Using a stereo pair of HiRISE images of a pole-facing crater slope at 38°S, 218°E, we measure topographic profiles along nine gullies. Typical slopes of the interior channel region (above the depositional apron) are ∼20°. We test the hypothesis that sediment transport on gully slopes occurs via fluvial transport processes by developing a numerical sediment transport model based on steep flume experiments performed by Smart (1984). At 20° slopes, channels 1 m deep by 8 m wide and 0.1 m deep by 3 m wide transport a sediment volume equal to the alcove volume of 6 × 105 m3 in 10 h and 40 days, respectively, under constant flow conditions. Snowpack melting cannot produce the water discharge rates necessary for fluvial sediment transport unless long-term (kyr) storage of the resulting meltwater occurs. If these volumes of water are discharged as groundwater, the required aquifer thicknesses and aquifer drawdown lengths would be unrealistically large for a single discharge event. More plausibly, the water volume required by the fluvial transport model could be discharged in ∼10 episodes for an aquifer 30 m thick, with a recurrence interval similar to that of Martian obliquity cycles (∼0.1 Myr).
 Martian gullies, first identified by Malin and Edgett , are of critical importance for understanding the evolution of the Martian hydrosphere and climate. These features probably required liquid water to form and are apparently young, suggesting that liquid water has been present at or near the surface of Mars in the recent past. However, there is currently no consensus on the quantities of water involved in gully formation, or indeed if water is actually required at all. In this paper we carry out a quantitative model of one proposed mechanism for gully formation, fluvial sediment transport, to estimate the fluid discharge rates required and thus to explore whether this mechanism is feasible.
 The morphology of Martian gullies (Figure 1) consists of an eroded alcove, incised channel, and a depositional apron. The distribution of gullies on Mars is limited to middle to high latitudes, with the most well developed occurring between 30 and 45 degrees [Bridges and Lackner, 2006]. Gullies are among the youngest features on Mars based on their superposition on relatively young features such as dunes and polygonal terrain, as well as a general scarcity of cratered gullies [e.g., Heldmann et al., 2007].
 On Earth, at least three mechanisms have been observed to form depositional fans: avulsing channelized rivers, sheet flows, and debris flows [Parker et al., 1998; Schumm, 1977; Blair and McPherson, 2009]. Although a depositional fan is usually accumulated by a combination of these processes, fans dominated by debris flows tend to have higher slopes than those dominated by fluvial processes [Harvey, 1984; Williams et al., 2006]. Debris and sheet flows tend to form oversteepened fronts, and, depending on their thickness, result in a convex-up topographic profile [Whipple and Dunne, 1992]. In many cases, deposition on alluvial fans occurs by a combination of debris flows, streamflows and intermediate hyperconcentrated flows [Sohn et al., 1999]. Determining which of these processes dominates often requires detailed fieldwork and facies analysis [e.g., Whipple and Dunne, 1992; Sohn et al., 1999] not generally possible on Mars.
 An Icelandic analog study by Hartmann et al.  found gully morphologies similar to Martian gullies with sinuous channels initiated by debris flows, and later scoured by stream flow. Howard et al. [2008a] have obtained estimates of the fluvial discharge rates needed to generate sinuous channels observed on distal fans on Mars based on theory from Ikeda et al. . Their work suggests an average discharge rate of 4.9 m2 s−1 per meter channel width with extrema at 0.4 and 13 m2 s−1 (section 3.1). However, the theory implemented by Howard et al. [2008a] assumes shallow slopes, small viscous forces, and that mean flow is steady and uniform downstream, assumptions that may not be appropriate for Martian gullies. In our study area at 38°S, 218°E (Figure 1), slightly sinuous channel forms are found in all gullies, a morphology consistent with fluvial or wet debris flow processes. Although some gullies show pronounced levees, suggesting that debris flow processes can dominate [Mangold et al., 2003], most gullies, including those in our study, do not.
 Gully slopes have been investigated hitherto using MOLA topography [Dickson et al., 2007; Heldmann and Mellon, 2004]. Slope data are important because the rate of sediment transport is a sensitive function of slope (see section 3.1). In this paper we obtain higher-resolution slope data using stereo pairs of HiRISE images. We combine our slope observations with a simple numerical model of fluvial sediment transport to determine the time scales and water volumes involved in gully formation, assuming that fluvial processes are taking place.
 The model results can be compared with proposed sources for water to determine if gully formation via fluvial processes is plausible. Similar numerical work has been done to determine erosion rates due to fluvial processes on Titan [Collins, 2005; Perron et al., 2006], while two recent laboratory studies have examined the hypothesis that Martian gullies are primarily fluvial- [Coleman et al., 2009] and debris flow- [Védie et al., 2008] related processes. The study by Coleman et al.  argues that water flows under Earth conditions can be scaled up and used as an analog to high-viscosity (1 Pa s) flows on Mars, although the morphology of fluvial features 1.5 m long generated in their flume at room temperature are somewhat different than the morphology of gullies on Mars. Debris flow experiments by the melting of near-surface ground ice in silty materials conducted by Védie et al.  in a cold room at −10°C closely reproduce a specific type of gully morphology found on Martian sand dunes consisting of a narrow, leveed channel terminating in the absence of a depositional apron [Reiss et al., 2007] suggesting seasonal variations in a periglacial environment may be responsible for gully formation of this specific type.
