Effect of plastic-viscous layering and strain softening on mode selection during lithospheric extension



[1] Factors controlling the selection of deformation modes during continental extension are investigated using analytical and numerical methods. We view the lithosphere as a laminate and examine a simple system with a uniform plastic layer overlying a uniform linear viscous layer. The rate of energy dissipation is analyzed for pure shear (PS), symmetric plug (SP), and asymmetric plug (AP) extension modes, and the analysis reveals that the primary control is the relative rate of dissipation in the two layers. A basic difference is that the plastic layer yield strength is independent of the strain rate, whereas the viscous stress depends on strain rate; therefore dissipation scales linearly and quadratically with extension velocity for these respective layers. When other parameters, e.g., extension velocity, and properties are held constant, minimum dissipation predicts that the modes AP, SP, and PS will be selected in this order with increasing viscosity of the lower layer. Transition viscosities between modes, ηT1 and ηT2 are 4 × 1021 Pa s and 8 × 1022 Pa s, respectively, for our parameters values. Numerical models confirm the analysis results, inferred mode controls, and order of mode selection when strain softening of the plastic layer occurs during extension. Implications for lithosphere that acts as a bonded plastic/viscous laminate include the following (1) asymmetric extension (AP mode) is preferred when the extension rate and/or effective viscosity is low or the viscous region temperature is high, (2) symmetric (SP mode) extension is preferred for intermediate combinations of parameters, and (3) overall pure shear (PS mode) may occur for opposite end-member parameter combinations.

1. Introduction

[2] Whether extension at the crustal or lithosphere scale may be symmetric or asymmetric [e.g., McKenzie, 1978; Wernicke and Burchfiel, 1982; Wernicke, 1985] is only part of the larger question concerning the factors that control the styles or modes of continental extension. Observations of major detachments along passive margins, as well as asymmetric half grabens and asymmetric exhumation of high-grade core complexes along single detachments indicate that extension may be asymmetric (Figure 1), [Boillot et al., 1992; Louden and Chian, 1999; Sibuet, 1992; Brun and Gutscher, 1992; Whitmarsh et al., 2001]. On the other hand, a range of rift basins as, for example, the Rio Grande Rift [Keller and Baldridge, 1999], Central Viking Graben [White, 1990], and core complexes with multiple opposite verging detachments (e.g., Menderes Massif [Gessner et al., 2001]) indicate that extension may also be symmetric at different scales (Figure 1). Although these observations demonstrate the importance of symmetry and asymmetry and the role of faults and ductile detachments, other modes including lithospheric scale uniform pure shear without focused deformation are also theoretically possible. To our knowledge this mode has not been reported except in a recent paper by Karner et al. [2003].

Figure 1.

Diagram illustrating simple end-member rifting modes discussed in the paper.

[3] Some insight exists concerning: (1) the factors that control asymmetry and symmetry during the early stages of extension and (2) the factors that cause a given mode to be abandoned during extension. Numerical models [Lavier et al., 1999; Frederiksen and Braun, 2001; Lavier and Buck, 2002; Huismans and Beaumont, 2002, 2003] indicate that strain softening (or strain rate softening [Behn et al., 2002]) may be a requirement for the initiation of asymmetric extension. Geometric hardening has been suggested as a main control that terminates asymmetric extension [Buck, 1988, 1993; Lavier et al., 1999; Lavier and Buck, 2002]. However, strain softening and geometrical hardening, for example, are probably universal lithospheric properties, and therefore extensional zones would have similar geometries if these were the only important controls.

[4] Here we propose at least a partial answer to the questions of the additional factors that control the modes of lithospheric extension, and which mode is selected in a particular circumstance. We base our approach on the view that the lithosphere is a composite laminated system, with viscous and plastic layers, in which the plastic layers weaken by focused deformation on faults and shear zones. We then use the principle of minimum rate of energy dissipation to predict the extensional mode that will be selected.

[5] Energy (dissipation) analysis of systems is a fundamental approach that allows system behavior to be predicted on the basis of the principle of minimum expenditure of energy (minimum work) [Bird and Yuen, 1979; Sleep et al., 1979; Masek and Duncan, 1998]. We use this technique to gain insight into which of the three modes outlined below and shown in Figure 2 will be selected. Although the minimum dissipation approach does not apply to all systems [Bird and Yuen, 1979], our main purpose is to compare the analysis with equivalent finite element models which are based on minimum dissipation.

Figure 2.

Diagram showing geometry and kinematics used for the dissipation analysis for (a) pure shear, (b) symmetric plug, and (c) asymmetric plug modes.

[6] To illustrate the approach, a simple two-dimensional, plane strain system comprising a uniform brittle-plastic layer overlying and bonded to a uniform linear viscous layer is analyzed approximately to estimate the rate of internal energy dissipation during the early stages of extension for three possible modes. These modes are (Figure 2) the PS mode, distributed uniform (pure shear) extension of the composite without focused deformation of the plastic layer or differential shear between the plastic and viscous layers; the SP mode, the symmetric single “plug” mode in which deformation in the plastic layer only occurs on two focused shears that bound the plug and there is differential shear between the plastic and viscous layers, and; the AP mode, the asymmetric single plug mode in which the plastic deformation is similar to that of the SP mode but is focused on only one of the shear zones that bounds the plug. The plastic layer may either be a von Mises or a frictional-plastic material characterized by the cohesion or internal angle of friction, respectively. Strain softening results in a decrease in the value of these parameters with increasing strain, thereby weakening the plastic layer when deformation is focused on shear zones.

[7] For this simple system there are two main contributions to the rate of internal energy dissipation. (1) From the plastic layer, this rate of dissipation is the same for each of the modes except when it is reduced by strain softening of the localized shears. (2) From the viscous layer, this rate of dissipation ranges from that owing to pure shear in PS to a combination of pure shear and shear of the boundary layer at the top of the viscous layer when there is differential extension between the two layers in SP and AP. Which mode is selected depends on the trade-off between the reduction of the plastic dissipation owing to strain softening in SP and AP versus the penalty of the increased rate of dissipation owing to the viscous boundary layer in SP and AP. By considering these trade-offs the mode selection can be predicted as a function of the layer properties and the rate of extension. The latter is important because the rate of dissipation in the viscous layer depends on the strain rate.

[8] The predictions of mode selection from the approximate rate of dissipation analysis is partly supported by an alternative force balance analysis. It is also confirmed by a series of finite element model experiments. These experiments range from the model used for the approximate analysis to models with more realistic rheological layering.

