We examine the Martian valley networks in the framework of topographic influences on precipitation. We use an analytical model and the Laboratoire de Météorologie Dynamique (LMD) early Mars global circulation model (GCM) to explore the local-scale distribution of orographically forced precipitation as a function of atmospheric pressure. In simulations with 500 mbar and 1 bar CO2 atmospheres, orographic lifting results in enhanced snowfall upslope of the observed valley network tributaries. Our framework also suggests that a 2 bar atmosphere cannot create the observed valley pattern at the highest-relief valley network, Warrego Valles. As in previous work, the GCM does not generate temperatures warm enough for rain or significant snowmelt in the highlands with CO2 greenhouse warming alone. Thus while transient periods of unusual warming are still required to melt the deposits and carve the valleys, our model predicts snow deposition in the correct locations.
 Because liquid water is unstable to freezing and sublimation under current Martian surface conditions, the valley networks incising some of the oldest landscapes on Mars are often cited as evidence that the Martian climate was once warm and wet. Whether they require that ancient Mars experienced high surface pressure or temperatures warm enough for rainfall has been debated, however [e.g., Wallace and Sagan, 1979; Carr, 1983; Squyres and Kasting, 1994; Craddock and Howard, 2002; Gaidos and Marion, 2003]. The geomorphology of the networks can provide some insight into the climate that created them [e.g., Craddock and Howard, 2002; Barnhart et al., 2009; Som et al., 2009; Irwin et al., 2011], but the influence of ancient Martian precipitation patterns on their distribution has not been investigated in detail.
 When moist air is lifted, it expands and cools, allowing saturation to occur and water to condense. Topography can generate precipitation [Giambelluca et al., 1986] or enhance background precipitation by channeling air along convergent paths, by controlling the distribution of solar heating, or by directly forcing air to rise as the ground rises beneath it [Roe, 2005; Smith, 2006; Houze, 2012]. If the valley networks were formed by precipitation, at a minimum their distribution should reflect enhanced water supply where orographic lifting of moist air is predicted.
 Orographic precipitation has been suggested to be an important influence on the distribution of water ice on recent Mars [Forget et al., 2006]. It should have been even more important for early Mars if the late Noachian–early Hesperian atmospheric pressure was at least a few hundred millibars: more water can precipitate from a warmer atmosphere, and the altitude dependence of the surface temperature (surface lapse rate) is greater in a thicker atmosphere due to increased thermal coupling between the atmosphere and surface, such that snow and ice deposits are more stable at high altitudes [Forget et al., 2013; Wordsworth et al., 2013].
 Topographic effects on insolation can exert a strong control on recent Martian fluvial activity by controlling the location of snow and ice stability and melt [e.g., Costard et al., 2002; Balme et al., 2006; Dickson et al., 2007; Williams et al., 2009; Morgan et al., 2010], but they are unlikely to have been an important factor in the distribution of the valley networks. Most of the valleys lie within 30° of the equator, where insolation is only weakly slope-dependent even today [Kreslavsky et al., 2008]. If the valleys were carved in an atmosphere with at least a few hundred millibars of CO2, stronger radiative and sensible heat coupling between the surface and atmosphere (notably because of scattering by CO2 and water ice clouds) could further homogenize surface insolation and the energy balance.
 Since it is not currently feasible to run general circulation models (GCMs) at a grid spacing that can resolve topography at the scales on which the density of the valley networks varies, a downscaling approach is necessary to investigate the influence of individual Martian ridges on precipitation patterns and amount. The orographic precipitation model of Smith and Barstad  uses large-scale climate fields and topographic data to calculate precipitation forced directly by orographic lifting, producing precipitation maps at the resolution of the topographic data. We used Mars Orbital Laser Altimeter (MOLA) topography data and climate fields from the Laboratoire de Météorologie Dynamique (LMD) generic model adapted to early Mars [Wordsworth et al., 2011; Forget et al., 2013; Wordsworth et al., 2013] as input to this model.
 We chose regions of study to satisfy three criteria. First, valleys in the region should be late Noachian-early Hesperian in age [Irwin and Howard, 2002; Fassett and Head, 2008; Hoke and Hynek, 2009]. Second, valley networks should appear relatively well preserved, unaltered by subsequent impacts, superposed lava flows, and erosion, ensuring that the current valley distribution reflects the distribution of emplacement as closely as possible. Finally, well-preserved valleys should be present on regional slopes facing in more than one direction so that valleys on different aspects can be compared. We found regions satisfying these criteria (Figure S1 in the supporting information) in Terra Cimmeria, Margaritifer Sinus (Paraná Valles and others), Terra Sabea (Tisia Valles and others), and Thaumasia Planum (Warrego Valles). We compared predicted orographic precipitation with maps of valley networks in these regions [Hynek et al., 2010] to which we added valleys newly visible in Mars Reconnaissance Orbiter Context Camera (CTX) and 100 m/pixel Mars Odyssey Thermal Emission Imaging System (THEMIS) daytime IR images.