 The rest of this paper is organized as follows. In section 2 we document our observations of gully morphology in our study area (Figure 1), while in section 3 we detail our numerical model and the parameters adopted. Section 4 presents the results, and section 5 discusses their implications.
2. Slope and Topography Measurements
Figure 1 shows our study area, located at 38°S, 218°E, which consists of a series of about a dozen ∼1 km long gullies incised into the pole-facing slope of a crater 20 km in diameter. Our work is focused on this area because all the gullies exhibit a morphology indicative of fluvial activity. By looking at a specific location, our goal is to limit the number of variables influencing gully formation such as differences in climate or regional hydrology. The morphology of these gullies is similar to those found in other midlatitude areas, but our topography measurements are limited to locations with HiRISE stereo coverage, and thus, we have focused our effort on this one location.
 The morphology of the gullies in this location is characterized by a sinuous interior channel (with a wavelength ranging between 30 to 60 m and a sinuosity of <1.04) incised into the alcove floor and depositional apron. Widths of the interior channel range from 1 to 5 m, and marginal channel levees commonly associated with debris flows [Johnson and Rodine, 1984] are absent in this location. These observations suggest that fluvial processes are at work. The variability in the presence of a channel on the fan, the depth of channel entrenchment on the fan, location of channel on the fan, and fan dimensions (ranging from incipient fan/alcove formation to fans ∼105 m2 in area) between neighboring gullies suggests a cyclic process of avulsion, entrenchment, and back-filling as is observed on terrestrial alluvial fans [Schumm, 1977; Blair and McPherson, 2009]. The Martian gully fans also lack large (approximately meter-sized) boulders which are present in the exposed alcoves, suggesting preferential downstream transport of smaller grains [Welty et al., 2008], or deposition by dry debris flows [Treiman, 2003].
 We use a stereo pair of HiRISE images to measure relative elevation changes between manually selected points employing a method described by Kreslavsky  (Figure 1). In this method, an observed parallax between pairs of points in a stereo pair of HiRISE images is used to determine a relative change in elevation using the camera viewing orientations. The length of the parallax vector describing the offset between identical points in the two images is used to determine the elevation change. After determining the elevation change, the orientation of the observed parallax vector can be compared with the orientation of an idealized parallax vector using the calculated change in elevation and camera orientations to quantify the error in the measured relief. Typically, we measure slopes between 9 or 10 points along the gully starting above the headwall of the alcove and ending beyond the extent of the depositional apron. Typical errors in slope using this method are between 0.5 to 2° over 100 m length scales. The error in the measured relief increases as the distance between the selected points decreases.
Figure 2a shows a close-up of one particular gully with the locations of stereo topographic profiles superimposed. Figure 2b shows the longitudinal profile, demonstrating a concave-up profile with slopes declining from 30° (58%) at the alcove headwall to 4° (7%) at the base of the apron. Figure 2c shows transverse profiles across the distal alluvial apron. The low relief of the apron makes the relative errors in topography larger, resulting in larger uncertainties in apron volume.
 Based on the topographic relief measurements made using the stereo technique, we estimate a maximum vertical offset of 25 m between the background topography and the floor of the channel. Approximating the alcove as a triangular prism 120 m wide, 400 m long and 25 m deep gives a volume of 6 × 105 m3. Based on the topography measurements of the distal apron shown in the Figure 2c, apron volume is comparable, at ∼105 m3 (± an order of magnitude) of eroded material, although this measurement is uncertain due to the large errors in measuring vertical relief over short distances (±5 m in elevation over 50 m length scales).
 The stereo measurements show that the background slope on which the gullies are located has a slope of 22° (40%). Out of the nine gullies we analyzed, all show a steadily decreasing slope from an average of 30 ± 4° (58 ± 10%) at the alcove headwall to 16 ± 2° (29 ± 4%) at the head of the apron (Figure 2d) [Parsons et al., 2008]. These measurements are in agreement with previous gully slope measurements done at MOLA resolution in a different region [Dickson et al., 2007]. However, these depositional apron slopes on Mars are significantly steeper than terrestrial fluvial alluvial fans, with slopes more closely resembling those of terrestrial clast-rich debris flows [Williams et al., 2006; Blair and McPherson, 2009; Stock and Dietrich, 2006]. On the other hand, the stresses arising from such slopes are lower on Mars because of the lower gravity (section 3.1). To proceed further, we assume that the gullies are formed by fluvial processes, and develop a simple numerical model to investigate the implications of making this assumption.
3. Fluvial Sediment Transport
 One of the major unknowns regarding the formation of Martian gullies is the degree to which water is involved in transporting sediment. Proposed gully formation hypotheses span the range from dry flows to wet debris flows to fluvial transport. Head et al.  suggest that the dominant sediment transport process changes with time as a gully evolves. Here, we focus on fluvial sediment transport as one end-member in this spectrum of hypotheses in order to determine whether it is consistent with the expected time scales and water volumes available for gully formation.
 There are a number of fluvial sediment transport predictors used to estimate sediment discharges for a given channel. These predictors utilize empirical data of sediment discharge and relate it to quantifiable physical properties of stream channels such as slope, sediment grain size, and channel depth. However very few predictors have been calibrated to slopes as steep as those of Martian gullies. An additional complication comes from not knowing the mechanism of erosion. For instance, when there is no shortage of sediment, alluvial processes of redistributing sediment dominate, whereas sediment-starved streams will erode via plucking or by particles impacting bedrock [Lamb et al., 2008; Sklar and Dietrich, 2004].
