Dynamic interactions between erosion, deposition, and three-dimensional deformation in compressional fold belt settings

Authors


Abstract

[1] This study investigates whether the three-dimensionality of erosion and deposition can influence three-dimensional folding, even when compression is purely convergent. This problem is studied by employing a three-dimensional mathematical model incorporating dynamic thin-plate deformation coupled with fluvial and hillslope sediment transport. The approach taken is to impose several fundamentally different sediment routing patterns (by changing the initial regional topographic slope, β, and the ratio of the characteristic timescales for deformation to fluvial surface processes, R) and to investigate how deformation responds. Numerical results indicate that when β and/or R are small (i.e., initial topography is regionally flat and/or rates of surface processes are slow compared to rate of imposed deformation), folding is two-dimensional and unaffected by surface processes since loads created by surface processes are imposed after deformation. This situation favors development of small-scale drainage networks transporting sediment from anticlines to adjacent basins. When β and/or R are relatively large, the development of a large-scale transverse drainage network, which can be maintained during folding, is able to strongly amplify and localize folding. This leads to interesting features where the largest transverse rivers coincide with the greatest structural and topographic amplitude of doubly plunging folds, which is caused by erosion-enhanced deformation. For intermediate parameter values, folding and surface processes interact in a complex manner, giving rise to three-dimensional en echelon structures that migrate in a transient manner. This study demonstrates that under some circumstances, one can anticipate major two-way interaction between surface processes and three-dimensional deformation.

1. Introduction

[2] Erosion and deposition are known to have a major influence on the mechanical and thermal evolution of compressional orogens. Mass redistribution related to surface processes modifies the stress state of the lithosphere, which may induce internal deformation and an overall isostatic adjustment [Beaumont et al., 2000]. In addition, prolonged erosion in rapidly deforming regions can lead to advection of hot weak rock close to the surface, leading to a positive feedback, giving rise to enhanced deformation, localization, and further exhumation and weakening [Zeitler et al., 2001a, 2001b; Koons et al., 2003]. The general effect of erosion and deposition is thus to enhance and localize deformation.

[3] Much of what we know about how surface processes influence deformation comes from numerically modeling two-dimensional (2-D) vertical sections (e.g., plane strain) through the crust coupled to 1-D eroding surface profiles [e.g., Beaumont et al., 1992; Avouac and Burov, 1996; Willett, 1999; Beaumont et al., 2000]. This approach is justified on the basis that many natural orogens form in response to nearly pure convergent tectonics. However, erosion and deposition act in three dimensions and generate complex topography which may bear little or no relation to the direction of convergence. This raises the question of whether the three-dimensionality of erosion and deposition can induce three-dimensional deformation, even under pure convergence.

[4] Erosion and deposition have been demonstrated to have an important influence on deformation at relatively large spatial scales (i.e., at the scale of natural orogens [e.g., Beaumont et al., 1992; Avouac and Burov, 1996; Willett, 1999]). However, the ability of erosion and deposition to modify deformation becomes less clear at scales less than 100 km, where the rock strength and rigidity are such that isostasy is no longer compensated locally. One argument against the possibility that three-dimensional erosion and deposition can induce three-dimensional deformation is that the short-wavelength loads related, for example, to erosion of closely spaced river valleys would be effectively filtered out due to the elastic rigidity of the crust (if the effective elastic thickness is on the order of 10 km or greater). Thus the three-dimensional loads related to surface processes at relatively small scales would not be experienced by the deforming crust. Some field studies appear to confirm this view [e.g., Gilchrist et al., 1994; Leonard, 2002]. However, an alternative possibility is that the crust may be deforming plastically, in which case even relatively short wavelength loads may induce local deformation. This type of behavior is characteristic of critical wedges, where small perturbations to the equilibrium taper (caused, for example, by erosion) may drive the local crust into a stress state where plastic failure is satisfied. Deformation must then take place in order to reduce stresses and maintain a stable stress state [Chapple, 1978; Stockmal, 1983; Davis et al., 1983; Koons, 1994]. Once again, some studies provide evidence in support of this possibility [Norris and Cooper, 1997; Pavlis et al., 1997; Simpson, 2004c]. The fact that both scenarios are possible reflects the importance of spatial scale, stress levels, and rheology in determining how strongly surface processes influence deformation.