[9] The overall results demonstrate that the mode preference is AP, SP and PS, and the modes will be selected in this order depending on the relative penalty of the viscous boundary layer rate of internal energy dissipation incurred by each mode. It is also demonstrated that for this system the external rate of energy dissipation, that is the rate of work done against gravity, and at free boundaries, plays a minor role in the mode selection during the early stages of extension. Finally, the results also provide a basis for defining the dominant and subordinate rheologies for each of these modes, a concept which is addressed in the discussion.

[10] In the following sections we present (1) the description of the approximate dissipation analysis for the simple system and the results; (2) the design of the finite element numerical models, and choice of material parameters and boundary conditions; (3) the results of the numerical experiments; (4) the comparison of the numerical model results with the predictions of the dissipation analysis, and (5) a discussion of the results, how they may be generalized, and their potential application to natural systems.

2. Energy Dissipation in Pure Shear, Symmetric, and Asymmetric Plug Modes

[11] We derive approximate velocity fields and corresponding rates of energy dissipation (termed dissipation for brevity) for each of the three modes outlined in the introduction and shown in Figure 2. Appendix A contains the equivalent force balance analysis. The criteria for the mode selection are based on the minimization of the dissipation.

[12] Following Bird and Yuen [1979], we define the total rate of energy dissipation as

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where equation image is half the rate of the internal viscous-plastic dissipation per unit volume, equation image is the rate of work done against gravity, equation image is the work done against external forces per unit length of free boundary, Γ, on which no velocity is specified, and U is the volume of the domain. For a viscous-plastic medium the total dissipation power 2equation image, as defined by Malvern [1969, p. 300] reduces to

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for an incompressible medium, where τij and equation imageij are the deviatoric stress and the strain rate tensors, respectively. The corresponding total rate of work against gravity is given by

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where ρ is density, g is the acceleration due to gravity, and Vz(x) is vertical component of velocity. Work done against gravity by raising material is considered positive and the converse when material is lowered.

[13] In the two-layer system (Figure 2) the upper uniform cohesive or frictional-plastic layer, thickness hp, is bonded to the underlying uniform linear viscous layer, thickness hv. The width of the system is 2L and it extends under velocity, V, applied uniformly with depth and equally and oppositely to both sides. The base is free to slip horizontally but does not move vertically, and there is a stress-free upper surface; therefore the last term in equation (1) can be neglected because work is not done against external forces on boundaries where velocities are unspecified.

[14] The criteria for the mode selection are based on the argument that at a transition between two modes the total dissipation in these modes is equal, i.e.,

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where modes 1 and 2 represent the AP, SP and PS modes. The analysis can be simplified considerably by demonstrating (see Appendix B) that the difference in the values of equation imageG between modes, i.e., Δequation imageG, is much smaller than the corresponding difference in Δequation imageI.

[15] For the analytical calculation we assume that there are no dynamical stresses, that deformation is incompressible and that the simple models have a uniform density. By considering mass conservation during extension and evaluating equation (3), equation imageG = −ρgVh2, for the PS mode, where h is the total thickness of the model and 2V is the total rate of extension (Figure 2 and Appendix B).

[16] In Appendix B cases in which the model thickness varies laterally, h = h(x), and the horizontal strain rate also varies laterally equation imagexx = equation imagexx(x) are considered as perturbations to the PS mode. The corresponding perturbations to equation imageGps, Δequation imageG are estimated for cases where the perturbations are larger than those that occur in the AP and SP modes. These estimates of Δequation imageG are in the range 10−4 to 10−2 of equation imageGps and are much smaller than Δequation imageI, the change in internal dissipation across either mode transition. These results are also supported by the estimates of equation imageG and Δequation imageG from the numerical models (section 3 and Appendix B).

[17] Under these circumstances, mode selection is only weakly dependent on the change in dissipation against gravity across the mode boundary even though the absolute values of this dissipation may be large for all of the modes. Therefore mode selection for the simple models considered here can be estimated with sufficient accuracy from the internal dissipation of the modes alone. We adopt this simplification because our goal is to demonstrate the relative order of mode selection, and not to estimate accurate values of the control parameters at the mode transitions. The latter is not possible because our analytical estimates of each of the terms in equation (4) are approximate and the errors in each of the estimates of the gravitational terms may be larger than their difference. In sections 2.12.4 we derive approximate analytical estimates of the internal dissipation of each of the three modes for both cohesive and frictional-plastic cases and then consider the implications for mode selection.

2.1. Pure Shear Mode, equation imageIps

[18] It is assumed that both the viscous and the plastic layers deform in plane strain by pure shear. In this case the viscous layer exerts no shear traction along the base of the plastic layer (Figure 2a). In pure shear deformation equation imagexz = 0. For linear viscous materials τij = 2ηequation imageij, and the viscous dissipation per unit length normal to the model plane becomes

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For pure shear equation imagexx = 2V/2L = −equation imagezz. Therefore

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and the viscous dissipation for the pure shear case is

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For a von Mises plastic layer (τxx − τzz)/2 = C(ε), where C(ε) is the cohesion which depends on strain ε resulting from strain softening. Using that equation imagexx = −equation imagezz, from incompressibility, gives the plastic dissipation:

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The total dissipation for the pure shear mode is therefore

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For the frictional-plastic case, hpρgsin(ϕ(ε))/2 is substituted for C(ε), which gives

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2.2. Symmetric Plug Mode, equation imageIsp

[19] In the symmetric plug mode the plastic layer extends in plane strain by localized shear along the inclined shear zones in the center of the model (Figure 2b). This rigid translation of the plastic layer is resisted by a shear traction in a viscous boundary layer below the base of the plastic layer.

[20] The viscous dissipation can be estimated assuming that the viscous layer deforms by a combination of boundary layer deformation and pure shear below the boundary layer. The horizontal differential velocity at the top of the boundary layer is given by

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We assume for simplicity that the boundary layer has a constant thickness, hb(x) = hb. Assuming symmetry about x = 0, a velocity field satisfying these conditions, the underlying pure shear, and incompressibility, but with a small error in mass conservation if hbhv, is given by

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for a local reference frame with z = 0 at the base of the boundary layer. The strain rate in the viscous boundary layer is therefore given by

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The dissipation in the viscous layer is given by

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We separate the dissipation into the boundary layer and underlying pure shear below and substitute equation imagexx, equation imagezz, equation imagexz. The dissipation in the boundary layer is given by

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The viscous dissipation for the pure shear layer is

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where hvhb is the residual thickness. The dissipation for a cohesive von Mises plastic layer deforming in a symmetric plug mode is given by

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where the slip rate on shears inclined at 45° is Vs = equation imageV. Note that the plastic dissipation in this case is equal to the plastic dissipation in the pure shear mode unless there is strain softening. The total dissipation for the symmetric plug mode is therefore given by

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and, equivalently, for the frictional-plastic case,

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We consider two end-members, SP1 where C(ε) or ϕ(ε) are not strain softened (Cnss(ε), ϕnss(ε)), and SP2 where C(ε) or ϕ(ε) are strain softened (Css(ε), ϕss(ε)).

