Global Analysis of Topographic and Climatic Controls on Drainage Basin Shapes

Drainage basins are fundamental units of Earth's surface, describing how flows accumulate across landscapes. They are direct expressions of how tectonics and climatic forces alter Earth's surface morphology. Here, we measure the width‐to‐length ratios (WLRs) of 386,931 drainage basins (average area ∼157 km2), covering all continents except Antarctica and Greenland. Global variations in WLRs are correlated with climatic aridity, whole‐basin slope, and local topographic roughness. Basins in arid landscapes tend to be narrower, potentially reflecting a higher prevalence of surface runoff and therefore a stronger slope‐parallel component of the transporting flow. Local topographic roughness is associated with wider basins, potentially reflecting greater dispersion of flow directions. Conversely, whole‐basin topographic gradients, potentially reflecting gradients in uplift, are associated with narrower basins. However, steeper basins are also often rougher, so revealing the effects of whole‐basin slope requires correcting for the confounding effects of roughness variations.

Basin evolution is governed by erosional and depositional processes in which both climatic and tectonic drivers play important roles (Hurst et al., 2019;Whittaker, 2012).Regional studies of drainage basin shapes have shown that basins may adapt to tectonically induced changes in regional surface slope (Castelltort & Simpson, 2006;Castelltort & Yamato, 2013;Castelltort et al., 2009;Jung & Ouarda, 2017;Jung et al., 2011;Phillips & Schumm, 1987;Zernitz, 1932).More precisely, weakly dissected river basins tend to be more elongated along regional topographic gradients, whereas more gently sloping surfaces yield wider basins (Castelltort & Simpson, 2006;Castelltort et al., 2009).Basin shapes are influenced by the initial conditions of slope and surface roughness.Thus, basins sharing similar shapes may have developed from topographically similar precursors.The relative chronology of river basin development in response to surface tilting is encoded in a basin's width-to-length ratio (WLR) and thus establishes a link between basin morphology and tectonics (Castelltort et al., 2009;Castelltort & Yamato, 2013).Structural complications, such as frontal thrust ramp anticlines, can also disrupt regularly spaced drainage patterns by introducing important nontransverse stream elements (Hovius, 1996).Sun et al. (1994) found a strong covariance between basin aspect ratios and the slopearea exponent θ in synthetic topography, and recently Shelef (2018) developed a model to relate θ to WLRs of high-order basins.
Basin shapes are influenced not only by regional slope but also by local surface roughness.Rough landscapes tend to randomize flow directions, leading to wider river networks (Castelltort & Yamato, 2013;Castelltort et al., 2009) and thus basins with higher WLRs.Using a landscape evolution model, Castelltort and Yamato (2013) demonstrated that surface roughness and regional slope play crucial roles in determining the equilibrium basin shape.Increases in regional surface slope or decreases in local roughness tend to make drainage basins narrower, whereas when local roughness dominates over regional slope, basins generally become wider (Castelltort & Yamato, 2013).However, we lack a comprehensive understanding of how the competition between regional slope and local roughness affects the shapes of natural drainage basins on a global scale and across different climatic zones.
River network branching angles (and, by implication, basin shapes) are correlated with climatic aridity, suggesting potential differences in erosion mechanisms across contrasting climatic zones (Getraer & Maloof, 2021;Seybold et al., 2017Seybold et al., , 2018;;Strong & Mudd, 2022;Yi et al., 2018).Infiltration rates are higher in humid regions (Berghuijs et al., 2022), potentially leading to greater dominance of diffusive processes such as groundwater flow and thus wider basins (Yi et al., 2018), compared to arid regions dominated by overland flow erosion, where surface tilting would be expected to lead to narrower basins.
Here we explore how topographic and climatic controls shape the WLR of 386,931 drainage basins across the globe.Local roughness and whole-basin slope have opposite effects on a basin's WLR, but they are strongly interrelated and also vary with climate.Thus, quantifying the roles of these drivers requires correcting for their confounding effects.