[5] This paper examines the effect of three-dimensional mass redistribution related to erosion and deposition on three-dimensional folding using a mathematical model. The approach taken is to impose several fundamentally different sediment routing patterns and to investigate how local deformation responds. The nature of the sediment routing system is varied by changing the initial regional topographic slope and the relative rates of surface processes and deformation. Relatively steep slopes and/or rapid surface processes relative to deformation favor the development of transverse river patterns, which transport mass efficiently out of the orogen, whereas relatively shallow topographic slopes and/or slow surface processes relative to deformation favor longitudinal river development and short-range sediment transport from anticlines to local internal basins. The differing pattern of mass redistribution between these cases is shown to exert an important influence on three-dimensional deformation and on the topography of fold belts.

[6] The mathematical model employed in this study consists of a quasi three-dimensional elastic-plastic crust overlying an inviscid (very weak) substrate and subjected to compressive tectonic deformation, erosion, and deposition. This model is described in detail by Simpson [2004b]. It is also reproduced in a more condensed nondimensional form in Appendix A for completeness. The mechanical model is based on thin-plate theory [Timoshenko and Woinowsky-Krieger, 1959]. Elastic thin-plate or thin-beam models are mostly used within the geological literature to determine the response of the lithosphere to vertical loads [Jordan, 1981; Karner and Watts, 1983]. Some studies have shown that in-plane (horizontal) loads acting on preexisting structures may generate significant additional vertical deflections, even when the applied stresses are well below the critical elastic buckling stress [Lambeck, 1983a, 1983b; van Balen et al., 1998; van Balen and Podladchikov, 1998]. In this study the in-plane loads actually generate buckling. This does not require excessive stresses because a plastic rheology (in addition to elasticity) is considered. This type of continuous deformation is known as plastic buckling and is especially well known in the engineering literature [e.g., Bloom and Coffin, 2001; see also Martinod and Davy, 1992; Zhang et al., 1996; Gerbault et al., 1999]. Note that the model does not permit development of discontinuous deformation (e.g., faulting). This limitation means that the model is applicable to deformation characteristic of fold belts (e.g., Zagros) at relatively small strains before faulting becomes the dominant mode of shortening. An alternative possibility to achieve in-plane deformation at reasonable stress levels would have been to include some form of viscous deformation [e.g., see Biot, 1961; Lambeck, 1983a, 1983b], which is not incorporated here. An elastic-plastic rheology is assumed here because this combination is typically assumed to dominate over viscous effects in the upper part of the crust.

[7] In previous work it has been demonstrated that surface processes only exert a significant influence on folding when the rates of surface mass redistribution are rapid compared to the rate of imposed deformation [Simpson, 2004a, 2004b]. In this case, erosion and deposition tend to enhance the ability to fold and to increase the fold wavelength relative to deformation in the presence of less efficient surface processes. For regionally flat initial surfaces it has been shown that the incorporation of surface processes does not generate local 3-D deformation [Simpson, 2004b]. In contrast to this previous work, the current paper identifies cases where the 3-D nature of erosion and surface can induce important local 3-D deformation (i.e., doubly plunging fold structures).

2. Coupled Mechanical–Surface Process Model

[8] The mathematical formulation of the model investigated is presented by Simpson [2004b] and is described only briefly here. The equations solved are also listed here in Appendix A in nondimensional form. The mechanical part of the model consists of a 3-D thin plate overlying an inviscid (very weak) substrate (Figure 1). The rheology of the plate is elastic-plastic (von Mises), depending on the local stress state. Physically, the plate is intended to represent the upper crust overlying a weak layer (e.g., lower crust or layer such as evaporites within the upper crust). The plate is pushed from one side and responds by the development of large-scale buckle folds. The surface topography is advected with the deforming plate and changes due to fluvial and hillslope sediment transport [Simpson and Schlunegger, 2003]. The mass redistribution resulting from erosion and deposition creates loads on the underlying plate, which induces a flexural-isostatic (passive) response and which influences subsequent internal deformation (dynamic response). The mass redistribution also modifies the plate thickness and therefore the depth-integrated strength distribution [see also Simpson, 2004a].

Figure 1.

Schematic of model consisting of an elastic-plastic plate overlying an inviscid substrate. A horizontal displacement is applied to one end of the plate, which causes folding. Erosion and deposition on the surface of the plate modify the distribution of vertical loads and the behavior of folding. The model is described more fully by Simpson [2004b].