2.3. Asymmetric Plug Mode, equation imageIap

[21] The velocity field in the asymmetric plane strain case is most easily described in a reference frame in which the right (footwall) side of the model (Figure 2c) remains stationary and the left side moves with velocity 2V with respect to S. This reference frame emphasizes the asymmetry of the system and velocity in the more usual reference frame is given by subtracting V everywhere. In this case the differential velocity in the boundary layer is

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The velocity field is

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and corresponding strain rate terms in the viscous boundary layer are

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With these terms the dissipation in the boundary layer can be calculated. Below the boundary layer pure shear is again assumed. The plastic dissipation is the same as that for the symmetric plug mode. The total rate of dissipation for the asymmetric plug mode is

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and, equivalently, for the frictional plastic case,

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2.4. Implications for Mode Selection

[22] The next step is to determine which is the minimum dissipation mode as the control parameters vary. The result of this analysis is summarized in Figure 3 which shows how the modes are distributed in relation to the plastic and viscous controls. NSS, PSS, and SS are the not strain-softened, partially strain-softened, and strain-softened strengths of the plastic layer, respectively, and ηT1 and ηT2 are transition viscosities that demark the mode boundaries. The mode space is labeled according to the minimum dissipation mode, PS, SP1, SP2 (where SP2 includes strain softening on the bounding shears), and AP. Arrows show how models typically evolve when the control parameters remain constant. For example, the path SP1 → AP is the normal one for models when the control viscosity is below ηT1, etc. We explain below how the preferred modes are selected and what determines the two transition viscosities.

Figure 3.

Mode space diagram. Arrows show evolutionary pathways for systems operating with constant control parameters, η, V, hp, hv. PS, pure shear; SP1, symmetric plug (not strain softened); SP2, symmetric plug (strain softened); AP, asymmetric plug; NSS, not strain softened; SS, strain softened; PSS, partially strain softened. The terms ηT1 and ηT2 are the viscosities at which the transition between the AP and SP1 and between the SP2 and PS modes, respectively, is predicted.

2.4.1. Plastic and Viscous Components of the Dissipation

[23] The SP mode (Figure 4a) can be used to illustrate the plastic and viscous contributions to the total dissipation. The plastic dissipation depends on the integrated strength of the plastic layer and the boundary velocity (e.g., equation (17)). As the shears bounding the plug strain soften, reflecting the strain dependence of cohesion, C(ε), or angle of internal friction, ϕ(ε), the plastic dissipation decreases from the SP1 to the SP2 value (Figure 4a). When other variables remain constant, the viscous dissipation (equations (15) and (16)) increases linearly with the viscosity (Figure 4a).

Figure 4.

Results of dissipation analysis. (a) Dissipation for SP1 and SP2 modes. (b) Comparison of dissipation for PS/SP1 mode and SP2 mode. The term ηT2 is the viscosity at which the transition between the SP2 and PS modes is predicted. (c) Comparison of dissipation for AP and SP1 modes. The term ηT1 is the viscosity at which the transition between the AP and SP1 modes is predicted.

[24] Consequently, at low viscosities the total dissipation is dominated by the plastic layer but at higher viscosities the viscous dissipation exceeds the plastic contribution. As shown by equations (15), (16), and (17), the crossover viscosity, at which the viscous dissipation becomes the larger component (Figure 4a), depends on the strength of the plastic layer, the degree of strain softening, the boundary velocity, and the horizontal, L, and vertical, hb, length scales of the viscous boundary layer. This general behavior with plastic dissipation dominating at low viscosities and the converse at high viscosities is true for all three of the modes.

2.4.2. Competition Between the Pure Shear and Symmetric Plug Modes

[25] Next we address the factors that control the selection of the pure shear and symmetric plug modes (Figures 3 and 4b). These modes have the same plastic dissipation prior to strain softening (equations (9) and (17)), i.e., when the SP mode is SP1. The corresponding PS viscous dissipation is, however, smaller than the SP value (Figure 4b and equations (7), (15) and (16)). Therefore, for low viscosities, there is no mode preference between PS and SP1 because the plastic dissipation begins to dominate, but as viscosity increases above the value at which viscous dissipation dominates, the lower-dissipation PS mode becomes increasingly preferred (Figure 4b). There is, however, no sharp transition between PS and SP1 at a particular viscosity.

[26] In contrast, when strain softening reduces the SP plastic dissipation (e.g., from SP1 to SP2, Figure 4b) there is a unique mode transition at ηT2 from SP2, preferred at low viscosities, to PS preferred at high viscosities (Figure 4b). This is the transition labeled ηT2 (Figure 3) between the fully softened, SP2 and the PS modes. The transition viscosity is estimated by equating the PS and SP2 dissipations (equations (9) and (18), where C(ε) is the SS value of the cohesion), which gives

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or for the frictional-plastic case,

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where ϕnss and ϕss are the corresponding unsoftened and softened internal angles of friction.

[27] The results imply that the PS mode will be preferred at high values of the control viscosity for both the NSS and SS cases, but there is no well defined mode boundary between PS and SP1. However, later in the evolution when strain softening occurs the mode selection is clearly delineated as SP1 → SP2, or PS → PS depending on whether the control viscosity is less than, or greater than, ηT2 (Figure 3).

2.4.3. Transition Between the Asymmetric Mode and the SP2 Symmetric Plug Mode

[28] In this case the question to be answered is whether at low control viscosities, lower than ηT2, the SP1 mode will transition to SP2 or the asymmetric plug mode, AP, during strain softening (Figure 3). Comparison of the total dissipations of the SP2 and AP modes (equations (18) and (32)) shows that for the same model parameter values SP2 and the strain-softened AP modes have the same plastic dissipation, but SP2 has the lower viscous dissipation. This result suggests that SP2 would always be selected. However, consideration of the path dependence of the total dissipation during the transitions SP1 to SP2, and SP1 to AP shows that the AP mode achieves the fully strain-softened state faster than the SP2 mode because all of the strain is focused on one shear.

[29] We therefore take this path dependence into account and compare the total dissipations in the SP and AP modes at the point where the AP mode shear zone is first fully softened, at which point the shears in the equivalent SP2 case are only partly softened. Equating equations (18) and (23) under these circumstances and solving for the transition viscosity for the von Mises case gives

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where Css(ε) is the strain-softened cohesion, and Cpss(ε) is the partly softened cohesion in the SP2 mode. For the frictional-plastic case the equivalent transition viscosity is given by

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Here ϕss(ε) is the strain-softened internal angle of friction, and ϕpss(ε) is the partly softened one.

