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Corresponding author: P. A. Carling, Geography and Environment, University of Southampton, Southampton SO17 1BJ, UK. (email@example.com)
 Landforms, morphologically similar to aeolian yardangs but formed by erosion of bedrock by currents on an estuarine rock platform, are described for the first time. The geometries of the “yardangs” are described and related to semi-lemniscate shapes that minimize hydraulic drag. The processes of bedrock erosion by the reversing sediment-laden tidal currents are described, and a semi-quantitative model for landform evolution is proposed. The model casts doubt on the “simple” role of the maximum in the two-dimensional vertical suspended sediment flux distribution and the consequent distribution of potential kinetic energy flux in the process of shaping the rock wall facing the ebb flow. Rather, although the kinetic energy flux increases away from the bed, the sediment becomes finer and abrasion likely is insignificant compared with coarse sand abrasion lower in the profile. In addition, the vertical distribution of sediment flux is mediated by topographic forcing which raises the elevation at which bed load intersects the yardang prow. Consequent erosion leads to ebb-facing caprock collapse and yardang shortening. In contrast, the role of ebb-flow separation is paramount in mediating the abrasion process that molds the rock surface facing the flood flow. The length of yardangs is the least conservative dimension, reducing through time more rapidly than the height and width. Width is the more conservative dimension which implies that once the caprock is destroyed, scour over the obstacle is significant in reducing body height, more so than scour of the flanks which reduces width. The importance of vertical fissures in instigating the final breakdown of smaller yardangs and their extinction is noted. Similarities to aeolian yardang geometries and formation principles and processes are noted, as are the differences. The model has implications for aeolian yardang models generally.
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 Yardang is a Turkmen word introduced by [Hedin1903] as a term for wind-abraded desert ridges that has been applied widely to aeolian-sculpted hillocks [Goudie, 1999]. The length scales of these streamlined landforms can range from centimeters (microscale), through meters (mesoscale) to kilometers (megascale: in the terminology of [Cooke et al. 1993]). Widely described in arid and semiarid environments on Earth [McCauley et al., 1977a; Laity, 1994; Livingstone and Warren, 1996; Breed et al., 1997], in the geological record [Tewes and Loope, 1992] and on Mars [Ward, 1979; Hynek et al., 2003; Carter et al., 2009; Zimbelman and Griffiths, 2009], the morphology is variable and can include long parallel ridges but at the mesoscale, the obstacle commonly consists of a streamlined form of finite length, which minimizes drag in the prevailing wind [Ward and Greeley, 1984], commonly with an overhanging upwind prow and rounded “whaleback” lee [see Goudie, 1999, Figure 8.2; McCauley et al., 1977b, Figure 14]. The overall shape of these mesoscale landforms is thought to be due primarily to aeolian abrasion by saltation and traction transport of sediment. Sediment is concentrated close to the bed and so erodes the lower portion of the upwind nose and flanks of the yardang more than those portions at a higher elevation (which are rounded by suspension load), thus resulting in the overhanging profile [Greeley and Iversen, 1985; Laity, 1987] and leeside rounding. In some instances, the plan view morphology is influenced by jointing patterns in the bedrock [e.g., Vincent and Kattan, 2006]. A harder caprock may characterize the top of soft-rock yardangs, and this structure can lead to a more pronounced overhang on the upwind side [de Silva et al., 2010]. At the simplest, theory has shown that the vertical location (the invert) of the maximum in the recession rate of the profile may be related to the maximum in the vertical sediment flux distribution and the consequent maximum in the distribution of kinetic energy flux of the approaching flow [Sharp, 1964; Anderson, 1986; Cooke et al., 1993; Bridges et al., 2005]. However, there are no field or wind tunnel experiments to demonstrate the relationship of this theory to the development of aeolian yardang forms in any detail.
 In this paper, estuarine bedrock landforms with streamlined shapes that are similar to aeolian yardangs are described. Because of the similarity of form to aeolian yardangs, these water-sculpted features are termed “yardangs” in the following text, in accord with the typology of Richardson and Carling . These features developed subaqueously, and so the primary aim of the paper is to demonstrate the relationships between sediment-laden aqueous flows, bedrock erosion, and the form of the bedrock obstacles. However, the morphological similarity of the estuarine landforms to aeolian mesoscale yardangs spurred the equally important secondary aim: to provide some generic observations from water flows that might be applicable to aeolian yardangs. A benefit of the estuarine environment is the high frequency of predictable tidal flows which allows a more detailed appreciation of the flow field in the vicinity of bedrock obstacles than has been reported for field examples of aeolian yardangs. Although there are important differences, the typical air-flow patterns and sediment transport pathways associated with wind fields around yardangs [Whitney, 1983; Ward and Greeley, 1984] broadly should be similar to those that occur around streamlined obstacles within water flows [Allen, 1982], so a degree of analogous development might be anticipated. In particular, the distribution of kinetic energy flux of the approaching flow is examined, and conclusions are drawn with respect to the erosion processes over both aeolian and subaqueous yardangs. Specifically, the hypothesis is tested that the heights of the erosional inverts in the yardang profiles are constrained by the vertical distribution of potential kinetic energy of suspended sediment.
