Journal of Geophysical Research: Earth Surface

A dynamic model to investigate coupling between erosion, deposition, and three-dimensional (thin-plate) deformation

Authors


Abstract

[1] This paper presents a mathematical model developed to investigate fully coupled interactions between erosion, deposition, and dynamic three-dimensional deformation within fold belt settings. The mechanical part of the model is an elastic-plastic (thin) plate overlying an inviscid substrate. This plate can be compressed from the side and responds by the development of buckle folds. The surface topography is advected along with the underlying deforming plate and can be modified through time by a combination of fluvial and hillslope sediment transport. The resulting mass redistribution creates loads on the plate which then feed back and influence subsequent deformation, erosion, and deposition. Preliminary results calculated with the model indicate that erosion and deposition have two main effects on deformation: The first is to decrease the amount of shortening required to initiate folding, while the second is to increase the fold wavelength. Both effects result from the ability of erosion and deposition to reduce the influence of gravity on deformation. In the extreme case of efficient erosion and deposition, deformation becomes localized on a single large-scale mega-anticline, as opposed to being distributed on a train of anticlines and synclines. This study indicates that erosion and deposition play a major role in governing the deformation and topography of fold belts and the sediment routing system acting within such a setting.

1. Introduction

[2] It is becoming increasingly recognized that surface processes such as erosion and deposition have an important influence on deformation associated with the formation of mountain belts. Nevertheless, much of our understanding regarding how erosion and deposition 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; Burg and Podladchikov, 1999; Beaumont et al., 2000]. This approach provides information on the spatial distribution of displacements in a vertical plane and estimation of the (mean) surface elevation. The vertical 2-D models do not, however, provide any information on how the 3-D deformation field and landscape morphology (including river valleys and ridges) evolve and interact with one another. Does erosion and deposition continue to have an important influence on deformation in 3-D given that erosion in 3-D is typically highly localized in narrow valleys? Can the 3-D nature of erosion and/or deposition induce 3-D deformation even when regional compression is purely two-dimensional? How are the drainage network, surface topography, and location of depocenters influenced by the 3-D nature of deformation? These are some of the questions which have motivated the development of the model presented in this study, which incorporates fully dynamic coupling between (quasi) 3-D deformation and surface erosion and deposition.

[3] A wide variety of modeling approaches have been taken in the literature to investigate coupling between 3-D surface erosion/deposition and deformation, the difference between which depends on the degree of coupling admitted and the intention (and scale) of the study. In the simplest case the 3-D deformation field (often only involving a nonzero vertical component) is assumed to be known (i.e., it is imposed kinematically) and completely uncoupled from evolving surface processes [e.g., Willgoose et al., 1991; Howard, 1994]. This rigid-substrate approach, although inconsistent with findings that deformation can be spatially inhomogeneous and that deformation can be strongly influenced by erosion, is most often applied at relatively small length scales where it is likely to be most valid and is useful for testing how topography and sediment flux respond to various different tectonic and climatic scenarios. In particular, this class of models has highlighted the importance of horizontal velocities in shaping topography [e.g., Willett et al., 2001] and the role of deformation in influencing drainage patterns, depocenters, and sediment fluxes [Tucker and Slingerland, 1994]. The next degree of complexity is achieved by introducing a flexural-isostatic response to erosion and deposition while assuming that the large-scale topography is formed kinematically, either as an initial condition [e.g., Kooi and Beaumont, 1994; Masek et al., 1994; Tucker and Slingerland, 1996] or evolving through time [Johnson and Beaumont, 1995; Garcia-Castellanos et al., 1997; Garcia-Castellanos, 2002; see also King and Ellis, 1990]. These models have demonstrated that introducing coupling of surface processes on flexural deformation tends to amplify vertical rock uplift rates from eroding regions (which acts to maintain topography for longer than had no flexural response been present) and increase the length scales over which erosion and deposition act (due to the elastic rigidity). Nevertheless, such models do not enable investigation of how surface processes influence internal deformation, which leads to formation of the orogen itself. In order for this to be possible, it is necessary to compute the deformation field either with a full 3-D mechanical model [e.g., Braun, 1993; Braun and Beaumont, 1995; Koons et al., 2003; see also Koons, 1994] or with simplified thin-sheet-type mechanical models from which an approximate solution can be obtained [e.g., England and McKenzie, 1982; Vilotte et al., 1982; Fletcher, 1991, 1995; Medvedev and Podladchikov, 1999a, 1999b]. To date, however, little effort has been made in coupling such mechanical models with surface processes.

