to uplift in the contractional domain. With an additional gravitational load of 1 cm (M11*; Figure 12c), maximum uplift values in the contractional brittle domain are smaller (compared to models M10E* and M10*) and eventually are negative in model M13*, where subsidence occurs (Figure 12d). While the viscous layer in model M11* still shows upward flow near the extensional far end, maximum Dz values are ∼0 mm in model M13*, where upward flow of the viscous layer is suppressed. However, model M13* shows increased vertical displacement in the viscous layer close to the topographic step at ∼50 mm from the rotation axis, which also affects vertical motions of the brittle layer (Figure 12d; see also Figure 9h).
4 Discussion
In this study, we used DVC in combination with 3D DIC to depict and understand the coupling of nearsurface rift development and deepseated crustal flow in a rotational rift setting. To simulate different initial crustal configurations, we used a series of brittleductile thickness ratios T_{BD}, where increasing values indicate increasing crustal strengths. Our results show that DVC applied to XRCT images provides a unique opportunity to gain insights on the timedependent internal deformation of analog models. This method allows for a fully quantitative description of both the surface deformation and the deformation of internal layers representing lower parts of the crust.
A previous study by Zwaan, Schreurs, and Adam (2018) reports DVC analyses performed on crustal scale analog rift models undergoing orthogonal extension. They showed that the interaction between two segmented rift branches caused substantial horizontal displacements in the viscous layer (lower crust analog), parallel to the direction of rift propagation. This shows that slight deviations (i.e., an offset between two parallel rift branches) from a cylindrical setup can result in considerable out of plane displacements. In the case of a rotational model setup, riftaxis parallel displacement components are even more pronounced. Zwaan et al. (2020) used the same experimental apparatus as this study and gave preliminary insights on riftaxis parallel displacement in a rotational system. Although they applied 2D DIC on a single longitudinal XRCT transect (Figure 10b in Zwaan et al., 2020), riftaxis parallel displacement patterns in our model M10* agree well with their findings. We also observe a similar enhanced horizontal Dy displacement in our model M10E* (see Figures 11a , 11b and 14b) from which we conclude that the contracting boundary sidewalls in our experimental apparatus do not influence displacement patterns in the extensional domain. Riftaxis parallel flow has also been documented in lithospheric scale numerical models (e.g., Le Pourhiet et al., 2018; Van Wijk & Blackman, 2005), highlighting the importance of out of plane displacement components even in models with reasonably cylindrical boundary conditions.
4.1 The Influence of RiftAxis Parallel Horizontal Flow on Rift Propagation
The presence of a horizontal pressure gradient (resulting from a topographic load at the contractional end of the model domain) influences the evolution of the rift structure. Figures 7 and 8 illustrate the delay in rift evolution in terms of rift tip propagation, the timing of the abandonment of rift boundary faults and successive initiation of faulting activity along intrarift faults (inward migration). To understand the cause of such delayed evolution, we consider deformation in the viscous layer. Figures 11 and 12 show the riftaxis parallel horizontal displacement Dy, and the vertical displacement component Dz, respectively. The subsiding brittle layer in the extensional domain is coupled with an upward flow of the viscous material underneath the rift axis (Figure 12). For increasing additional gravitational loads, the Dz upward flow component appears to be perturbed and overprinted by a strong riftaxis parallel horizontal flow component Dy (Figure 12d). T. C. Schmid, Schreurs, and Adam (2022) propose that intrarift faults initiate early as antithetic faults but remain largely passive while slip is accommodated along the rift boundary faults. It is only after they reach the brittleductile interface, where shear stresses are high (Corti et al., 2010), that intrarift faults accommodate considerable displacement and concomitant fault activity along the rift boundary faults decreases. Hence, the upward flow of the viscous material increases and facilitates further activation of intrarift faults. Zwaan, Schreurs, and Adam (2018) showed the inhibiting effect of synrift sedimentation on the ascent of viscous material below the rift structure. Although we do not consider the effect of sedimentation, the presence of a large additional gravitational load induces a dominant riftaxis parallel horizontal displacement component Dy, which is strong enough to overprint or inhibit vertical upward flow of the viscous material. This effect delays rift evolution, as expressed by retardation of rift tip propagation, inward fault migration, and activation of intrarift faults.
