This paper examines cylindrical pothole growth on streambeds using empirical analyses of field data and geometric constraints. Pothole depths (d) and average radii () at three localities have the relationship = kd, where k and ɛ are regression coefficients (R2 ≥ 0.72). Observed ɛ (0.57, 0.67, 0.85) translate to d increasing faster than r at all localities. The strong correlations and absence of potholes with very low or high ratios of depth/diameter suggest that small concavities act as pothole seeds and enlargement is quasi-systematic. Exploiting the power relationship, growing potholes can be represented as deepening and radially expanding cylinders. Absolute and relative distributions of erosion can be calculated for floors and walls using this geometrical approach. Volumetrically, more substrate is eroded from pothole walls than floors during growth for ɛ > 0.5. Among sample populations, as much as 70% more material is eroded from walls than floors (ɛ = 0.85). Wall and floor surface areas differ by 1 or more orders of magnitude for observed ɛ, and as a result, erosion rates are fastest atop floors. Differences in erosion rates may reflect the efficacy of erosion phenomena. Low-angle impacts of tools on walls presumably have low erosion efficiencies. Efficacies are presumably influenced by substrate properties, and floor and wall erosion rates are most comparable in the weakest observed strata, although substantially more material is removed from walls at this locality (ɛ = 0.85). Additional data is needed, but quantifiable relationships may exist between geometries, substrates, and erosion phenomena.
 Incision of bedrock streambeds has recently become the subject of numerous process-oriented studies conducted at multiple spatial and temporal scales [Hancock et al., 1998; Whipple et al., 2000a, 2000b; Stark and Stark, 2001; Springer and Wohl, 2002; Stock et al., 2004; Whipple, 2004], but numerous questions concerning discrete erosion phenomena remain unanswered [Whipple, 2004]. For instance, particle versus bed interactions driven by persistent, nontransient vortices in potholes or other sculpted forms may locally dominant channel incision, but relationships between vortex-driven incision and variables applicable to modeling whole river system behavior are ill defined. However, it has been demonstrated that sculpted forms can be quantitatively tied to stream hydrology [Blumberg and Curl, 1974; Curl, 1974]. Currently, focus has turned to stream potholes because they are interpreted to play key roles in some streams (Figure 1) [Hancock et al., 1998; Whipple et al., 2000a, 2000b], despite their presence often being limited to short stream reaches. Ideally, defining relationships between pothole dynamics and phenomena of greater spatial and temporal scales will lead to improved landscape modeling capabilities [Whipple, 2004].
 Potholes grow by a combination of wall and floor erosion. Sediment particles involved in erosion are generically called tools (Figures 1a, 1b, and 1d) [Gilbert, 1877]. Tools entrained within pothole vortices, but above the floor may contribute to erosion by impacting wall surfaces. Tools residing on pothole floors may roll, slide, or skip. Bed load–size tools that become entrapped in a pothole are referred to as grinders (Figure 1a). Grinders are swept around pothole floors and against potholes walls, which causes a variety of erosive effects such as abrasion. Grinders have been perceived to be largely responsible for pothole growth, but Whipple et al. [2000a] report that abrasion by suspended sediment is probably responsible for coalesced potholes in a rapidly incising stream. Nonetheless, bed load is 2 to 3 orders of magnitude more erosive than fine sediment in vortex-driven, experimental flows [Sklar and Dietrich, 2001]. By inference, bed load is probably more effective at eroding pothole floors than suspended sediment when it is present.
 Differences in erosion phenomena acting on pothole floors and walls should be expressed as differences in the spatial distribution of surface erosion rates. Quantifying the spatial distribution of erosion phenomena, as represented by wall and floor volumes, allows for calculation of spatial variations in surface erosion rates (i.e., Vw/Sw, where Vw is volume eroded from a wall and Sw is wall surface area). The rates should differ if total floor and wall surface areas are unequal or if volumetric losses are unequal. Intuitively, both scenarios should apply and the spatial distribution of surface erosion rates may be instructive as to the nature and effectiveness of erosion phenomena in potholes.
 The main objective of this paper is to develop a numerical description of pothole growth that will aid future studies. Field observations and data justify geometric formulations that describe potholes. Three data sets consisting of pothole geometries measured at three locales are individually and collectively characterized using empirical techniques. A power relationship between pothole dimensions is empirically defined, and the relationship is used to explore the dynamics of pothole growth.