 In all of our GCM simulations, as in prior work [Forget et al., 2013; Wordsworth et al., 2013], the surface temperature is too cold to allow sustained liquid water. If these simulations represent the typical state of early Mars, the distribution of the valley networks is a function of two climates [Halevy and Head, 2012; Wordsworth et al., 2013]: a cold “background” climate, warmed only by H2O and CO2 gas and clouds as in the GCM simulations, during which snow deposits would have been emplaced; and an intermittent warm climate, warmed by one or more additional greenhouse gases, during which they would have melted. In this paper, we investigate this cold end-member scenario. Therefore, the model was run until ice reached an equilibrium distribution in the highlands, where it would have melted upon the arrival of a transient warm era [Halevy and Head, 2012; Wordsworth et al., 2013].
 The LMD generic climate model [Wordsworth et al., 2011] was run in early Mars mode with a pure CO2 atmosphere at surface pressures of 0.5 bar, 1 bar, and 2 bar, at 25° obliquity. Present-day GCM-scale topography was used. The grid spacing was 5.625° in latitude and 11.25° in longitude. All other run parameters were set as described in Wordsworth et al. .
 The gridded MOLA topography used with the analytical model has 128 pixels per degree resolution. We leveled superposed fresh craters in the MOLA topography using the Photoshop CS5 heal tool [Matsubara et al., 2011] but otherwise left the topography unaltered from its present state.
 The analytical orographic precipitation model used in this study has been validated in numerous regions of the Earth [e.g., Barstad and Smith, 2005; Smith et al., 2005; Crochet et al., 2007; Minder et al., 2008; Hughes et al., 2009]. It is based on general equations of fluid dynamics and thermodynamics [Smith and Barstad, 2004] and is therefore applicable to a broad range of planetary climates. In addition to topography and wind, the model requires as input an environmental atmospheric lapse rate, a saturated adiabatic lapse rate, and a surface reference temperature. From these variables, a condensation coefficient Cw, vertical wavenumber m of topographically forced waves, moist layer depth Hw, and falling timescale τf can be calculated. Condensation timescales τc were calculated as in the GCM [Montmessin et al., 2002] but assuming saturation in order to remain consistent with the orographic precipitation model. Falling timescales were calculated assuming that hydrometeors are similar in shape to terrestrial snowflakes and fall at terminal velocity. The environmental lapse rate was averaged over the region of study and over a Martian year. The saturated adiabatic lapse rate was calculated for condensing water in a pure CO2 atmosphere using the annual average surface pressure and temperature in the region of study. Wind speeds and reference temperatures used in the precipitation model were taken from ~145 m above the surface.
 The central model equation, which relates Fourier transformed topography to Fourier transformed orographic precipitation , is
where . After performing an inverse Fourier transform of , negative values of precipitation are set to zero. For a detailed explanation of the model, see Smith and Barstad .
 As a consequence of its mathematical formulation, the model delivers a “snapshot” of precipitation assuming constant atmospheric properties across its domain, rather than calculating a time-evolving condensation field where precipitated water is removed from the atmosphere, as a GCM would. As such, precipitation amounts are likely to be excessive downwind of topographic highs.
 To satisfy the validity criteria of the orographic precipitation model [Smith, 2006], the nondimensional parameter UNm−1h−1 must be greater than 1, where U is the wind speed and h is the height of the obstacle. Nm is the moist buoyancy frequency given by
where g is the gravitational acceleration at the Martian surface in m/s2, Γe is the environmental lapse rate in the region of study in K/km, Γm is the moist adiabatic lapse rate in K/km, and T is the temperature at the reference level where climate variables are calculated [Smith, 2006]. In other words, surface winds should have sufficient kinetic energy to flow over regional topographic obstacles rather than around them.
3.1 Orographic Precipitation
 Analytical model results are consistent with a strong role for orographic precipitation. The assumption of saturation means that the modeled orographic precipitation is an overestimate and should not be considered quantitatively. As a more qualitative comparison, in our regions of study, it is largely of order 10−3 to 10−2 mm/h at 1 bar compared to average synoptic-scale precipitation of order 10−4 to 10−3 mm/h.