4.2 Effect of DeepSeated Deformation on Surface Topography and Rift Evolution
This enhanced lower crustal channel flow is ultimately expressed by increased vertical motions in models with an intermediate to low T_{BD} (Figures 9 and 13). Regardless of the brittleductile thickness ratio, local contraction results in a zone of uplift in models with flat initial topography (Figures 9a–9e and 13a). However, in models with an additional gravitational load (Figures 9f–9j and 13–13e), the intensity of vertical motion at the surface increases in experiments with thick viscous layers (i.e., T_{BD} = 0.5). While models with thin viscous layers (i.e., T_{BD} = 2) still yield net uplift in the contractional domain (Figures 9f and 13b), models with viscous layers of intermediate thickness (i.e., T_{BD} = 1) show a net subsidence (Figures 9g–9j and 13c–13e). In these models the channel thickness is sufficient to maintain a riftparallel, horizontal channel flow component which transports material away from the contractional domain, counteracting the tectonic uplift caused by contraction (Figures 9g–9j and 13c–13e). This results in a net subsidence in the contractional domain and, where the horizontal flow component decreases, the excess material in the lower crust enables synchronous uplift during extension while rifting occurs. In the extreme case (i.e., thick viscous layer; T_{BD} = 0.5), uplift raises the propagating rift tip above its original elevation, resulting in net uplift of the rift system when crossing the uplift barrier (Figures 9j and 13e).
4.3 Conceptual Model for RiftAxis Parallel Flow
The insights from the experiments on how lower crustal flow interacts with upper crustal deformation allow us to develop a flow model for our rotational extension experiments. In our experiments, riftaxis parallel horizontal displacement velocities calculated from DVC analyses are systematically higher compared to theoretical values obtained from Equation 1. At the central longitudinal transect (X_{3}) of model M13* (i.e., 1 cm additional gravitational load) for example, maximum Dy values at the final stage occur close to the topographic step and are ∼30 mm (Figure 14a). However, theoretical values for Poiseuille flow (Equation 1) yield a maximum Uy velocity of ∼3.75 mm hr^{−1} resulting in maximum Dy values of ∼15 mm. For model M11* (i.e., 1 cm additional gravitational load) measured maximum Dy values are ∼15 mm, whereas Equation 1 yields a maximum Uy velocity of ∼1.25 mm hr^{−1} that translates to a maximum Dy value of ∼5 mm at the final stage. For models without an additional gravitational load, measured maximum Dy values are ∼5 mm and Equation 1 predicts 0 mm. Interestingly, the differences between measured and theoretical values are systematically larger with increasing additional gravitational loads. This discrepancy can be attributed to two causes: First, the rotational opening of the experimental apparatus provokes riftaxis parallel flow away from the rotation axis even without an additional gravitational load (Figures 10, 11 and 14). Second, the strong riftaxis parallel displacement component measured at the model surface (gray arrows Figures 9g–9j) indicates that the contact between the brittle and viscous layers is not a fixed boundary (with respect to the viscous layer) and therefore allows higher flow velocities in the viscous domain than expected in the case of pure Poiseuille flow.
Figure 14b shows Dy profiles through the brittle and viscous layers at distinct positions along the rift axis. The effect of the pressure gradient (i.e., parabolic shape of the displacement profile) is evident in the viscous layer near the topographic step and decreases farther away from the rotation axis in the extensional domain. The brittle layer shows uniform Dy values through the entire layer thickness indicating that the brittle layer moves as a rigid block. It is only in a thin layer at the brittleductile interface, where a displacement gradient in the granular material occurs. This transition is likely due to energy dissipation and is expressed by enhanced shear strain, which reduces displacement values in the brittle layer compared to the viscous layer (Figure 14c). The displacement gradient dDy/dz reveals anticlockwise rotation at the brittleductile transition indicating a toptoleft shear sense (red arrows; Figure 14c).