3. Study Area
 The data sets used in this paper originate from field measurements of natural pothole dimensions, specifically depth (d) and radius (r), made in three different reaches of the Orange River, Northern Cape, South Africa. The 2350 km long Orange River is the major drainage from southern Africa into the Atlantic Ocean, covering a catchment of >892,000 km2 (Figure 2) [Tooth and McCarthy, 2004]. Flow throughout the Orange River drainage basin is highly erratic, but perennial with major flooding occurring during the austral spring and summer.
 Coarse bed load is notably sparse at all three study sites (Figures 1b–1d), with grinders absent from the majority of potholes and little evidence of extensive gravel or boulder bars. Observed bed load clasts vary in lithologic composition, but bedded ironstones are an important component at the study localities. Ironstones are dense and extremely hard and thus presumably important tools for eroding potholes and other channel features [Sklar and Dietrich, 2001]. By contrast, fine sediment, particularly sand, is present in reaches adjacent to the study sites at Augrabies and Boegoeberg (Figure 1b) and there are large expanses of sculpted bed forms downstream of the Kakamas site (Figure 1d). Collectively, (1) local presence of suspendable sediment; (2) polished pothole surfaces, (3) coincident sculpted forms, and (4) fast water velocities as inferred from sufficient stream competence to maintain large areas of bare bedrock as opposed to bed load bed forms are strong evidence for suspended sediment playing a nontrivial role in local channel processes.
3.1. Augrabies Falls National Park
 The potholes examined at Augrabies Falls National Park lie in the southernmost (river left) anabranch of a 2-km wide belt of bedrock channels upstream of the two principle waterfalls that mark the termini of a 150-km long anabranching river segment in the Orange River (Figure 2) [Tooth and McCarthy, 2004]. There is no local gauging station and discharge is subdivided among many channels (anabranches). The potholes are scattered along a ∼40 m long (longitudinal) reach of an inner channel that is inset within a 400-m wide anabranch (Table 1). The potholes occupy a prominent, 6-m high knickpoint. The coordinates of the center of the reach are 28°35′47″S, 20°20′41″E. Potholes penetrate virtually all knickpoint surfaces and are forming a vertical slot within the bed of the inner channel, which bisects the vertical knickpoint face. Schmidt hammer and Selby scores are indicative of resistant bedrock (Table 1). Bed load is sparse in the anabranch, although gravel- to cobble-size tools are present in the majority of the deepest potholes. Dense packing of potholes has led to coalescence of potholes, unlike the other two study locales.
Table 1. Summary of Reach Characteristics, Pothole Statistics, and Erosion Ratios
 Boegoeberg Dam on the Orange River is located approximately 170 km upstream from Upington, Northern Cape (Figure 2) and are widely scattered over a 150 m, longitudinal length of the 550-m wide channel (Table 1). Potholes are in the immediate vicinity of a small gauging station located ∼2.5 km downstream of the dam. The gauging station is on river left and 150–200 m downstream of a prominent weir. The coordinates of the midpoint of the site are 29°01′32″S, 22°1′00″E. The potholes are across an inner channel from the gauging station on a channel-spanning, 0.5 km long strath. Although sediment is extremely sparse on the strath, large islands and deposits of sand lie immediately downstream of the strath terminus. Flow is perpendicular to strike and beds dip 30° in the upstream direction. Plumbing failed to determine the full depth of the inner channel, although a minimum depth of ∼13 m was established. Potholes are spaced many radii apart.
3.3. Upstream of Kakamas
 Potholes were examined ∼6 km upstream of Kakamas (Figure 2). The potholes are located on the river right shoulder of a 60-m wide inner channel. The potholes are found atop and adjacent to a 50-m long segment of the inner channel, which is identifiable by the presence of large potholes up to 4 m deep (Table 1). The coordinates of the site are 28°46′26″S, 20°41′35″E (Figure 2). Foliation dips ∼30° and strikes subparallel to the channel. Although the rock hardness score for the phyllitic quartzite is similar to that at Boegoeberg Dam (Table 1), weathering of micas along the gneissic foliation causes rock surfaces to be friable except where erosion rates are (apparently) fastest and clean, fresh (hard) bedrock surfaces are maintained by abrasion. Bed load is largely limited to gravel and cobble bars in small anabranches to the river left of the inner channel. Potholes are spaced greater than a few radii apart.