[4] The aim of this study is to present a new model developed to investigate fully dynamic coupling between erosion, deposition, and (quasi) 3-D deformation. A 2-D (thin-beam) version of this model was studied recently by Simpson [2004]. The mechanical part of the model consists of a thin elastic-plastic plate overlying an inviscid (i.e., very weak) substrate. The plate can be compressed from the side and responds by the development of folds. The surface topography is advected along with the deforming plate and changes in response to coupled fluvial and hillslope sediment transport [Simpson and Schlunegger, 2003]. This mass redistribution creates loads on the plate that can influence subsequent deformation. Note that vertical deformation can result from both a flexural-isostatic response to erosion and/or deposition (i.e., passive response) and internal deformation (i.e., dynamic response).

[5] The model developed is intended to be applicable to relatively small-scale (≤200 km) fold belts developing on the margins of compressive orogens (Figure 1a). This scale is small enough such that individual folds can be resolved and also small enough such that the rigidity due to elasticity must be considered finite (i.e., isostacy is not locally compensated). The simple one-layer rheological stratification of the model is intended to represent the elasto-plastic upper crust (or some other competent unit) overlying a very weak lower crust (or other weak horizon such as evaporites) which is completely decoupled from the mantle (or underlying units). This one-layer model is simpler and implies a smaller scale than the orogen-scale models of Medvedev and Podladchikov [1999a, 1999b] and Garcia-Castellanos [2002], which include more complex compositional stratification (e.g., including crust, mantle) and rheologies (including ductile deformation). The advantage of the model over full 3-D calculations [e.g., Braun, 1993; Braun and Beaumont, 1995] is reduced computation cost at the expense of obtaining an approximate solution. In addition, the formulation presented here does not permit the formation of faults. The main advantages of the model over the classic thin-sheet model [i.e., England and McKenzie, 1982] are (1) the development of folding instabilities; (2) the inclusion of elasticity (known to be important in controlling the influence of surface processes on deformation in the upper crust at relatively small spatial scales [e.g., Gilchrist et al., 1994; Simpson, 2004]) in addition to plasticity; and (3) the gradients in crustal thickness (e.g., that may be induced by erosion and deposition) need not be small.

Figure 1.

Schematic showing the (a) geological setting and the (b) corresponding mathematical model described within this study. The model consists of an elastic-plastic plate (representing either the entire upper crust or a competent package within the upper crust) overlying an inviscid (i.e., very weak) substrate (representing either a weak lower crust or a weak layer such as evaporites within the upper crust). The weak layer is implied to completely decouple the plate deformation from any other deformation occurring at deeper levels. 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 can be applied to the investigation of interactions between erosion, deposition, and 3-D deformation at small to intermediate spatial scales (e.g., <200 km) within fold belts, such as seen on the margins of compressional orogens.

[6] Initial calculations presented here indicate that erosion and deposition have an important influence on deformation, especially in increasing the wavelength of deformation and in reducing the amount of convergence before deformation is initiated. These effects, in turn, strongly influence the location and timing of subsequent erosion and deposition. However, because this is one of the first studies to investigate the problem of coupling between surface processes and dynamic 3-D deformation, these results are preliminary and rather general in nature. The reader is referred to G. D. H. Simpson (Dynamic interactions between erosion, deposition and three-dimensional deformation in compressional fold belt settings, submitted to Journal of Geophysical Research, 2004, hereinafter referred to as Simpson, submitted manuscript, 2004) for a more complete presentation of model results.

2. Mathematical Formulation

[7] The model consists of a thin elastic-plastic plate underlain by an inviscid substrate and subject to surface erosion and deposition (Figure 1b). The plate is referred to rectangular cartesian coordinates x, y, and z, where x and y lie in the midplane of the plate and where z is measured perpendicularly from the midplane and is positive upward (see Table 1).