4.4 Coupling Between Brittle and Viscous Layers During RiftAxis Parallel Displacement
The horizontal displacement field at the surface of models with an additional gravitational load is systematically different to those of all models with an initial flat topography (Figure 9). Such modified horizontal displacements indicate a riftaxis parallel overprint, similar to those observed in longitudinal DVC transects (Figures 11 and 14). However, as discussed above, the riftaxisparallel displacement component at the model surface is reduced compared to the displacement component in the viscous part of the model suggesting a degree of brittleviscous decoupling between the two model layers (sensu Zwaan et al., 2019).
To further investigate the mechanical coupling of brittle and ductile deformation during rifting, we use the quantitative results for models with T_{BD} = 1 and compare the evolution of the riftaxis parallel mean values in the brittle (Figure 15a) and viscous domain (Figure 15b). We restrict the investigation to a 10 cm wide central area (i.e., close to the X_{3} longitudinal transect; Figures 11 and 14) in the extensional domain to exclude displacement effects due to rotational opening.
The riftaxis parallel displacement in the brittle domain (Figure 15a) shows positive and negative trends for models with (i.e., M11* and M13*) and without (i.e., M10* and M10E*) an additional gravitational load, respectively. However, for model M10*, this negative trend is negligible, within the error. Note that positive and negative values indicate mean displacements toward and away from the location of maximum opening, respectively. In the viscous domain, mean values all show positive trends indicating riftaxis parallel flow toward the location of maximum opening. Plotting the evolution of the ratio between mean values from the brittle and viscous domain (Figure 15c) reveals the degree of brittleviscous coupling between the two model layers with respect to riftaxis parallel displacements. If the mean Dy values of the brittle and viscous layer in one experiment are totally coupled, this ratio should be 1. Any ratio below 1 indicates an increasing contrast between the riftaxis parallel displacement in the brittle and viscous layer. Positive ratios <1 indicate displacement directions in the viscous domain are larger than those in the brittle domain. Any ratio <0 indicates opposing displacement directions in the brittle and viscous layers. Note that absolute values of mean displacements in the viscous domain are always larger than in the brittle domain and hence, all ratios plot between −1 and 1.
In the case of an additional gravitational load (i.e., M11* and M13*), both displacement directions are away from the rotation axis toward the extensional domain. The proportional increase in both brittle and viscous Dy values is expressed by a constant ratio of ∼0.2. This suggests that for both experiments, riftaxis parallel displacements in the brittle domain are reduced by a factor of 5 in contrast to the viscous flow below the rift. Interestingly, this ratio only increases by a small amount for a larger gravitational load. The increased flow in the viscous domain seemingly affects the riftaxis parallel displacement in the brittle domain equally, regardless of the gravitational load.
For models without an additional gravitational load, negative ratios indicate opposing displacement directions in the brittle and viscous layers. However, the evolution documents an early phase (up to ca 100 min) with continuously increasing displacement contrasts suggesting a first phase of increased decoupling (Figure 15c). After this early phase, ratios increase for both models as Dy mean values in the viscous domain increase faster than Dy mean values in the brittle layer decrease (i.e., become more negative). We speculate that in this later phase the intensified viscous flow (away from the rotation axis) is due to the increased differential topography along the rift axis which counteracts the opposing Dy displacement in the brittle domain. Eventually, the viscous flow likely diminishes the negative Dy mean values in the brittle layer such that it flips the direction and hence, increases the brittleviscous coupling.
Rift tip propagation governed by growth of the rift boundary faults occurs during the early stages of model runs (within 20%–30% of the model runtime; Figure 7; see also Jackson et al., 2017; Rotevatn et al., 2019; T. C. Schmid, Schreurs, & Adam, 2022). In our experiments M10E* and M10* the corresponding rift tip propagation does not deviate substantially from those of other experiments (Figure 7). It therefore appears that the increase in coupling between Dy mean values in the brittle and viscous layers (after ca 120 min) occurs too late during the model runs to significantly influence lateral rift propagation. Nevertheless, the evolution of the coupling between Dy mean values in the brittle and viscous layers in models M10E* and M10* highlights that it is not a static but rather dynamic process which may change during progressive rotational extension.
