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B01411 HILLEY AND STRECKER: STEADY STATE EROSION OF CRITICAL COULOMB WEDGES processes (2). This acceleration in erosion reduces the rate at which the wedge widens with time. This particular model assumes that wedges grow in their mechanically critical state. However, other models [e.g., Willett, 1992] may be used to estimate the path of points within this diagram when the wedge does not grow in its critical condition. In these cases, the trajectory of points may deviate from the mechanical stability lines [Davis et al., 1983; Dahlen et al., 1984; Willett, 1992]. In the case that the wedge grows selfsimilarly (‘‘self-similar growth’’; fixed alpha and beta) and in its mechanically critical state, the point representing the wedge geometry will not move from a single point on this diagram. Instead, increases in the depth to the basal decollement with time causes the vT/K base 10 logarithm contours to rotate counterclockwise in Figure 13 until the erosional flux balances the tectonic accretion rate. Finally, the wedge geometry will approach its steady state form (3) in which erosion is sufficient to remove all material accreted to the wedge. This study shows the ultimate form of the wedge when it is in its erosional steady state form (Figure 13, C, dashed lines). When the wedge is in mechanical and erosional steady state, the orogenic wedge geometry plots at the intersection of the vT/K contour and the mechanical stability line appropriate for a specific orogenic wedge (Figure 13, solid circles). Using these diagrams, the development and ultimate geometry of orogenic wedges that gain material by accretion and lose material by bedrock fluvial incision may be understood. [42] Finally, it is important to acknowledge the assumptions of our estimated values of vT and our model formulations. For the Taiwanese orogenic wedge, vT was estimated by Suppe [1981] to be 500 m 2 /yr based on balanced cross sections that sample accretion over several million years. However, in the Himalaya, wedge accretion rates were based on geodetic measurements and the thickness of accreted foreland material and the potential lower crustal Tibetan channel (Appendix A). While the geodetic rates may only sample instantaneous accretion rates, no further information exists regarding the material fluxes entering the Himalayan wedge. In addition, the application of geodetic rates to wedge accretion rates over longer timescales has been successfully employed previously [e.g., DeCelles and DeCelles, 2001]. [43] For simplicity, the formulations of Davis et al. [1983] and Dahlen [1984] assume that the wedge has a simple triangular cross section and deforms brittle-elastically at the Coulomb failure limit of the wedge. In real orogens, plastic deformation within the wedge and deviations in this simple geometry may confound the simple relationships presented. In addition, we neglect isostasy in our analysis and assume that heterogeneities in mechanical and erosional properties within the wedge may be represented by average, homogeneous values. Isostasy may increase the decollement angle and reduce the surface slope to maintain the inverse relationship predicted between fault dip and surface slope [Davis et al., 1983]. Both isostasy and heterogeneity within orogens are important factors, and so their incorporation into our models provides a fruitful direction of future research. However, as shown by Davis et al. [1983], Dahlen [1984], and DeCelles and DeCelles [2001], simple wedge models apparently capture the first-order features of the structures of many orogens. 14 of 17 B01411 [44] In addition, it is important to note that all terms related to the mechanical properties of the wedge have vanished from our formulation (equation (7)), as long as we assume that the wedge maintains a triangular cross section. More complex rheologies may lead to the violation of this geometry [e.g., Willett, 1999]; however, where an approximately triangular wedge geometry can be demonstrated, our model can be employed. Therefore, because we estimate the geometry of the wedge a priori, the steady state geometry is independent of the rheological properties, and isostatic adjustments should be encapsulated within the derived estimates of vT/K, allowing us to discard many of the mechanical and isostatic processes and parameters that may be difficult to constrain by field data. [45] Two of the largest uncertainties in our analysis include the assumption of mechanical and erosional steady state, and the broad application of the bedrock power law incision model to an entire orogen. First, few wedges likely exist that are in this steady state condition, and the Taiwan