[9] The model setup considered consists of an initially square domain, which is compressed from one edge and blocked from the opposite boundary. Both of these boundaries are also constrained to have zero vertical deformation. The lateral boundaries are constrained against lateral movement. Thus imposed collision is purely two-dimensional, and the collision belt is in plane strain at the scale of the model domain. Simulations have also been performed where the lateral boundaries are unconstrained in a direction perpendicular to the compression direction; however, this has no influence on observed fold geometries. The plate has an initially uniform thickness. The initial topography consists of a planar surface sloping uniformly toward the compressed boundary. The topography at the lower boundary has a fixed elevation, whereas the boundary conditions at the lateral and upper boundaries are zero sediment flux perpendicular to the boundary. A uniform distribution of small-amplitude random noise is added to the initial topography and initial plate midplane in order to provide some irregularity.

[10] Nondimensional parameter values used in calculations are listed in Table 1. The meaning of these parameters and characteristic scales is discussed in Appendix A [see also Simpson, 2004a, 2004b]. The parameters are chosen so that the model results are most applicable to deformation in a fold belt setting at small to intermediate spatial scales (i.e., ≤200 km). Thus, for example, if the characteristic length scale L (the initial length of the model domain) is chosen to be 180 km, a ratio for h0/L of 0.0556 (Table 1) corresponds to a plate thickness of 10 km. Similarly, if the characteristic stress E (Young's modulus) is chosen to have a reasonable value of 1 × 1011 Pa, a ratio for σ0/E of 0.001 (Table 1) implies a plastic yield stress of 100 MPa. This stress level represents the likely maximum order of magnitude for differential stress in the upper crust [see Brace and Kohlstedt, 1980]. The fact that the plastic yield stress is chosen to coincide with the likely upper limit for stresses in the crust means that the influence exerted by surface processes on deformation is conservative. The most important parameter with respect to how strongly surface processes influence deformation is the ratio of characteristic timescales for imposed deformation and fluvial erosion (see equation (A21)). Because this parameter is poorly constrained and is anticipated to exhibit considerable variation depending on the tectonic and climate setting, simulations are performed for a variety of values.

Table 1. Governing Nondimensional Parameters and Their Values Used in Calculationsa
ParameterValue
equation image1.76 × 10−5
equation image4.06 × 10−2
equation image4.76 × 10−2
ν0.3
equation image0.0556
equation image0.001
n2
equation image2.78 × 10−4
equation image2.85

3. Model Results

[11] In order to determine how mass redistribution due to erosion and deposition influences three-dimensional deformation, a series of numerical experiments have been performed where the nature of the sediment routing system is varied by changing the regional topographic slope, β (= H/L, where H is initial maximum absolute change in topographic elevation across the model domain of length L), and the relative rates of surface processes versus deformation, R. Simulations for three different slopes (0%, 1%, and 2%, all other parameters are held constant) are presented. These simulations were chosen because they demonstrate the complete range of behavior observed in a larger number of simulations performed. In addition, one simulation where erosion and deposition are omitted is also presented, which serves as a reference case. Note that all results are presented in nondimensional form. Thus, for example, all variables with length as a dimension are scaled relative to L (the initial length of the model domain), and all times are scaled relative to the characteristic deformation timescale, L/equation image (see Appendix A). Note also that the numerical value for the total amount of shortening reported in Figures 23456 is also equal to the nondimensional time.

Figure 2.

Contour map (light shades are high and dark shades are low) of topography in the absence of surface erosion and deposition as deformation progresses with increasing shortening (from top to bottom). The plate is pushed from the lower boundary toward the upper boundary. The plate initially deforms into a series of quasi-cylindrical, low-amplitude folds and then later buckles into a train of large-amplitude cylindrical folds.

Figure 3.

Maximum deflection plotted as a function of increasing shortening. Results from four different simulations are shown, however, results from two simulations plot on the same curve (the case when erosion and deposition is omitted is identical to the case when the initial topography is regionally flat and erosion and deposition are included). The initial buckling instability is indicated on each curve with an arrow. Any deformation prior to buckling is flexural-isostatic related to erosional unloading of the crust. Note that as the initial topographic slope is increased, the onset of buckling is decreased, and the rate of vertical deflection increases. This is due to the ability of mass redistribution to enhance deformation, caused both by the greater power of rivers and the larger mass above base level available to be eroded and applied as a load on the underlying crust in steeper margins.

Figure 4.

Contour map (light shades are high and dark shades are low) of topography which had no initial regional slope as deformation progresses with increasing shortening (from top to bottom). Since there is no regional slope, the sediment routing system that develops entirely reflects the growth of local structures. Note that even though significant erosion and deposition take place, the deformation is identical to the case when surface processes were omitted (see Figure 2).

Figure 5.

Plots of the maximum, mean, and minimum topography (determined by averaging in the strike-parallel direction across the entire model domain) for three simulations with different initial regional topographic slopes and shown with increasing shortening (from top to bottom). As the topographic slope increases, deformation becomes progressively more localized in the region where dissection is the greatest (i.e., near the upper boundary). For intermediate initial slopes, the profiles are complex due to the en echelon nature of structures.