 Our first study area, in Terra Cimmeria, centers on a broad ridge that extends ~650 km northwest from Herschel Crater (Figure 1). Two large, dendritic valley networks dissect most of its northeast slope; smaller dendritic networks cover part of its southwestern slope. A northwesterly wind causes maxima of orographic precipitation to occur where most of the observed valleys head (Figure 1a), and in all of our GCM simulations, such winds are dominant at the nearest grid point to this location (Figure 2). For comparison, if the model is run with a wind direction that is uncommon in the GCM, very few of the valleys head near predicted precipitation maxima (Figure 1b).
 We also applied the models to the valleys of Margaritifer Sinus and Terra Meridiani (Figure S2). Valleys in the region preferentially form on north or west facing slopes, and westerly winds are predominant in the region in all of our GCM simulations, followed by northwesterly in all but the 2 bar case (Figure S3). The maxima of drainage density, including Paraná Valles and the valleys on Newcomb Crater, all lie on maxima of predicted orographic precipitation in a westerly wind. Several lows are much more heavily dissected on one side, and the precipitation model generally reproduces this pattern as well (for example, in the boxed region in Figure S2).
 In our region of study in Terra Sabea (Figure S4), valleys preferentially form on northward and eastward facing slopes, and the orographic precipitation distribution fits observed valleys well when calculated with a northeasterly wind. Winds in this region have a northerly or easterly component for more than 90% of the year in all of our simulations (Figure S5).
3.2 Dynamical Constraints on the Noachian Climate
 The valley networks typically head at or near topographic divides or peaks [e.g., Hynek and Phillips, 2003; Hynek et al., 2010; Mest et al., 2010; Irwin et al., 2011], but drainage density in our regions of study does not show any clear altitude dependence. In the regions discussed above, small, dendritic or subparallel valleys incise the interior rims and walls of older craters. Since these are often densest near the maxima of the larger valleys on crater exteriors, and in many cases appear embayed by Hesperian volcanic crater fill, we interpret them as coeval with the larger valleys. If the valley networks were emplaced in a climate with slow average near-surface winds, orographic precipitation would reach a maximum partway up the slope because the humid low-level air would lack sufficient kinetic energy to ascend the entire slope. Unless synoptic or large-scale convective precipitation was so strong that it overwhelmed the signature of orographic precipitation, windward slopes would be most heavily incised well below their summits in a U < Nmh regime, and leeward slopes would experience very little erosion.
 The relief of the incised topography at our first three sites is not great enough to allow us to constrain the early Mars CO2 surface pressure. Median winds are fast enough at all three sites and all three surface pressures to surmount the tallest crater rims (Table 1). However, the Warrego Valles network is incised into a ~5 km rise on the Thaumasia Plateau, making it the only Noachian–Hesperian valley network suitable to test for flow regime differences between climates. Researchers using more recent imagery [Ansan and Mangold, 2006; Hynek et al., 2010] have noted that its west facing slopes are more heavily dissected than its south and east facing slopes and that the south and east facing valleys are overwhelmingly restricted to lower altitudes.
aNm is rounded to the nearest 0.001 s−1, wind speed to the nearest 0.1 m/s, and ridge height to the nearest 100 m. Ridge height is measured from the head of the lowest tributary on a slope to its tallest point; it does not correspond with absolute elevation, but the elevation change forced on a rising air parcel.