1.1 Location and Site Description
 The yardangs are on Hills Flats, an intertidal bedrock platform in the Severn Estuary, UK (Figure 1) [Allen and Fulford, 1996]. The geology is the Triassic Mercia Mudstone Group [Welch and Trotter, 1961], parallel-bedded layers, 0.5 to 1 m thick, dipping a degree or two to the northwest. Gray sandstone forms the platform, above which higher elevation beds, brick-red sandy marl with thin intercalations of gray sandstone, are now truncated further to landward by small vertical cliffs (0.5 m to 1.5 m high). Each yardang, clustered around 51°40′42.6″N: 2°32′26.2″W, has a sandstone base but consists of approximately 0.5 to 1 m thick unit of marl commonly exhibiting a 0.08 to 0.16 m thick cap of sandstone.
 Two intersecting sets of vertically orientated fissures occur in the beds [Allen and Fulford, 1996] which are <0.02 m wide to hairline and extend a few meters only with 0.5 to 1.0 m spacings. Yardangs show no alignment with fissures, but their structural integrity is affected by the presence of the fissures, as is shown in section 3. A sample of 58 fissure orientations showed that there are two distinct populations. One group, with a range of strikes between 265° and 287°, has an average strike of 276°. The other group range between 145° and 222°, with an average strike of 183°.
 On the platform, there is a thin, patchy veneer of shale gravel, sand, and mud (Figure 1b). Thus, a sparse coarse bed load passes individual yardangs. Although the bed load moves up estuary with the flood tide, residual sediment transport is down estuary with the ebb tide [Carling et al., 2005].
 The tidal flat experiences a semidiurnal, macro-tidal regime with marked diurnal and lunar-monthly inequalities. At Avonmouth, (Figure 1a), the extreme astronomical tidal range of 14.8 m ranges from a low of −0.2 m below chart datum (−6.7 m Ordnance Datum (O.D.)) to a high of 14.6 m (8.1 m O.D.). Mean high-water Spring tides lie at 13.2 m (6.7 m O.D.), and mean high-water Neap tides lie at 9.8 m (3.3 m O.D.) [Allen and Duffy, 1998]. The yardang swarm is subtidal during Neap tides but is exposed for brief periods during low water Spring tides.
 In subtidal channels, suspended fine sediment including approximately 23% quartz grains [Bryant and Williams, 1983] is present at high concentrations for all tidal states [Hydraulics Research Station, 1981; Crickmore, 1982; Kirby, 1986] and the lower 2 m of the concentration profile is well mixed [Kirby and Parker, 1983]. Deposited sand constitutes between 5 and 10% of material on the flats, with 11 to 12% clay and the remainder chiefly silt and organic detritus [Allen and Dark, 2007]. The platform is not subject to intense wave action; Allen and Duffy  calculated <0.4 m average wave heights which raise the grain size of suspended silt but the grain size of suspended sand is not affected [Allen and Dark, 2007].
 The study required an integrated investigation of hydrodynamics, rock hardness, and erosion rates as well as the geometry of the yardangs. For clarity, the nomenclature applied to define different parts of the yardangs is defined in Figure 2a. A low angle (1° to 3°) pediment may exist between the horizontal platform and the pedestal but is frequently absent. The pedestal is that part of the landform between the pediment (or platform) and the caprock (where present) and may display a distinct invert (at height ź above the platform) between the lower pedestal and the upper pedestal at which point the upper profile becomes overhanging. The invert is usually best developed on the ebb-facing facet with less well-defined inverts along the flanks. Inverts occasionally occur on the flood-facing facets, but commonly, the invert, the upper pedestal, and the caprock are absent on these latter facets which then have whaleback forms.
2.1 Yardang Geometry
 The locations of 41 yardangs were mapped using handheld GPS accurate to within ± 4 m on the survey days. The surveyed yardangs represent the majority of the well-defined landforms. Less distinct examples forming the scabland to the west were not measured, and additional examples may exist in subtidal locations further to the north. The greatest length (L), the greatest width (W) orthogonal to L, and the greatest height (H) of each yardang were measured using a graduated tape, meter rule, and spirit level. The alignment of the long axis in degrees relative to magnetic north was determined using a sighting compass sighted along an axis determined by two ranging poles positioned at the ends of the greatest length of each feature. The yardangs appear to have semi-lemniscate planforms. Consequently, the plan view was approximated as Joukowski semi-lemniscate, or ellipsoid, bodies (Tables 1 and S1) as such approximations have been used to define the plan view of streamlined bodies [Baker and Kochel, 1978; Clark et al., 2009]. Assuming a semi-lemniscate body, a dimensionless shape parameter (k) can be defined:
where A is the plan view area of the yardang.
Table 1. Summary Statistics for Hills Flat Yardangs
aSetting the yardang length to equal unity, the relationship is given between the observed ratios at Hills Flats and the expected ratios for minimized obstacle drag (the latter after Halimov and Fezer ).
where a and b are the semi-lengths (i.e., L/2 and W/2) and Pe is the perimeter length. The values of parameters k, A, and P are included in Table 1 to demonstrate the typical plan view properties of the yardangs.
2.2 Rock Hardness and Rock Erosion Rate
 A Schmidt hammer (Mastrad Ltd., Douglas, UK), range 10 to 70 N mm−2, was used to obtain compressive strengths of (1) caprock of gray sandstone, (2) red marl pedestals, and (3) sandstone platform. The sandstone tends to weather by grain or flake removal. The appellation “marl” [Welch and Trotter, 1961] is imprecise; the rock has the appearance of a shale, sometimes exhibiting faint horizontal lamination but which weathers into 1000 to 5000 mm3 “cuboids,” rhombs, or irregular-shaped blocks of rock rather than flaking along bedding planes. The marl is a brittle material and should be subject to erosion by individual grain removal and propagation of fine networks of cracks rather than by scratching which characterizes erosion of ductile materials [Bitter, 1963a, 1963b]. Debris, when only partially detached, can remain in situ as a patchy surfaced weathered layer (less than approximately 10 mm thick), but areas of unweathered rock, from which weathered material has been eroded and removed, occur widely. All reported data consist of readings obtained from flat clean surfaces of unweathered rock away from the edges of the yardangs to avoid a resonant response.