[30] An estimate of ηT1 can be obtained by assuming that when the AP shear becomes fully softened the SP1 shears will not have softened significantly. In this case ηT1 is given by equations (27) and (28) with Cpss and ϕpss replaced by their non strain-softened (nss) values. This value is an upper bound estimate of the transition viscosity, ηT1 between the AP and SP2 modes. More generally, the value of ηT1 depends on the difference between Cpss and Css (equation (27)), and therefore it will decrease from the estimate given above as this difference becomes smaller. AP will be the preferred mode when the viscosity is lower than ηT1 (Figure 3).

[31] This behavior can be interpreted physically in the following manner. Any noise in the system will lead to the onset of strain softening on one of the SP bounding shears before the other, and given the strong positive feedback this shear will preferentially soften and reach its fully strain-softened state first. Therefore, everything being equal, the preferred transition is SP1 → AP because this is the path that decreases the plastic dissipation most rapidly. However, this analysis ignores the competing effect of the viscous dissipation which increases during the SP1 → AP transition. The mode that is selected is the one that decreases the total dissipation most rapidly and this will depend on the control viscosity. Below ηT1 the path of the AP mode decreases the total dissipation the most rapidly, and therefore this mode will be selected despite the final higher viscous dissipation in this mode (Figure 3).

2.4.4. Model Evolution and Pathways in Mode Space

[32] Returning to Figure 3, it was stated above that the arrows link the normal mode progression for models when the control parameters remain constant. It can now be seen that there are three main regions of the mode space separated by the transition viscosities ηT1 and ηT2 and there are three corresponding normal mode progressions. In models where the control viscosity is greater than ηT2, PS is the only available mode and the others are forbidden because they have a higher total dissipation even when strain softening of the plastic layer occurs. For control viscosities between ηT1 and ηT2 the mode evolution is SP1 → SP2. That is, the deformation remains symmetric. For control viscosities less than ηT1 the mode evolution is SP1 → AP, so that the deformation becomes asymmetric as the plastic layer strain softens.

[33] These model pathways in mode space provide the basic insight concerning why the different modes are selected and why the style may be symmetric or asymmetric and even pure shear if the control viscosity is large enough. An interesting result is that the SP2 and AP modes are clearly separated even though both modes have the same plastic dissipations. The symmetry versus asymmetry in these cases is determined by the viscous layer properties, not by the plastic layer. This last result demonstrates that minimum dissipation is not necessarily the sole criterion for a mode to be selected. Mode selection also depends on the accessibility of the minimum dissipation mode and this depends on the dissipation path among the modes. The strain softened AP mode is a metastable mode from the standpoint of total dissipation. The dissipation path can lead to the AP mode, which makes the lower-dissipation SP2 mode inaccessible because the system must pass through a higher level of dissipation, softening the second shear, in order to reach the SP2 mode. The accessibility and selection of modes is addressed in section 3 using numerical models.

3. Numerical Model Results

[34] The next step is to present four series of finite element model experiments designed to test the approximate analytical theory. Series 1 and 2 are particular examples that correspond to the two-layer lithosphere with either a strain softening, cohesive (series 1) or frictional-plastic (series 2) layer, underlain and bonded to a uniform linear viscous layer. Series 3 introduces a more realistic depth-dependent rheology that does not prescribe the layer thicknesses. Series 4 builds on series 2 by replacing the free slip basal boundary condition with an underlying region of low-viscosity asthenosphere.

3.1. Model Description

[35] An Arbitrary Lagrangian-Eulerian (ALE) finite element method for the solution of mechanical plane strain, incompressible, viscous-plastic creeping flows [Fullsack, 1995; Willett, 1999; Huismans and Beaumont, 2003] is used to investigate extension of models with a simple geometry (Figure 5) and rheological structure (Table 1). Plastic deformation has a pressure-dependent Drucker-Prager yield criterion

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where (J2)1/2 = (equation imageτijτij)1/2 is the second invariant of the deviatoric stress, C is the cohesion, ϕeff is the effective internal angle of friction, and p is the dynamical pressure. With appropriate choices of C and ϕ this yield criterion can approximate noncohesive frictional sliding (C = 0, model series 2), or von Mises yielding (ϕeff = 0, model series 1) in rocks. In particular, ϕeff is defined to include the effects of pore fluid pressure such that, p sin(ϕeff) = (ppf) sin (ϕ), where ϕ is the dry internal angle of friction and pf is the pore fluid pressure. A ϕeff of 15° is a midrange value consistent with hydrostatic pore fluid pressure conditions in the following experiments.

Figure 5.

Geometry of the numerical models showing upper plastic layer and lower viscous layer thicknesses, the weak seed, and the velocity boundary conditions, V. Extension is seeded by a small plastic weak seed. The model has a free top surface, and the other boundaries have zero tangential stress (free slip). Sedimentation and erosion are not included in the model. There is, however, a small amount of numerical surface diffusion. The Eulerian grid has 401 and 121 nodes in the horizontal and vertical directions, respectively. Strain softening is achieved in a simple manner by a linear decrease of the cohesion or the angle of internal friction with deviatoric effective strain (I2).

Table 1. Model Parameters
ϕnssangle of Internal Friction, unsoftened15°
ϕssangle of internal friction, softened
Ccohesion0, 30, 230 MPa
ηviscosity1021, 1022, 1023 Pa s
η0scaling viscosity for model series 31.5 × 1023 Pa s
γdepth dependence viscosity for model series 30.25 km−1
ρdensity3000 kg/m3
Vextension velocity1, 10 cm/yr
hpbrittle layer thickness60 km
hvviscous layer thickness60 km
Lhorizontal length scale600 km

[36] Strain softening is introduced by linear decreases of C(ε) and ϕeff(ε), respectively, over a range of strain, where ε represents the second strain invariant, (I2)1/2 (Figure 5 and Table 1). As briefly summarized by Huismans and Beaumont [2003], frictional-plastic faults and brittle shear zones may be weakened by high transient or static fluid pressures or by gouge or mineral transformations [e.g., Sibson, 1990; Streit, 1997; Bos and Spiers, 2002]. For frictional-plasticity we use an unsoftened ϕeff = 15° and reduce this value linearly to 2° over the range 0.5 < ε < 1.5. As in section 2, we now refer to ϕeff as ϕ, but it should be recalled that the values take account of the pore fluid pressure. For cohesion softening we reduce the unsoftened C = 230 MPa linearly to C = 30 MPa over the same strain range. The unrealistic high values of C were chosen so that both series 1 and series 2 models have the same integrated strength of the plastic layer.

