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The role of weathering in the formation of bedrock valleys on Earth and Mars: A numerical modeling investigation

2011; American Geophysical Union; Volume: 116; Issue: E11 Linguagem: Inglês

10.1029/2011je003821

2156-2202

Jon D. Pelletier, Victor R. Baker,

Planetary Science and Exploration

Journal of Geophysical Research: PlanetsVolume 116, Issue E11 Free Access The role of weathering in the formation of bedrock valleys on Earth and Mars: A numerical modeling investigation Jon D. Pelletier, Jon D. Pelletier [email protected] Department of Geosciences, University of Arizona, Tucson, Arizona, USASearch for more papers by this authorVictor R. Baker, Victor R. Baker Department of Geosciences, University of Arizona, Tucson, Arizona, USASearch for more papers by this author Jon D. Pelletier, Jon D. Pelletier [email protected] Department of Geosciences, University of Arizona, Tucson, Arizona, USASearch for more papers by this authorVictor R. Baker, Victor R. Baker Department of Geosciences, University of Arizona, Tucson, Arizona, USASearch for more papers by this author First published: 19 November 2011 https://doi.org/10.1029/2011JE003821Citations: 23AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract [1] Numerical models of bedrock valley development generally do not include weathering explicitly. Nevertheless, weathering is an essential process that acts in concert with the transport of loose debris by seepage and runoff to form many bedrock valleys. Here we propose a numerical model for bedrock valley development that explicitly distinguishes weathering and the transport of loose debris and is capable of forming bedrock valleys similar to those observed in nature. In the model, weathering rates are assumed to increase with increasing water availability, a relationship that data suggest likely applies in many water-limited environments. We compare and contrast the model results for cases in which weathering is the result of runoff-induced infiltration versus cases in which it is the result of seepage- or subsurface-driven flow. The surface flow–driven version of our model represents an alternative to the stream-power model that explicitly shows how rates of both weathering and the transport of loose debris are related to topography or water flow. The subsurface flow–driven version of our model can be solved analytically using the linearized Boussinesq approximation. In such cases the model predicts theater-headed valleys that are parabolic in planform, a prediction broadly consistent with the observed shapes of theater-headed bedrock valleys on Mars that have been attributed to a combination of seepage weathering and episodic removal of weathered debris by runoff, seepage, and/or spring discharge. Key Points Weathering plays a major role in bedrock valley development Theater-headed valleys appear to require seepage In idealized cases, theater-headed valleys are parabolas; data confirm this 1. Introduction [2] Weathering of rock (used in this paper to refer to the breakdown of rock into unconsolidated material by physical and/or chemical means) is an important process in landscape development. Recent studies have made great strides in quantifying the controls of weathering rates on hillslopes. Available data indicate that water acts as a catalyst for rock breakdown in many physical (e.g., freeze-thaw fracturing) and chemical (e.g., hydrolysis) processes in water-limited environments. A water-limited environment in this context refers to an environment for which the mean annual precipitation is less than the mean annual potential evapotranspiration [Budyko, 1974]. To the extent that biological materials (e.g., plant roots) also enhance rock weathering, the rate of rock breakdown can also be expected to increase with increasing water availability in water-limited environments because biomass and water availability are highly correlated in such environments [e.g., Whittaker and Niering, 1975]. Recent advances in cosmogenic radionuclide (CRN) analysis suggest that bedrock weathering rates on Earth increase systematically with increasing water availability. Riebe et al. [2004] measured rates of chemical weathering and inferred rates of physical erosion and chemical denudation using a chemical depletion fraction approach in granitic terrain over a wide range of climates. Pelletier and Rasmussen [2009] used the data of Riebe et al. [2004] to document a systematic positive correlation between soil production rates and mean annual precipitation, assuming a steady state balance between soil production rates and physical erosion rates. Soil production in this context refers to the rate of breakdown of bedrock into transportable material. Pelletier and Rasmussen [2009] inferred values for P0, the maximum soil production rate (the maximum value occurring on a bare bedrock surface and decreasing with increasing regolith