4. Field Methods
 Concavities are numerous in bedrock streams; therefore a determination must be made as to what constitutes a pothole. Concavities with approximately elliptical apertures and smooth internal surfaces whose appearances suggested polishing were considered to be potholes (Figures 1a–1c). As a general rule, sampled potholes have smooth walls (Figures 1b and 1c) with no spiraling grooves (e.g., Figure 1a). Small potholes have slightly concave floors and larger potholes have slightly convex floors. Variables depth (d) and radius (r) were measured for as many potholes as possible using graduated tapes. As a result, the pothole numbers (n) reported in this paper are representative of >80% of potholes in each reach and the relative numbers of potholes between the three study reaches. Measurements are for maximum d. Tools were cleared from pothole floors to obtain d whenever necessary. Less than 10% of all potholes contained tools, and individual tool accumulations were volumetrically small compared to pothole volumes. Two perpendicular radii (r1 and r2) were measured. Average radii () are reported herein. Rudimentary sketches were made of pothole perimeter outlines and internally visible fractures. Dimensions and derivative data are interpreted using standard statistical methodologies.
5. Empirical Analyses of Field Data
 Plots of pothole dimensions show a consistent relationship between d and at all three localities (Figure 3). Distributions are log linear; a simple power law describes relationships between d and :
Regressions to obtain k and ɛ produce the trend lines and results in Figure 3, and values in Table 1. Regression residuals are normally distributed for all three data sets, but residual values increase as pothole size increases at Augrabies. The variance problem could stem from application of the wrong transformation (log), but other transformations do not yield better results and the log transformation works well for data from the other localities. Alternatively, the increasing residual values may reflect coalescence of potholes, which is only frequent at Augrabies. The effect of coalescence is a substantial change in d, r1, and/or r2 concurrent with a lesser change in one or two of the other dimensions. The resulting pothole would probably have a different relationship between d and than the average relationship characteristic of the overall population. A test was performed on the Augrabies data to determine what effect the potholes with high residuals have upon the regression values. Data more than two standard deviations from the mean were pruned. Regression performed upon the pruned data set yields k = 2.3 and ɛ = 0.57. The effect of pruning the data is small and regression values obtained using the complete data set are used herein.
 Four of six individual d and distributions are lognormal as evaluated using the chi-square goodness-of-fit test (α = 0.05), although skewed tails are visible in several histograms (Figure 4). Pothole d at Boegoeberg and at Kakamas fail tests for lognormality, although the significance of the Boegoeberg χ2 statistic is 0.05, which is equal to α. The lack of lognormality in the two sample populations must be considered when interpretations are made using the field data.
6. Geometrical Description of Pothole Growth
 The systematic relationships between d and can be exploited to describe pothole growth. Conceptually, pothole can be calculated (equation (1)) for the range of d values in a sample (Table 1). The resulting sets of d and define the basic geometries of average cylindrical potholes for each of the three sites during pothole growth. Knowledge of the dimensions of potholes during growth provides the means to calculate volumes and other geometrical variables as functions of pothole scale, explore the effects of scale upon relationships between variables, and compare pothole growth among study sites.
 The justification for using the power relationship for representing pothole growth includes the following facts. The relationship is observed at three widely separated sites among potholes developed within three different lithologies. Regressions yield large R2 values that are highly significant. The well-defined relationships between d and (Figures 3a–3c) are strong evidence against excavation of narrow or broad aperture potholes followed by accelerated enlargement in the dimension that was not favored during initial growth (Figure 3d). Such a scenario would lead to a broad scatter of points (potholes) to the left or right of observed distributions (Figure 3d). Hence pothole growth must be progressive with small concavities acting as seeds for small potholes, which grow along paths with average dimensions described by equation (1). (Figure 1c). The ubiquity of this scenario is indeterminable until more studies are conducted, but in this case equation (1) is an expression of how sampled potholes grow and therefore the processes responsible for pothole growth.
 Measured pothole dimensions also provide some insight about growth cessation and pothole removal. The lack of data points above (Figure 3d) of the observed distributions indicates that potholes are not removed by incision of adjacent bed surfaces (truncation). Progressive or persistent truncation of potholes at a locality would result in potholes with comparatively large r but small d. Absent such potholes, the inference must be drawn that potholes are removed abruptly without opportunity to skew observed distributions. Possible mechanisms include coalescence or collapse of the growing potholes and detachment of the host block(s). Weakening of channel substrates by potholes may precipitate the latter event.
 In summary, empirical data and related observations provide qualitative and quantitative support for describing pothole growth using equation (1). Exponents may differ between sites, but there is an underlying similarity of scaling between study sites.