Table 1. Notation
SymbolUnitDefinition
wmplate deflection
u, vmin-plane displacements of midplane
smtopographic elevation
qm2 s−1fluvial fluid flux accumulated from rainfall
x, ymin-plane spatial coordinates
zmcoordinate perpendicular to plate
lx, lyminitial horizontal dimensions in x and y directions
tstime
Wminitial plate deflection
TPa min-plane stress resultant vector
MPa m2bending moment vector
SPastress vector (negative in compression)
σ0Paplastic yield stress
E total strain vector
Ep in-plane membrane strain vector
Kb bending strains
Ee, Ep elastic and plastic strain vectors, respectively
λ plastic multiplier
hmplate thickness (=h0/2 + sw)
h0minitial uniform plate thickness
ν Poisson's ratio of plate
EPaYoung's modulus of plate
ρ1kg m−3density of overlying medium (air)
ρ2kg m−3density of underlying medium
ρ0kg m−3density of material eroded/deposited
κm2 s−1erosional diffusivity
c(m2 s−1)/(m2 s−1)nfluvial erosion coefficient
n exponent for dependency of sediment transport on fluid discharge
αm s−1rate of rainfall
gm s−2acceleration of gravity
equation imagem s−1imposed horizontal boundary velocity

2.1. Deformation of Elastic-Plastic Thin Plate

[8] The mechanical part of the model is based on conventional thin-plate theory, including in-plate deformation [e.g., Timoshenko and Woinowsky-Krieger, 1959]. The formulation presented closely follows that laid out by Zienkiewicz and Taylor [2000]. The objective of thin-plate theory is to reduce a three-dimensional problem to an approximate two-dimensional one. Classical (Kirchhoff) thin-plate theory is derived in terms of the following simplify assumptions: (1) Normals to the undeformed midplane are assumed to remain straight and normal throughout deformation (so that transverse normal and shearing strains may be neglected in deriving the plate kinematic relations) and (2) transverse normal stresses are assumed to be small compared with other normal stress components (so that they can be neglected in the stress-strain relations).

[9] As a consequence of the first assumption, the components of displacement of any point in the plate (u1, u2, u3) may be expressed in terms of the corresponding midplane quantities (u, v, w; i.e., the 3-D solid is reduced to 2-D surface located along the middle of the plate, see Figure 1b) by the relations

equation image

where 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). These can be used to compute total strains E (with components εxx, εyy, γxy = 2εxy) as

equation image

where the in-plane membrane strains Ep are given by

equation image

the changes in curvature caused by bending Kb are

equation image

and where W is the initial deflection of the plate prior to loading. These deformations are subject to the limitations that strains are small compared to unity, rotations of the midplane relative to x and y are moderately small, and rotation of the midplane relative to z is negligibly small. An important feature of this formulation is that the membrane strains are coupled to bending strains, and thus the two problems of in-plane and transverse deformation cannot be dealt with separately. The nonlinear terms responsible for this coupling (e.g., (w,x)2 in equation (3)) are vital because they enable the evolution of folding to be determined. If these terms are neglected, as is commonly the case [e.g., Turcotte and Schubert, 2002], one can determine the conditions for folding instability but not the magnitude of the deflection itself. Variations in strains (required in a variational formulation such as that presented) are given by

equation image

and

equation image

where δ represents the increment.

[10] The rheology of the plate is assumed to be elastic-plastic. Moreover, it is assumed that total strains can decompose additively into elastic and plastic parts according to

equation image

where subscripts e and p refer to elastic and plastic, respectively. The relationship between stresses and elastic strains Ee is given by

equation image

where the material matrix De for an isotropic homogeneous plate is

equation image

where E is Young's modulus and ν is Poisson's ratio and the stress vector S is

equation image

Note that σzz is zero (plane stress), which is a consequence of assumption 2 above. Plasticity is assumed to be governed by a von Mises-type yield criteria:

equation image

where σ0 is the yield stress. The von Mises criteria is used because it is simple and because it avoids the problem of defining pressure (which is not clearly defined for a plate because the vertical stress is assumed to be zero) that would be necessary if a more standard pressure-sensitive criteria (e.g., Mohr-Coulomb) was used. 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

By integrating stresses through the plate thickness, one obtains stress resultants related to bending,

equation image

and to in-plane (membrane) deformation,

equation image

[11] The governing equations for the thin plate can be completed by writing the virtual work expression

equation image

where δΠ is the change in the potential energy of the system, the displacement increments are

equation image

the load vector b is

equation image

and integration is carried out over an area A (in the xy plane). The term bz accounts for normal loads on the upper and/or lower surface of the plate and is given in this study 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 = h0/2 + sw). The first two terms on the right-hand side of equation (19) 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. Note that when there is no erosion and deposition, s = w + h0/2, h = h0 (i.e., the topography mimics the deforming plate), and the load terms in equation (19) reduce to −(ρ2 − ρ1) gw, as anticipated [see, e.g., Turcotte and Schubert, 2002].