 A further complication is determining the mode by which sediment is transported. Within the fluvial transport regime, sediment can be transported as dissolved load, suspended load, or as bed load. Here we implement an empirical sediment transport capacity prediction function developed by Smart . This predictor is referred to as a sediment transport capacity predictor because, in Smart's steep slope experiments, sediment reached up to the flow surface and, at very steep slopes, occasionally left the flow altogether and became airborne for a short time. It is therefore more appropriate to regard the proposed equation as a formula giving the transport capacity, in the absence of fine suspended material, for alluvial materials with mean grain size greater than about 0.4 mm, in channel slopes of up to 11° (20%) in the absence of bed armoring.
 The channel flow depths in Smart's experiments range between 1 and 10 cm. These shallow flow, steep slope conditions make Smart's sediment transport predictor particularly relevant to Martian gullies. Due to the lack of laboratory experiments at steeper slopes, we have applied Smart's transport predictor to the 15–20° slopes of Mars. However, because of the reduced gravity on Mars, a 0.1 m deep flow on a 18° slope applies the same stress to the channel bed (described below) as the stress in Smart's experiments (a 0.06 m deep flow on a 11° slope).
 The dimensionless Einstein transport parameter (ϕ) is related to the discharge rate
where ρs, ρ, g, and D50 are the sediment density, fluid density, gravity, and the median sediment diameter, respectively. In Smart's work, ϕ and the nondimensionalized shear stress, or Shield's parameter (τ*), are empirically related
where S is the local slope, τ = ρgh sin(S) is the bed shear stress, h is the channel depth and τ*c is the slope-corrected critical Shield's stress (the stress at which sediment transport is initiated) [Smart, 1984]. Cs is a factor inversely related to friction (≈4.5) given by the ratio of the shear velocity () to the mean flow velocity and is taken to be a constant.
Figure 3a shows τ*c and contours of τ* as a function of grain size and channel depth for a slope of 20°. It also plots τ*s, the critical stress at which sediment transport begins via the suspension of particles of a given grain size. τ*s is the stress at which the shear velocity () equals the settling velocity for a particle of a given grain size [Dietrich, 1982]. The quantity τ* increases with increasing channel depth and decreasing grain size, as expected. Figure 3a suggests that bed load sediment transport is dominant because τ* < τ*s for much of the parameter space, and, due to the steep slope assumed, the applied stress is much greater than the critical Shields stress (τ* ≫ τ*c). Smart's experiments (indicated by the shaded region in Figure 3a) resulted in sediment transport as both bed load and suspended load, and equation (1) represents the total flux from these two modes of transport.
 The sediment and water discharge rates are contoured in Figure 3b using the same axes as Figure 3a. Because τ* ≫ τ*c, Figure 3b shows that the water and sediment discharge rates scale as h1.5 and are almost independent of grain size, as expected from equations (1)–(3). The thick black line corresponds to a channel depth to grain size ratio of 10:1 which is representative of Smart's experiments. For slopes representative of Martian gullies, as long as h exceeds the grain size D50 by a factor of more than about 10:1, the results are independent of grain size. This affords a considerable simplification, since the relevant grain size for Martian gullies is poorly constrained. If this criterion is not satisfied, however, the sediment transport rate will depend on the mean grain size.
 The sediment transport rate's independence of grain size is due to the assumption that the grain friction factor (Cs) is constant with grain size. Theoretically, Cs should decrease with increasing grain size (if channel depth remains constant) resulting in slower sediment discharge rates at larger grain sizes. The mean measured value of Cs from Smart's experiments is 6.2 ± 1.9, but seems to decrease with increasing grain size (as one would expect due to increased bed roughness) [Postma et al., 2008; Vollmer and Kleinhans, 2007; Shvidchenko and Pender, 2000]. Cs also seems to decrease with increasing slope [Smart, 1984]. We have assumed Cs = 4.5 which corresponds to grains 10 times smaller than the channel depth at slopes of 11° (the steepest in Smart's experiments). Our assumption of Cs = 4.5 gives relatively small sediment and water discharge rates resulting in conservatively large time scale estimates. Increasing Cs by a factor of 2 would decrease the formation time scale by the same factor. Note that changing Cs will not affect the sediment concentration because both qs and qw (see below) are linearly dependent on Cs.
Figure 3c shows how both the sediment and water discharge rates (per unit channel width) vary with slope for a median grain size of 10 cm and a channel depth of 1 m. The water discharge rate shown in Figures 3b and 3c is calculated assuming a constant friction factor Cs = 4.5 and using
where qw is the water discharge rate per unit width [Smart, 1984]. The box and error bar gives the mean and extrema in water discharge estimates from Howard et al. [2008a] for the distal portions of Martian gullies based on slope, width, depth, and channel sinuosity measurements. There is a large uncertainty in the predicted discharge based on the range of possible channel dimensions and the application of terrestrial sinuosity-discharge relationships at low slopes to the steep slopes and lower gravity of Martian gullies. However, these estimates are in general agreement with the discharge rates we calculate using equation (4).
3.2. Numerical Model
 For a two-dimensional geometry, the evolution of channel elevation z as a result of lateral sediment transport is governed by a continuity equation:
where ϕ′ = 0.64 is the sediment packing density [Parker et al., 1998] (1 − porosity) and x is the horizontal coordinate. We implement a numerical version of equation (5) in which gully and apron geometry in the third dimension is crudely accounted for (Table 1).