6.2. Formulation and Erosion Metrics
 Enlargement of a pothole, as constrained by equation (1), requires simultaneous retreat of the pothole floor (downward) and sidewalls (outward). The contact between floor and wall will simultaneously move downward and outward in concert with the two surfaces. Movement of the contact along the z-y plane (Figure 5) is described by equation (1), which can be expressed as y = kz. The area below this curve represents material eroded from the bed. The region above the curve, but within the pothole, represents material eroded from the wall. The total volume eroded from a pothole floor (Vf) between any two steps (n, n + 1) can be obtained by integrating equation (1) along the z axis,
where z is the position of the pothole floor for the step in subscript. The total volume eroded from pothole walls (Vw) during the same interval includes substrate eroded above and below the position of the floor at step n (boundary between regions a and b in the Figure 5 inset). The volume of material eroded from above region a is simply the difference of two cylinders,
where y is the radius of the pothole for the step in subscript. The volume of material associated with region b in the Figure 5 inset can be calculated by subtracting Vf from the volume of a disk with radius yn+1 and thickness zn+1 − zn, which is simply the increase in depth between steps. The formula is
Vw is the sum of equations (3) and (4). The total volume of material (VT) eroded between any two steps is, VT = Vw + Vf.
 The formulation provides the means to define differences in erosion volumes and relative surface erosion rates between pothole floors and walls. The ratio Vw/Vf can be used to evaluate normalized differences in erosion volumes. However, choice of a metric of surface (S) erosion rates (i.e., Vw/Sw and Vf/Sf) is complicated by the fact that floor and wall surface areas increase between steps. Therefore a reference surface area must be chosen. Sn+1 is chosen as the reference surface for floors and walls. Sn is rejected, because its use to calculate Vf/Sf would produce values that are greater than the actual depth increase.
6.3. Results of Geometrical Formulation
 The ratio Vw/Vf is greater than 1.0 for ɛ > 0.5 (Figure 6 inset). The difference is especially great for potholes with ɛ ≤ 0.7. However, actual erosion rates expressed in the traditional manner of V/S [e.g., Hancock et al., 1998; Whipple et al., 2000a] are substantially different (Figure 7). Sidewall and floor erosion rates are only comparable where potholes have especially large apertures relative to depth (ɛ ≥ 0.9) (Figures 6 and 7). Even at Kakamas, where potholes have large apertures relative to depths (Figure 3c), surface erosion rates differ by approximately tenfold. The common power relationship masks significant diversity in geometrical growth patterns.
 Pothole dynamics are very similar at the three sample locations despite significant differences in geomorphology and substrate properties. Potholes scale as power functions and d and are strongly correlated with one another at each locality (Table 1 and Figures 3a–3c). Actual pothole geometries, specifically relationships between d and , differ significantly between localities and ranges of d and also differ (Figures 3 and 4 and Table 1) Nonetheless, adherence to the power relationship and observed values of ɛ translate to intersite consistencies of erosion phenomena. Greater volumes of substrate are eroded from pothole wall than floor surfaces (Vw/Vf > 1) and erosion rates (V/S) are highest atop floor surfaces in all cases. Consistency of geometrical scaling properties and erosion dynamics, expressed as spatial distributions of relative erosion volumes and rates in multiple rock types and different channel settings (i.e., knickpoint versus strath surface) can be interpreted as evidence for a single set of underlying growth mechanisms.
 Flow strength and lines determine tool impact effects upon pothole surfaces. Downward spiraling vortices presumably direct tools toward pothole walls because of centripetal motion, but flow lines (and tools) must be approximately parallel to pothole wall surfaces or coherent vortices could not persist. Flow indicators in lateral potholes record just such conditions [Springer and Wohl, 2002]. Oblique tool impacts are less erosionally effective than more acute impacts [Hancock et al., 1998; Sklar and Dietrich, 2001, 2004] and this may explain disparities between wall and floor erosion rates determined in this study (Figures 7 and 8). Coarse tools may roll, slide, and saltate along pothole bottoms, which on the basis of experiments with sediment in cylinders stirred by a propeller, are more effective at eroding substrate than fine sediment [Sklar and Dietrich, 2001]. Erosion by suspended sediment may be enhanced by turbulence associated with the switch from descending to ascending flow, which would cause tools to strike surfaces at more acute angles than tools descending along pothole walls. The relative contributions of walls and floors to total erosion are presumably a function of local geology, geomorphology, and hydrology. A lack of knowledge concerning interplay of these factors, particularly the actual flux of bed load through potholes, prevents definitive statements from being made about erosion of pothole floors.