HydroBASINS
Our global analysis is based on the HydroBASINS database (Lehner & Grill, 2013).HydroBASINS provides globally consistent basins and sub-basins, organized into 12 hierarchical levels using the Pfafstetter coding system (Verdin & Verdin, 1999).Because whole-basin slope and topographic roughness are scale-dependent, our study uses level-12 basins (∼157 km 2 ) to restrict the range of scales being analyzed.Other scales yield similar results (Text S1 in Supporting Information S1).We also limit our analysis to complete headwater basins, which encompass all channel segments upstream of the basin's outlet (Text S2 and Figure S1 in Supporting Information S1).As basins are a snapshot of Earth's current topography, they exhibit varying degrees of disequilibrium between uplift and incision.

Width-To-Length Ratio
To quantify the shapes of drainage basins, we used the WLRs of the minimum-area bounding boxes enclosing them (Figure 1).Compared with methods that rely on channel length (Rigon et al., 1996;Shelef, 2018;Sun et al., 1994), this approach yields a distinct metric which only relies on the location of the ridge lines; it is independent of the basin outlet's location and the methodological choices involved in stream network extraction (Castelltort & Yamato, 2013;Hovius, 1996).
WLRs reflect the average convergence of flow directions within a basin, with lower WLRs indicating narrower basins with smaller flow convergence angles (Castelltort & Yamato, 2013).

Climatic and Topographic Metrics
The aridity index (AI = P/PET) describes the balance between precipitation (P) and potential evapotranspiration (PET), and is often used as an indicator for climatic conditions.For our global analysis, we use 30-year (1970-2000) mean AI from the 30-arc-second Global Aridity and PET Database (Trabucco & Zomer, 2018), averaged over each basin.Note that higher AI values indicate more humid conditions.We expect that the broad spatial patterns of aridity in this 30-year record also correspond to the distribution of relatively humid versus relatively arid regions over geomorphologically relevant time spans (Riebe et al., 2004), even if global average aridity has changed over geological time.
Similar to the work of Castelltort and Yamato (2013) we use whole-basin slope and local roughness as possible topographic drivers.Whole-basin slope (S b ), sometimes also called regional slope, is calculated by fitting a plane to the basin's topography.A steeper whole-basin slope indicates a more pronounced tilting of the whole basin.In contrast, local roughness (R), estimated by the basin-averaged pixel-to-pixel slope in degrees, indicates the average variability in elevation between neighboring pixels (at a scale of ∼90 m), compared to the basin scale (∼10 km).For both slope calculations we used the MERIT-DEM data set (Yamazaki et al., 2017), which provides a homogenized digital elevation model for most of Earth's land surface.In particular it also covers latitudes above +60°north which are not captured by SRTM.Basins and topography coevolve over geological time, so presentday basin shapes could reflect the processes that have shaped them over long timescales.To reveal regional patterns in basin shapes and their controls across the globe, we spatially aggregated our WLRs, as well as topographic and climatic variables across equally sized hexagons of 5,000 km 2 (similar results were also obtained for hexagons of 10,000 km 2 ; Text S3 and Figure S2 in Supporting Information S1).