 To monitor erosion, on the 26 February 2004, a rectangular pit (40 mm by 40 mm in plan and 35 mm deep) was cut in the marl of the invert of an ebb-facing prow. A similar pit was cut in the platform 0.20 m up estuary from the prow. The site was revisited frequently, and by 10 July 2008, the platform pit was undetectable and the invert pit was extremely faint. The erosion rates are obtained by considering the total periods of tidal inundation as determined in section 2.3.
 In order to provide information on Spring tidal flow speeds, flow directions, flow depths, periods of inundation, and suspended sediment concentration data in the yardang field for a series of Spring tides, an S4 InterOcean Systems current meter (threshold: 0.03 m s−1) and a McLane Phytoplankton sampler were deployed at 0.40 m above the bed at location P3 (Figure 1a). The meter recorded current speed and direction, and water depth at 2 s intervals from 2 December 2005 for a series of six typical Spring tidal cycles. The phytoplankton sampler was modified to sample suspended sediment (2 mm > D50 < 0.7 µm) and calibrated in the laboratory to give concentrations with less than 10% error of target values. The sampler collected twenty-four 1 l samples at 20 min intervals throughout a 12.88 m tide. To determine the grain size distribution of sediment above the bed, suspended sediment was collected in six vertical 80 mm diameter cylinders with openings located at 0.25 m, 0.35, 0.45, 0.55, 0.65, and 0.75 m above the bedrock surface. It was not possible to sample closer to the bed as turbulence re-suspended sediment from cylinders shorter than 0.25 m. A bed load sample was obtained by amalgamating samples of the gravel on the bedrock surface during low-tide exposure. The size distribution of the sediment was determined by settling through a 2 m column [Amos et al., 1981] using the method of Dyer .
3.1 Yardang Alignment
 The average alignment for 41 yardangs is 244° with a SD of 21.4° (Figure 1c). Occasionally, two or rarely three closely spaced yardangs occur in line, which alignment is interpreted to represent the breakup of one long yardang into several shorter examples.
3.2 Yardang Shape
Halimov and Fezer  proposed that a plane surface subject to unidirectional flow can be dissected progressively to produce flat-topped mesas at first, elongated ridges and finally low streamlined whaleback landforms that present semi-lemniscate shapes to the flow and thus minimize drag. The optimal W/L ratio for such bodies is 0.27 [Fox and McDonald, 1979], as longer bodies have excessive skin drag. Setting the length equal to unity, the general scaling relationship for a smoothed elongate body is ~L/W/H = 1/0.2/0.1 using aeolian yardang data obtained from both field measurements and laboratory experiments [Fage et al., 1929; Halimov and Fezer, 1989; Ward, 1979] and L/W = 1/0.33 to 1/0.25 using data from experimental water flows [Hoerner, 1965]. Broadly similar L/W water flow scaling ratios are obtained from analyses of streamlined “islands” [Baker and Kochel, 1978; Komar, 1983, 1984; Wyrick, 2005], but in the case of islands, the landforms are not submerged.
 Table 1 and Figure 3 summarize the dimensional characteristics of the yardangs. Fifteen yardangs showed distinctive inverts in the ebb-facing prows ( = 0.16 m; SD = 0.015 m) with inclined lower pedestals of angle, α, facing the ebb flow ( = 33°; SD = 4.4°) and overhangs above the inverts (Table S2). Note that the position of the invert is not usually at the interface between the pedestal marl and the sandstone capstone but commonly is within the marl. The height histogram is strongly skewed because the caprock limits the height of yardangs to less than 0.7 m. In Table 1 and Figure 3, it is evident that the Hills Flat yardangs exhibit geometries that are not dissimilar to the streamlined forms reported by [Halimov and Fezer1989] but tend to be broader for given length. [Halimov and Fezer1989] noted that as aeolian yardangs became smaller, they tended to retain the expected shape ratios noted in Table 1. The same principle does not apply to the Hills Flat yardangs; despite scatter, the width is reduced linearly as the length is decreased until residual small forms remain that have elongate whaleback forms with W/L ratios approaching 0.25 (Figure 4a and Table 1) or are near conical. In addition, there is no clear relationship between W/L and the presence or absence of a caprock. In Figure 4a, only yardangs with a height of ≥0.6 m have caprock present, and as W/L reduces, the height does not reduce systematically nor is there any threshold response to W/L once the caprock is lost.
 Figure 4b shows that few yardangs exhibit L/H ratios in excess of the expected value of L = 10H (Table 1) for a streamlined form and the majority have L/H ratios which are low. These results indicate that height is more conservative than length as yardangs erode. Figure 4c shows that only a minority of yardangs exhibit W/H ratios less than the expected value of W = 2H (Table 1) for a streamlined form and the majority of ratios are too high. These results indicate that width is more conservative than height as yardangs erode.