Figure 6.

Contour map (light shades are high and dark shades are low) of topography which had a 1% initial regional slope as deformation progresses with increasing shortening (from top to bottom). The (a) initially developed transverse drainage system becomes abandoned once (b) buckling begins leaving wind gaps and causing reversal of some drainage segments. Note that even though the applied compression is purely compressive, (c, d) the deformation eventually becomes fully three-dimensional. This stems from interference between deformation and the heterogeneous distribution of loads related to erosion and deposition. The result is the formation of complex en echelon structures, which change position through time. This has a major impact on the drainage system and the location of depocenters (compare Figures 6c and 6d).

3.1. Deformation in the Absence of Erosion and Deposition

[12] Deformation in the absence of erosion and deposition is characterized by the initiation, amplification, and locking of a regular train of buckle folds (Figure 2). These folds begin as isolated basins and ridges localized on initial irregularities in the plate. With increasing shortening these basins and ridges spread laterally and link up, forming a quasi-cylindrical array of low-amplitude (<5.5 × 10−4), relatively short wavelength (∼0.1667) folds. True buckling does not occur until after a total shortening of almost 0.11 (Figure 2c) and is marked by an abrupt increase in fold amplitude (Figure 3). The resulting structures are nearly perfectly cylindrical and have a wavelength of ∼0.28 and amplitudes of 0.0056–0.0111. Buckling is immediately followed by lockup, which is marked by a progressive decrease in the rate of amplitude growth with increasing shortening. This postbuckling lockup phase involves passive folding as opposed to folding in response to a dynamic instability which occurs when buckling is initiated. The end result is a series of cylindrical (two-dimensional) folds that have a nearly uniform amplitude in the across-strike direction.

3.2. Erosion, Deposition, and Deformation of an Initially Flat Surface (β = 0)

[13] When erosion and deposition are included on the surface of the initially flat plate, a sediment routing system develops in response to local fold structures (Figure 4). However, since the surface is regionally flat, sediment is simply eroded off anticlines and deposited in neighboring synclines. This results in a simple drainage system on anticlines consisting of a closely spaced network of uniformly spaced rivers sourced near structural culminations and a relatively uniform depocenter distribution in synclines (Figures 4c and 4d). Although the mass redistribution related to erosion and deposition influences stresses in the underlying plate, this proves insufficient to significantly influence deformation (compare Figures 2 and 4). This is due to three factors. First, for the model parameters used the rates of surface processes are slow compared to the rate of rock uplift generated during active buckling. Thus the load created by erosion and deposition does not contribute to the dynamic folding instability but is imposed during the passive (lockup) folding phase, when rock uplift rates have significantly decreased. Second, since the load is created after active folding, when stresses have been somewhat relieved, the crust where the load is imposed is largely elastic (with the exception of plastic fold hinges) and is relatively rigid. Third, since there is no regional topographic slope, all erosion and deposition that takes place must do so in response to existing folds. This implies a time lag between the timing of folding and the timing of sediment transport. Once again, the load due to sediment transport is imposed after active folding, making it difficult for surface processes to influence folding. The end result is a train of cylindrical folds which have a nearly uniform amplitude in the across-strike direction (Figure 5c and 5d) and which, although influenced (in terms of topography) by surface processes, appear very similar to the structures observed when surface processes had been absent.

3.3. Erosion, Deposition, and Deformation of a Gently Sloping Surface (β = 0.01)

[14] When the initial topographic surface has a finite regional slope, the sediment routing system that forms reflects competition between the regional gradient and the gradient generated in response to local folding. In the period before any folding instability is initiated a transverse drainage system becomes established on the sloping surface (Figure 6a). This results in long-range mass transport out of the model domain, which causes the elastic-plastic plate to be bent upward into a quasi-cylindrical flexure (Figure 5e). The rock uplift rates associated with this erosion-induced deformation, related to flexural-isostatic rebound, are small, and rivers can easily maintain their course across the actively growing flexure. This is not the case when the plate begins to buckle in response to tectonic compression after total shortening of ∼0.028 (Figures 5f and 6b). The rock uplift rate on anticlines outpaces the rate of downcutting in rivers, causing transverse rivers to become abandoned, leaving wind gaps in their place and causing deposition of sediment in internal basins. Subsequent deformation is controlled by the buckling of a plate subjected to strong laterally varying surface loads, which were imposed on the plate mainly during the initial period when the transverse drainage became developed. The combined result of these two features (i.e., buckling and heterogeneous loads) is that the cylindrical folds break up into a series of en echelon structures (Figures 6c and 6d). This complex three-dimensional deformation strongly influences drainage patterns and depocenters, leading to an even more heterogeneous distribution in surface loads, which further modify deformation. Commonly observed features include exhumation of previously buried basin sediments, burial of previously exposed anticlines, wind gaps, and transverse drainage systems which cut across anticlinal structures (compare Figures 6c and 6d). These results demonstrate that very complex structures may arise from interactions between three-dimensional deformation and three-dimensional mass redistribution related to erosion and deformation.