2.2. Surface Erosion and Deposition

[12] Erosion and deposition influence the deforming plate by modifying the distribution of vertical surface loads (via equation (19)) and by changing the plate thickness and therefore the strength distribution. The surface evolution model considered here is a simple transport-limited model that 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, and q is the surface fluid flux, which is determined from the mass balance relation

equation image

where α is the rainfall rate (in excess of infiltration) and n is a unit vector pointing down the local slope of the topographic surface. The first bracketed term in equation (20) represents erosion/deposition, whereas the second term accounts for topography uplifted or downwarped by deformation of the plate. When equation (20) is written in nondimensional form [see Simpson, 2004], an important parameter controlling model behavior is the ratio of deformation to fluvial erosion characteristic timescales, defined as

equation image

where L is the characteristic length scale of the problem (assumed here to be ly), equation image is the imposed boundary displacement rate, the deformation timescale τd = L/equation image, and the fluvial erosion timescale τe = L2−n/(cαn). 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.

[13] Surface evolution governed by equations (20) and (21) 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 [see, e.g., Tucker and Whipple, 2002, and references therein]. While the incorporation of these effects is desirable in future work since this will lead to significantly different topographic morphologies, the overall pattern and magnitude of mass redistribution and the resulting tectonic response to erosion and/or deposition is unlikely to be drastically modified.

3. Solution Technique

[14] Solutions to the mechanical governing equations were computed by the Galerkin finite element method on a three-node triangular mesh. Shape and weighting functions used were linear for in-plane deformation and cubic (three-node, 9 degrees of freedom nonconforming functions) for bending deformation [see Zienkiewicz and Taylor, 2000]. Integration was performed numerically using three-point Gauss quadrature in the xy plane and six-node Gauss-Lobbato quadrature through the plate thickness. The general solution procedure consists of the following steps: (1) Apply a boundary load increment and estimate displacement increments; (2) use the total displacements to determine the strains using the relations for Ep and Kb; (3) determine the corresponding elastic stresses; (4) adjust stresses to account for plasticity, if necessary; (5) evaluate the stress resultants T and M; (6) evaluate the residual force vector based on the variation equation; (7) resolve the system loaded by the residual (out of balance) force vector; and (8) iterate until convergence. The tangent stiffness matrix required for performing the nonlinear Newton-Raphson iterations was reformed at the beginning of each load step and again after the first iteration. This combination was relatively inexpensive and lead to rapid convergence of the solution.

[15] The mechanical part of the code was tested against numerical and analytical solutions, where possible, and yielded good results for various simple engineering test problems (e.g., large deflection vertical loading of an elastic plate [Kawai and Yoshimura, 1969]; large deflection vertical loading of an elastic-plastic plate [Papadopoulos and Taylor, 1991]; buckling stress prediction of an elastic plate [Timoshenko and Woinowsky-Krieger, 1959]). Nevertheless, few solutions exist (especially for elastic-plastic buckling), and further work is needed in benchmark development and testing. All mechanical solutions presented were carried out with a mesh resolution of 41 × 41 nodes (total of 3240 finite elements). This resolution was sufficient to achieve convergence on the relatively small spatial domains investigated here (i.e., 120 × 120 km) but would be insufficient for very large domain sizes, where the deformation becomes more complex.

[16] Solutions to the surface process governing equations were obtained using the Galerkin finite element method (with linear shape and weighting functions) for the surface topography and the finite difference method for the surface fluid flux. The mesh resolution in both cases was 164 × 164 nodes, which is a factor of 4 greater than resolution on the mechanics mesh.

[17] After each loading step, displacements are interpolated from the mechanics mesh to the erosion mesh, and the coordinates of both meshes are updated in a Lagrangian manner. Following the computation of changes in surface topography, a corresponding vertical load increment is determined from the average over all erosion elements contained within each mechanics element (yielding a constant vertical load increment within each mechanics element).