 None of the yardangs exhibit any evidence of horseshoe-shaped scoured furrows cut into the platform around the obstacles. Such furrows commonly form readily around bluff obstacles [Greeley et al., 1974] but are absent when erosion by suspended sediment is dominant, in contrast to dominance by bed load-induced erosion [Allen, 1982, p. 185]. However, once streamlined bodies have developed, furrow scour is minimized [Allen, 1982, p. 186]. Note that the majority of yardangs have overhanging prows of sandstone facing into the ebb flow and have well-rounded whalebacks of the marl facing into the flood flow (Figure 2). However, some of the larger landforms have overhanging prows at both ends of the obstacle. Both ebb and flood prows may collapse if sufficiently prominent (Figure 2b), a process aided by vertical transverse fissures (Figure 2b).
3.3 Rock Hardness and Rock Recession Rate
 Twenty-nine readings of unweathered caprock hardness yielded an average value of 16 N mm−2 (SD = 3.68) within a range of 12 to 26 N mm−2. Ten readings of the pedestal rock hardness yielded an average value of 13 N mm−2 (SD = 3.24) within a range of 10 to 20 N mm−2. Ten readings of the platform hardness yielded an average value of 27 N mm−2 (SD = 3.41) within a range of 22 to 30 N mm−2. All weathered surfaces returned values <10 N mm−2. The recession rate (R) of the ebb-facing prow averaged 0.0219 mm d−1, which, although a single-point estimate, provides some information on the rapidity of yardang extinction.
 The recession rate, R, of an impacted near-vertical surface is
where Å is the mass of the target material removed from the surface area per unit time, ρt is the density of the target material, T is the susceptibility of the target material to detachment by the kinetic energy, and qke is the instantaneous flux of kinetic energy to the unit surface area per unit time perpendicular to the flow [Rosenberger, 1939; Anderson, 1986]. Thus, for a given rock mass, the rate of erosion is directly proportional to the applied kinetic energy, an issue considered in section 4.
 Tidal flow data are presented here, in some detail, as these data are used in section 4 to explain aspects of the form of the yardangs. Figure 5a presents the flow direction data for a tide of predicted height at Avonmouth of 12.88 m. The characteristics of this tide are similar to the other five Spring tides sampled and so are considered representative of median Spring tides. The flood tide has an average bearing of 52° (SD = 6.82°; n = 1167) over the period of strongly accelerating and decelerating flood tide between time increments 636 and 2970 s. As the flood tide weakens (Figure 5b), the bearing of the current swings toward the north. The period of high “slack” water lasts 6 min during which the flow direction is variable. At the beginning of the ebb tide, the flow has a northerly bearing directly toward the main ebb-dominated channel before assuming a steady average bearing of 236° (SD = 4.96°; n = 3100) over the period of strongly accelerating and decelerating ebb tide between time increments 5600 and 11,800 s before low “slack” water is approached.
 The maximum water depth (hmax) above the yardang field was 8.38 m with maximum flood velocity of 1.65 m s−1 and maximum ebb velocity of 1.85 m s−1, during which Reynolds numbers (Re = Uh/ν) ranged between 2 × 107 and 4 × 107 and between 4 × 107 and 8 × 107 during the period of strongest flood and ebb currents, respectively. The yardangs are deeply submerged through 95% of the tidal cycle (Figure 5c). The duration of the ebb tide is around 1.7 times the duration of the flood tide and velocities during the ebb are sustained at more than 0.5 m s−1 for twice as long as during the flood. Thus, the yardang field is ebb dominated both in terms of sustained current speeds and duration. All flows are subcritical with maximum Froude numbers approaching 0.3 on both flood and ebb. The shear stresses applied to the bed were not measured at location P3 but at locations P2 and P1 (Figure 1a) Williams et al. [2006, 2008] measured peak average Spring tidal current speeds between 1.30 m s−1 and 1.79 m s−1 and average stresses of between 0.89 and 4.9 N m−2, with occasional excursions to 10 N m−2, with the flood peak typically being 1.6 N m−2 and the ebb peak being 4.9 N m−2. Location P1 is slightly to landward of the yardang field and in shallower water (hmax = 5.48 m) during Spring tides. Thus, the slightly higher velocities at the yardang field are as expected, and the range of shear stresses imposed on the bed within the yardang field should cover the same range, although including more occasions with the higher values reported by [Williams et al. 2006, 2008]. In general, ebb-flow parameters (excluding depth) at P3 have twice the values of the flood tide, and thus, the ebb is assumed to be approximately twice as erosive.
 Suspended sediment concentrations, at 0.40 m above the bed, increased monotonically during the flood tide from an initial value of 148 mg l−1 to a value of 654 mg l−1 just before high slack water. Concentrations fell at high water, spiked on the first ebb-flow to 749 mg l−1 (presumably as slack-water settled sediment was resuspended from the bed) and then oscillated during the ebb flow (average ebb concentration: 145 mg l−1; SE = 22 mg l−1). The grain size data in Figure 6 show that whilst fine gravel is transported predominately as a traction load, there is a significant coarse sand/granule component in suspension up to 0.45 m above the bed. Above this elevation, the grain size distribution is uniformly similar at all elevations ranging from silt to 0.25 mm sand but with a variable, random, component of coarser grains at each elevation.
4.1 General Consideration
 The alignments of the yardangs (average 244° with a SD of 21.4°) are consistent with their being eroded and aligned by the rectilinear reversing Spring tidal flow (flood tide: 52° (SD 6.82°); ebb tide: 236° (SD 4.96°). The bearing of the flood tide is essentially 180° out of phase with the ebb tide, and so it cannot be stated that the yardangs are better adjusted in terms of alignment to the bearing of the ebb or flood tide. However, other morphological characteristics, as discussed below, indicate an ebb dominance in term of form. The difference between the average yardang alignment and the average bearing of the two fracture patterns is 32° and 61°, and so there is no structural control on the yardang alignment. However, fractures transverse to the long axes are important foci for enhanced erosion that sometimes leads to the breakup of individual long yardangs into two or more aligned smaller yardangs and in the erosion of yardangs generally. These results indicate that the yardangs develop their distinctive form and orientation due to tidal abrasion. Similar observations of lack of structural control have been made with respect to some aeolian yardangs [Breed et al., 1989].