Table 1. Definitions and Measured or Theoretical Values or Range of Values for Parameters Used in the Numerical Simulations
Width increases by a factor of 3.65 downslope in order to conserve water volume.
 For the gully geometry, we assume that the channel, or alcove, flanks remain at the angle of repose. Thus, as the channel downcuts into the steep background slope, additional material from the channel flanks must be transported downstream (Figure 4). The eroded volume is equal to the time and width-integrated sediment flux divided by the packing density. The finite difference continuity equation in this case is given by
 Here ΔV is the change in alcove volume in a grid cell of length Δx over a time step Δt, and ΔHa is the corresponding change in alcove depth H. The channel width is w, Δq is the difference in discharge rates qs between the upstream and downstream sides of the cell (see equation (5)), and α is the apparent slope of the valley wall in the direction perpendicular to the channel (α = 22° dipping perpendicular to a 25° surface gives a true dip of 33°) (Figure 4). It may be verified that equation (6) reduces to a finite difference representation of equation (5) if α = 90°, that is, the channel is two-dimensional.
 Solving for ΔHa using the quadratic equation gives
Sediment conservation on the depositional apron requires using the Exner equation to calculate elevation change [Parker, 1991a, 1991b]. Assuming the fan is radially symmetric with an opening angle, θ of 60° (based on observations from our study location), the elevation change on the depositional fan due to a change in sediment flux is given by
where r is the horizontal radial distance downstream from the fan apex and θ is expressed in radians.
Equations (7) and (8) are used to update the channel profile z(x) every time step. The location of the fan apex changes with time, and is the point at which the elevation of the channel bed exceeds the elevation of the original uneroded surface.
 The initial model topographic profile consists of a 1.33 km long segment with a slope of 25° which shallows to a slope of 2° over an additional 0.67 km span. Water is allowed to flow into the model 300 m downslope from the top of the steep slope and occupies an initial channel that is 10 times smaller than the initial channel dimensions. The channel depth and width increase linearly (in proportion to each other) to their fixed initial values over a downslope distance of 20 m. This “ramping up” of the channel dimensions is done in order to maintain numerical stability. The distance over which the channel is initiated has no physical significance, although we are effectively assuming that all the water involved in forming the gully is released over this 20 m reach of channel. As the slope shallows, the water velocity decreases (equation (4)). Because water volume is conserved in our model, the channel cross-sectional area must therefore increase downslope due to the decrease in velocity resulting from a shallowing slope. For simplicity, we assume that the increase in cross-sectional area is accommodated by an increase in channel width [Finnegan et al., 2005]. In our model, the channel width increases by a factor of 3.65 from the steeply sloping portion to the shallow sloping portion. At the downslope boundary (after flowing over the break in slope and depositing most of its sediment), water is allowed to flow out of the simulation, carrying a small portion of sediment with it. During the simulations mass conservation is checked by calculating and comparing the volume of the alcove to the volume of the depositional apron. Some sediment in lost (due to the flow leaving the model), but the volumes of the alcove and the apron are equal to within about 5%. The simulations are run until a volume of material (6 × 105 m3) has been removed from the alcove region, consistent with the observations (section 2).
3.3. Parameters and Assumptions
 In order to determine the range of time scales and water volumes needed to form gullies via fluvial erosion, we must prescribe values for two unknowns: channel depth and sediment grain size. Channel depth is estimated by measuring channel widths in HiRISE images (observed channel widths range from 3 to 25 m) and assuming a width to depth ratio. Howard et al. [2008a] use this approach to estimate discharge rates for gullies assuming a channel depth to width ratio of 1:8. Finnegan et al.  give examples of how a river's depth to width ratio depends on the substrate in which the channel is developed, and can vary from 1:5 for bedrock channels to 1:60 for gravel channels. To explore the range of time scales associated with variations in channel depth, we simulate two cases: one with initial channel dimensions of 1 m deep by 8 m wide, and one with a 0.1 m deep by 3 m wide initial channel.
 As shown in Figure 3b, changing the grain size does not greatly influence the rate of erosion as long as grains are significantly smaller than the channel depth. However, the grain size is important in determining the regime (bed load vs suspension) that dominates fluvial sediment transport (Figure 3a). Previous studies have assumed a grain size of 10 cm for Martian channels [Kleinhans, 2005], although it remains relatively unconstrained with measured mean grain sizes ranging from 1 mm to 30 cm based on lander and rover imagery (ignoring dust) [Golombek et al., 2003; Herkenhoff et al., 2004]. The grain size frequency distribution is likely to be bimodal, especially in a fluvial environment, and may have peaks in the size frequency at 1 mm and 10 cm [Kleinhans, 2005]. We will assume that the grains are sufficiently small that the results are independent of grain size and that sediment is transported as a mix of suspended and bed load material in a fashion similar Smart's experiments. If transport via suspension is more efficient than Smart's predictor (resulting in higher sediment concentrations), then our derived formation time scales will be overestimates.
 Our model also assumes that water loss processes (freezing, boiling, infiltration) are negligible over the time scale of gully formation [cf. Heldmann et al., 2005]. Heldmann's simulations regarding the stability of liquid water flows on the surface of Mars found that if the flows contain soluble salts at a concentration of 0.02 mole fraction (twice that of terrestrial seawater) flows traveling down an 18° slope could extend for tens of kilometers due to the suppression of the brine solution's vapor pressure. Based on work by Carr , modest, pure water flows 10 cm deep could persist on the Martian surface under current climate conditions for a few hours before completely freezing. We assume that the concentration of sediment suspended in the flow is sufficient to neglect freezing and evaporation over the 2 km domain of our numerical simulations.