3.2. Model Series 1, Two-Layer, Cohesion Softening

[37] In series 1 a strain-softening cohesive plastic layer overlies a uniform constant linear viscous layer, and the viscosity ranges from 1021 to 1023 Pa s (Figures 6a–6c). The results (Figure 6) are shown after 40 km total extension at 1 cm/yr. At low viscosity (Figure 6a) the AP mode is selected. Strain softening has focused the plastic deformation onto one shear and the extension is highly asymmetric. When η = 1022 Pa s (Figure 6b) asymmetry is suppressed and extension in the plastic layer occurs along two symmetric shears in the SP2 mode. At high viscosity (Figure 6c) local deformation in the plastic layer is completely suppressed and extension of both layers occurs by pure shear, the PS mode.

Figure 6.

Model series 1. Two-layer models of lithosphere extension. Upper layer strain-softening cohesive-plastic and its cohesion C (ε) decrease from 230 to 30 MPa as second invariant of strain (I2)1/2 = ε = 0.5–1.5 increases, lower layer uniform viscosity. (a)–(c) Different values of viscosity. Vertical side boundaries extend at 0.5 cm/yr. Extension is 40 km. Weak seed, C = 30 MPa, in center at base of cohesive-plastic layer seeds deformation.

3.3. Model Series 2, Two-Layer, Frictional-Plastic Softening

[38] In series 2 a strain-softening frictional-plastic layer overlies a uniform constant linear viscous layer, and the viscosity ranges from 1021 to 1023 Pa s. The results (Figure 7) are also shown after 40 km total extension at 1 cm/yr. The results (Figure 7) exactly parallel those of model series 1; the AP, SP2, and PS modes are selected with increasing viscosity.

Figure 7.

Model series 2. Two-layer models of lithosphere extension. Upper layer strain-softening frictional-plastic angle of internal friction ϕ(ε) decreases from 15° to 2° as second invariant of strain (I2)1/2 = ε = 0.5–1.5 increases; cohesion C = 0, lower layer uniform viscosity. (a)–(c) Different values of viscosity. Vertical side boundaries extend at 0.5 cm/yr. Extension is 40 km. Weak seed, ϕ = 2°, in center at base of frictional-plastic layer seeds deformation.

3.4. Model Series 3, Two-Layer, Depth-Dependent Rheology

[39] Models series 3 attempts to address one limitation of model series 1 and 2. The brittle-ductile transition in series 1 and 2 is inherently fixed to the material interface between the upper (brittle) and lower (viscous) layer. This inhibits variation with depth of the brittle ductile transition resulting from variations in viscosity or strain rate. In this model series we now allow for depth-dependent rheology in the lower layer, that is it may behave either in a viscous or frictional-plastic manner. The viscous deformation in the lower layer depends on depth through η = η0 exp [−γ(zzL/2)], where zL = 120 km is the lithosphere thickness, and γ = 0.25 km−1 controls the depth dependence. This viscosity relation gives a linearized approximation of a dry olivine power law rheology [Karato and Wu, 1993] at midlithospheric depths (Figure 8); η0 is chosen such that the brittle-ductile transition is at 60 km depth for the reference case.

Figure 8.

Linearized depth-dependent viscous rheology employed in model series 3. The viscous deformation in the lower layer depends on depth through η = η0 exp [−γ (zzL/2)], where zL = 120 km is the lithosphere thickness, and γ = 0.25 km−1 controls the depth dependence. This viscosity relation gives a linearized approximation of a dry olivine power law rheology [Karato and Wu, 1993] at midlithospheric depths. For this type of rheology and for reference thermal conditions, scaling viscosity μ0 ∼ 1.5 × 1023 and γ ∼ 0.25 km−1. The η0 is chosen such that the brittle-ductile transition is at 60 km depth for the reference case.

[40] In addition to varying the nominal scaling viscosity of the lower layer we include one model where we vary equivalently the boundary velocity by one order of magnitude. Figure 9a shows the reference case, with a frictional-plastic strain-softening upper layer and a lower layer with the same frictional plastic parameters ϕ = 15–2° and a scaling viscosity of η0 = 1.5 × 1023 Pa s. Deformation of the plastic layer is highly asymmetric. When the velocity is increased to 10 cm/yr (Figure 9b) the extension mode is symmetric. Viscous shear stress along the base of the plastic layer inhibits the development of the asymmetric mode. Figure 9c shows the equivalent case where the scaling viscosity is increased to η0 = 1.5 × 1024 Pa s. Clearly, increasing viscosity or the strain rate by the equivalent amount gives the same result.

Figure 9.

Model series 3. Two-layer models of lithosphere extension. Upper layer strain-softening frictional-plastic angle of internal friction ϕ(ε) decreases from 15° to 2° as second invariant of strain (I2)1/2 = ε = 0.5–1.5 increases; cohesion C = 0, lower layer either frictional-plastic with same parameters as upper layer or with viscosity depending exponentially on depth through η = η0 exp [−γ (zzL/2)]. Vertical side boundaries extend at 0.5 cm/yr. Extension is 40 km. Weak seed, ϕ = 2°, in center at base of frictional-plastic layer seeds deformation.

3.5. Model Series 4, Three-Layer, Viscous Layer Underlain by Low-Viscosity Asthenosphere

[41] The series 4 models are the same as those of series 2 with the addition of a 480 km thick, 1021 Pa s uniform linear viscous asthenospheric reservoir below the viscous lower lithosphere. The reservoir is passive and is only subject to a uniform low-velocity inflow through the side boundaries that mass balances the system for the extension of the overlying lithosphere. Introduction of the asthenosphere facilitates an overall isostatic balance and replaces the fixed base. It also introduces tractions, owing to the coupling between the lithosphere and asthenosphere, which is more realistic than the free slip boundary condition. Series 4 therefore provides a link between the simple models presented here and fully thermomechanically coupled ones [Huismans and Beaumont, 2002, 2003].

[42] In the results shown (Figure 10) the lower lithosphere viscosity ranges from 1021 to 1023 Pa s. The results are essentially the same as those of the series 2 models except the extending system is isostatically compensated by flow in the asthenosphere and not by flow in the viscous lower lithosphere. The same mode progression, AP, SP2, and PS, is selected with increasing viscosity. The results demonstrate that the same behavior in regard to mode selection occurs in the more realistic isostatically compensated models, and that the lack of isostasy in the simple analysis, and in the series 1–3 models, is not a overriding factor in the mode selection. The series 4 models compare well with the coupled thermomechanical model results [Huismans and Beaumont, 2002, 2003].

Figure 10.

Model series 4. Three-layer models of lithosphere extension, including low-viscosity asthenosphere. Upper lithosphere layer strain-softening frictional-plastic angle of friction ϕ(ε) decreases from 15° to 2° as second invariant of strain (I2)1/2 = ε = 0.5–1.5 increases; cohesion C = 0, lower lithosphere layer uniform viscosity. Asthenosphere layer uniform viscosity η = 1 × 1021 Pa s. (a)–(c) Different values of “lithosphere” viscosity. Vertical side boundaries extend at 0.5 cm/yr. Extension is 40 km. Weak seed, ϕ = 2°, in center at base of frictional-plastic layer seeds deformation.

