Viscous Rheology Research Articles

Penetration of self-affine fractal rough rigid bodies into a model elastomer having a linear viscous rheology

The penetration of a rigid body with a randomly rough, self-affine surface in a half space filled with a linearly viscous elastomer is studied numerically using the method of boundary elements. Using Radok's principle of functional equations, it is shown analytically that this problem is closely related to the recently investigated problem of contact of self-affine surfaces with an elastic half space. We show that the penetration velocity occurs to be a power function of the applied force and time, the corresponding exponents depending only on the Hurst exponent. For comparison, the same problem is solved using the method of reduction of dimensionality. Both three-dimensional numerical results and the method of reduction of dimensionality support the analytical predictions provided by general scaling arguments.

A Parametric Numerical Study of the Interaction between Seismic Waves and Landslides for the Evaluation of the Susceptibility to Seismically Induced Displacements

This study presents the results of a numerical analysis investigating the influence of the interaction between seismic waves and slopes on seismically induced displacements of existing landslides. Three step-like slope geometries with dip angles varying from 10 degrees to 45 degrees and two landslide mechanisms were considered; 54 multi-frequential dynamic equivalent signals were applied to the models and were derived according to the LEMA_DES approach based on the accelerometric records of the European Strong Motion and COSMOS databases. The simulations were performed using a finite-difference approach that assumed different viscous rheologies and considered different values for both the dynamic properties and the mechanical parameters. The interaction between seismic waves and landslides was analyzed using frequency ratios defined for characteristic periods associated with the landslide dimensions (i.e., thickness and length) and the seismic input. A regression analysis was also performed to derive correlations between seismically induced displacements and the frequency ratios, the values of the Arias intensity, and the critical accelerations. The obtained results reveal that (a) the seismically induced displacements increase with the seismic amplification induced by the landslide mass; (b) the coupling of a 2D interaction between seismic waves and slope with the 1D resonance of the landslide mass, which is strictly related to the high-impedance contrasts between the landslide mass and the substratum, significantly affects the seismically induced displacements; and (c) the seismically induced displacements increase more intensely with increasing slope angle for a rototranslational mechanism with respect to a translational one. Moreover, the seismically induced displacements that can be expected from the here-obtained results can significantly differ from those computed by the sliding-block stability analyses based on Newmark's method.

Dynamics of interplate domain in subduction zones: influence of rheological parameters and subducting plate age

Abstract. The properties of the subduction interplate domain are likely to affect not only the seismogenic potential of the subduction area but also the overall subduction process, as it influences its viability. Numerical simulations are performed to model the long-term equilibrium state of the subduction interplate when the diving lithosphere interacts with both the overriding plate and the surrounding convective mantle. The thermomechanical model combines a non-Newtonian viscous rheology and a pseudo-brittle rheology. Rock strength here depends on depth, temperature and stress, for both oceanic crust and mantle rocks. I study the evolution through time of, on one hand, the brittle-ductile transition (BDT) depth, zBDT, and, on the other hand, of the kinematic decoupling depth, zdec, simulated along the subduction interplate. The results show that both a high friction and a low ductile strength at the asthenospheric wedge tip shallow zBDT. The influence of the weak material activation energy is of second order but not negligible. zBDT becomes dependent on the ductile strength increase with depth (activation volume) if the BDT occurs at the interplate decoupling depth. Regarding the interplate decoupling depth, it is shallowed (1) significantly if mantle viscosity at asthenospheric wedge tip is low, (2) if the difference in mantle and interplate activation energy is weak, and (3) if the activation volume is increased. Very low friction coefficients and/or low asthenospheric viscosities promote zBDT = zdec. I then present how the subducting lithosphere age affects the brittle-ductile transition depth and the kinematic decoupling depth in this model. Simulations show that a rheological model in which the respective activation energies of mantle and interplate material are too close hinders the mechanical decoupling at the down-dip extent of the interplate, and eventually jams the subduction process during incipient subduction of a young (20-Myr-old) and soft lithosphere under a thick upper plate. Finally, both the BDT depth and the decoupling depth are a function of the subducting plate age, but are not influenced in the same fashion: cool and old subducting plates deepen the BDT but shallow the interplate decoupling depth. Even if BDT and kinematic decoupling are intrinsically related to different mechanisms of deformation, this work shows that they are able to interact closely. Comparison between modelling results and observations suggests a minimum friction coefficient of 0.045 for the interplate plane, even 0.069 in some cases, to model realistic BDT depths. The modelled zdec is a bit deeper than suggested by geophysical observations. Eventually, the better way to improve the adjustment to observations may rely on a moderate to strong asthenosphere viscosity reduction in the metasomatised mantle wedge.