 The relationship between W and L is described by
 Despite variance, the data trend in Figure 4a is well represented by the linear function (equation ((5))) but power functions have been used widely in other studies of streamlined bodies to describe W/L relationships so equation ((6)) also is considered here. Equations ((5)) and ((6)) are developed using W/L data for individual yardangs (Figure 4a). If the ergodic principle is adopted and the smaller yardangs are assumed to have evolved from larger ones, it is evident that the trend line reflects evolution of large yardangs, which are relatively narrow (e.g., equation ((5)): predicted W/L for 7 m long yardang is 0.24), to smaller yardangs which are relatively broad (e.g., equation ((5)): predicted W/L for 1 m long yardang is 0.80). Baker and Kochel , Kehew and Lord , and Wyrick  obtained power functions relating W to L for streamlined islands, and their regression coefficients usually are not statistically different from 1.0 [see also Komar, 1984], indicating proportionate changes in W/L across the data range. However, their data represent bodies that are orders of magnitude larger than the yardangs, and island shapes are subject to channel-width constraints and channel bend migration effects—which effects do not apply to isolated submerged bluff bodies in wide flows. A regression with the coefficient constrained to equal unity does not follow the trend of the yardang data and is statistically significantly different to equation ((5)) (Figure 4a). The form of equation ((5)) implies that larger obstacles are better streamlined than smaller bodies. This observation implies that the mass of each of the larger yardangs is sufficient that despite erosion, the bodies can survive long enough to become optimally streamlined. However, small yardangs, formed by the structurally influenced divisions of larger yardangs, do not have the body mass to survive long enough to become optimally streamlined. Equation ((5)) also suggests that although the flanks of the yardangs are eroded through time, the bulk losses through time are greatest in terms of length, as has been suggested with aeolian yardangs [Ward and Greeley, 1984]. Given an ebb tidal flow dominance and field evidence for caprock collapse primarily at the up estuary end of yardangs, the length losses are greatest due to ebb flows. The difference in the yardangs and streamlined islands results may reflect scale issues or, more likely, the immersed versus non-immersed status of the yardangs versus islands, respectively.
4.2 Drag Force of Yardangs
 A drag force results from the presence of a yardang in the tidal flow. This drag can be minimized by tidal scour eroding the obstacle to provide a body shape that minimizes flow resistance. The total drag is composed of (1) the skin drag, which is proportional to the surface area of the body, and (2) the form drag which is dependent on the shape of the body and the consequent degree of flow separation around the body. Streamlining will reduce the wake zone and thus reduce the form drag [Schlichtling, 1960, p. 269]; an approximately exponential reduction in width and height of the distal portion largely minimizes separation [Patterson, 1938; Chow, 1981] which explains the whaleback form.
 The drag force is the product of the dynamic pressure and the reference area (A) which is typically defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of fluid motion [Hoerner, 1965]:
where the product of the maximum values of the width (W) and the height of the obstacle (H) are taken to represent A and ρ is the sediment-laden fluid density. As the drag varies as the velocity squared, drag is maximized during peak flow speeds, so U is defined as the maximum velocity of the fluid for flood or ebb (aligned with the long axis of the body) and Cd is a non-dimensional drag coefficient.
 Typical values of the drag coefficient are 0.47 for a sphere settling in water, 1.05 for a cube settling in water, and 0.04 for a streamlined “tear drop” settling in water [Prandtl, 1949]. For a Hills Flat yardang, the most appropriate similitude is a half tear drop fixed to a plane surface (Figure 2a) which has a drag coefficient of 0.09 [Hoerner, 1965] when the bluff end is facing into the ebb flow; the drag increases if the flow is angled against the tear drop obstacle [Andreas, 1995]. No data have been identified to determine the drag coefficient of a half tear drop with the tail facing into the flood flow. Reviewing geometries similar to tear drops, [Andreas1995] concluded that Cd for the bluff case was 0.1 and three times this value where the flow was reversed by 180°. The bluff case Cd values reported by Hoerner  and Andreas  agree within 10%, and consequently, Andreas's multiple (0.09 × 3) is applied to yardangs subject to flood flows.
 Solutions of equation ((7)) for each of 41 yardangs with significant bluff profiles were obtained assuming ebb-flow dominance (Cd = 0.09) and using a maximum flood velocity or maximum ebb velocity and fluid densities based upon suspended sediment concentrations reported in section 'Hydrodynamics'. The average value of drag for the 41 cases has a standard deviation of 37% which largely reflects the variation in W and H between yardangs. For the flood tide, Fd = 279 N; for the ebb tide, Fd = 98 N. Thus, in low-density ebb flows, fluid drag is minimized. The enhanced drag in flood flows is as expected [see Hoerner, 1965, p. 2–3] due to augmented wakes that develop downstream of the bluff (ebb eroded) noses that face up estuary.