3.4. Erosion, Deposition, and Deformation of a Steep Initial Surface (β = 0.02)

[15] Increasing the regional topographic gradient has two potentially important means of influencing deformation and the evolving sediment routing system. First, it increases the power of rivers to cut across actively growing structures, which can affect the pattern of mass redistribution. Second, it increases the total mass of rock above base level which can be eroded and thus applied as a load to the underlying plate. Both of these effects are observed in simulations with relatively steep (≥2%) initial topographic surfaces. Initially, one observes the development of a transverse drainage system which generates flexural-isostatic rebound and the formation of a quasi-cylindrical upwarp (Figures 7a and 5i). This upwarp propagates progressively up the surface as rivers cut deep canyons in their headwaters. When buckling related to tectonic compression is initiated (after total shortening of 0.022), most rivers are able to maintain their course across actively growing folds and cut deep gorges (Figures 7b and 5j). This unloading dramatically amplifies and localizes deformation (Figures 7c and 5k). The localization of deformation prevents deformation from breaking up into more complex en echelon structures, such as observed for more gently sloping margins. The amplification of deformation results in folds where the largest amplitudes coincide broadly with the greatest depth of incision. Owing to the significant regional slope, this location tends to be near the upper boundary, where a megaantiform is developed (Figures 5l and 7d). This megaantiform is dome shaped (doubly plunging) and is transected at its axial culmination and highest topographic position by a large transverse river (Figure 7d). As discussed in section 4, this remarkable feature is also observed in some natural orogens. Once again, this simulation indicates that the three-dimensional nature of erosion and deposition can exert an important influence on three-dimensional deformation.

Figure 7.

Contour map (light shades are high and dark shades are low) of topography which had a 2% initial regional slope as deformation progresses with increasing shortening (from top to bottom). (a) Much of the initially developed transverse drainage system is able to maintain its course across (b) growing structures during buckling. This causes long-range mass transport out of the system, which dramatically localizes and amplifies deformation, particularly where the depth of incision is the greatest near the upper boundary. Note that the largest river system in Figure 7d cuts the largest antiform transversely at its highest position in a series of deep gorges. Similar structures are observed in natural orogens.

3.5. Importance of the Relative Rates of Surface Processes and Deformation

[16] In addition to the regional slope β, the relative rates of surface processes and deformation (i.e., R) has a strong control on the nature of the sediment routing system developed and thus on the influence exerted by surface mass redistribution on folding (Figure 8). For any given topographic slope, transverse drainage has a greater chance of being maintained during folding, and surface processes have a greater chance of strongly influencing folding, if the rates of surface processes are fast relative to the rate of imposed deformation (i.e., large R (see equation (A21) and Figure 8)). On the other hand, if surface processes are very slow relative to the rate of deformation (low R), transverse drainage becomes abandoned during folding, resulting in a local drainage system that exerts no influence on folding. As the regional topographic slope is increased, transverse drainage can be maintained for smaller ratios of R. These model observations are consistent with conceptual models developed to explain drainage patterns in fold and thrust belts, which are typically understood in terms of competition between rates of fluvial incision, aggradation, and rigid block uplift [e.g., Burbank et al., 1996]. Note that as the regional topographic slope is decreased and the rate of surface processes relative to deformation is increased (R is increased), the possibility for developing complex 3-D deformation is reduced (i.e., the “intermediate” field in Figure 8 becomes smaller). This result is consistent with the interpretation that the complex 3-D deformation in the model results from interaction between folding and a complex load distribution related to a former transverse drainage distribution, which is more difficult to achieve as the regional topographic slope approaches zero.

Figure 8.