or soil thickness) across a range of climates. Pelletier and Rasmussen [2009] showed that P0 could be quantified using an energy-based variable that combines mean annual precipitation and temperature and increases from several centimeters per thousand years in semiarid climates to more than several meters per thousand years in humid climates. Rasmussen and Tabor [2007] also showed similar relationships among soil thickness, clay content, and water availability on stable upland surfaces in the Sierra Nevada. In essence, the work of Rasmussen and Tabor [2007] and Pelletier and Rasmussen [2009] are updates on the classic work of Jenny [1941], documenting precipitation as a principal soil-forming factor in which increased precipitation leads to an increased rate of soil formation. [3] Although most available weathering-rate data come from hillslopes, the breakdown of rock by physical and chemical processes also occurs in many ephemeral bedrock channels [Büdel, 1982; Wohl, 1993; Howard, 1998; Whipple et al., 2000; Wohl and Springer, 2005; Murphy et al., 2009]. Büdel [1982], for example, showed that the beds of ephemeral bedrock channels in Svalbard, Norway, weather vigorously by freeze-thaw action and that the principal role of flooding during the snowmelt season is to transport the weathered debris out of the channel or valley. On the basis of available data from hillslope studies, we hypothesize that in water-limited environments the rate of rock breakdown in bedrock valleys likely increases with the average water content in the rock or in rock fractures, provided that the depth of flow from runoff, seepage, or spring discharge is not so great that the flow acts to thermally buffer the bedrock from atmospheric temperature variations that drive freeze-thaw and other weathering processes. This hypothesis suggests that zones of flow concentration in incipient valleys may act as microenvironments in which weathering is enhanced because of greater water availability, just as the rate of rock breakdown on hillslopes is enhanced across regions from arid to more humid climates. Enhanced weathering may lead to further flow concentration in a positive feedback that leads to bedrock valley development, provided that runoff, seepage, and/or spring discharge are capable of episodically transporting loose debris out of the valley. [4] Despite the fact that weathering is an essential process in the development of many bedrock valleys, numerical models for the development of bedrock valleys generally do not include weathering explicitly. In the stream-power model, for example, erosion by plucking is quantified as a power law function of stream power [Howard and Kerby, 1983; Whipple and Tucker, 1999]. Plucking is a combination of the weathering of bedrock into loose debris and the transport of that debris out of the valley [Whipple et al., 2000]. The rate of debris transport can be related to the shear stress or stream power of the flow, but no linkage between weathering and stream power (or similar controlling variables related to topography and/or water flow) has been clearly established in the context of quantitative models for bedrock valley development, such as the stream-power model. A key goal of this paper is to establish such a linkage and show that, when weathering is explicitly included, the model predicts the formation of bedrock valleys morphologically similar to those in nature. [5] A model that explicitly distinguishes weathering and transport has three important advantages over models that do not distinguish weathering and transport. First, such models explicitly show how weathering is linked to stream power and/or other controlling variables related to topography and/or water flow. Second, such models explicitly quantify the thickness of the loose debris layer above bedrock (herein termed soil). This is especially important because the rate of colluvial transport depends on the thickness of the loose debris layer. As such, distinguishing weathering from transport in the fluvial component of the model is essential for proper modeling of the colluvial component. Third, models that explicitly include weathering enable the effects of weathering caused by seepage to be more clearly distinguished from the effects of weathering caused by runoff-induced infiltration. In this study we exploit this third advantage in particular to clarify the role of seepage weathering in the development of theater-headed valleys. [6] Bedrock valleys can be divided into theater-headed (i.e., valley heads that are smooth and U shaped in planform) and nontheater-headed types. Of course, there is considerable diversity in these valley types, and weathering plays an important role in both types of valley. In this paper, however, we pay particular attention to the role of weathering in theater-headed valleys because they have been the subject of recent debate. In the classic groundwater-sapping model