Lateral fold growth and fold linkage

We use a three-dimensional finite element model for viscous flow to investigate the lateral linkage mode of two initially isolated fold segments. Depending on the separation distance between the fold hinge lines, four different fold linkage scenarios are observed. Linear linkage yields a sub-cylindrical fold with a saddle at the location where the two initial folds linked. Oblique linkage produces a fold resembling a type II refold structure. Oblique–no linkage results in two curved folds with fold axes plunging in opposite directions. Linear–no linkage yields a fold train of two separate subcylindrical folds with fold axes plunging in opposite directions. Representative numerical models for detachment and matrix folding for both power-law and linear viscous rheologies exhibit the same four fold linkage modes. The transition from linkage to no linkage occurs when the fold separation between the initially isolated folds is slightly larger than one-half of the low-amplitude fold wavelength. We demonstrate a natural example of lateral fold linkage from the Kurdistan region of Iraq, and point out the economic interest of saddle points linking the initially isolated fold segments.

Impact of grain size on the convection of terrestrial planets

This article presents a set of simulations of mantle convection, using a new model of grain size‐dependent rheology. In the present paper, it is shown that this rheology behaves in many ways as a visco‐plastic rheology. I use a model of grain size evolution which has been calibrated on experimental data in a previous paper. In this physical model, the grain size is directly related to the stress state, following a temperature‐dependent piezometric law. The rheology used here allows both diffusion and dislocation creep, depending on the grain size. At low stress, the grain size is high and forces the rheology to be dislocation dominated. For sufficiently high stresses, the equilibrium regime reached by the grains is located in the diffusion creep. In this case, the viscosity is linked to a stress‐dependent grain size, which actually makes the rheology more non‐Newtonian than it is in dislocation creep. This experimentally calibrated model allowed me to perform a set of numerical experiments of convection in which the rheology may be diffusion or dislocation creep dominated, depending on the state of the stress tensor. Then, The stress exponent varies from 3 to 5 because of grain size, which has a large impact on the temperature dependence of the viscosity. The present paper shows the impact of this new model on the convection regimes of terrestrial planets. In particular, for a wide range of parameters, I observe the episodic regime which is thought to govern the dynamics of Venus. This process of episodic resurfacing was obtained in previous simulations using visco‐plastic rheologies is a tight range of parameters. I obtain it here without using an ad hoc plasticity law, only using a viscous rheology based on laboratory measurements. In these simulations, I show that the cooling rate of the terrestrial planets may be largely modified by the consideration of a grain size‐dependent rheology.

Mechanical versus kinematical shortening reconstructions of the Zagros High Folded Zone (Kurdistan region of Iraq)

This paper compares kinematical and mechanical techniques for the palinspastic reconstruction of folded cross sections in collision orogens. The studied area and the reconstructed NE–SW trending, 55.5 km long cross section is located in the High Folded Zone of the Zagros fold‐and‐thrust belt in the Kurdistan region of Iraq. The present‐day geometry of the cross section has been constructed from field as well as remote sensing data. In a first step, the structures and the stratigraphy are simplified and summarized in eight units trying to identify the main geometric and mechanical parameters. In a second step, the shortening is kinematically estimated using the dip domain method to 11%–15%. Then the same cross section is used in a numerical finite element model to perform dynamical unfolding simulations taking various rheological parameters into account. The main factor allowing for an efficient dynamic unfolding is the presence of interfacial slip conditions between the mechanically strong units. Other factors, such as Newtonian versus power law viscous rheology or the presence of a basement, affect the numerical simulations much less strongly. If interfacial slip is accounted for, fold amplitudes are reduced efficiently during the dynamical unfolding simulations, while welded layer interfaces lead to unrealistic shortening estimates. It is suggested that interfacial slip and decoupling of the deformation along detachment horizons is an important mechanical parameter that controlled the folding processes in the Zagros High Folded Zone.