4. Comparison of the Dissipation Analysis and the Numerical Model Results

[43] The results of the numerical models are in qualitative agreement with the predictions of the dissipation analysis (Figure 3) in that the modes selected for the strain-softened, SS, state change from AP to SP2 to PS as the viscosity of the lower lithosphere increases. The results also indicate that the transition viscosities ηT1, and ηT2 fall between 1021 and 1022 Pa s, and 1022 and 1023 Pa s, respectively, for the parameter values (Table 1) used in the numerical series 1, 2 and 4 models.

[44] The transition viscosities cannot be predicted solely from the simple analysis because the theory does not predict hb and L (e.g., equations (25) and (27)). However, the theory and numerical models can be shown to be consistent by substituting values of hb and L estimated from the numerical models into the equations for ηT1 and ηT2. For the PS and SP2 cases, for model series 1 and 2, hb approaches 60 km and L ∼ 300 km. For these parameter values and those from Table 1 the total internal dissipation of the SP2 mode diverges from that of the PS mode for η > 1022 Pa s. This is consistent with the numerical model behavior where the SP2 mode is replaced by PS when η = 1023 Pa s. Equivalently, in the AP mode, for model series 1 and 2, hb ∼ 5–10 km and L ∼ 300 km. Using these values and those of Table 1 gives ηT1 ∼ 2 × 1021 Pa s, again in good agreement with the corresponding mode change in the numerical models. Of course these results are only approximate estimates, but they provide a strong indication that the simple theory explains the characteristic behavior of the numerical models.

[45] The numerical results are also interesting in regard to the effect of the weak seed. The seed is designed to “seed” the localized deformation in the numerical models and was not needed in the simple theory because the position of the plug was specified a priori and not determined dynamically. The simple theory corresponds to the numerical models in the limit that the seed becomes infinitely small. Even the finite sized seed creates only a minor reduction in the integrated strength of the plastic layer and therefore has a minor influence on the overall results. It is in this regard an interesting result that when the viscosity of the lower layer is 1023 Pa s the seed is ignored and no local deformation occurs, as predicted by the theory. This result is a nice demonstration that inherited weaknesses may not be reactivated during lithospheric extension.

5. Discussion

5.1. Summary of the Dissipation Analysis and Numerical Model Results

[46] In section 2 it was demonstrated that mode selection in models of lithospheric extension can be predicted based on the minimization of rate of energy dissipation. However, our analytical descriptions of the modes may be incomplete, or there may be other lower dissipation modes that we have missed. The dissipation analysis shows only which of the analyzed modes is favored. That the series 1 and 2 finite element model experiments (section 3) confirm the approximate analysis (section 2) indicates that these modes are also accessed as the system evolves, and provides some reassurance that the approximate treatment is acceptable and that other missed modes are not strongly preferred. Moreover, it appears that the basic analytical mode predictions, confirmed by numerical model series 1 and 2, also apply equally to the numerical models with the more advanced rheologies (model series 3) and more realistic basal boundary conditions (model series 4).

[47] The overall results can be summarized by describing the final mode selection in relation to the viscosity of the viscous layer. When the viscous dissipation is negligible, the asymmetric plug (AP) mode is chosen because this is the fastest route to minimize the plastic dissipation by strain softening only one, not two, shear zones during partial strain softening. As viscous dissipation increases, the mode selected will change from asymmetric plug (AP) to strain softened symmetric plug (SP2) at the transition viscosity ηT1. Equation (27) when recast as

display math

can be interpreted in terms of the trade-off between two differential forces, the differential “viscous penalty force” incurred by deforming in the AP as opposed to SP2 mode; and the differential “plastic gain force”, the advantage of the fully strain-softened AP mode over the partially strain-softened SP2 mode. The mode switches at ηT1, when the penalty equals the gain.

[48] There is no critical transition viscosity between the non strain-softened symmetric plug (SP1) and pure shear (PS) modes, but PS becomes progressively advantaged as the penalty of the differential viscous force between PS and SP1 modes increases. This implies that PS will be selected when the viscous dissipation is high, as demonstrated by the numerical model results (section 3). There is, however, a critical transition viscosity, ηT2, between the SP2 and PS modes (Figure 3).

[49] The physical insight provided by the analysis is that the system behavior is determined not by the intrinsic properties of either the plastic or viscous layers but, instead, by the trade-off in their combined behavior necessary to minimize the dissipation, or, equivalently, the trade-off between differential penalty-gain forces.

5.2. Generalization of the Results

[50] The analysis should be expandable to more complex models of the lithosphere comprising several plastic and viscous layers if the analytical modes for these systems can be described in relatively simple terms. In addition, there are other modes, for example, those comprising multiple symmetric and/or asymmetric plugs, that can be added to the analysis. Wijns et al. [2003] have already shown that the viscosity of the lower layer influences the multishear band mode.

[51] The multiple plug modes may help in an assessment of the conditions that favor reactivation of inherited weak faults or shear zones in the upper plastic crust during lithospheric extension. The gain in reactivation is that the plastic dissipation can be reduced by comparison with the SP mode by any multiplug mode that utilizes and reactivates the weak, nominally SS, shears. However, there is an associated “viscous penalty” from the small but finite boundary layer viscous shear between the plastic and viscous layers. Equations equivalent to (27) and (30) can therefore be used to predict when reactivation will and will not lead to minimum dissipation [Buiter et al., 2004, also personal communication, 2004]. If the multiplug, fault reactivation mode results in a differential viscous penalty force that is greater than the differential plastic gain force achieved by reactivation, this mode will not be chosen. The opposite end-member behavior is that the PS mode will be chosen and the whole system will extend by pure shear without fault reactivation, even though the upper crust may be riddled with faults. Under these circumstances there is a clear transition viscosity between the PS and other modes. This is consistent with previous modeling results showing the control of viscosity of the lower layer on the transition between narrow and wide rift modes [Buck et al., 1999].

[52] An additional mode (L. Moresi, personal communication, 2004) in which a plastic shear forms at the base of the upper layer, thereby decoupling it from the viscous layer, can be shown to require higher dissipation than the modes considered here, which explains why it is not selected in the numerical models and probably is not favored in the lithosphere.