Global Patterns in Basin Shape
Globally, the mean, median, and 10th and 90th percentile WLRs of level-12 complete basins are 0.57, 0.56, 0.32, and 0.87, respectively.When these ratios are aggregated over equal-area hexagons, clear regional patterns emerge, indicating a tendency toward wider basins in mountainous regions which are characterized by rougher surfaces (e.g., the Rocky Mountains, Andes or eastern Tibetan Plateau, Figure 2c).However, regions with relatively high surface roughness also tend to be relatively steep (Figures 3a,3b,3d,and 3e).This suggests that in many mountainous regions, the impact of whole-basin slopes on WLRs may be masked by local roughness.Similar behavior has been shown in model simulations, in which basins become wider when local roughness dominates over whole-basin slope (Castelltort & Yamato, 2013).Tectonic uplift gradients shape regional slopes and locally modify surfaces through folding and faulting.Sediments eroded from mountain belts are typically deposited in adjacent basins.Drainage basins in such depositional zones tend to be relatively narrow; examples include the US Great Plains (Figure 2a), Tian Shan in China (Figure 2b), the Himalayan foreland (Figure 2f), and the Eastern Andes foreland basins.Here, river networks and their associated basins reflect regional topographic gradients generated by deposition close to the mountain front.Regional topographic gradients can also reflect dynamic tilting caused by underlying mantle flow (Mitrovica et al., 1989).An illustrative example is the North American Great Plains, which have experienced dynamic vertical motion resulting from the subducting Farallon plate (Willett et al., 2018).
Basins in humid climates (AI ≥ 0.65) also tend to be wider (higher WLR) than those found in arid regions (AI < 0.65) (Figures 2 and 3c).The narrowest basins (indicated by red in Figure 2) are predominantly concentrated in arid and semi-arid regions (Figure 3c).Some arid and hyper-arid basins may be influenced by wind-blown dune fields which dictate the steepest descent directions and thus basin shapes (e.g., in the Kalahari Desert; Figure 2e).In humid regions like the Amazon Basin, by contrast, basins are typically wider (Figures 2d  and 3c).  Figure 3d clearly shows that WLRs systematically increase with increasing local roughness (Spearman rank correlation ρ = 0.22, p < 0.0001; note that all rank correlations are calculated on the raw hexagon-based data, without binning).Figure 3d also shows the covarying effect of whole-basin slope with roughness, as indicated by the color gradient (ρ = 0.90).The unexpected positive correlation between whole-basin slope and WLR (ρ = 0.12, p < 0.0001) shown in Figure 3e may be the result of this covariation, with the surface tilting effect being masked by the stronger impact of roughness (Because pixel-to-pixel gradients are typically much larger than the whole-basin slope, calculating the local roughness using topographic data that are detrended by the whole-basin slope does not affect our results; see Text S4 and Figures S3 and S4 in Supporting Information S1).

Climatic and Topographic Controls
Figure 3f indicates that WLRs systematically increase with increasing AI (ρ = 0.28, p < 0.0001), suggesting a tendency for basins to be wider in humid climates than in arid ones.This relationship could be explained by a change in the relative dominance of different flow processes in different climatic settings.While arid landscapes are predominantly influenced by surface runoff, diffusive processes and groundwater-surface water interactions are more prevalent in humid landscapes (Freund et al., 2023;Seybold et al., 2017;Yi et al., 2018).However, humid basins also tend to have higher local roughness (ρ = 0.33, p < 0.0001; see color scale in Figure 3f); this covariation could potentially strengthen the correlation between aridity and WLR.Overall, our study suggests potential climatic influences on drainage basin shapes, emphasizing the necessity for further investigations to elucidate the underlying physical processes.
To disentangle the effects of whole-basin slope and local roughness on WLRs, we classify our data set into three classes of whole-basin slope and three classes of topographic roughness.Regardless of the whole-basin slope, WLRs always show the same positive relationship with surface roughness, reflecting the tendency for flow directions to be more variable in rougher terrain (Figure 4a).However, the positive correlation between WLR and whole-basin slope shown in Figure 3e is reversed when we control for roughness effects.More precisely, basins in each the two highest roughness classes tend to become narrower as whole-basin slope increases (Figure 4b).This behavior is consistent with numerical studies (Castelltort & Yamato, 2013) and demonstrates that the narrowing effect of surface titling may be masked by the global co-variation between whole-basin slope and local roughness.However, in the lowest roughness class (blue symbols in Figure 4b), WLR increases with whole-basin slope in flat basins (whole-basin slope <∼0.3°), suggesting that local roughness may still dominate over whole-basin slope in these cases (Figure 4b).We quantify the joint effects of whole-basin slope and local roughness on WLRs using a multiple regression model, which reveals that the effect of whole-basin slope is weaker than, and of opposite sign from, the effect of local roughness (Text S5 and Table S1 in Supporting Information S1).