 Flow separation will occur around a bluff body once Re ≥ approximately 50 to 100 [Hoerner, 1965], and so at Hills Flats, flow separation is universal given the recorded Re numbers and enhanced drag will occur in high Reynolds number flows (Re > 107) unless the W/L ratios are less than ~0.4 [Baker and Kochel, 1978]. Bodies do not elongate much beyond W/L = 0.33 because the skin friction increases the resistance to flow. About half the yardangs have ratios > 0.4, and consequently, these less-streamlined bodies are subject to significant flow separation. To minimize the value of Fd for any given flow conditions, the drag coefficient would need to be reduced. Such a reduction can be accommodated by minimizing the ratio of W/L → 0.27 [Fox and McDonald, 1979]. However, it is evident from Table 1 and Figure 4a that the average ratio is not optimal; in part, the reason is the effect of scour associated with wakes noted above. Only four yardangs approach the optimal ratio (i.e., 0.18, 0.22, 0.23, and 0.23) as yardangs with optimum W/L ratios become structurally unstable due to the presence of fissures.
 Fissures play a fundamental role in altering rock resistance on shore platforms [Naylor and Stephenson, 2008], and in the majority of yardang cases, both pedestals and caprock exhibit occasional vertical fissures, such that the long, narrow yardangs breakdown readily. In the present examples, long yardangs can become “dissected” once the caprock breaks up to form two yardangs each of shorter length whilst each maintains a width similar to the “parent” body and thus do not exhibit optimal streamlining. Further, it is evident that there is a scale dependence inasmuch as smaller, or narrower, yardangs appear more susceptible to structural breakdown than larger bodies. Once the caprock is destroyed, rapid erosion of the exposed pedestals occurs and yardang extinctions follow.
4.3 Model of Yardang Erosion
 The peak fluid stresses across Hills Flats during Spring tides are typically in the range 0.89–10 Nm−2 [Williams et al., 2006, 2008]. Fluid stresses act as a tractive force on the bedrock surfaces, and the critical value for erosion is proportional to, but less than, the rock hardness [Sunamura and Matsukura, 2006]. [Sunamura and Matsukura2006] measured the erodibility of mudstones in sediment-free and sediment-laden flows of similar character to those at Hills Flats. With reference to their results and the rock hardness data, the fluid stresses alone are inadequate to erode the weathered or unweathered bedrock; rather, the impacts of bed load and suspended grains must be accounted. The tidal flows (z = 0.40 m) typically can have suspended sediment concentrations of <654 mg l−1, which translate into a volume concentration of <0.03%. Such concentrations are typical of sand-laden fluids [Bagnold, 1963; Hanes, 1986] in a dilute flow rather than a grain collisional regime. In dilute flows, fine grains such as silts and clays behave as the fluid phase and produce minimal values of shear stresses on any surfaces [Anderson, 1986; Nakagawa and Imaizumi, 1992; Laity and Bridges, 2009]. Therefore, only the high concentrations of suspended load of sand in the unmeasured near-bed parts of the tidal flows (Figure 6) and any bed load transport should be responsible for abrading the yardangs. A simple conceptual model of this process is developed below.
 The shape results show that in plan view, the intertidal yardangs are broader than the ideal for minimization of drag (Table 1 and Figure 4a). Thus, as a first approximation, flow around the flanks of the broader subaqueous yardangs and in the lee should be similar in behavior to flow around a conical obstruction. Flow slows immediately upstream of such an obstacle [Paola et al., 1986], and coherent vortices accelerate along the flanks [Maunder and Rodi, 1983] before forming a series of less coherent but highly turbulent vortices in the lee between the main flow and the wake [Calvert, 1967; Werner et al., 1980]. The approaching flow abraded those facets directly facing upstream [Richardson and Carling, 2005], while the vorticity along the lateral margins and in the lee abrades the flanks and the leeside platform. The possibility of enhanced scour in the lee [see Whipple et al., 2000a] is in contrast to conditions in air, where the low viscosity and low density of air compared with water results in poor entrainment of sediment in leeward vortices and consequent aeolian leeside deposits are common [Greeley and Iversen, 1985]. In water, deposition may still occur in the wake [Paola et al., 1986] but is minimized downstream of optimally streamlined obstacles and no wake deposits were ever seen at Hills Flats during periods of platform exposure.
 In principle, high suspended sediment concentrations and hence increase in fluid density and viscosity [Julien, 1995] might suppress turbulence intensity [Best and Leeder, 1993; Wang and Larsen, 1994; Baas and Best, 2002] and so reduce pressure drag, such that W/L ratios should be larger in turbid flows than clear-water flows [Wyrick, 2005]. However, despite this possibility, fluid densities are not greatly augmented by the measured suspended sediment concentrations and turbulence levels are high (Re ~ 107 to 108) and so an increased incidence of leeside or stoss-side scour in subaqueous environments, in contrast to aeolian environments, is the preferred explanation for the larger, less-streamlined, W/L ratios recorded for the intertidal yardangs (Table 1 and Figure 4a) as such strong scour will result in greater truncation of lengths rather than reduction in widths.
 Obstruction of the flow on the upstream side of a bluff body causes localized vertical downflow and scour along the upstream flanks for obstacle cones with slopes >15 ≤ 30° [Okamoto et al., 1977; Paola et al., 1986; Schär and Durran, 1997]. On cones with shallower slopes, the scouring effect is small but flow deceleration can cause sediment deposition at the nose of the obstacle which would blanket the bedrock and prevent further basal scour. Steeper brittle faces abrade faster than shallow ones due to the angle-dependent flux of sediment [Pugh, 1973, p. 151] against the bedrock surface as well as nonlinear effects of momentum transfer efficiency and rebound effects that are proportional to incident angle [Bridges et al., 2004, 2005]. As explained below, these principles are mediated by the structure of the velocity profile and the sediment concentration profile upstream of the obstacle.