5.3. Relationship to Previous Modeling Results: Dominant and Subordinate Rheologies

[53] Huismans and Beaumont [2003] introduced the concept of dominant and subordinate rheologies to explain some aspects of numerical model results. When either the plastic or viscous rheology becomes the main load bearing part of the system it can be considered to be the dominant rheology and may force the weaker, subordinate part of the system into a compliant style of deformation. This compliant style may differ from the intrinsic mode that would have been selected in the absence of forcing by the dominant rheology. This concept can now be quantified in terms of the minimum dissipation analysis and, in particular, in terms of the trade-off between the differential penalty and gain forces.

[54] When the system operates at a subcritical η, that is a value below ηT1 (e.g., equation (27), the AP mode, the intrinsic plastic layer minimum dissipation mode is selected and the plastic rheology dominates. When, with increasing viscous dissipation the SP mode is chosen, the viscous differential penalty force is now large enough that the plastic layer is forced by the dominant viscous layer into a mode that is not the one that would be chosen by the plastic layer alone. At even higher levels of the viscous differential penalty force the viscous rheology becomes even more dominant and forces the subordinate plastic layer to select the PS mode. Thus one rheology can be said to dominate the other. The dominant and subordinate characteristics of the rheologies are determined by the relative sizes of the associate differential penalty and gain forces.

[55] This explanation can be applied to the results of models 2 and 9 [Huismans and Beaumont, 2003] in which asymmetric, AP mode extension occurs when the rifting velocity is 0.3 cm/yr but exactly the same system exhibits SP mode extension when the rifting velocity is 10 cm/yr. The high-velocity increases the viscous dissipation and the differential viscous penalty force more than the differential plastic gain force (equation (30)). The viscous rheology dominates and the mode switches. Essentially the same result is shown in Figure 10a. For even lower extension velocities, e.g., model 8 [Huismans and Beaumont, 2003], asymmetric AP mode extension is selected throughout the rifting, implying that the differential viscous penalty force is always sufficiently small that the viscous rheology is subordinate and therefore the lithosphere extends asymmetrically under the control of the dominant plastic rheology.

5.4. Implications for Natural Systems: Symmetric and Asymmetric Lithospheric Extension

[56] The use of the minimum dissipation principle to analyze lithospheric extension models, as employed here, is valid because we analyze a simple system which has a few well-defined properties, the strain-softening plastic rheologies and linear viscosity. Furthermore, the minimum dissipation approach is not used to construct the full incremental finite deformation of the system as, for example, by Masek and Duncan [1998]. Instead, it is used in an approximate analysis, as by Burbidge and Braun [2002] to choose among a few well-defined modes.

[57] It is also to be expected that equivalent finite element model experiments will give the same results as the approximate analysis, provided that our approximations of these modes (section 2) are reasonably accurate, and that the modal dissipations rank in the same order despite these approximations. The results should be the same because the velocity-based viscous-plastic finite element formulation that we have used [Fullsack, 1995] is based on the self-consistent Galerkin method which also minimizes the rate of energy dissipation.

[58] One conclusion is that this approach is a useful way to analyze and make physical interpretations of numerical model results, and that the same approach may be useful when analyzing physical laboratory model behaviors. However, as Bird and Yuen [1979] have pointed out, there is no guarantee that the dissipation analysis employed here can be directly extrapolated to natural lithospheric extension. The primary reasons are as follows: (1) only a subdomain of the Earth is chosen for analysis and it is not clear if this region can be treated in isolation (for example, a particular restriction is that we analyze a simplified two-dimensional system) and (2) other variations in properties not included in the current models may allow even lower dissipation modes to exist. For example, viscous strain softening could be more important than plastic strain softening. If so, the minimization should also include the variation of the functional (equation (1)) with respect to the dependence of viscosity on strain and, more generally, power law flow and temperature [Bird and Yuen, 1979]. In addition, an equivalent analysis of elastic-viscous-plastic behavior is required to determine the importance of elasticity.

6. Primary Conclusions

[59] We started this paper with the general problem of the controls on the modes of lithospheric extension and under what circumstances is extension asymmetric, symmetric, or pure shear without fault reactivation. The results presented illustrate the application of the principle of minimum rate of dissipation to a simplified model of the lithosphere comprising a cohesive, or frictional-plastic upper layer bonded to an underlying linear viscous lower lithosphere. The main conclusions (Figure 11) apply to these simple models but the principles may also apply to the natural system. Figure 11 illustrates the predicted mode selection pathways during strain softening of the plastic upper lithosphere (NSS → PSS → SS) and the control of the mode selection by the viscosity of the underlying viscous layer. These conclusions are derived from an approximate analysis and are supported by the finite element experiments.

Figure 11.

Diagram summarizing main results of dissipation analysis and numerical model results of simple two-layer models of extension.

[60] Depending on the viscous dissipation the mode that is chosen during the early phase of extension, before the onset of strain softening (i.e., the NSS regime), will be the symmetric plug, SP1, or pure shear, PS, if the viscosity of the lower layer (equation (25)) is greater than the critical value, ηT2. Later, during partial strain softening, PSS, systems with viscosities greater than ηT2 will ignore the possibility of strain softening on the shears bounding the plug and remain PS as they evolve. Systems with viscosity lower than ηT2 will adopt either the asymmetric plug, AP mode, or the strain-softened symmetric plug, SP2 mode, if the viscosity is greater than the critical value, ηT1 (equation (27)).

[61] The mode selection depends on the minimization of the internal rate of energy dissipation which is used to derive equations (25) and (27). The processes can be understood physically in terms of the trade-off between the gain of reduced dissipation by selecting a lower-dissipation mode in the plastic layer versus the “penalty” of higher viscous dissipation incurred in the viscous layer. This trade-off can be expressed through the differential penalty and gain forces (equation (30)) by the concept of dominant and subordinate rheologies (section 5.3).

[62] The potential application to lithospheric extension is that the AP mode is preferred when the dissipation in the viscous crust or lower lithosphere is small, for any of the reasons that make η subcritical with respect to ηT1 (equation (27)), for example, extension is slow and the strain rates are low, or the effective viscosity is intrinsically low because the viscous part of the lithosphere is hot and weak. The AP mode is therefore likely to be preferred for slow extension of viscously weak lithosphere, but there are other possibilities that make ηT1 greater than η (equation (27) or (30)), for example, factors that make ηT1 large; so the selection is always a relative one among the factors that make ηT1 large.

[63] Equivalently, the SP2 mode is predicted for faster extension of colder, higher effective viscosity lithosphere but this is not unique, any factor that makes ηT1 less than η and ηT2 greater than η (equation (25)) will select this mode. Again, the choice is relative.

[64] Whether the PS mode is observed on the Earth is debatable. If it is not, it follows that lithospheric systems are subcritical with respect to ηT2. By implication, the differential viscous penalty force is never large enough to outweigh the differential plastic gain force for this mode. That is, the viscous rheology never dominates so strongly that fault reactivation, or plastic stain softening on focused shears is suppressed. This conclusion is probably wrong in an absolute sense in regard to natural systems but it offers insight into the system behavior and the mechanisms that control the system.