Folding in power-law viscous multi-layers

We study high-amplitude folding in layered rocks with two-dimensional numerical simulations. We employ the finite-element method to model shortening of an incompressible multi-layer with power-law viscous rheology. The Lagrangian numerical mesh is deformed and re-meshed to accurately follow the layer interfaces. Three settings are considered: (i) pure shearing of a confined multi-layer, (ii) simple shearing of a multi-layer above a detachment, and (iii) slump folding owing to gravity sliding. In our pure shear simulations, finite-amplitude folds always develop despite confinement and thin weak interlayers. The fold shapes can be significantly irregular, resulting from initial geometrical heterogeneities that are perturbations of the layer interfaces and differences in layer thickness. The bulk normal viscosity of the multi-layer decreases significantly with progressive folding. This structural softening decreases the bulk normal viscosities by a factor of 2-20. For simple shear, the multi-layer does not develop asymmetric fold shapes significantly. Fold axial planes in the multi-layer are mostly curved and not parallel. For slump folding, fold shapes can be significantly asymmetric exhibiting strongly curved fold axial planes and overturned fold limbs. The rheology of the competent layers has a major impact on the fold shapes for gravity-driven multi-layer folding.

Simple shear is not so simple! Kinematics and shear senses in Newtonian viscous simple shear zones

AbstractThis work develops an analytical model of shear senses within an inclined ductile simple shear zone with parallel rigid boundaries and incompressible Newtonian viscous rheology. Taking account of gravity that tends to drive the material downdip and a possible pressure gradient that drives it upward along the shear zone, it is shown that (i) contradictory shear senses develop within two sub-zones even as a result of a single simple shear deformation; (ii) the highest velocity and least shear strain develop along the contact between the two sub-zones of reverse shear; (iii) for a uniform shear sense of the boundaries, a zone of reverse shear may develop within the top of the shear zone if the pressure gradient dominates the gravity component; otherwise it forms near the bottom boundary; (iv-a) a ‘pivot’ defined by the intersection between the velocity profile and the initial marker position distinguishes two sub-zones of opposite movement directions (not shear sense); (iv-b) a pivot inside any non-horizontal shear zone indicates a part of the zone that extrudes while the other subducts simultaneously; (v) the same shear sense develops: (v-a) when under a uniform shear of the boundaries, the shear zone remains horizontal and the pressure gradient vanishes; or alternatively (v-b) if the shear zone is inclined but the gravity component counterbalances the pressure gradient. Zones with shear sense reversal need to be reinterpreted since a pro-sheared sub-zone can retro-shear if the flow parameters change their magnitudes even though the same shear sense along the boundaries is maintained.

Comparing thin-sheet models with 3-D multilayer models for continental collision

Various models have been proposed to explain tectonic deformations during continent collision. A frequently applied model is the thin viscous sheet model which is however not fully 3-D and assumes a priori diffuse thickening as the dominant deformation style. We compare a fully 3-D multilayer numerical model with a corresponding thin viscous sheet numerical model for the scenario of continent indentation. In our comparison we focus on the three basic viscous deformation styles thickening, buckling (folding) and lateral crustal flow. Both numerical models are based on the finite element method (FEM) and employ either a linear or power-law viscous rheology. The 3-D model consists of four layers representing a simplified continental lithosphere: strong upper crust, weak lower crust, strong upper mantle and weak lower mantle. The effective viscosity depth-profile in the 3-D model is used to calculate the depth-averaged effective viscosity used in the thin-sheet model allowing a direct comparison of both models. We quantify the differences in the strain rate and velocity fields, and investigate the evolution of crustal thickening, buckling and crustal flow resulting from the two models for two different phases of deformation: (1) indentation with a constant velocity and (2) gravitational collapse after a decrease of the indenting velocity by a factor of 5. The results indicate that thin-sheet models approximate well the overall large-scale lithospheric deformation, especially during indentation and for a linear viscous rheology. However, in the 3-D model, additional processes such as multilayer buckling and lower crustal flow emerge, which are ignored in the thin-sheet model but dominate the deformation style in the 3-D model within a range of a few hundreds of kilometres around the collision zone and indenter corner. Differences between the 3-D and thin-sheet model are considerably larger for a power-law viscous than for a linear viscous rheology. Buckling and lower crustal flow are significant in the 3-D model with power-law viscous rheology. For example, fibre strain rates due to buckling can be several hundred per cent different to the depth-averaged strain rate and the lateral mass flow of lower crustal material can be up to six times more than the flow of upper crustal material. Our results also show that the horizontal velocity fields of the upper crust and upper mantle remain nearly identical in the 3-D model during indentation despite their mechanical decoupling due to an intermediate weak lower crust. This result questions the validity of using similarities between velocities from the surface global positioning system (GPS) and mantle shear wave splitting data as evidence for a mechanically coupled lithosphere. 3-D multilayer models provide a more complete picture of continental collision than thin-sheet models as they enable studying the timing, locality and relative importance of different processes simultaneously which is especially important for the hundreds of kilometre scale around the collision zone and indenter corners. 3-D models are, however, still computationally challenging and we, therefore, also present results of a computational performance test of several solution algorithms.