 Simulations are performed with a grid size of Δx = 2 m, with 1000 total points, and a time step of 0.025 and 0.1 s for the 1 m and 0.1 m deep channels, respectively. This time step ensures that the Courant criterion is easily satisfied for typical sediment velocities of ∼1 m s−1.
 The result from a typical sediment transport simulation are shown in Figure 5. This model assumed a channel 1 m deep by 8 m wide with a sediment grain size of 10 cm, and resulted in an alcove volume of 6 × 105 m3 after 14 h of continuous flow. Figure 5a shows a perspective rendering of the final model geometry, which resembles the gullies seen in Figures 1 and 2. The insert provides a visual comparison between the model result and the gully shown in Figure 2a shown at the same scale. The overall agreement is striking; the principal difference is that the rear of the alcove in the model is steeper than actually observed. This difference arises because material upslope of where the channel is initiated is not allowed to move downslope in the model and results in a steep cliff forming above the point of channel initiation. Figures 5b–5d show the evolution of channel elevation, slope and downcutting with time. The rate of downcutting decreases with time, because as the channel deepens more material has to be removed from the channel flanks (Figure 4). The apex of the fan starts at the base of the steep slope, but propagates upslope with time as the local slope changes, resulting in a depositional fan extent that is in close agreement with observations.
 The main result from this particular model is that gully formation can take place extremely rapidly, primarily because the steep slopes, even with relatively small channels, result in rapid sediment transport rates. The roughly 10 h time scale associated with an 1 by 8 m channel increases to 40 days when the channel is 0.1 by 3 m. In the 1 by 8 m channel case, the maximum water discharge rate is 45 m3 s−1 and the total volume discharged is 1.8 × 106 m3 for the entire simulation. In the 0.1 by 3 m channel case, the maximum water discharge rate is 0.5 m3 s−1 and the total volume of water involved in eroding the gully is also 1.8 × 106 m3 based on the water discharge rate from equation (4). This total volume implies a sediment:water ratio of 33% by volume (equivalent to a sediment concentration of 25% by volume) for both channels. In section 5, we will discuss the significance of this finding, and compare our results with the expected discharge rates from plausible sources for liquid water.
 These water, sediment volumes and time scale results depend on our assumptions made in section 3.3. In general, our assumptions have been conservative, giving results that give longer formation time scales. For instance, we have used relatively small channel sizes, but if we were to increase our modeled channel width by a factor of two, then our time scale would decrease by the same factor. Similarly, if we chose a larger value for Cs, then the time scale would be shorter than the estimates given above.
 The results presented above suggest gully formation time scales, assuming continuous discharge, of days to months, and peak water discharge rates of ∼45 m3s−1 or ∼0.5 m3s−1 for channels 1 by 8 m and 0.1 by 3 m, respectively. Quantitative estimates of discharge rates by Heldmann et al.  and Howard et al. [2008b] give values of 30 m3 s−1 and 28 m3 s−1, respectively, similar to our value for the larger channel. Heldmann et al.  also obtained flow durations of ∼103 s, somewhat shorter than our estimates (but see below). If gullies are indeed fluvial features, there are two hypothesized sources of water: either groundwater discharge [e.g., Mellon and Phillips, 2001]; or snowpack melting [e.g., Christensen, 2003]. Below we examine both hypotheses in the light of our numerical model results.
5.1. Groundwater Discharge
 Groundwater discharge was the first explanation proposed for the formation of Martian gullies [Malin and Edgett, 2000]. We test this hypothesis using the duration and water volume results from our numerical model above, together with theoretical calculations of groundwater discharge from a permeable aquifer. The simplified physical situation is shown in Figure 6a: fracture of a surface cap (e.g., an ice plug) perhaps due to increased aquifer pressure [Mellon and Phillips, 2001] permits water to flow out of an aquifer at a rate determined primarily by the permeability. The flow rate will decline with time as drawdown of the aquifer occurs, until flow eventually ceases (e.g., due to freezing).
 As described in Appendix A, an approximate expression for the resulting discharge rate can be obtained assuming that the channel depth is small compared to the aquifer thickness. In a 2-D Cartesian geometry this discharge rate (per unit width) may be approximated by:
Here T is the aquifer thickness, κ is the permeability, ρ = 1000 kg/m3 is the density of water, g = 3.71 m s−2, porosity (ϕ) is 0.2, t is time, and μ = 10−3 Pa s is the dynamic viscosity of water. For a given value of T, equation (9) actually overestimates the discharge rate by a factor of 4 or more (see Appendix A). Integrating this function over time gives the total volume discharged
where wa is the width of exposed aquifer over which discharge occurs. Figure 6b plots the required aquifer thicknesses and permeabilities (solid line) needed to match the model time scales and total discharge using equation (10). In this plot, wa is assumed to be 100 m, comparable to the alcove width. As the aquifer permeability increases, the aquifer thickness required to deliver the inferred water volume over the inferred duration decreases. Longer flow durations permit lower permeabilities for the same aquifer thickness. Keep in mind that we are underestimating the actual required aquifer thickness because equation (9) overpredicts the discharge rate.