Discussion
Local roughness and whole-basin slope are strongly correlated with WLR but are also with each other.Castelltort and Simpson (2006) suggested that these two topographic variables may be combined as a measure for the "relative roughness" of the surface R rel = R/S b , where R is roughness and S b is whole-basin slope.Relative roughness correlates strongly with WLR (ρ = 0.23, p < 0.0001, Figure 5a), and is also weaklier correlated with either local roughness or whole-basin slope (ρ = 0.14, p < 0.0001 and ρ = 0.28, p < 0.0001 respectively) than those two topographic metrics are with each other (ρ = 0.90, p < 0.001).This suggests that relative roughness may be a good metric for topographic influences on WLRs.
To investigate how relative roughness affects WLRs under different climatic conditions, we split our data set into humid (AI ≥ 0.65) and arid (AI < 0.65) regions (UNEP, 1997).In humid landscapes, WLRs increase monotonically with increasing relative roughness, reaching an approximate plateau above R rel ≈ 7 (Figure 5b).In arid regions, by contrast, WLRs peak around R rel ≈ 7 and decline as R rel increases further.As Figure 5c shows, binned averages of WLR increase almost linearly with log 10 (AI) in arid regions but are nearly constant in humid regions.Multiple regression analysis reveals that relative roughness effects on WLRs are similarly strong in both humid and arid conditions; however, AI effects are much weaker in humid climates than in arid ones (Text S5 and Table S1 in Supporting Information S1).To explore how the spatial scale of the underlying basins affects the correlations between WLR, local roughness, whole-basin slope and AI, we repeated our analysis for larger (level-9 and level-8) basins (Text S1 and Figures S5 and S6 in Supporting Information S1).Across a large variation in basin scales, ranging from level 12 (∼157 km 2 ) to level 8 (∼1,000 km 2 ), our findings described in Section 3.2 remain robust.Specifically, WLRs remain strongly correlated with local roughness, whole-basin slope and AI, and the narrowing effects of whole-basin slope are masked by co-variation with local roughness.As basins increase in size larger than level 8, one would expect that the variability in climate and topography would tend to average out.Larger basins may also incorporate a wider range of external factors, and are by definition less numerous, further complicating efforts to statistically discern the influence of climatic aridity, local roughness and whole-basin slope on basin shapes.

Conclusions
Our global analysis of 386,931 level-12 basins (Figure 2) reveals that WLRs vary in response to both topographic and climatic drivers (Figure 3).Our analysis confirms, at global scale, inferences about topographic drivers that have previously been drawn from simulation modeling and studies of selected individual basins and regions (Castelltort & Yamato, 2013;Castelltort et al., 2009).Basin WLRs are higher (i.e., basins are wider), on average, where local roughness is greater (i.e., landscapes are more deeply dissected; Figure 3d).One would expect basins to have lower WLRs (i.e., to be narrower) where, all else equal, whole-basin slopes are steeper (i.e., where basins are draped over regionally sloping surfaces).This relationship is masked, in single-variable plots like Figure 3e, by a strong correlation between whole-basin slope and local roughness.When one controls for the confounding effects of roughness variations, however (Figure 4b), the expected negative relationship between whole-basin slopes and WLRs is revealed.
Our analysis also reveals climatic influences on global variations in basin shape.Basins are wider, on average, in more humid climates (Figure 3f), but local roughness is again a confounding factor because humidity and roughness are correlated.When the correlations between climate, local roughness, and whole-basin slope are accounted for using multiple regression (Equation S1 in Supporting Information S1), climatic aridity is revealed to be almost an equally strong control as whole-basin slope across the whole data set.Climate also modulates the effect of topography; the relationships between relative roughness and WLR are different in arid and humid landscapes (Figure 5b).Likewise, WLRs are a strongly increasing function of climatic humidity in arid regions, but do not vary systematically with humidity in humid regions (Figure 5c).These results may help to inform conceptual models of drainage basin formation under the joint influence of climate and topography.whole-basin slope, is available from Yamazaki et al. (2017).The hexagon-averaged data set is available at Li et al. (2024).