orogen is probably the best example of this phenomenon [Suppe, 1981]. Because this steady state condition is likely the exception to the rule, the direct applicability of our models to other orogens may be limited. However, our results provide important information about the state toward which mountain belts tend to evolve. In many areas of the world and ancient orogens, information on steady state behavior does not exist. However, as this study and those of others [e.g., Willett, 1999; Hilley et al., 2003] have shown, erosion may nevertheless exert important controls on deformation within and topographic form of many orogens. Our results provide a valuable heuristic guide for interpreting changes in orogenic structure with respect to these erosional factors in settings where tectonic and erosional rates may not be available. [46] Second, we assume that the rate-limiting process in orogenic wedges is bedrock fluvial incision [Whipple et al., 1999]. Other erosional processes may be important in denuding the orogenic wedge; however, many of these other processes (e.g., glacial transport) are less well understood than bedrock erosion. Instead of including less well constrained processes, we followed the approach of many [e.g., Whipple et al., 1999; Whipple and Tucker, 1999; Kirby and Whipple, 2001; Whipple and Tucker, 2002] by using the power law as our rate-limiting orogen-scale erosion process. However, there are several important limitations of the bedrock power law incision model that warrant discussion. The erodibility parameter (K) may be influenced by rock type [Stock and Montgomery, 1999], the distribution of fractures within the bedrock [e.g., Whipple et al., 2000], allometric changes within the channel, the relations between catchment area and effective discharge in the channel [Whipple and Tucker, 1999; Sklar and Dietrich, 1998], the power law exponents [Sklar and Dietrich, 1998], and sediment input from hillslopes [Sklar and Dietrich, 1998, 2001; Whipple and Tucker, 2002]. Many of the factors that control the value of K remain only qualitatively or poorly understood, limiting the predictive capability of the bedrock incision model. As an example, Sklar and Dietrich [1998, 2001] found that sediment input from hillslopes may exert a strong control on the rate of bedrock incision, and this sediment input may be modulated by the effects of climate and uplift rate [Montgomery and
B01411 HILLEY AND STRECKER: STEADY STATE EROSION OF CRITICAL COULOMB WEDGES Dietrich, 1994; Fernandes and Dietrich, 1997; Whipple and Tucker, 2002]. Also, incision rates may be related to the amount of excess transport capacity that these bedrock channels may contain after sediment has been transported (the ‘‘over-capacity’’ model of Kooi and Beaumont [1994]). However, we do not consider these models because they tend to predict unrealistic stream profile geometries relative to those observed in nature [Whipple and Tucker, 2002]. Stock and Montgomery [1999] analyzed a series of bedrock channels in different parts of the world that were cut into various types of bedrock and found that the primary controlling factor of K appeared to be lithology. The consistency between our inferred K values and those estimated for similar rock types indicates that while there are a plethora of factors included in K, temporal averaging over the timescales applicable to orogen-scale development may fold many of these factors into a bulk lithology factor in which specific and perhaps covarying erosional mechanisms may be captured by a single effective rock type. Therefore, while our formulations presented will change with greater understanding of these erosional processes, the heuristic results of our analysis may be unaffected. 6. Conclusions [47] We developed a coupled erosion-deformation model that is used to understand how deformation within and erosion of an orogenic wedge are linked. The model assumes that the wedge (1) has a triangular cross section, (2) gains mass either from underplating of foreland sediments or injection of material into the back of the wedge, (3) looses mass removed by erosional processes whose rates are limited by fluvial bedrock incision, (4) is in a steady state form in which all material that is tectonically introduced into the wedge is removed by erosion, and (5) is at its Coulomb failure limit. Under these assumptions, the morphology of the steady state wedge is controlled by its mechanical properties (m, mb, l, lb), the basin hydrologic parameters (ka and h), the erosional exponents (m and n), the sole-out depth of the basal decollement (D), and the ratio of the flux of tectonically added material to the erosional constant (vT/K). [48] In general, increasing