 Because the aquifer thickness cannot exceed the total slope relief of roughly 300 m (Tmax), the aquifer permeability must exceed roughly 10−9 m2 for the 1 by 8 m channel case, and 1.5 × 10−11 m2 for the 0.1 by 3 m channel case. Although the aquifer permeability is not well constrained, work by Manga  suggests a regional permeability of crust containing basaltic lava flows of ∼10−9 m2. However, other authors suggest a less permeable substrate, averaging ∼10−12 m2 [Harrison and Grimm, 2009] for the Martian regolith. Our results for a scenario involving a single groundwater discharge event lie within the range of plausible permeabilities (from 10−12 to 10−9 m2) for the 0.1 by 3 m channel case. However, the 1 by 8 m channel requires aquifer permeabilities that are probably too high to be reasonable.
 As flow proceeds, the aquifer will experience drawdown, where the lateral drawdown distance δ is roughly δ ∼ (Qtot/(Tϕ))1/2. Since the gullies are separated by a few hundred meters, drawdown distances in excess of this value will result in one gully cannibalizing the water supply of another. At present it is not clear whether multiple gullies formed simultaneously or not. If they did, then δ must be less than half the gully separation distance (δmax ≈ 150 m). Figure 6b also plots δ as a function of permeability for the calculated value of T. As the permeability increases, so too does the drawdown distance, as expected.
 If gullies form simultaneously from a single groundwater discharge event, then there is no permeability that satisfies both the aquifer thickness and drawdown length restrictions (indicated by the shaded gaps in Figure 6b). Based on the results shown in Figure 6b, either gully water is discharged from a thick (300 m), permeable (10−11 m2) aquifer as a single event lasting about 40 days, drawing water from a radial distance of ∼200 m (beyond the maximum value of 150 m suggested by the observations), or gully formation takes place over many episodes of short-lived (approximately hours to days) groundwater discharge events from a thinner aquifer.
 Although it is hard to assess the likely thickness of potential aquifers, clay-bearing layered deposits in Mawrth Vallis appear to be a few tens of meters thick [Wray et al., 2008], while extensive volcanic layers in Valles Marineris are typically 5–50 m thick [McEwen et al., 1999]. We conclude that the gullies in our study region are unlikely to have formed by a single discharge event; more plausibly, they were formed by multiple, short-lived discharges from a relatively thin aquifer. Heldmann et al.  also concluded that short-lived (∼103 s) flows were likely responsible for gully formation.
 Assuming groundwater sourced from a 30 m thick aquifer is responsible for fluvial activity in gullies, then it would take 10 discharge events, drawing water from a radial distance of 150 m with a porosity of 20% to account for 2 × 106 m3 of water. If each discharge event lasted 7 days, the permeability required is then 10−9 m2, consistent with the estimate by Manga . During each discharge event, this scenario would give discharge rates comparable to that of our 0.1 by 3 m channel simulation. Lower permeabilities or thinner aquifers would give water discharge rates substantially smaller than those determined from our model, and would become too small to transport a significant volume of sediment due to fluvial processes.
 Since at least some gullies are only a few Myr old [e.g., Schon et al., 2009], this multiple discharge hypothesis requires a recurrence interval of ∼100 kyr, similar to the periodicity of the Martian obliquity cycle. This age also implies a recharge rate of ∼1 m3/yr, or about 10−4 m/yr. The obliquity time scale has been suggested to drive the episodic release of groundwater into gullies by Mellon and Phillips  due to pressurization of a saturated, confined aquifer by the propagation of a freezing front. As we will address in section 5.2, melting of a snowpack due to climate variations may provide the aquifer recharge necessary to make a groundwater-fed gully hypothesis feasible.
5.2. Snowpack Melting
 Direct runoff from melting of an ice-rich mantle is an alternative to a groundwater source for explaining fluvial activity in gullies. Furthermore, snowpack melting and subsequent storage in an aquifer is another way in which water may be delivered to gully alcoves. Observations in support of a snowpack melting hypothesis include the global distribution of gullies along midlatitude bands [Dickson et al., 2007; Heldmann et al., 2007], their occurrence on pole-facing slopes at midlatitudes in the southern hemisphere [Heldmann and Mellon, 2004], their presence on isolated massifs [Christensen, 2003], and their association with an ice-rich mantling unit [Bridges and Lackner, 2006]. At high obliquity, polar water ice will sublimate during summer and precipitate at lower latitudes [Mellon et al., 1997; Mellon and Jakosky, 1995]. If the seasonal snowpack (∼1 m thick) is unprotected by dust, it will sublime and melt over 1 to 2 Martian years [Williams et al., 2008, 2009]. Alternatively, if the ice is protected by dust, it could potentially survive to the present day, resulting in an ice-rich mantling unit located in areas of minimum solar insolation. This mantle could potentially produce meltwater runoff at the present day, suggesting gullies could be currently active [Christensen, 2003]. However, the location of gullies on pole-facing crater slopes at midlatitudes is not consistent with locations of maximum melting according to [Williams et al., 2008].
 Based on energy balance models of a snowpack undergoing melting and sublimation, a small amount of meltwater available for runoff can be produced for a few days each Martian year under a wide range of atmospheric and obliquity states. Based on work by Clow  and Williams et al. , 1–2 mm of runoff per m2 can be accumulated over a few days during each Martian year. The differences between these models is the location and timing of the predicted melting. Regardless of the model, the maximum discharge rate predicted from snowpack melting is small. A melting rate of 0.25 mm h−1 per m2 integrated over a gully alcove 6 × 104 m2 in area gives a maximum discharge rate of 0.005 m3 s−1. Such fluid discharge rates are orders of magnitude smaller than the discharge rates we inferred in section 4, and will not result in significant sediment transport as either bed load or suspended load.