[65] Analysis of the simple two-layer system in terms of dissipation and force balance leads to some equivalent results but it appears that dissipation is the more sensitive analysis tool for systems composed of both viscous and plastic components. This difference occurs because dissipation in the viscous part of the system is proportional to the square of strain rate, whereas forces are proportional to strain rate.

Appendix A:: Force Balance Estimate of Mode Switching

[66] Here we give a parallel method to estimate mode changes using a comparison of the total force required for extension of the two-layer system under consideration. We assume that the mode which requires the least total force will be the mode in which the systems deforms. The force balance approach is complementary to the dissipation analysis. The results of the force balance analysis are consistent with the mode transition between PS and SP predicted by the dissipation analysis. The force balance approach appears, however, not sensitive enough to discriminate between SP and AP modes. In calculating the total force we use the same geometry and quantities as in the dissipation analysis (Figure 2). The total force combines a viscous and a plastic component: F = Fv + Fp. The plastic force is given by the vertically integrated strength of the plastic layer:

display math

The total viscous force is the sum of the integrated stress τxx on a vertical section near or at the boundary and the integrated shear stresses along the horizontal boundaries at top (and bottom) of the viscous layer:

display math

In the viscous pure shear case τxx = 2ηequation imagexx and τxz = 0. As equation imagexx = V/L, the total force needed to extend the two-layer system in pure shear mode is given by

display math

for x > 0. In the symmetric plug mode case shear terms in the boundary layer with thickness hb are given by equation imagexx = V/L (1 − equation image) and equation imagexz = equation imageV(1 − equation image)/hb (equation (13), for x > 0 and local reference frame with z = 0 at the base of the boundary layer), while pure shear affects the remaining lower part of the layer with thickness (hvhb). Thus the total force needed to extend the two-layer system in symmetric plug mode is approximately given by

display math

In the asymmetric plug mode case shear terms in the boundary layer with thickness hb are given by equation imagexx = 2V/L(1 −equation image) and equation imagexz = V(1 − equation image)/hb (equation (22)), while pure shear affects the remaining lower part of the layer with thickness (hvhb). Thus the total force needed to extend the two-layer system in asymmetric plug mode is approximately given by

display math

A comparison with the results for the total dissipation analysis in the pure shear mode, the symmetric plug mode, and asymmetric plug mode cases, respectively, shows strong similarities. Minor differences in multipliers of the respective terms relate to differences in integration constants and the quadratic versus linear strain rate terms in the integrands.

Appendix B:: Evaluation of the Role of Gravity

[67] We first derive an analytical expression for the rate of work against gravity for the PS mode. We then derive an expression for the perturbation in the rate of work against gravity for other cases and use estimates of its value to place bounds on the rate of work against gravity for SP and AP modes. The work against gravity (equation (3)) is defined as

display math

For uniform density and incompressible flow mass conservation during pure shear gives

display math

where h = hp + hv is the total model height (Figure 2, section 2). For pure shear extension (Figure 2a),

display math

and the rate of work against gravity is

display math

We can treat other modes as perturbations of the pure shear mode and write

display math

Let Vz = Vz(x, z) and h(x) = h + Δh(x), where h is the height in the pure shear case. The perturbation must conserve mass:

display math

We now assume that the vertical velocity Vz(x, z = h) is proportional to the height of the perturbation. Then

display math

where ΔVz(x, z) = −∣ΔVz/L. Using mass conservation within any region of the model of width ΔL and assuming overall pure shear, the perturbed velocity field becomes

display math

and the perturbation is

display math

Functional forms of Δ(equation imagexx) and Δh(x) can be calculated for specific modes. Here we use a more general function comprising a series of “box cars” for both functions, amplitudes α/2 and Δh/2 such that

display math

The result is independent of the number of box cars, or their width, provided the perturbation conserves mass. The perturbation of the rate of work against gravity is then

display math

where the pure shear case is taken to be already perturbed. For the case where a narrow region, width L1, is subsiding between two long-wavelength flank uplifts, total width L2, α/2 has amplitude α1/2 over the region of width L1 and α2/2 over the region of width L2, where L1 + L2 = 2L. The perturbation of the rate of work against gravity becomes

display math

Mass conservation requires, L1α1/2 = L2α2/2, and the resulting Δequation imageG is the same as in equation (B11). Although these estimates of Δequation imageG are based on small perturbations in h and Vz, the results are also correct for perturbations in h that are not small, provided that the change in h during the time interval for which Δequation imageG is estimated is small.

[68] We now compare Δequation imageG with the work against gravity for the pure shear case e.g., equation imageGps = −ρgVh2. At a lithospheric scale, for the case of a reference height, h = 100 km, with perturbation in topography, Δh ∼ 1 km, and for variations in vertical velocity, α/2 = 1%, the perturbation of work against gravity is of the order Δequation imageG = 10−4equation imageGps. For the case of Δh ∼ 10 km and variations in vertical velocity α/2 = 10%, Δequation imageG = 10−2equation imageGps.

[69] If for the symmetric or asymmetric plug mode Δh is of the order 5–10 km, and vertical velocities in the plug area deviate order 10–20% from the pure shear case, the associated range of the fractional perturbation Δequation imageG/equation imageGps is [4 × 10−3–1.6 × 10−2].

[70] We have calculated the rate of work against gravity in the numerical models of model series 2. The rate of work against gravity estimated for the pure shear case compares very well with the corresponding theoretical prediction and is 68.5 × 103 W/m. The difference in the rate of work against gravity for the transition between PS and SP modes, and for the transition from SP to AP modes is at most 800 W/m and in general less than that. It is clear that the difference in rate of work against gravity compared to the absolute value for the pure shear case is of the order 10−2 or less. This is consistent with the estimate from the perturbation analysis above. The difference in internal dissipation for the respective mode transitions is in the order of Δ equation imageIPS–SP ∼ 2200 W/m, and Δequation imageISP–AP ∼ 3000 W/m. The contribution of the differential rate of work against gravity (equation (4)) is therefore small compared with the corresponding rate of the internal dissipation differential.


[71] We wish to thank the associate editor Adrian Lenardic and the reviewers, Luc Lavier, Louis Moresi, and an anonymous reviewer. We acknowledge Philippe Fullsack and Glen Stockmal for stimulating discussions. This research was funded through an ACOA-Atlantic Innovation Fund contract, and an IBM-Shared University Research grant. C. Beaumont was funded by the Canada Research Chair in Geodynamics. Beaumont also acknowledges support through the Inco Fellowship of the Canadian Institute for Advanced Research. Numerical calculations used software developed by Philippe Fullsack.