The neutral lines in buckle folds

The neutral line in a buckle fold, dividing areas of outer-arc extension from areas of inner-arc shortening, is a fundamental concept in structural geology. In the past, folds have been constructed kinematically from a given neutral line geometry using the tangential longitudinal strain pattern. In this study, a mechanical finite element model is used to numerically buckle single-layer folds with Newtonian and power-law viscous rheology. Two neutral lines can be distinguished, the incremental neutral line (zero layer-parallel strain rate) and the finite neutral line (zero finite layer-parallel strain). The former develops first and migrates through the layer from the outer towards the inner arc ahead of the latter. Both neutral lines are discontinuous along the fold and terminate either at the bottom or top interface of the layer. For decreasing viscosity ratio between layer and matrix and for decreasing initial amplitude, the neutral lines develop later during folding and, for some cases, no neutral line develops. The dynamical behaviour of the neutral lines is similar for Newtonian and power-law viscous rheology if the viscosity ratio is large, but substantially different for small viscosity ratios. The results are discussed in light of fold-related structures, such as outer-arc-extension structures and inner-arc-shortening structures.

Deformation of an elliptical inclusion in two-dimensional incompressible power-law viscous flow

An iterative, semi-analytical solution is derived for deformation of an elliptical (in cross-section) power-law viscous inclusion within an infinite linear viscous matrix undergoing a general 2D incompressible flow. Finite-element numerical models are used to extend the analysis to that of a power-law viscous matrix. The general behaviour of a deformable elliptical inclusion is not dramatically changed by power-law viscous rheology, but the effective viscosity is now a function of the orientation and axial ratio of the inclusion. Overall, the effect is similar to a markedly increased viscosity ratio for a stronger inclusion, or a decreased ratio for a weaker inclusion, when compared to the linear viscous case. As a result, rather low reference state viscosity ratios between inclusion and matrix (e.g., 2 to 3, determined at the same effective strain rate for both materials) can produce marked differences in behaviour for the range of power-law stress exponents established experimentally for many minerals and rocks (typically 3–6). Even for very high strain within a shear zone ( γ > 100), initially nearly circular inclusions ( R < 2) can maintain low axial ratios ( R < 2–3) and widely variable orientations. These inclusions deform internally and are not rigid, but continue to rotate or oscillate without strong elongation.

Regional stress fields under Tibet from 3D global flow simulation

Tibetan area is the most active continental collision zone on earth. Several major earthquakes occurred around the boundaries of Tibetan plateau and caused massive damages and casualties. The dynamics of this area is not well understood due to the complex structure of Tibet and its surrounding area. In this study, a 3D global flow simulation with only viscous rheology is applied to studying the stress distribution in this area, and the interaction between Tibet and its surrounding areas is investigated. Finally, the possibility of combining regional modeling with global models is also discussed.

Effects of a non-coaxial flow rule on shear bands in viscous-plastic materials

A class of non-coaxial plasticity models is revisited in which the plastic deformation is carried by one (single slip) or two (double slip) slip systems that are inclined at \({\pm (\pi/4+\nu/2), 0\le \nu \le \phi}\) to the less compressive principal stress axis; ν is the angle of non-coaxiality and \({\phi}\) is the internal angle of friction of the material. The non-coaxial model is combined with a Newtonian viscous rheology to form a viscous-plastic model. The model is applied to model strain localisation in extension and compression on a time scale typical for deformation processes in the lithosphere. We would like to dedicate this article to the memory of Ioannis Vardoulakis who always took a keen interest in bifurcation theory and shear banding for granular materials.