 In the work by Williams et al. , the obliquity must be high enough for atmospheric precipitation of snow to occur (requiring obliquities larger than ∼35°). As the snow seasonally melts, runoff is maximized at locations with the warmest regolith, and the thickest snowpack. The locations at which melting is maximized are equator-facing slopes at 50°S, or pole-facing slopes at 20°S. In order to generate the total amount of water predicted from our sediment transport modeling, these high-obliquity snowpacks would have to seasonally melt for ∼15 kyrs assuming 2 mm of runoff every Martian year, and an alcove area of 6 × 104 m2. This water would need to be stored and then released rapidly in order to generate the discharge rates necessary for sediment transport via fluvial processes. Since the obliquity in recent (<3 Ma) cycles can exceed 35° for up to a cumulative period of 20 kyr [Laskar et al., 2002], such an eventuality cannot be dismissed entirely. Nonetheless, for this mechanism to work, it requires a fairly complicated series of events: near-surface melting followed by downward fluid infiltration and storage for tens of kyr, followed by sudden, rapid release.
5.3. Alternative Sources of Water
 Another possible source of aquifer recharge is meltwater produced from geothermal heating of surface or subsurface ice. Geothermal melting of surface ice is an unlikely water source because it would require thick ice sheets (>1 km) and an elevated heat flux (0.1 W m−2) [Russell and Head, 2007] to generate liquid water. The estimated thickness of a recently deposited ice-rich mantle is two orders of magnitude smaller (between 1 and 10 m) [Mustard et al., 2001] than the ice thickness required to generate liquid water. Alternatively, melting could occur at the base of a subsurface ice table if it is covered by a thick (hundreds of meters), very low thermal conductivity (0.045 W m−1 K−1) material with a moderate heat flux of 0.03 W m−2 [Mellon and Phillips, 2001]. This insulating layer could be composed of dry unconsolidated regolith covering an ice table.
 Hydrothermal circulation is another possible mechanism for delivering water to gully alcoves. The heat needed to drive circulation requires an elevated geothermal flux, either from a magmatic intrusion or by impact cratering. However, gully activity has not been observed to correlate with recent volcanic activity, nor can intrusions reproduce the observed distribution of gullies along midlatitude bands. Hydrothermal systems generated by newly formed impact craters would likely be short-lived (<104 yrs) for craters 20 km in diameter exposed to freezing conditions at the surface [Barnhart et al., 2010], and are unlikely to direct fluid flow up to the rims of craters, unless guided by permeability structures.
5.4. Alternative Sediment Transport Processes
 As mentioned in section 3.1, this paper focuses on a transport capacity predictor from Smart  that is applicable in the absence of fine suspended sediment and in the absence of bed armoring for slopes up to 11° in Earth-like gravity. Due to the lack of laboratory experiments at steeper slopes, we have applied Smart's transport predictor to the 15–20° slopes characteristic of Martian gullies. The lower gravity on Mars means that transport on these slopes is treated appropriately by the Smart predictor, at least for shallow channels (Figure 3a); nonetheless, further laboratory experiments [e.g., Coleman et al., 2009] would be helpful. Alternatives to the sediment transport processes involved in Smart's experiments involve flows with much higher concentrations of sediment such as hyperconcentrated flows, wet debris flows, and dry debris flows (see below).
 We have neglected fine, suspended sediment transport in our model. If this mechanism is important, then the gullies could have formed even more rapidly. This in turn would require a thicker aquifer and/or higher permeability (equation (10)), making a fluvial origin less plausible. Alternatively, if the channel dimensions are smaller than the ones used in our simulations, gully formation could have taken place over the same time periods given above (if suspended sediment transport is accounted for), but potentially involving less water. However, subaerial water flows transporting sediment via suspension could persist for tens of hours under current Martian conditions [Heldmann et al., 2005; Kuznetz and Gan, 2002] resulting in sediment deposition at shallower slopes than sediment transported as bed load, and likely forming a thinner, more extensive depositional apron than what is observed on Mars [Parker, 1999].
 Another assumption we have made in the simulations is that the regolith is unconsolidated. This assumption is based on the incoherent appearance of the dusty surface into which gullies tend be incised and the lack of cliff-forming units at our study location. However, the Martian regolith may have some strength due to the presence of bed rock, indurated regolith, or an ice table. If this is the case, then fluvial transport is even less likely because the depth of water in the channel must be large enough to overcome the cohesive strength of the regolith before transport can begin. This will mean that more water is required for fluvial transport compared to the unconsolidated case that is assumed in our numerical work. This assumption is made in order to estimate the minimum water volume and time scale required for gully formation, and because the shear strength of regolith in gully environments is not observationally constrained. If the substrate has strength, then more water would be required to carve the gullies than the volumes suggested in section 4, making it harder to provide a water source, suggesting that debris flow mechanisms (or dry mechanisms) are to be preferred. Future studies to determine the strength of the regolith in gully environments are needed to determine the mechanism(s) responsible for gully erosion.