4. Boundary Conditions, Initial Conditions, and Parameter Values

[18] The system of governing equations is solved for the five unknowns u, v, w, s, and q subject to the following boundary conditions: (1) simple support conditions (i.e., w = 0) at y = 0 and y = ly; (2) imposed displacement increment v0 for v at y = 0; (3) zero v at y = ly; (4) zero u at x = lx/2 on the y = 0 and y = ly boundaries; (5) zero w,x at x = 0 and x = lx; (6) fixed elevation for s at y = 0 and y = ly; (7) zero s,x at x = 0 and x = lx; (8) fluid flow is periodic at the x = 0 and x = lx boundaries; and (9) fluid flux is zero at the y = 0 and y = ly boundaries if inflow occurs; otherwise, the fluid can flow out unconstrained. These conditions are summarized in Figure 1b. Physically, these boundary conditions state that the plate is being compressed from one side at a constant horizontal velocity while it is blocked from moving in the direction of compression at the opposite boundary. Moreover, the loaded boundary and the opposing blocked boundary are constrained to have no vertical deformation. The lateral boundaries are free to deform in the vertical direction but are constrained against horizontal movement in the direction parallel to the orogen. The topography is fixed at a constant elevation at the loaded and opposing blocked boundary, while it is constrained to have no slope parallel to the orogen axis on the lateral boundaries (i.e., zero sediment flux).

[19] The initial plate deflection W was assumed to be given by the following simple isotropic function:

equation image

where w0 and a are the maximum amplitude and characteristic length scale of the initial deflection, respectively. All computations were performed with w0 = 10 m and a = 5000 m. The initial topography is given by

equation image

where β is a sequence of random numbers with a uniform distribution over the range [0 1]. For convenience, the topography throughout the remainder of this study is referred relative zero and not to the top of the deflected plate. Note, however, that the topography is actually located a distance h0/2 above the midplate deflection.

[20] Although the preliminary results presented within this study are intended to be general in nature, the scale of the model is chosen to be broadly applicable to a fold belt setting. Accordingly, the initial model domain is 120 × 120 km with a 10-km-thick elastic-plastic layer. The plastic yield stress is chosen to be 100 MPa. This stress level represents the likely maximum order of magnitude for differential stress in the upper crust [see Brace and Kohlstedt, 1980]. Other parameters used in calculations are listed in Table 2.

Table 2. Parameter Values Used in Calculations
ParameterValue
E, Pa1011
ν0.2
h0, m104
lx, ly, m1.2 × 105
σo, Pa1 × 108
g, m s−29.8
ρ1, kg m−31
ρ0, kg m−32300
ρ2, kg m−32700
n2
c, s m−2100–500
α, m s−13.17 × 10−8
κ, m2 s−13.17 × 10−8
equation image, m s−16.34 × 10−10

5. Example of Model Results

[21] In order to demonstrate the basic features of the model and the influence exerted by surface processes on deformation, a series of simulations are presented where the importance of erosion and deposition is varied via the nondimensional parameter R (see equation (22)). Large values of R imply rapid erosion and deposition relative to deformation, whereas small values imply that the rate of imposed deformation is rapid relative to the rate of surface processes. Results of four representative simulations, three of which were performed with different values of R (i.e., 19.01, 47.53, and 95.01) and one where erosion and deposition were omitted, are presented in Figures 234.

Figure 2.

(a–x) Computed topography from four different simulations where the efficiency of erosion and deposition relative to deformation is varied, shown after various amounts of shortening. The plate is pushed from the lower boundary and eventually responds by the development of folds. The parameter R is defined in equation (22) and represents the ratio of deformation to fluvial erosion characteristic timescales (with large values of R implying relatively short fluvial timescales or efficient fluvial erosion). The initial condition is defined in equation (24). Parameter values used in the calculations are listed in Table 2. The total shortening (Δ y), maximum topographic elevation (max(s)), and minimum topographic elevation (min(s)) are indicated under each contour plot (10 contour intervals are plotted in each case). Anticlines and synclines are indicated on the left-hand side of each contour plot in the standard manner. As the relative efficiency of erosion is increased (going from the left column to the right column), the deformation becomes more localized, and folding is initiated earlier.