4.5 The Role of RiftAxis Parallel Crustal Flow in Natural Settings
Our XRCT scanned models show a substantial amount of riftaxis parallel displacement in the viscous domain (Figures 10, 11, 14 and 15). Also in the absence of a gravitational load, we observe riftparallel, central flow channels with maximum Dy values up to 5 mm (Figure 11) and mean values of ∼1 mm (Figure 15) during 40 mm of rift extension. Based on our scaling, this translates to maximum and mean flow velocities of 2 and 0.4 mm/a in nature, respectively. Such values may seem minor when compared to the orthogonally oriented divergence velocity (i.e., 8.8 mm/a) but are they negligible? Various studies have documented, an outofplane flow component in the lower crust and its implications on rift evolution (e.g., Amato et al., 2004; Gautier et al., 2008; Little et al., 2011; MacCready et al., 1997; P. D. Clift, 2015).
Based on gravitational potential and buoyancy considerations, Clift (2015) proposes riftparallel lower crustal flow from the continental crust toward the oceanic rift propagator in the Woodlark Basin as well as in the South China Sea, similar to observations in our analog models. In addition, Little et al. (2007, 2011, 2013) conclude for the Woodlark basin that, based on structural data from the D'Entrecasteaux Islands, regional, lower crustal flow parallel to the riftaxis must have been largely decoupled from the general NNWSSE extension direction between the Woodlark microplate and the Australian plate (Little et al., 2011). The exhumed gneiss domes on the D'Entrecasteaux Islands show depthdependent trends in stretching lineation orientation, indicating riftaxis parallel lower crustal flow toward the Papuan Peninsula in contrast to a dominant riftaxis normal stretching lineation in upper crustal levels (Little et al., 2011). The proposed flow direction (based on shear fabrics) for riftparallel flow is opposite of what we observe in our rotational models and may be explained by asthenospheric inflow into the thinned crust (Abers et al., 2002; Little et al., 2011; Mondy et al., 2018), which we do not consider in our models. The rotational experiment by Mondy et al. (2018) documents a prominent riftaxis parallel flow component in the asthenosphere following the tip of the propagating rift toward the rotation axis. Simultaneously, vertical upward flow compensates for the thinning crust and impedes riftaxis parallel flow in the lower crust away from the rotation axis (Mondy et al., 2018). Interestingly, in early stages of the experiment presented in Mondy et al. (2018), the riftaxis parallel flow component of the asthenosphere is directed toward the extensional far end. However, the authors do not state its effect on lower crustal flow. In our models, we use a rigid basal setup and therefore cannot account for the effect of the asthenosphere on the general model evolution. In this regard, one could argue that our models represent an early stage of continental rifting, where the role of the asthenospheric flow is negligible.
Despite these two contrasting suggestions from Clift (2015) and Little et al. (2011) for the riftaxis parallel lower crustal flow direction, field data from gneiss domes in core complexes on the D'Entrecasteaux Islands is key for uncovering coeval but orthogonal extension directions at varying crustal levels (i.e., riftaxis parallel lower or mid crustal flow and riftaxis perpendicular stretching in the upper crust). The documentation of two orthogonally oriented stretching lineations in such gneiss domes also led other authors to propose the existence of riftaxis parallel lower crustal flow, orthogonal to the regional extension direction (e.g., Amato et al., 2004; Gautier et al., 2008; MacCready et al., 1997). MacCready et al. (1997) document riftaxis parallel northward flow of the middle crust in the Ruby Mountains, Nevada as a response to a riftaxis parallel extension gradient with increasing upper crustal stretching values toward the north. Similarly, Gautier et al. (2008) document in the Nigde Massif, Turkey, riftaxis parallel, lower crustal flow (perpendicular to the extension direction) and determine a northerly flow because of an extension gradient with increasing values toward the northern end of the basin. In both cases, the driving mechanism of regional flow seems to be an extension gradient in which riftaxis parallel lower crustal flow compensates greater thinning of the upper crust. Both settings have kinematic boundary conditions that cause riftaxis parallel lower crustal flow toward areas with higher extension, identical to what we document in our models. Gautier et al. (2008) describe such riftaxis parallel flow as regionalscale lateral channel flow, in contrast to a more localscale lateral inward flow toward the rift axis, which compensates thinning of the upper crust parallel to the extension direction (Block & Royden, 1990). Similarly, we observe riftperpendicular (Figure 10d) and riftaxis parallel (Figures 10e and 11) horizontal flow components in addition to vertical upward flow (Figures 10f and 12) in our models. Eventually, the combination of all displacement components (as documented in nature as well as in our models) give rise to complex 3D flow patterns which can transport a considerable amount of material in or out of a 2D plane. This effect clearly must be considered when estimating crustal extension from 2D riftperpendicular cross sections (P. D. Clift, 2015).