Terra Cimmeria, 0.5 bar
NW, 5.3 m/s
Terra Cimmeria, 1 bar
NW, 4.9 m/s
Terra Cimmeria, 2 bar
NW, 3.3 m/s
Margaritifer Sinus, 0.5 bar
W, 7.0 m/s
Margaritifer Sinus, 1 bar
W, 6.1 m/s
Margaritifer Sinus, 2 bar
W, 4.9 m/s
Terra Sabea, 0.5 bar
NE, 4.1 m/s
Terra Sabea, 1 bar
NE, 3.1 m/s
Terra Sabea, 2 bar
NE, 1.6 m/s
Thaumasia Plateau, 0.5 bar
W, 6.9 m/s
Thaumasia Plateau, 0.5 bar
E, 3.3 m/s
Thaumasia Plateau, 1 bar
W, 6.9 m/s
Thaumasia Plateau, 1 bar
E, 3.0 m/s
Thaumasia Plateau, 2 bar
W, 2.5 m/s
Thaumasia Plateau, 2 bar
E, 1.9 m/s
 Our GCM results (Figure 3) provide a robust explanation for this pattern. In our 500 mb and 1 bar simulations, the wind at the nearest grid point is predominantly from the west and would have strongly enhanced precipitation on that side of the mountain. About 41% of westerly winds are strong enough to surmount the plateau in the 1 bar climate. Southerly and easterly winds are less common and have median speeds near 3 m/s in both the 500 mb and 1 bar simulations. Rather than flowing over it, the slower southerly and easterly winds would almost always have been forced to flow around the mountain after ascending partway, resulting in nearly all precipitation being emplaced on the lower half of the slopes (Figure 4). The meridional temperature gradient, the wind speed, and the extent of the Hadley cell decrease with increasing surface pressure in our simulations, as in Kerber et al. , due to the higher heat transport capacity of thicker atmospheres. In the 2 bar simulations, even the yearly maximum wind is not strong enough to produce the highest tributaries on either side, and westerly winds are rare at Warrego Valles (Figure 3) because the winter hemisphere jet no longer reaches 40°S [Kerber et al., 2013]. Since the 2 bar wind strength and direction are both inconsistent with observations, the distribution of Warrego Valles casts doubt on the possibility of an early Mars surface pressure of 2 bars or greater. Since no other late Noachian-early Hesperian valley networks span as broad a range of altitudes, Warrego Valles is the only region where the U > Nmh criterion can be used to distinguish between climates. Crater dating indicates that the network formed contemporaneously with the majority of valley networks [Ansan and Mangold, 2006; Fassett and Head, 2008], however, suggesting that this sole available example can inform us of the general conditions at the time.
 Recent climate modeling studies [Wordsworth et al., 2013] have indicated that on large scales, water in a thicker early Martian atmosphere migrates to the highlands because adiabatic cooling creates a cold trap at high altitudes. Here we have shown that additional precipitation forced by topography at scales below GCM resolution enhances this effect. Where late Noachian–early Hesperian valley networks are well preserved, we have shown that their observed maxima can be predicted by an orographic precipitation model driven by GCM wind and temperature data. This confirms that topography exerted a significant control on early Martian precipitation patterns and reinforces previous evidence [e.g., Craddock and Howard, 2002; Hynek and Phillips, 2003; Barnhart et al., 2009; Hoke and Hynek, 2009; Luo and Stepinski, 2009; Som et al., 2009; Hynek et al., 2010; Mest et al., 2010; Irwin et al., 2011] that the water that flowed through the valley networks originated in the atmosphere, not through melting of ground ice [Gulick, 2001]. Recently, Matsubara et al.  showed that valley networks with the observed morphometry can form in a more arid climate if precipitation is orographically enhanced at high altitudes; in light of our results, this may indicate that ancient Mars was more arid than previously thought.
 This work has helped to constrain the CO2 inventory of ancient Mars: since valley distribution is consistent with emplacement by orographic snowfall, our results require a late Noachian–early Hesperian atmosphere with surface pressure of at least a few hundred millibars to create a surface lapse rate, allowing preferential preservation of highland snow deposits for long intervals between melt episodes [Wordsworth et al., 2013], but are most consistent with surface pressure below 2 bars. Erosion rates in this scenario would have been controlled by the characteristics of the intermittent warm climate required to cause melting. Future GCM studies including the greenhouse effects of warming by impacts [e.g., Segura et al., 2002] or sulfur volatiles [e.g., Halevy and Head, 2012] will constrain plausible melting rates and the possible role of transient rainfall.
 Previous studies [Burr et al., 2009; Luo and Stepinski, 2009; Grant et al., 2011; Irwin et al., 2011; Grant and Wilson, 2012] have suggested that orographic precipitation could have been important in the early Martian climate, but ours is the first to demonstrate this quantitatively. We have shown that on an early Mars warmed only by CO2, orographic lifting would have strongly enhanced snowfall where the densest valleys are observed, providing local-scale evidence for a precipitation origin for the valley networks. Our results at Warrego Valles, where topographic relief is sufficiently high that flow regimes differ between climates with different CO2 surface pressures, support a late Noachian surface pressure below 2 bars. Our results build on the icy highland model [Wordsworth et al., 2013] by describing the effects of this climate at the scale of geological analysis.
 Thanks are extended to W. Tennant for the wind rose GrADS script and B. Hynek for valley network GIS shapefiles. We gratefully acknowledge the NASA Graduate Student Researchers Program (GSRP) grant to KES from the NASA Langley Research Center and a grant from the NASA Mars Data Analysis Program (MDAP) to JWH. We thank two anonymous reviewers for thorough and constructive reviews that improved the paper.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.