Phase diagram showing how the ability of surface processes to influence deformation depends on two critical parameters: the influence of the initial regional topographic slope, β, and the ratio of the characteristic timescale for imposed deformation to the characteristic timescale of fluvial erosion, R (see equation (A21)). Dotted lines show the approximate boundary separating different styles of deformation, determined on the basis of different simulations (shown as circles, crosses, and pluses). Surface processes exert no influence on folding when β and R are relatively low (e.g., see Figure 4), whereas they strongly amplify and localize deformation when β and R are relatively large (e.g., Figure 7). For intermediate values, complex en echelon fold patterns are observed (Figure 6), which result from the three-dimensional folding of a plate with strongly heterogeneous loads (created by surface mass redistribution).

4. Discussion and Implications

[17] Model results indicate that the ability of surface processes to influence folding depends on the pattern of the sediment routing system, which itself depends on the regional topographic slope β and the rates of surface processes relative to the rate of imposed deformation R (Figure 8). When β and/or R are small, folding in response to pure compression is purely two-dimensional and is effectively uncoupled from surface processes. Folding under these conditions leads to the formation of linear basins and ridges characterized by a simple drainage system lacking large, throughgoing transverse rivers. At the other extreme, when β and/or R are large, transverse rivers are able to maintain their course across growing structures and can actively amplify and localize folding. For intermediate parameter ranges, surface processes and deformation interact to generate en echelon structures which migrate laterally with time (i.e., they do not remain localized). This situation favors complex drainage patterns consisting of transverse and lateral rivers, wind gaps, and highly transient features such as exhumation of previously buried basin sediments and burial of previously exposed anticlines.

[18] The important influence exerted by the nature of the sediment routing pattern on folding is related to the length scale over which sediment transport takes place. If conditions are such that transverse drainage and large-scale connectivity of the drainage network can be maintained, then the correlation length of sediment transport is equal to the size of the system, implying long-range sediment transport. Thus mass is efficiently transported from regions undergoing rock uplift to regions undergoing subsidence, which has the effect of amplifying and localizing folding [see Simpson, 2004a]. If, on the other hand, transverse drainage cannot be maintained (i.e., the drainage network cannot remain connected on a large scale), the correlation length scale of the drainage network is reduced to half the fold wavelength. This reduces the ability of surface processes to influence deformation because at relative small scales, loads related to mass redistribution become buffered by the finite elastic rigidity of the plate. Note that whether the drainage network is connected or disconnected on a large scale will be important for how sensitively deformation responds to large-scale variations in climate.

[19] An important result of this study is that the three-dimensional nature of erosion and deposition can introduce three-dimensional effects during deformation, even when convergence is purely two-dimensional. This deformation is three-dimensional in the sense that fold structures exhibit variations in plunge along strike, while overall collision remains plane strain. Erosion and deposition redistribute mass, which creates a heterogeneous planform distribution of loads and depth-integrated strength. During compression, when the rate of surface processes are of the same order as the rate of applied deformation and for finite regional slopes, folding can interact with this heterogeneous field to cause a complex three-dimensional deformation pattern, consisting of doubly plunging, en echelon folds, which bears no obvious relationship to the sediment routing pattern. These three-dimensional structures are most commonly developed during pure compression, when heterogeneities related to mass redistribution exist prior to deformation. The only cases when this condition is not satisfied in simulations are when surface processes are absent or when the plate topography is initially regionally perfectly flat and therefore had no preexisting sediment routing system (i.e., rivers could not be antecedent). However, note that three-dimensional structures can also develop in response to purely two-dimensional convergence in the absence of surface processes [see also Fletcher, 1995; Audoly, 1999; Bloom and Coffin, 2001; Audoly et al., 2002; Simpson, 2004b]. Together, these results indicate that caution must be exercised when inferring the nature of the displacement field on the basis of the form of deformed structures.

[20] Three-dimensional structures can also develop under pure compression if large-scale connectivity of the drainage network can be maintained in the cross-strike direction due to unloading by large transverse rivers. However, in this case, surface processes do not simply interact with folding, causing interference; they strongly influence the location and 3-D form of individual folds. This may lead to the situation where large transverse rivers intersect doubly plunging fold structures at their highest structural and topographic position (i.e., at the axial culmination). The fact that such transverse rivers are antecedent is important because it means that erosion takes place contemporaneously with fold growth, and therefore unloading can contribute directly to the dynamic folding instability. It is also important because antecedent drainage is anticipated to develop preferentially within structural and topographic lows, not highs as observed. The mechanism of erosion-induced amplification of folds [Simpson, 2004c] provides an explanation for the coincidence of transverse rivers and structural/topographic highs observed in natural orogens [e.g., see Oberlander, 1985; Alvarez, 1999; Zeitler et al., 2001a, 2001b; Finlayson et al., 2002; Koons et al., 2002].