 For conditions at Hills Flats, where there is an unlimited supply of fine and coarse suspended sediment, the suspended sediment concentration, and thus, the particle impact rate should scale with the fluid velocity [Whipple et al., 2000a, 2000b]. As was noted in section 'Introduction' for aeolian systems, abrasion by bed load, including traction and saltation, should dominate close to the bed [Sklar and Dietrich, 2004] with abrasion by suspension loads being important for bedrock outcrops, such as yardangs, protruding high into the flow [Hancock et al., 1998; Whipple et al., 2000a; Richardson and Carling, 2005]. Thus, below, the hypothesis is tested that the heights (ź) of the erosional inverts in the yardang profiles are constrained by suspension abrasion processes, and the conclusions derived should equally apply to aeolian and subaqueous yardangs.
 Although the variation in the velocity distribution in the vertical at the study site has not been explored, it is reasonable to assume that during the strongly accelerating phases of the tide, the vertical flow profile immediately above the planar platform is logarithmic, with velocity increasing with height [Soulsby and Dyer, 1981; Lamb et al., 2008]:
where Uz is the reference velocity at a height z in the water column, u* is the shear velocity, zo is the roughness length, and κ is von Kármán's constant (0.40).
 In the same vein, it might be expected that the relative concentration of grains (S) being well mixed [Kirby and Parker, 1983] and the relative grain size (D), being fine, will accord with the Rouse profile [Vanoni, 2006] and approach 1.0 near the platform surface (Figure 7). The shear velocity and the roughness length are estimated using the quadratic stress law:
where Cdref for a gravel-strewn rock bed is in the range 0.0025 and 0.004 [Sternberg, 1972; Thompson et al., 2004] for the peak ebb tidal reference current speed of 1.85 m s−1 and
 Allowing for unsteadiness in the tidal flow [Soulsby and Dyer, 1981; Dyer, 1986] and using equations ((9)) and ((10)), u* ranges between 0.07 and 0.14 m s−1, typically between 0.08 and 0.12 m s−1 (consistent with the shear stress measurements of Williams et al. [2006, 2008] at P1 and P2), and zo = O 0.001 m ± 60%.
 The Rouse equation describes the vertical profile of a suspended sediment size class in water flows:
where Ws is the estimated settling velocity of the grains [Jimémez and Masden, 2003, equation (12)] and Sz is the concentration at a given height, z, relative to Sa, the concentration at reference height, a [Vanoni, 2006]. The reference height relates to the shedding of suspended sediment upward from a bed load layer, and the reference height in this case is the height of the top of the bed load layer from the fixed bed [Robinson and Brink, 2005]. The coefficient β is usually close to one (ranging between 0.2 and 1.5).
 If a static or near-static granular layer exists on the platform, then the granular bed is dominated by frictional forces (S ~ 1.0) with no flux or small flux of grains, little kinetic energy expenditure, and consequently little or no abrasion [Sklar and Dietrich, 2004]. Above the bare platform or sediment bed level, active bed load transport will occur in traction and in saltation such that S < 1, D reduces with height above the bed, collision forces dominate grain interactions and kinetic energy expenditure, and potential abrasions thus are maximized at some height above the platform. As saltation is replaced by suspension dynamics at higher levels, the concentration profile and the average size of the constituent grains will reduce further (S < <1) such that abrasion should decrease with increased height. In water, as in air, finer sediment has less abrasive power, even in the faster flows higher in the velocity profile [Nakagawa and Imaizumi, 1992; Laity and Bridges, 2009]. It might be expected therefore that the vertical distribution of the kinetic energy flux imposed on the bedrock surface of a pedestal and prow will exhibit a peak in the region of collisional forces.
 Susceptibility to abrasion is proportional to the kinetic energy of an impacting particle [Rosenberger, 1939, p. 65; Greeley and Iversen, 1985]. The power expended per unit time is equal to the product of the kinetic energy and the number of impacting grains [Momber, 2008, p. 33]. Thus, assuming the speed of the suspended particle, Up, is similar to the speed of the flow, Uz:
where the number of spherical grains of diameter, D, is N = S/[ρs(4/3)π r3], where r is D/2.
 Calculated vertical mass suspended sediment fluxes for sand-sized and finer sediment do not demonstrate any peak in the individual grain-size profiles. However, granule-sized sediments (10 mm > D > 1 mm), which are found in suspension below 0.35 m height (Figure 6), do exhibit peaks in the mass flux profiles in the range 0.08 m < z' < 0.2 m for all reasonable values of β. Figure 7 shows two examples for granules and for 1 mm sand; peakedness is increased as the grain size coarsens. It is evident that the mass flux integrated over a range of coarser grain sizes peaks in the vicinity of, or just below, the heights of the inverts. The level of the maximum kinetic energy flux will fluctuate as tidal state, concentration, and grain size vary and will rise or fall as any bed layer of gravel aggrades or deflates. However, a fixed bedrock surface (with a small bed load flux) is present for most of the time, and so the average and the variance of the heights of the inverts noted above as = 0.16 m (SE = 0.015 m) in the upstream facing facet of yardangs (relative to the platform level) should demarcate the typical height range over which the height of the kinetic energy flux maximum varies in a suspension dominated flow [cf. Sharp, 1964; Anderson, 1986: Cooke et al., 1993; Bridges et al., 2005]. However, for the suspended sediment at Hills Flats, there is no peak in the distribution of the potential kinetic energy for any suspended grain size; rather, the kinetic energy increases with height above the bed (Figure 8) as is the case for the potential abrasive power of the coarser suspended grain-size fractions (Figure 9).