A simple analytical solution for slab detachment

An analytical solution is presented for the nonlinear dynamics of high amplitude necking in a free layer of power-law fluid extended in layer-parallel direction due to buoyancy stress. The solution is one-dimensional (1-D) and contains three dimensionless parameters: the thinning factor (i.e. ratio of current to initial layer thickness), the power-law stress exponent, n, and the ratio of time to the characteristic deformation time of a viscous layer under buoyancy stress, t/ t c . t c is the ratio of the layer's effective viscosity to the applied buoyancy stress. The value of t c / n specifies the time for detachment, i.e. the time it takes until the layer thickness has thinned to zero. The first-order accuracy of the 1-D solution is confirmed with 2-D finite element simulations of buoyancy-driven necking in a layer of power-law fluid embedded in a linear or power-law viscous medium. The analytical solution is accurate within a factor about 2 if the effective viscosity ratio between the layer and the medium is larger than about 100 and if the medium is a power-law fluid. The analytical solution is applied to slab detachment using dislocation creep laws for dry and wet olivine. Results show that one of the most important parameters controlling the dynamics of slab detachment is the strength of the slab which strongly depends on temperature and rheological parameters. The fundamental conclusions concerning slab detachment resulting from both the analytical solution and from earlier published thermo-mechanical numerical simulations agree well, indicating the usefulness of the highly simplified analytical solution for better understanding slab detachment. Slab detachment resulting from viscous necking is a combination of inhomogeneous thinning due to varying buoyancy stress within the slab and a necking instability due to the power-law viscous rheology ( n > 1). Application of the analytical solution to the Hindu Kush slab provides no “order-of-magnitude argument” against slab detachment and, therefore, supports existing studies suggesting a currently ongoing slab detachment in the Hindu Kush slab.

Dynamic seismic ruptures on melting fault zones

[1] We present a physical model that describes the behavior of spontaneous earthquake ruptures dynamically propagating on a fault zone and that accounts for the presence of frictional melt produced by the sliding surfaces. First, we analytically derive the solution for the temperature evolution inside the melt layer, which generalizes previous approximations. Then we incorporate such a solution into a numerical code for the solution of the elastodynamic problem. When a melt layer is formed, the linear slip-weakening law (initially governing the fault and relying on the Coulomb friction) is no longer valid. Therefore we introduce on the fault a linearly viscous rheology, with a temperature-dependent dynamic viscosity. We explore through numerical simulations the resulting behavior of the traction evolution in the cohesive zone before and after the transition from Coulomb friction and viscous rheology. The predictions of our model are in general agreement with the data from exhumed faults. We also find that the fault, after undergoing the breakdown stress drop controlled by the slip-weakening constitutive equation, experiences a second traction drop controlled by the exponential weakening of fault resistance due to the viscous rheology. This further drop enhances the instability of the fault, increasing the rupture speeds, the peaks in fault slip velocity, and the fracture energy density.

Why subduction zones are curved

[1] We give an explanation for the polarity, localization, shape, size, and initiation of subduction zones on Earth. By considering a soft, thin, curved lithospheric cap with either elastic or viscous rheology supported by a thick, nearly incompressible mantle, we find two different characteristic subduction geometries arise depending on boundary conditions: (1) plate boundaries where subduction results primarily from the gravitational body force (free subduction) have characteristic plate lengths and form arc-shaped dimpled segments resulting from the competition between bending and stretching in edge buckling modes of thin spherical shells, and (2) subduction zones due to localized applied loads that push one slab of thin, positively buoyant lithosphere beneath an overriding plate (forced subduction) form localized straight segments, consistent with the deformation of indented spherical shells. Both types of subduction are nonlinear subcritical instabilities, so small perturbations in the mechanical properties of the lithosphere have pronounced effects on subduction initiation and evolution. Yet in both cases, geometric relationships determined by the shape of the Earth itself play the most critical role in controlling the basic morphology and characteristic length scales of subduction zones.