Figure 1 .
Figure 1.Calculation of width-to-length ratios (WLRs) from widths (W) and lengths (L) of minimum-area bounding rectangles.Basins in (a)-(d) are denoted by blue polygons; the bounding boxes are shown by dashed black lines.Basins (a)-(d) are situated in diverse settings characterized by various values of local roughness (R), whole-basin slope (S b ) and aridity index (AI).

Figures
Figures 3d-3f illustrate how variations in local surface roughness (R), whole-basin slope (S b ), and climatic aridity (AI) influence WLRs.In order to make these relationships visible we aggregated our data in 50 equal-frequency bins.The color scales in (d)-(f) indicate how each x-axis variable co-varies with a potentially confounding variable.

Figure 2 .
Figure 2. Global patterns of hexagon-averaged width-to-length ratios (WLRs), with examples of basins in the (a) North American High Plains, (b) Tian Shan, (c) eastern Tibetan Plateau, (d) Amazon Basin, (e) Kalahari Desert, and (f) Himalayan foreland.The color gradient visually depicts the variation of WLRs between wide basins (blue) and narrow basins (red).Notably, basins in depositional zones at mountain fronts (a, b, f) and in arid regions (e) tend to have narrower shapes.Conversely, wide basins are predominantly observed in flat and humid areas (d) and mountainous regions with high local roughness (c).

Figure 3 .
Figure 3. Maps of hexagon-averaged (a) local roughness, (b) whole-basin slope, and (c) aridity index (AI).Panels (d)-(f) display the relationships between width-tolength ratios (WLRs) and (d) local roughness, (e) whole-basin slope, and (f) AI.Error bars indicate standard errors.We binned the data in 50 equal-frequency bins.The color scales in (d)-(f) indicate the variation of whole-basin slope (d) and local roughness (e, f), as potentially confounding co-variates in the plotted relationships (Note that in panels (d)-(f), the binning procedure retains most of the variability on the horizontal axes, but averages out much of the variability on the vertical axes, and in the color scales.).WLR increases with increasing local roughness and whole-basin slope.Steeper whole-basin slopes (e) are also associated with greater local roughness, potentially masking the expected negative relationship between whole-basin slope and WLR.WLR increases with increasing climatic humidity, as does local roughness (see color gradient in panel (f)).

Figure 4 .
Figure 4. Relationships between width-to-length ratio and (a) local roughness for different whole-basin slope classes and (b) whole-basin slope for different local roughness classes, with standard error bars.The entire data set is divided into three equal-sized classes for each scenario.We binned the points in each class in 20 equal-frequency bins.Within each wholebasin slope class, basins tend to widen as local roughness increases (a).Within each roughness class, basins tend to become narrower with increasing whole-basin slope (b).

Figure 5 .
Figure 5. Relationships between width-to-length ratio (WLR) and (a) relative roughness (R rel ) for the whole data set, (b) relative roughness in arid and humid regions (red and blue symbols, respectively), and (c) aridity index (AI).Error bars indicate standard errors.The data are aggregated in 50 equal-frequency bins.Note that the binning procedure retains most of the variability on the horizontal axes, but averages out much of the variability on the vertical axes.The color scale in panel (a) indicates the variation of AI.(a) WLR correlates strongly with relative roughness.(b) The relationship between R rel and WLR is non-monotonic in arid regions (AI < 0.65, red circles), and monotonic in humid regions (AI ≥ 0.65, blue triangles).(c) The relationship between AI and WLR is steeply increasing in arid regions and nearly constant in humid regions.