values of vT/K produce steady state orogenic wedges that are wider than their low vT/K counterparts. Increases in this parameter may result from increasing material flux introduced into the wedge, the exposure of less-erodible rock types within the wedge, and/or decreased erosional power due to reduced precipitation and runoff. Our results highlight the role of erosion, and by inference, the importance of climate and rock type exposure in determining orogenic structure. While a steady state condition may not be satisfied in most orogens, our analysis provides important heuristic results that may be used to interpret both active and ancient orogens in terms of the accretion and erosion of material from the wedge where information about the controlling processes and their rates may be unavailable or erased. [49] We applied these models to the Taiwan and Himalayan orogenic wedges, which may be in tectonic and erosional steady state. In the case of Taiwan, previous work has found that erosion rates inferred from fission track analyses are consistent with those required to explain wedge steady 15 of 17 B01411 state geometry [Dahlen and Barr, 1989]. However, these studies do not allow direct comparison of the inferred rock resistance to fluvial bedrock incision with those exposed within the island. Our analysis demonstrates that large values of vT/K are required to explain the structural and topographic relations in the Himalaya relative to Taiwan. However, independent estimates of the material flux entering the Himalaya relative to Taiwan are substantially lower and require an order of magnitude smaller K acting to erode this wedge. This reduced K value in the Himalaya is consistent with the lower precipitation and exposure of less erodible rock types within this wedge relative to Taiwan. Our calculated K values are consistent with those determined independently for similar rock types. Therefore differences in rock type and climate apparently lead to key differences in the erosion, and hence, structure of these two mountain belts. Appendix A: Estimation of Material Leaving Retrowedge in Taiwan [50] To calculate the range of material fluxes eroded from Taiwan’s eastward verging retrowedge, we calculated the velocity of the Luzon arc remnants relative to the prowedge and retrowedge drainage divide using Global Positioning System measurements [Yu et al., 1997] to determine the velocity of material moving out of the wedge at its eastern boundary. We found that material moves out of the wedge’s eastern boundary at a horizontal velocity of 36–48 mm/yr. We assumed a maximum rock uplift rate of 10 mm/yr throughout the retrowedge. This value was chosen as a conservative maximum estimate of rock uplift rates, and so a reduction in this value slightly reduces the material flux leaving the retrowedge and increases that eroded from the prowedge. Using these values, we determined the velocity component perpendicular to the wedge surface and integrated this velocity along its length. By assuming that all material moving through the orogenic system is removed by erosion, leaving the cross-sectional geometry of the orogen unchanged, these calculations indicate that between 168 and 224 m 2 /yr of material moves through the surface slope of the retrowedge. By assuming that all material moving through the orogenic system is removed by erosion, we equated erosional removal with this amount. The total estimated accreted material flux was estimated to be 500 m 2 /yr; therefore between 276 and 333 m 2 /yr is removed from the prowedge before reaching the boundary between the prowedges and retrowedges. This estimate is consistent with fluxes calculated from GPS-derived decollement slip rates beneath the western portion of the island [Hsu et al., 2003]. [51] Acknowledgments. G.H. acknowledges support from the Alexander von Humboldt Foundation and the DAAD IQN program for support of his research at Potsdam. M.S. acknowledges support by the German Research Council (DFG) grant STR 373/-3. Also, G.H. thanks Vince Matthews for introducing him to the critical Coulomb wedge and astutely suggesting its utility in understanding orogen-scale tectonic and landscape development. We gratefully acknowledge Edward Sobel for commentaryonanearlydraftofthismanuscript.Inaddition,FritzSchlunegger, Terence Barr, and Jean Braun all provided valuable reviews that improved the manuscript. Finally, we would like to acknowledge comments on the manuscript by Roland Bürgmann that improved our analyses. References Barr, T. D., and F. A. Dahlen (1989), Brittle frictional mountain building: 2. Thermal structure and heat budget, J. Geophys. Res., 94, 3923–3947.
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