Fluid flow during unbending: Implications for slab hydration, intermediate-depth earthquakes and deep fluid subduction

We calculate the tectonic stress profile and associated direction of fluid flow during unbending and dehydration of oceanic plates, for a range of critical parameters that affect their combined elasto-plastic and viscous behaviour, such as bending curvature, age, pore fluid pressure and viscous flow laws. In all models, negative pressure gradients are established at Moho depths, down to the base of the slab elastic core. Fluids released at these depths flow downward across the plate, “wetting” or further hydrating the underlying dry levels, and ultimately accumulate at the base of the elastic core, increasing the pore fluid pressure and triggering deep seismicity. The thickness of the “wet” layer increases for low bending curvatures, low pore fluid pressure, old slabs and dry viscous rheologies. “Wetting” of the upper 10–30 km of the slab has important implications for its rheological and anisotropic structure. Unbending favours the redistribution and trapping of significant amounts of fluids in subducting oceanic plates that will subsequently be released at the base of the upper mantle.

Viscosity estimates of salt in the Hormuz and Namakdan salt diapirs, Persian Gulf

AbstractThe parabolic surface profiles of the Hormuz and Namakdan salt diapirs in the Persian Gulf suggest that they have been extruding with Newtonian viscous rheologies for the last 104 years. We derive velocity profiles for these diapirs, neglecting gravitational spreading and erosion/dissolution while assuming incompressible Newtonian rheology of the salt. Fitting known rates of extrusion at specific points in its elliptical cross-section, the dynamic viscosity of the salt of the Hormuz diapir is found to range between 1018 and 1021 Pa s. Approximating its sub-circular cross-section to a perfect circle, the range of viscosity of the salt of the Namakdan diapir is obtained as 1017–1021 Pa s. These calculated viscosities fall within the range for naturally flowing salts elsewhere and for other salt diapirs but are broader than those for salts with Newtonian rheology deforming at room temperatures. The salts of the Hormuz and Namakdan diapirs are expected to exhibit a broader range of grain size, which matches the limited existing data.

Heat partitioning in terrestrial planets during core formation by negative diapirism

We investigate the gravitational heat partitioning between silicate and metallic phases during core formation by negative diapirism in terrestrial planets. We model the dynamic evolution of an iron diapir, sinking through a solid silicate proto-mantle. Our calculations include viscous heating and strong temperature and composition-dependence of viscosity. We show that the diapir mean temperature evolves towards an asymptotic value T d∞, which depends on the efficiency of viscous heating and heat transfer between the diapir and its surroundings, and on the viscous rheology. We derived a simple analytic model that captures the thermal evolution observed in our numerical experiments. This model can be applied to determine the heat distribution within terrestrial planets with a large number of diapirs, during their growth and early differentiation. We show that the partitioning of core formation heating during negative diapirism depends strongly on the size distribution of the iron diapirs. A large number of small sinking iron diapirs favors metal–silicate heat exchange and leads to a relatively hot core, which allows a sustainable dynamo, as on Earth. A small number of large diapirs have the opposite effect and could correspond to a Mars-like planet with an early dynamo that is no longer active. In all cases, negative diapirism tends to leave the lowermost mantle significantly hotter than its upper part, which could trigger large amounts of melting in the early deep mantle.

Mechanics of basin inversion: Finite element modelling of the Pannonian Basin System

This finite element modelling study addresses the contraction of rheologically layered and laterally heterogeneous lithosphere representative for the Pannonian Basin and its surroundings. The time interval and strain rate adopted in the experiments reflect the Pliocene–Quaternary inversion of the basin. Several sets of fully coupled elastoviscoplastic thermo-mechanical, plane strain 2D models explore the evolution of buckling, and stress/strain changes across the lithosphere. The viscous rheology of the asthenosphere allows for a detailed simulation of isostatic rebound during deformation. The numerical models predict the successive development of surface undulations, caused by crustal and/or lithosphere folding, at three different characteristic wavelengths. Among these, the longest wavelength folds occur systematically at the rim of the basin as marginal bulges, while the short wavelength folds overprint the earlier folds and are observed at a later stage during compression. The thermo-mechanical evolution of the lithosphere in response to progressive horizontal contraction is described in terms of a characteristic delay of the changes in the stress regime and a reduction of elastic strain in the strong crustal layers. The models predict a change in stress state along the flanks of the basin, caused by the development of weak basin lithosphere in their vicinity. Comparing the modelling results with tectonic features of the Pannonian Basin has resulted in the identification of three different stages in the Pliocene–Quaternary basin inversion of the Pannonian–Carpathian system.