Figure 3.

Variation of (a) maximum topographic elevation and (b) plate deflection as a function of total shortening. (c) Curves for the maximum topographic elevation plotted against the normalized shortening, which is defined as the shortening divided by the value of shortening at buckling. Note that when erosion and deposition are absent or inefficient, buckling is initiated much later than if surface processes are relatively efficient. The arrow pointing to the “no erosion” curve in Figure 3b indicates the point when secondary buckling is initiated, causing the breakup of cylindrical folds into doubling-plunging structures (see Figure 2f).

Figure 4.

(a–i) Maximum and minimum topographic profiles taken parallel to the convergence direction for the simulations presented in Figure 2. Note the localization of deformation on a single antiform as the efficiency of erosion is increased. Note also the dramatic incision (lowering of the minimum topographic profile) that has occurred (between Figures 4h and 4i) on the flanks of the main antiform (in Figure 4i), which occurred in response to infilling of the flank synclines and the establishment of fully transverse drainage.

[22] In the case where erosion and deposition are absent, deformation can be subdivided into three main evolutionary phases that occur during progressive shortening: prebuckling, synbuckling, and secondary buckling. During the prebuckling phase, deformation spreads outward from the initial circular imperfection and organizes into a series of low-amplitude buckle folds with wavelengths of ∼40 km and axes perpendicular to the direction of compression (Figures 2c and 2d). The amplitude of these folds grows gradually until the synbuckling phase is entered, which is initiated after ∼24 km of shortening. The beginning of the synbuckling phase is signaled by an abrupt increase in fold amplitudes from <100 m to ∼1 km during ∼1 km of shortening (Figure 3). The resulting folds are cylindrical in nature, with no asymmetry between synclines and anticlines, and they show no evidence for the initial imperfection from which they formed (Figure 2e). Folds continue to grow rapidly during the synbuckling phase but at ever decreasing rates as the build up of membrane stresses causes folds to lock up. Eventually, if the amount of applied shortening is sufficient, the plate may undergo secondary buckling, which is marked by a renewed rapid increase in fold amplitudes (Figure 3b) and by the breakup of cylindrical folds into doubly plunging structures (Figure 2f). This latter folding stage is of considerable interest because it indicates that truly three-dimensional structures can result from pure two-dimensional compression.

[23] When erosion and deposition are present but their rates are slow relative to the rate of imposed deformation (i.e., R = 19.01), deformation of the elastic-plastic plate proceeds largely as if surface processes had been absent. Buckling is initiated after the same amount of imposed shortening (Figure 3) and generates folds with the same wavelength as the case where erosion and deposition were omitted (compare Figure 2k with Figure 2e). The only difference that occurs is that the maximum and minimum topographies are reduced by erosion and deposition, respectively, which creates an asymmetry in the topography between anticlines and synclines (Figures 2l and 4a–4c). Even though this mass redistribution generates significant loads (both positive and negative) on the plate (e.g., ∼1 km of erosion has occurred after ∼30 km of shortening), both the magnitude of this load and the timing of when this load is applied relative to buckling are such that surface processes have no noticeable influence on deformation.

[24] When the efficiency of erosion and deposition relative to deformation is increased by a factor of 2.5 (i.e., to R = 47.53), deformation of the plate is significantly modified by the presence of surface processes. During the prebuckling phase, mass is eroded from the initial imperfection and is transported externally toward the compression-directed boundaries, where it is deposited in shallow synclines (Figure 2o). This mass redistribution has two important effects on subsequent deformation. First, it effectively reduces the influence of gravity by unloading uplifting regions while loading downwarping ones. This effect tends to enhance the ability to fold and increase the wavelength of deformation [see also Simpson, 2004]. Secondly, because loading is imposed prior to buckling (in addition to during buckling), it acts as an additional initial imperfection, which also tends to reduce the amount of loading required to initiate buckling [Bloom and Coffin, 2001]. The combined result is that the plate buckles after only ∼8 km of shortening (as opposed to ∼24 km (see Figure 3)) at an increased wavelength of ∼65 km (compared to ∼40 km), consisting of two major anticlines separated by a single basin (Figures 2p and 4d–4f). Continued shortening causes deformation to become progressively more localized, a situation that eventually leads to a single major asymmetrical anticline with a maximum topographic elevation of ∼2000 m after ∼14 km of convergence (Figures 2r and 4f).