 On Earth, the characteristics of the local colluvium can determine whether debris flow or streamflow processes dominate [Blair, 1999]. On Mars, the permeability of the regolith is likely to play a similar role. Furthermore, the deposition of atmospheric dust and/or the potential presence of an ice table in the near surface may influence the permeability and therefore the mechanism by which sediment is transported.
 The time-averaged sediment concentration of 25% in our model are equivalent to the maximum concentrations in Smart's  experiments, and suggest that the transport processes for Martian gullies may fall in the transition region between streamflow and hyperconcentrated flow [Vallance, 2000]. It is possible that fluvial processes are not the primary mode of sediment transport in gullies, but may play a secondary role in modifying the surface [Head et al., 2008] with wet or dry debris flows being responsible for the majority of sediment transport, as is the case in steep upland channels on Earth [Lancaster and Casebeer, 2007; Hartmann et al., 2003]. As noted in section 1, it is very hard to distinguish streamflow- and debris flow-dominated fans purely by remote sensing, although the absence of pronounced levees argues against the latter process [cf. Costard et al., 2002]. On the other hand, it has been argued that the slope-area relationships observed in the few Martian gullies for which we have high-resolution (∼1 × 1 m grid spacing) topography suggest debris flow processes are at work [Lanza et al., 2010]. From a purely theoretical viewpoint, debris flows are attractive because they require much less water to transport the same sediment volume. Theoretical modeling of debris flows has received less attention than that of streamflows, but some work has been done [e.g., Stock and Dietrich, 2006] and should certainly be applied to Mars in the future.
 In this paper we have focused on the end-member hypothesis that fluvial processes are responsible for gully formation. Fluvial sediment transport on steep gully slopes is rapid. Channels 1 by 8 m and 0.1 by 3 m in dimension transport a sediment volume of 6 × 105 m3 in 10 h and 40 days, respectively, under constant flow conditions. Both the 1 by 8 m channel and the 0.1 by 3 m channel require 1.8 × 106 m3 of water, resulting in a sediment:water volume ratio of 33% (equivalent to a sediment concentration of 25% by volume).
 Except in the unlikely event that aquifer thicknesses exceed 300 m, a single event of groundwater release is not plausibly responsible for gully formation. About ten discharge episodes could generate the correct gully morphology assuming a ∼30 m thick aquifer and permeabilities that are consistent with other estimates [Manga, 2004]. In the case where multiple episodes of discharge occurs, the maximum interval between events is similar to the periodicity of obliquity cycles which were shown by Mellon and Phillips  to result in pressurization of a hypothetical confined aquifer. The required aquifer recharge rates can be met by high-obliquity seasonal melting of snow packs (perhaps aided by the presence of salts), or, less likely, by melting due to geothermal heat assuming a thick (∼100 m), low thermal conductivity (0.045 W m−1 K−1) surface material [Mellon and Phillips, 2001]. Although there are no observational constraints on the gully formation time scale, there are examples of episodicity of fan deposition [Schon et al., 2009], consistent with the multiple-discharge hypothesis.
 Snowpack melting cannot produce the water discharge rates necessary for fluvial sediment transport, unless long-term (tens of kyr for a single discharge event) storage of the resulting meltwater occurs [cf. Christensen, 2003]. Also, basal melting of an ice sheet due to geothermal heat would require ice thicknesses on order of 1 km [Russell and Head, 2007], and, thus, is unlikely to be a source for gully water in the recent past.
 In reality, gully formation may include a mixture of streamflow and debris flow processes. If the substrate has strength, then more water would be required to carve the gullies than the volumes suggested in section 4, giving support to debris flow mechanisms (or dry mechanisms). Differentiating between these end-member processes by remote sensing is difficult, illustrating the need for future Mars rovers. High-resolution topography and geological and hyperspectral mapping may help to test some aspects of the fluvial end-member hypothesis, for instance the estimated number (∼10) and recurrence interval (∼100 kyr) of discharge events [cf. Schon et al., 2009], the thickness of candidate aquifers, and the strength of the regolith. From a theoretical standpoint, applying terrestrial models of debris flows to Mars [e.g., Stock and Dietrich, 2006] will permit this end-member hypothesis to be tested against current or future observations.
Turcotte and Schubert  give the following linearized expression for the evolution of the phreatic surface h in an unconfined aquifer:
where h0 is the height of the phreatic surface prior to uncapping of the aquifer, x is the distance from the aquifer cap and other variables are defined in section 5.1. If the height of water in the channel is h1 then the solution to equation (5) is
where the similarity variable z = (μϕ/κρgh0t)1/2x/2.
 The Darcy velocity u is given by
 Together with the definition of z, equations (A2) and (A3) may be used to derive the Darcy velocity at the aquifer cap (z = 0). Setting h1 = fh0, the discharge rate per unit width qw is given by
 Taking the factor f(1 − f) to be of order unity then results in equation (9) where we have identified h0 with the aquifer thickness T. In practice, f(1 − f) ≤ 0.25, which means our discharge rates overestimate real values by at least a factor of 4, resulting in an under prediction of aquifer thicknesses. Note, however, that the original assumption of linearity (equation (5)) is violated if f ≪ 1.
 The authors would like to thank Maarten Kleinhans, Jonathan Stock, and Noah Finnegan for their helpful discussions and suggestions which improved this work. Thanks are also due to Misha Kreslavsky for his assistance in making slope measurements with HiRISE stereo. A thorough review by Keith Harrison was extremely helpful in preparing the manuscript for publication as was the feedback from an anonymous reviewer.