for the formation of theater-headed bedrock valleys, layers at the base of a cliff or steep slope wear away more rapidly, undercutting overlying materials and leading to their collapse [e.g., Wentworth, 1928, 1943; Baker, 1990]. In this process, the water table is drawn to the low potential represented by the valley bottom. Seepage, which need not be continuous or visible over short timescales (e.g., it may be related to past climatic conditions), concentrates preferentially at the valley head. Because weathering is generally enhanced by increased water availability, seepage weathering can trigger removal of the lower portions of the valley slopes more rapidly than the upper portions, triggering collapse of the upper portions. Transport of weathered debris may occur by gravity (dominant in transporting debris from the top of the valley headwall to the bottom of the headwall), runoff, seepage, and/or spring discharge (collectively dominant in transporting debris away from the base of the headwall and out of the valley), resulting in back wearing of the valley headwall. Laity and Malin [1985, p. 207], emphasized the fact that seepage or spring discharge needs not the only agent that transports loose debris out of the valley in the groundwater-sapping model, i.e., continued retreat of a valley's head and walls following seepage weathering "[require] removal of talus accumulated at the base of the slope… moved by a combination of surface wash, subsurface flow, and undermining, gravity fall and wind action." In this paper we use the term groundwater sapping to refer to this combination of seepage weathering and subsequent transport of the weathered debris away from the valley head by mass movements, fluvial entrainment, and/or wind action. [7] In this paper we use the term theater-headed valleys rather than amphitheater-headed valleys because the designation of a U-shaped valley head as an amphitheater by Hinds [1925] and many subsequent workers was recognized as incorrect by Flint [1947, p. 93, note 19]. Flint [1947] was concerned that the cirque-like heads of valleys were being described with this term. The prefix amphi (from Greek) means two sided or all around. Thus, the classical Greek amphitheater consists of having two U-shaped theaters (Greek theotron) facing each other to form a complete arena enclosed all around by steeply rising rows of seats. As such, amphitheaters are oval in planform, while theaters are U shaped. [8] Figures 1 and 2 illustrate several classic examples of theater-headed valleys on Earth and Mars that have been attributed to groundwater sapping. On Earth, the theater-headed valleys of the Colorado Plateau [Laity and Malin, 1985] (Figures 1a and 1b) are particularly compelling examples of sapping because extensive, well-developed theater-headed valleys occur where the contact between an aquifer (e.g., the Navajo Sandstone) and an aquiclude (e.g., the Kayenta Formation) is exposed at the surface. This association is difficult to explain without a model that includes groundwater seepage. Figure 1c illustrates examples of theater-headed valleys in Canyonlands National Park, where headwalls act as narrow divides between adjacent drainages. These examples illustrate that theater-headed valleys are not limited to immature drainage networks with large contributing areas (i.e., low drainage-density networks incised into vast surrounding plateaus). Figure 1Open in figure viewerPowerPoint Examples of theater-headed valleys on Earth, illustrated with shaded relief images of U.S. Geological Survey Digital Elevation Models (DEMs). (a, b)Theater-headed valleys occur on the east side of the Escalante River in southern Utah where the contact between the Navajo Sandstone and the Kayenta Formation promotes groundwater seepage. (c) Theater-headed valleys in Canyonlands National Park, Utah. Figure 2Open in figure viewerPowerPoint Examples of theater-headed valleys in western Valles Marineris, Mars, illustrated with a shaded relief image of Mars Orbiter Laser Altimeter (MOLA) gridded topography and images from the Mars Express High Resolution Stereo Camera (HRSC). [9] The Ius Chasma region of Valles Marineris is home to many of the best examples of theater-headed valleys on Mars (Figure 2). The morphological similarity between valleys in Figures 1 and 2 has been used as a basis for extending the sapping model of theater-headed valley formation from Earth to Mars in cases for which no evidence of catastrophic flooding exists in the region [Gulick and Baker, 1989; Gulick, 1998; Malin and Carr, 1999; Grant, 2000; Malin and Edgett, 2000; Grant and Parker, 2002]. [10] Lamb et al. [2006] argued that theater-headed valleys are not uniquely consistent with groundwater sapping. This is an excellent point. However, it is important in the context of this debate to differentiate between different types of theater-headed valleys and between theater-headed valleys and channel