[25] The dramatic influence that surface processes can have on deformation is epitomized if the efficiency of erosion and deposition is increased relative to deformation by a further factor of 2 (i.e., R = 95.01). Although the amount of convergence required to initiate buckling is not further reduced (Figure 3), the wavelength generated during the initial folding instability increases such that the model domain is occupied by a single mega-anticline flanked on either side by two shallow synclines (Figures 2v and 4g). These flank synclines become progressively more shallow with increased shortening and eventually become completely filled with sediment eroded from the central anticline (Figure 4i). This filling marks an important event for subsequent deformation because it enables mass to be transported directly from the central anticline out of the system (i.e., fully transverse drainage is established), which leads to dramatic unloading and enhanced tectonic and uplift on the central anticline. This leads to major and abrupt reincision of deposits flanking the anticline (Figure 2x) and enhanced uplift of the anticline itself (Figure 3a).

6. Discussion and Conclusions

[26] This study has presented a mathematical model that enables investigation of fully coupled interactions between erosion, deposition, and 3-D deformation at the scale of a fold belt (e.g., ≤200 km). The model considers a single elastic-plastic plate overlying an inviscid (i.e., very weak) material. Depending on the application, the plate is intended to present either the upper crust detached over a very weak lower crust or an upper competent unit detached over a weak horizon such an evaporite (all entirely within the upper crust). The plate can be compressed from one or more lateral boundaries and responds by the development of buckle folds, which are expressed topographically as anticlines and synclines. This topography is advected along with the underlying deforming plate and can be modified through time by a combination of fluvial and hillslope erosion and deposition. The resulting mass redistribution creates loads on the plate that can then feed back and influence subsequent deformation, erosion, and deposition.

[27] Preliminary results computed from this model confirm conclusions reached in other studies focussing on 2-D problems at larger scale [e.g., King and Ellis, 1990; Beaumont et al., 1992; Willett, 1999; Beaumont et al., 2000], which have shown that erosion and deposition have a major impact on deformation. When the dominant mode of deformation in nature is folding, this study indicates that erosion and deposition will tend to have two main effects: The first is to decrease the amount of shortening required to initiate folding, while the second is to increase the fold wavelength. In the extreme case of relatively efficient erosion and deposition, deformation becomes localized on a single large-scale mega-anticline, as opposed to distributed on a train of anticlines and synclines. Thus erosion and deposition play a major role in governing the deformation and topography in fold belts and in the sediment routing system acting within such a setting. The origin of these effects lies mainly in the ability of both erosion and deposition to decrease the influence of gravity during deformation by unloading anticlines and loading synclines [Simpson, 2004]. In addition, mass redistribution resulting from erosion and deposition that occurs prior to buckling tends to have the same effect as an initial imperfection, which also tends to reduce the amount of loading required to initiate buckling [Bloom and Coffin, 2001]. Effects due to changes in the plate thickness are of secondary importance, except for advanced stages of convergence (and erosion), when they may begin to dominate.

[28] An important motivation for studying interactions between erosion and deformation is to resolve the long-standing debate concerning how climate change modifies the height of mountain ranges [see, e.g., England and Molnar, 1990]. If surface processes are envisaged to be dynamically uncoupled from deformation (as is assumed in most surface evolution modeling), enhanced erosion due to increased precipitation will lead to a decrease in mean elevation [e.g., Bonnet and Crave, 2003], although mountain peaks may actually rise due to isostatic rebound [England and Molnar, 1990]. If, on the other hand, deformation is dynamically coupled to surface processes, enhanced erosion under some circumstances may localize deformation, leading to actual growth of (mean) topography at the orogen scale [Avouac and Burov, 1996; Zeitler et al., 2001]. Which of these two very different scenarios occurs in nature will depend on stress state, rheology, and the spatial scale of interest. This study shows that the maximum topography decreases as the efficiency of surface processes are increased, indicating that a change to a more erosive climate should not increase the height of topography at the scale investigated (i.e., ≤200 km). However, model results do demonstrate that the height of topography is strongly influenced by the ability of rivers to transport mass efficiently away from elevated regions. For example, it is shown that if rivers, which previously deposited sediment within local internal basins, manage to reorganize and flow to more distal regions, the resulting change in the length scale of mass redistribution can cause abrupt rock and surface uplift and reincision of previously deposited sediments. Burbank [1992] discussed field evidence from the Himalayas that appears to indicate very similar behavior. Once established, such “open systems” characterized by dominantly long-range sediment transport are likely to be relatively sensitive to changes in the magnitude of mass transfer and therefore also to climate change. Thus the conclusion made by Avouac and Burov [1996] that erosion-enhanced deformation could increase the height of mountain ranges appears possible only at large spatial scales.