 The results shown in Figures 7-9 demonstrate that although the heights of the inverts correlate approximately with the peak in the mass flux of coarser grains, it is not possible to relate the invert geometry in any simple manner to potential kinetic energy or potential abrasive power expenditure of the suspended load when the distribution of these parameters are considered as two-dimensional vertical profiles alone. The primary explanation is that the profiles of potential expenditure do not allow for whether particles detach from the flow (coarser grains) and impact the surface or follow the flow field (finer grains). The driver for invert development must consider only the kinetic energy of the suspended load which impacts the surface together with the bed load (both traction and saltation load) all of which are mediated by the complex fluid stressing that occurs around the prow of any bluff body in fluid flow [e.g., Kiya and Arie, 1972, 1975; Gowda et al., 1985; Nigro et al., 2005]. The development of the pediment and lower pedestal through time should allow bed load (traction and saltation) to be advected to higher elevations thus raising the elevation of the peak expenditure of kinetic energy due to the unmeasured bed load. Similarly, Udo et al.  also concluded that the effect of a saltation load on the overall sediment concentration profile had been underrepresented in appreciations of near-bed aeolian concentration profiles. Thus, in air, as well as in water, the distribution of abrasive power expenditure will be significantly conditioned by the ability of coarser sediment particles to decouple from the flow and abrade the surface whilst fine sediment follows the deflecting flow lines around the yardang. Further study of the complex bluff body sediment-laden flow field would be required to determine the exact controls on invert elevation.
 Retreat of the steep-faced, near-vertical invert will cause the lower pedestal to reduce in angle, reducing erosion across the lower pedestal [Schoewe, 1932; Bridges et al., 2004] and enhancing the propensity for occasional burial by bed load. In the same fashion, the upper pedestal overhang above the invert will reduce in angle as the invert retreats faster than the leading edge of the harder capstone. Nevertheless, the progressive removal of pedestal rock mass will weaken the support given to the prow such that caprock disintegration will occur in due course due to gravitational failure and by “jacking” of caprock slabs and flakes induced by turbulent uplift pressures [Clement, 1988]. The collapse of the prow then results in a steep facet being exposed again but without the protective presence of the cap. At this stage, the distinct invert in the upstream facet is lost (Figure 2b).
 Aeolian yardangs frequently present a whaleback form downwind of the prevailing wind direction [Greeley and Iversen, 1985, p. 135], but this form has not be considered beyond noting that the form minimizes drag [Ward and Greeley, 1984]. Thus, in reversing tidal currents, an interesting problem is why some yardangs have developed whaleback forms for their down estuary extremities but not at all for their up estuary extremities. This orientation is additional evidence for the dominance of the ebb flow in molding the yardang form by erosion. The whaleback form, in the present examples, requires breakup of the caprock. Whereas the ebb-facing prows of caprock are commonly undermined by erosion of the softer pedestals, as detailed above, the flood caprock is rarely undermined significantly. Rather, the whaleback morphology implies strong turbulent flow is responsible for the caprock erosion within the wake formed by the ebb flow. The shape and flow characteristics of wakes downstream of cones of various angles are largely self-similar [Calvert, 1967]. Calvert showed that the plan views of the wakes of both steep (60°) and shallow (20°) angle cones in fully turbulent flow are characterized by two free shear layers that develop above the lower slopes of the cones, enclosing a separation bubble in which reverse flow and high values of turbulence parameters persist. Thus, it is probable that the turbulence intensity in the wakes of the yardangs is sufficient to break up the caprock such that the developing whalebacks are subject to sustained abrasion by the flow patterns over their backs.
 The existence of water-eroded landforms at Hills Flats of similar form to aeolian yardangs is evidence of convergent evolution of form in response to geophysical forcing by a fluid. The well-formed yardangs originated by extension of gullies by tidal scour between promontories in the cliff lines until promontories are isolated. The body geometries are semi-lemniscate and minimize hydraulic drag. The yardangs decrease in size through time in a largely predictable fashion: the lengths being the least conservative dimension reducing through time more rapidly than height or width.
 Importantly, for both air and water, the model casts doubt on the simple role of the maximum of potential kinetic energy flux of suspended sediment in the two-dimensional vertical as the controlling process for shaping the overhanging rock wall facing the dominant (ebb) flow. Instead, although a peak may occur in the vertical concentration profile, and despite the potential kinetic energy flux increasing away from the bed, the sediment becomes finer with height and the majority of these finer particles may not impact the surface, so abrasion decreases. In addition, the distribution of total sediment flux, including the bed load component, will further be mediated by topographic forcing, and this also will influence the height of inverts. Turbulent ebb-flow separation, symmetrical in plan view, occurs along and above the distal flanks of the yardangs resulting in the distinctive whaleback forms due to sustained abrasion over their backs. By analogy, the results of this investigation of the development of yardang-shaped bed forms in water flows may aid interpretation of the formation mechanisms of aeolian yardangs more generally.
 Nick Lancaster is thanked for providing copy of McCauley et al. [1977a]. Jon J. Williams (University of Plymouth) is thanked for the loan of the S4 current meter. Paul Bell (Proudman Oceanographic Laboratory) is thanked for pre-processing tide data, and David Jones (POL) is thanked for the loan of the suspended sediment sampler. Jo Nield (University of Southampton) is thanked for Figure 2b. The editor and associate editor provided constructive guidance in the review process. The comments of Edward Anthony and four anonymous referees assisted greatly in focusing the main arguments.