knickpoints. Channel knickpoints, for example, are common along channels influenced by rapid base level fall and/or structurally controlled erosion [e.g., Pelletier et al., 2009]. Alcoves may form at the base of such channel knickpoints [e.g., Haviv et al., 2010]. Weathering associated with water splashing up from the plunge pool may also widen the alcove to a width slightly greater than that of the channel. In any case, however, alcoves associated with channel knickpoints are comparable to or slightly wider than their associated channels. In contrast, most theater-headed valleys that have been attributed to groundwater sapping on Earth and Mars are many times larger than the channels that exist both above and below their valley heads. The Niagara River, an example that Lamb et al. [2006] used to argue that theater-headed valleys may form as a result of waterfall erosion, occupies the entire valley floor both above and below Niagara Falls. As such, the Niagara River near Niagara Falls is simply a wide channel with a knickpoint, a situation fundamentally different from those of most theater-headed valleys that have been attributed to groundwater sapping on Earth and Mars. Laity and Malin [1985, p. 207] emphasized this distinction, stating, "field evidence suggests that the scarps associated with theater heads do not result from waterfall erosion. The notches through which runoff flows at the top of the headwall are generally very narrow and represent an insignificant fraction of the total relief and breadth of the theater head." In this paper we argue, using a combination of numerical modeling and morphological analyses of theater-headed valleys, that groundwater seepage is likely an important influence on the formation of many theater-headed valleys. More specifically, we argue that valleys that are predominantly U shaped in cross section and are much wider than the channels that occupy the valley floors are most likely to have been influenced by groundwater-sapping processes. We agree with Lamb et al. [2006] that some rounding of valley headwalls can occur in valleys that are V shaped in cross sections even in the absence of seepage, a point to which we will return to in the Discussion section of this paper. [11] The subsurface flow–dominated version of our model is conceptually similar to the models employed by Howard [1995] and Luo and Howard [2008]. Howard [1995] proposed a model in which the rate of escarpment retreat was assumed to be a function of seepage discharge. Howard [1995] showed that theater-headed valleys could form in his model. However, the model of Howard [1995] did not explicitly differentiate weathering and transport, making the precise role of seepage difficult to infer. The relationship between process and form in theater-headed valleys also remains poorly quantified. For example, what variables control the widths or curvatures of theater-headed valleys? In order to determine the controlling variables of theater-head morphology and to further test whether or not groundwater sapping is required to form theater-headed valleys in cases in which alternative mechanisms (e.g., catastrophic flooding) can be ruled out, it is necessary to compare the morphologies of valleys produced by numerical models that include groundwater seepage with those that do not. This paper seeks to partially fill that gap, building from previous modeling work. Luo and Howard [2008] differentiated weathering from transport and modeled soil production and weathering and transport on Martian landscapes subject to surface and subsurface water flow. They showed that seepage weathering in combination with transport of weathered debris by runoff could form theater-headed valleys. It is not straightforward to apply the results of Luo and Howard [2008] to terrestrial cases, however. 2. Methods Surface Flow–Dominated Weathering Model [12] We consider two end-member weathering models in this paper: one in which weathering occurs in relation to the amount of runoff-driven infiltration and the other in relation to the amount of seepage. Both end-member models transport weathered debris by colluvial processes (e.g., mass movements, creep, bioturbation) and slope wash or fluvial entrainment. Except for the source of water that drives weathering, we have designed the two models to be as similar as possible (both in terms of the mathematical components of each model and the specific parameter values adopted in the two models) in order to isolate the effects of the presence or absence of seepage weathering on the output topography. Both models track the elevation of the bedrock surface, b(x,y,t), and the thickness of a mobile layer of soil or weathered material, h(x,y,t), via the conservation of mass equations: where z(x,y,t) is the elevation of topography, x and y are distances along the two spatial dimensions, t is time, ρb is the bedrock density, ρs is the bulk density of soil, P is the