[29] Results for the 3-D plate (i.e., this study) differ significantly from the 2-D beam model studied recently by Simpson [2004], even considering the difference that the plate studied here is underlain by an inviscid substrate, whereas a quasi-viscous substrate was included in the beam model. In the case of a beam, folds grow more slowly and reach smaller amplitudes than for folds in a plate with similar properties. This difference reflects the tendency for a plastic beam to be significantly weaker than a plastic plate because when part of a beam undergoes plastic deformation, the entire section perpendicular to the beam is also assumed to be plastic, which is unlikely to be the case in a plate and in reality. Thus the conclusion made by Simpson [2004] on the basis of a beam model that erosion-enhanced deformation could increase the height of mountain ranges (for certain parameter ranges) appears unlikely at the spatial scale (i.e., ≤200 km) investigated. Other features (e.g., such as the localization of deformation on a single mega-anticline when erosion and deposition are efficient) are observed in both beam and plate models and appear to be robust.

[30] An interesting feature demonstrated by the model results presented here is that fully three-dimensional structures (i.e., those exhibiting variation along strike) can develop in response to purely two-dimensional convergence [see also Fletcher, 1995; Audoly, 1999; Bloom and Coffin, 2001; Audoly et al., 2002]. This can occur because primary buckling only releases stresses in the transverse direction. However, residual stresses in the longitudinal direction, if high enough, may induce “secondary buckling” and the formation of complex 3-D structures consisting of doubly plunging anticlines and synclines which form a basin-and-dome-type topography. Interestingly, three-dimensional erosion and deposition do not generate significant 3-D deformation (i.e., the deformation remains largely cylindrical). This is probably because, for the simple setup investigated, rivers and valleys are distributed homogeneously along strike and are spaced at relatively small distances. The high-frequency loads related to narrowly spaced river valleys are effectively filtered out (due to the strength of the plate), leaving only the long-wavelength load to influence deformation, which is uniform in the strike-parallel direction. Thus although the sediment routing system displays a complex structure, the resulting load experienced by the plate is largely two-dimensional.

[31] Because the main purpose of this paper has been to present a new model formulation, the results presented are preliminary and rather general in nature. Although initial results are promising, many questions, such as those posed in section 1, remain to be answered and are currently under investigation (see, e.g., Simpson, submitted manuscript, 2004). While the model is deficient in that it solves for an approximate three-dimensional deformation field (due to the thin-plate assumptions), it does account for the main two-way feedbacks between surface processes and internal deformation. Most previous 3-D studies have adopted kinematic tectonic models (arguing that explaining the geometry of natural orogens is more important than predicting the internal deformation field) and then use surface process models to determine mass redistribution, from which a vertical flexural-isostatic component is determined [e.g., Johnson and Beaumont, 1995; Garcia-Castellanos et al., 1997; Garcia-Castellanos, 2002]. However, as the current dynamic model illustrates, although surface processes do induce a vertical flexural component of deformation, by far the greatest influence arises from the way in which surface processes modify the dynamic deformation instabilities (in this case, folding) leading to major changes to the pattern of deformation and the overall geometry of the orogen. This point indicates that fully dynamic modeling of three-dimensional interactions between surface processes and deformation is necessary to understand the morphology and evolution of topography and the pattern of deformation itself in natural orogens.

Acknowledgments

[32] Yuri Podladchikov, Alex Densmore, Fritz Schlunegger, Mike Ellis, and Luca Jay are thanked for providing inspiration and encouragement and for participating in many stimulating discussions during the course of this project. Reviews by Mike Ellis and an anonymous reviewer helped clarify a number of issues and are greatly appreciated. This work was partially supported by the Swiss National Science Foundation (grant 620-57863).

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