[21] The model results indicate that interactions occurring between surface processes and folding are truly fully coupled (at least under some circumstances, as outlined in section 3.5). This point is already widely appreciated at the orogen scale, where surface processes have been demonstrated to strongly influence deformation and the topography, which then feeds back to modify the patterns of surface processes [e.g., Willett, 1999]. However, at smaller scales it is more common to neglect the influence of surface processes on deformation and treat the substrate as a rigid template onto which surface processes etch topography. This approach may justified if the crust behaves elastically and has an elastic thickness on the order of 10 km or greater because it will behave relatively rigidly [e.g., Gilchrist et al., 1994; Leonard, 2002]. However, if the crust is being subjected to compression and is actively deforming in a plastic manner, even relatively closely spaced loads related to surface mass redistribution may influence deformation. This possibility is consistent with some observations [Norris and Cooper, 1997; Pavlis et al., 1997; Simpson, 2004c] but is yet to be fully appreciated.

[22] Finally, the model results demonstrate that topography may vary in a highly transient manner in response to complex internal deformation which has nothing to do with changing external conditions. Observed model features include exhumation of previously buried basin sediments, burial of previously exposed anticlines, drainage diversion around laterally propagating structures, and drainage reversal. These feature are most likely to be observed in nature when rates of surface processes are fast enough, relative to rates of deformation, to modify deformation but are not fast enough to dominate it.

5. Conclusions

[23] The most important conclusions of this study can be summarized as follows.

[24] 1. The ability of surface processes to influence 3-D folding depends on the pattern of the sediment routing system, which itself depends on the magnitude of the regional topographic slope β and on the ratio of the characteristic timescale for imposed deformation to the characteristic timescale of fluvial erosion, R (see Figure 8).

[25] 2. Large values of β and/or R (i.e., steep regional topographic gradients and/or rapid surface processes compared to rates of imposed deformation) favor (1) long-range sediment transport related to a transverse drainage network and (2) strongly amplified and localized folding. Under these conditions it is common to observe large antecedent transverse rivers intersecting the axial culmination of doubly plunging fold structures. This remarkable feature results from erosion-induced fold amplification.

[26] 3. Small values of β and/or R favor (1) development of “normal” cylindrical, two-dimensional folds unaffected by surface processes and (2) small-scale, “disconnected” drainage systems connecting local anticlines with adjacent basins.

[27] 4. For intermediate values of β and R, one observes (1) doubly plunging, en echelon folds that migrate transiently through time and (2) complex sediment routing systems. These complex features result from the buckling of a plate subjected to strong laterally varying surface loads created by erosion and deposition.

Appendix A:: Model Formulation

[28] The mathematical model employed in this study consists of a thin elastic-plastic plate overlying an inviscid substrate subject to surface erosion and deposition. The model is described in detail by Simpson [2004a, 2004b, 2004c]. The governing equations are listed here in nondimensional form (note, however, that no special symbols are used to indicate that variables are nondimensional). To nondimensionalize the governing equations, the following characteristic scales are adopted [see also Simpson, 2004b]: time, L/equation image, where L is the initial length of the square model domain and equation image is the imposed horizontal boundary velocity; lengths, thickness, and displacements, L; stress, E, where E is Young's modulus; surface fluid flux, αL, where α is the steady state rate of rainfall; bending stress resultant, L2E; and in-plane stress resultant, LE. Solutions to the governing equations were obtained with the finite element method.

A1. Mechanical Model

[29] The mechanical part of the model is based on conventional thin-plate theory and includes finite in-plate deformation [e.g., Timoshenko and Woinowsky-Krieger, 1959]. The formulation presented follows that laid out by Zienkiewicz and Taylor [2000]. The starting point for the formulation is the virtual work expression

equation image

where integration is carried out over an area A (in the x-y plane), δΠ is the change in the potential energy of the system, δu are the displacement increments, δEp are the variations in-plane membrane strains, δKb are the variations in curvature caused by bending, M are the stress resultants related to bending, T are the stress resultants related to in-plane deformation, and b is the load vector, defined respectively as

equation image
equation image
equation image
equation image
equation image
equation image

In these relations, u, v, and w are the displacements of the plate midplane parallel to the x, y, and z axes, respectively, h is the plate thickness, W is initial deflection of the plate prior to loading, S is the stress vector defined as

equation image

and bz accounts for normal loads on the upper and/or lower surface of the plate and is given here by

equation image

where ρ1 is the density of the overlying medium (air), ρ2 is the density of the underlying inviscid medium, ρ0 is the density of eroded/deposited material, s is the surface topography, h0 is the initial uniform plate thickness, and h is the actual plate thickness (h = h0/(2L) + sw). The first two terms on the right-hand side of equation (A9) account for loads created by replacement of material on either side of the plate as it deforms (i.e., hydrostatic restoring pressure), and the third term accounts for loads created by erosion and deposition. The notation w,x indicates differentiation with respect to the subscripted term after the comma (e.g., w,x = ∂w/∂x and w,xx = ∂2w/∂x2).