soil production rate measured normal to the surface, θ is the slope angle, U is the rock uplift rate, and E is the erosion rate. Soil production occurs in the model via the exponential production function of Heimsath et al. [1997, 1999, 2001, 2006]: where P0 is the maximum soil production rate (which in the exponential production function occurs for bare bedrock conditions) and h0 is a constant equal to approximately 0.5 m [Heimsath et al., 1997, 1999, 2001, 2006]. [13] In the surface flow–dominated model the maximum soil production rate P0 is a function of the average water content of the rock and hence is related to total infiltration. Infiltration is modeled using a Green-Ampt approach in which infiltration is proportional to the time-averaged depth of overland or channel flow. Our model does not resolve individual flow events, but instead assumes that infiltration into the bedrock increases linearly with a characteristic flood flow depth, l, i.e., where P0s is a characteristic maximum soil production rate (units of L1T−1) for hillslopes in a particular climate (e.g., ∼0.01–0.1 m kyr−1 in arid to semiarid climates) and ls is a characteristic overland flow depth on hillslopes during runoff events (e.g., ∼1 cm). Equation (5) assumes that lateral redistribution of infiltrated water is negligible at scales over which the model is applied, i.e., tens of meters to kilometers. Manning's equation and the observed power law width-area relationship of bedrock channels [Whipple, 2004] enable (5) to be recast in terms of contributing area A via a power law relationship with an exponent of 3/8: [14] The values of P0s and As are constrained so that P0 is of the order of a few centimeters per thousand years on hillslopes, where contributing areas are on the order of 1–10 m2. This constrains P0s to be ∼0.01–0.1 m kyr−1 and As to be ∼1–10 m2. The weathering model of this paper does not explicitly include structural responses to erosion, i.e., exfoliation jointing. Instead, we consider exfoliation jointing to be implicitly part of the weathering process because exfoliation jointing introduces weaknesses into the rock that weathering agents must exploit in order for slope failure to occur as adjacent or overlying rock is exhumed. It is this combination of exfoliation jointing and subsequent weathering that makes mass movements from the valley headwall possible. [15] The transport of unconsolidated debris, where it exists, occurs by colluvial and slope wash or fluvial processes. Colluvial transport is modeled using a nonlinear depth- and slope-dependent transport relationship: where κd is a transport coefficient (units of L1 T−1) [Roering, 2008] and Sc (unitless) is the tangent of the angle of stability of unconsolidated debris, equal to approximately 1 as a reference value. The value of κd is ∼1 m kyr−1 based on available data for the effective diffusivity of soil-mantled landscapes in semiarid climates (i.e., approximately 1 m2 kyr−1) [Hanks, 2000], assuming a characteristic soil thickness of 1 m. The erosion rate of unconsolidated material is equal to the divergence of colluvial sediment flux (i.e., (7)) plus a slope wash or fluvial transport term that depends linearly on unit stream power, i.e., where S is the local gradient of the hillslope or valley, K is an erodibility coefficient (units of L1 T−1), and E is defined to be positive if material is being removed. The conditional statement in (8) restricts erosion to occurring if and only if unconsolidated debris is available for transport. The contributing area in our model is computed using the D∞ algorithm of Tarboton [1997]. The first term in (8) is solved using upwind differencing while the second term is solved using an explicit method that computes the erosion or aggradation attributable to colluvial processes using the difference between fluxes calculated explicitly between each pair of pixels in the x and y directions. The time step of the model was forced to obey the Courant stability condition for each term in (8). It should be noted that Pelletier [2010] proposed an alternative to (8) that incorporates subgrid-scale variations in the effective width of overland or channel flow explicitly in both terms on the right-hand side of (8). In that approach, variations in the grid-resolution dependence of multiple-direction flow routing on hillslopes versus valleys is used to differentiate between hillslopes, where sheetflow or rill flow occurs throughout a pixel, and valleys, where flow is confined to a valley-floor channel with width smaller than that of a pixel. Adopting such an approach is crucial if the goal is to precisely model the transition from hillslopes to valleys (in order to quantify controls on valley density, for example). In this paper we adopt the simpler and more common approach of not explicitly including effective width in (8). We tested an alternative of the model of this paper with the Pelletier [2010] approach included and verified that the results of this paper were not qualitatively sensitive to whether or not effective width was explicitly included. [16] Given representative values for P0s ∼ 0.01–0.1 m kyr−1, As ∼ 1–10 m2, κd ∼ 1 m kyr−1, and Sc ∼ 1, the behavior of the surface flow–dominated model is determined principally by the ratio K/P0s. In the limit that K/P0s goes to zero, soil builds up on hillslopes and in valleys, slowing erosion and forming soil-mantled landscapes. In the opposite limit, unconsolidated debris is transported out of the valley as quickly as it is formed. In this limit, the model results are qualitatively independent of K/P0s. That is, as long as K/P0s is sufficiently large that unconsolidated material does not build up on hillslopes, the erosion of the landscape will be dictated by P0s and As, parameters that are reasonably well constrained with available data. Subsurface Flow–Dominated Weathering Model [17] In the subsurface flow–dominated model, bedrock weathering is driven entirely by groundwater seepage. As such, this model assumes that weathering of the landscape above valley heads (a low-relief plateau is assumed in the models of this paper) can be neglected and that weathering and collapse of the upper portions of valley headwalls are controlled by the rate of removal of rocks below them that are subject to seepage weathering. In nature, the lower portions of valley headwalls are directly subject to seepage weathering if seepage is present. The upper portions of valley headwalls are not directly influenced by seepage but nevertheless retreat at a rate comparable to that of the lower portions of the headwall because of undermining. If rocks from the upper portion of the headwall were removed at rates significantly lower than those of the seepage-influenced rocks below them, an ever-growing overhang would form that, given sufficient time, would guarantee collapse. Transport of debris from the headwall to the valley floor takes places primarily by colluvial transport and mass movements, i.e., headwall slopes in the subsurface flow–dominated model form in excess of Sc, and hence soil is moved off the headwall primarily by mass movements. [18] Removal of debris from the base of talus slopes and from the valley floor requires transport of debris by episodic flood events since seepage is unable to transport significant amounts of debris in many terrestrial cases. As such, in the subsurface flow–dominated model we use the same runoff-driven transport equation that we used for the surface flow–dominated case (i.e., (8)). On Mars, it may be that transport of weathered debris by seepage and/or wind is responsible for 100% of the removal of weathered debris from theater-headed valleys. Without more quantitative constraints on past seepage discharges, sediment grain sizes, etc., on Mars it is difficult to evaluate this hypothesis. On Earth, however, seepage discharges are too small (at least under present conditions) in many theater-headed valleys for seepage alone to be responsible for the removal of debris, as recognized by many studies [e.g., Laity and Malin, 1985; Lamb et al., 2006; Irwin et al., 2008]. [19] The subsurface flow–dominated model uses the Boussinesq equation to compute the height of the groundwater table, i.e., where Sy is the specific yield (unitless), k is the hydraulic conductivity (units of L1 T−1), and η(x, y, t) is the water table height. Equation (9) is solved using a finite difference scheme analogous to that used to solve (8) with boundary conditions η = 0, where z < zb (zb is the elevation of the seep above the base level of erosion (defined to be z = 0)), η = η0 at the upslope boundary of the model, and the initial condition η = η0. The weathering rate in the subsurface flow–dominated model is proportional to seepage discharge at the valley headwall and sides, i.e., where is the unit normal vector of the valley headwall and sides and the product η∇η is evaluated on the upslope sides of the portions of the landscape where seepage occurs, i.e., the interface defined by z = zb. The variable c (units of T−1) in the subsurface flow–dominated model plays a role analogous to that of the variable P0s in the surface flow–dominated model, i.e., both quantify the relationship between the rate of potential soil production and the flow of water that drives soil production. The transport of unconsolidated debris out of the canyon is modeled using the same relationship (i.e., (8)) as in the surface flow–dominated model. [20] Analytic solutions for the shapes of theater heads in the subsurface flow–dominated model can be obtained by linearizing the Boussinesq equation. This analytic solution, while it approximates the nonlinear groundwater flow equations with a linear equation, complements the numerical results we will present in the next section and provides a simple testable prediction for theater-head morphology that can be