 Available data and geometrical results do allow for identification of factors that may be responsible for some of the differences among potholes of the three localities. Examining the two extremes present in the data set, floor erosion rates at Augrabies (ɛ = 0.57) are 3 orders of magnitude faster than wall erosion rates, whereas the difference is only an order of magnitude at Kakamas (ɛ = 0.85) (Figure 7). Compressive strength is higher at Augrabies than at Kakamas (Table 1). Foliation planes in the phyllitic quartzite at Kakamas are tightly spaced, which decreases the mechanical strength of the substrate relative to a massive granite gneiss with widely spaced joints such as is found at Augrabies. Furthermore, the quartzite at Kakamas is phyllitic and therefore (presumably) weathers more easily than the quartz-rich granite gneiss of Augrabies. Recognizing that bedrock is significantly more resistant (and shows less variability) at Augrabies, we postulate that the low ɛ value is an adjustment to rock hardness. Lower values of ɛ translate to smaller r for a given d. Limiting radius has the effect of increasing the ratio of wall surface area to floor surface area and tool action is focused on a smaller floor area in potholes with low ɛ. Thus potholes that are deep and narrow may focus tool action on small floor areas because distribution of work across larger areas would provide insufficient intensities to enable effective erosion. In such cases, oblique tool impacts should have erosion efficiencies that are quite low because of bedrock properties, which would serve to minimize surface erosion rates. The relationship may be nonlinear because of the effect of velocity upon tool kinetic energy. Extending this line of reasoning to Kakamas, the substrate has lower compressive strength, possesses fine discontinuities (foliation planes), and is more variable. Presumably, the more favorable conditions for erosion cause necessary work intensities associated with walls and floors to be less disparate. Oblique impact of tools have greater effects and enhance wall erosion, which are expressed as larger ratios of Vw/Vf (Table 1). The postulate requires further exploration, but the general lack of bed load at all localities favors it and the implication that bedrock resistance may be directly expressed in pothole geometry.
 The results of this study show that (1) observed pothole populations can be described by a single geometrical formulation, whereby pothole radius can be described as depth raised to an exponent (ɛ). (2) A functional, geometrical description of pothole growth can be constructed using the basic geometric rules for cylinders and (3) numerical results indicate that more material is eroded from wall than floor surfaces during pothole growth for observed values of ɛ. (4) Wall surface areas are larger than floor areas and as a direct result actual erosion rates (volume/surface area) are highest atop floors. (5) Inferred erosion rates are compatible with expectations concerning erosion processes believed important for pothole excavation. (6) Speculatively, the properties of individual pothole sample populations may vary as a function of substrate resistance.
 Potholes can be important components of channel incision [Wohl, 1993; Hancock et al., 1998; Whipple et al., 2000a, 2000b]. Results made possible by the combination of field and geometrical analyses provide new insight into pothole dynamics and show that these and other methods can elucidate the relationship between pothole geometries and controlling factors. Ultimately, such advances will provide extremely valuable information to geomorphologists who seek to understand the nature and controls of long-term channel incision [see Whipple et al., 2000a; Whipple, 2004].
pothole depth, cm.
regression coefficient; describes pothole size for a given geometry.
pothole radius, cm.
mean pothole radius, cm.
surface area of pothole floor, cm2.
surface area of pothole wall, cm2.
volume of substrate eroded from pothole floor, cm3.
volume of substrate eroded from pothole wall, cm3.
distance from center of aperture to rim (r), cm.
distance from pothole aperture to floor pothole vertical axis, cm.
 Research was supported by National Science Foundation grants EAR-0228876 (G.S.S.) and EAR-0228853 (E.E.W.). Additional research funding was provided by grant 2002-33 of the Ohio University Research Council (G.S.S.) and grant BA-04-03 of the Ohio University Baker Fund (G.S.S.). We thank De Beers Africa Exploration for providing logistical support and financial assistance (S.T.) and Augrabies Falls National Park, RSA, for unrestricted access to the Orange River at Augrabies Falls, RSA Department of Water Affairs for access to the Boegoeberg reach, and David Spandenberg for access to the Kakamas site. The manuscript benefited greatly from insightful reviews by Kelin X. Whipple and two anonymous reviewers.