[30] The rheology of the plate is assumed to be elastic-plastic. Accordingly, the total strains E are split into elastic and plastic components, i.e.,

equation image

where subscripts e and p refer to elastic and plastic, respectively. The total strains (with components εxx, εyy, γxy = 2εxy) can be computed from the relation

equation image

where the in-plane membrane strains Ep are given by

equation image

and the changes in curvature caused by bending Kb are

equation image

The relationship between stresses and elastic strains Ee is given by

equation image

where the nondimensional material matrix De for an isotropic homogeneous plate is

equation image

where ν is Poisson's ratio. Plasticity is assumed to be governed by the following von Mises–type yield criteria:

equation image

where σ0 is the yield stress (which is assumed to be constant throughout the plate). Plastic yielding occurs when F > 0 by an associative flow law

equation image

where λ is the plastic multiplier and δEp are the plastic strain increments. Stresses of yielding points are required to remain on the yield surface as specified by the consistency condition

equation image

A2. Surface Process Model

[31] Erosion and deposition can potentially influence the deforming plate by modifying the distribution of vertical surface loads (via equation (A9)) and by changing the plate thickness h and therefore the depth-integrated stress distribution (via equations (A5) and (A6)). The surface evolution model considered here is a simple transport-limited model which incorporates two main sediment transport processes: concentrative fluvial transport dominant in channels and dispersive transport dominant on hillslopes [Simpson and Schlunegger, 2003]. The model is governed by two equations for two unknown functions, the topographic elevation s and the fluid flux q due to runoff from rainfall. The equations can be written as

equation image

where ∂w/∂t is the uplift/subsidence rate, κ is a hillslope erosional diffusivity, c is a fluvial erosion coefficient, n is a fluvial erosion power exponent, α is the rainfall rate (in excess of infiltration), and q is the surface fluid flux which is determined from the mass balance relation

equation image

where n is a unit vector pointing down the local slope of the topographic surface. The first bracketed term in equation (A19) represents erosion/deposition, whereas the second term accounts for topography uplifted or downwarped by deformation of the plate.

[32] Surface evolution governed by equations (A19) and (A20) is one of the simplest models that naturally leads to the development of incised topography and diffusive hillslopes. This model neglects many effects known to be important in governing evolution of mountainous topography such as landsliding and supply-limited conditions associated, in particular, with bedrock incision that are incorporated in more complete models [e.g., see Tucker and Whipple, 2002, and references therein].

A3. Governing Parameters

[33] The above equations are governed by a total of nine nondimensional parameters, listed in Table 1. The most important of these with respect to coupling between surface processes and folding is the ratio of the characteristic timescale for imposed deformation (τd = L/equation image) to the characteristic timescale of fluvial erosion (τe = L2−n/(cαn)), defined as

equation image

which appears in equation (A19). A large value of R (relative to unity) corresponds to a situation where the rates of fluvial processes are fast relative to the rate of imposed deformation, whereas the reverse is true for a small value of R. Note that the second erosion parameter appearing in equation (A19) (i.e., equation image/(equation imageL), the ratio of the deformation to hillslope erosion timescales) is generally small (because the hillslope diffusivity is so small) and is not important for deformation. The ratio σ0/E (appearing in equation (A16)) is an indication of how close the behavior of the plate corresponds to the elastic limiting case. Relatively large values of σ0/E (more elastic behavior) favor rapid fold growth and long fold wavelengths. As the behavior of the plate becomes more elastic, the influence exerted by surface processes on folding is decreased. The remaining nondimensional parameters important in controlling plate deformation are those appearing in equation (A9) (e.g., ρ2gL/E and ρ0gL/E), which control how strongly gravity influences the folding processes. Large ratios of these parameters inhibit folding and reduce fold wavelengths [Simpson, 2004a].

Acknowledgments

[34] This work has benefited greatly from discussions with Yuri Podladchikov and Alex Densmore and from reviews by Philip Davy, an anonymous reviewer, and the Associate Editor, Mike Ellis. Funding was partially supported by Fritz Schlunegger and the Swiss National Science Foundation (grant 620-57863).

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