5 Conclusion

When compared to models with an initial flat topography, gravitational loading influences the timing of inward migration of faults and retards activation of intrarift faults. Hence, models with an additional gravitational load show less advanced rift maturity after an identical run time when compared to flat topography models. However, this does not affect lateral rift propagation, regardless of the brittleductile thickness ratio T_{BD}.

Models with initial flat topography develop rift structures with subsiding basins in the extensional domain and uplifted areas confined by thrusts in the contractional domain. These patterns are identical for all brittleductile thickness ratios. Additional gravitational loading, however, enhances vertical displacements and changes patterns of subsidence and uplift. Subsidence due to gravitational loading drives horizontal flow in the viscous model layer, which results in uplifted areas in the extensional model domain.

For thicker lower viscous layers, flow channels are larger and hence provide more space for enhanced horizontal ductile channel flow. If an additional gravitational load is applied, this creates more pronounced uplift in the extensional domain.

Enhanced riftaxis parallel flow in the viscous model layer overprints vertical upward flow near the rift axis. Upward flow originally helps to activate intrarift faults and drives inward fault migration. Hence, viscous riftaxis parallel delays inward migration of activity along faults and enables boundary faults to be active for a longer time. This results in a comparatively less mature rift stage at a given time.

Riftaxis parallel flow in the viscous model domain is driven by a pressure gradient due to differential topography (Poiseuille flow). Viscous flow drags the brittle upper layer as a rigid block. Shearing occurs along the brittleductile interface where energy dissipation results in reduced Dy displacement values compared to Dy displacement values in the viscous layer.

DVC depicts enhanced horizontal riftaxis parallel flow patterns in the viscous domain for models with additional gravitational loads. Correlation between growing mean values of riftaxisparallel displacement components in the viscous and brittle layer, shows different degrees of coupling between these two domains. For models with an additional gravitational load, stable mechanical coupling (of some degree) between the brittle and ductile layer is expressed by identical displacement directions toward the extensional domain. In models without additional gravitational loads, mechanical coupling is transient and changes during the model run, showing that this is a dynamic process, rather than a static one.

Riftaxis parallel ductile flow, documented in rift settings with an extension gradient, results in material transport out of the rift transect. When scaled to nature, displacements from our models confirm substantial outflow of lower crustal material parallel to the rift axis. This mechanism may explain discrepancies between subsidence estimates based on upper crustal extension and subsidence in profiles and emphasizes the three dimensionality of rifting.
Acknowledgments
The authors thank Sandy Cruden and Anouk Beniest for thorough reviews of the original manuscript. The DIC analyses are part of an EPOS TCS MSL TransNational Access project and were performed at the HELTEC lab at the GFZ Potsdam. The authors thank Matthias Rosenau and Michael Rudolf for providing access to and assistance in operating DaVis 8.4 and thank Nicole Schwendener for operating and assisting us with the XRCT machine. The authors are grateful for helpful discussions with Frank Zwaan and Peter Clift. This project is supported by the Swiss National Science Foundation (Grant No. 200021_178731). The authors thank the Berne University Science Foundation for financial support of the rotational extension machine (Grant No. 201519). Open access funding provided by Universitat Bern.
Open Research
Data Availability Statement
Rheological measurements of the brittle and viscous materials used in this study are available in the form of open access data publications via GFZ Data Services (T. Schmid et al., 2020a, 2020b; Zwaan, Schreurs, Ritter, et al., 2018, respectively). An additional open access data publication on the GFZ Data Service (T. C. Schmid, Rudolf, et al., 2022) provides additional images and movies of our models. Links to these data sets are provided in the reference list.