[1] Under projected scenarios of sea-level rise, subsidence, and sediment starvation many deltas around the world are expected to drown. Delta growth dynamics, which determine the ability of a delta to adapt to these changes, are poorly understood due to the difficulty of measuring change in slowly evolving landscapes. We use time-series imagery of experimental, numerical, and field-scale deltas to derive four laws that govern the growth of river-dominated deltas. Land area grows at a constant rate in the absence of relative sea level change, while wetted area keeps pace, maintaining a constant wetted fraction over the delta surface. Scaling of edge-lengths versus areas suggests delta shorelines are nonfractal, even though the channel network is fractal. Consequently channel-edge length, which provides critical habitat, grows more rapidly than delta area. These laws provide a blueprint for delta growth that will aid in delta restoration and help predict how existing deltas will evolve.

[2] Efforts to combat land loss in coastal deltas focus on restoring the natural ecological and sedimentary processes [Kim et al., 2009]. Over geologic time, overbank deposition and channel switching (avulsion) distribute sediment across deltas [Roberts, 1997], building and maintaining land area in the face of relative sea level rise. Human activities have upset this balance in modern deltas, increasing effective sea level rise [Ericson et al., 2006] by enhancing naturally occurring subsidence and inhibiting delta plain deposition [Syvitski et al., 2009]. In the case of the Mississippi Delta, upstream sediment loss due to dams may preclude full restoration of the delta [Blum and Roberts, 2009], but diversions of the river could create significant land [Kim et al., 2009] and coastal habitat [Day et al., 2007].

[3] Restoration planning requires prediction of relative changes in the size and shape of the channel network and shoreline as land area increases. Here we analyze this problem using an analogy to allometry in biology [Huxley, 1932; Huxley and Teissier, 1936], which studies the size and scaling of body parts relative to overall body size, either during growth of an individual or between species [Gould, 1966]. Geomorphologists have long noted the similarities between evolving landscapes and growing organisms [Bull, 1975], but have primarily focused on the “inter-species” allometry of steady-state landscapes [Galloway, 1975; Syvitski and Saito, 2007; Turcotte, 1997]. Here we aim to provide a quantitative framework for allometry of growing deltas.

2. Data and Measurements

[4] To analyze allometric scaling in growing deltas we use a dataset documenting planview evolution of three river-dominated deltas (Figure 1), taking advantage of new experimental and numerical techniques which allow simulation of delta dynamics in unprecedented detail [Hoyal and Sheets, 2009; Edmonds and Slingerland, 2010]. The Mossy delta began forming ca. 1929 after the 1870s avulsion of the Saskatchewan River [Smith et al., 1998]. Subsequent evolution was captured by six aerial images spanning 60 years (1–10 m resolution). The experimental delta was created in the SAFL Delta Basin facility using a weakly cohesive sediment mixture to promote formation of stable distributary channel networks, and flow was dyed blue to facilitate mapping of wetted areas (DB07 progradation phase [Martin et al., 2009]). The delta prograded for 100 hr, with 1 mm overhead photos taken every 30 s (866 total). The numerical delta was simulated with Delft3D [Edmonds and Slingerland, 2010; Lesser et al., 2004], a coupled hydrodynamic-morphologic model, with a steady input discharge of 1000 m^{3}/s, a moderately cohesive sediment mixture, and no waves or tides (run p [Edmonds and Slingerland, 2010]). During the 5 yr simulation, output was sampled every ≈14 days (127 images total) over the 25 m simulation grid. For all deltas, sediment supply and base level are approximately constant through time. (See auxiliary material for complete dataset documenting evolution of the three deltas.)

[5] We characterize delta morphology by creating wetted-area maps that partition each image into wet vs. dry pixels (Figure 1). All wetted-area maps include channels, and for the experimental and numerical deltas they also include floodplain lakes and coastal wetlands (these are excluded for the Mossy due to poor land-cover contrast). We then partition wet pixels into channel vs. open water using the opening angle method (OAM) [Shaw et al., 2008], which provides a consistent approach to mapping shorelines on complex deltaic coasts, where it is difficult to distinguish shoreline from channel banks. We map shorelines with opening angles of 75° (Mossy and numerical deltas) and 90° (experimental delta). Finally, from the wetted area maps and OAM shoreline we compute land area, A_{land}, shoreline length, L_{shore}, wetted area, A_{wet}, and wetted edge-length, L_{wet} (Figure 2).

3. Land Growth

[6] Land area time series A_{land}[t] show a remarkably linear trend (Figure 3a), suggesting a constant growth rate throughout delta evolution. To understand this we use a simple box model for conservation of sediment mass in growing deltas [Wolinsky et al., 2010], d[cA_{land}H]/dt = fQ_{s}, where H is spatially averaged deposit thickness, c is deposit sediment concentration (1-porosity), Q_{s} is sediment supply, and f is sediment capture ratio (fraction of sediment supply captured on the delta plain). The box model shows that linear growth (constant dA_{land}/dt) is the default condition for constant effective supply fQ_{s} in the absence of aggradation and compaction (i.e., constant H and c), conditions that appear to hold for the Mossy and experimental deltas (Figure 3a). Aggradation driven by sea level rise, subsidence, and/or significant delta-plain (basin-floor) slopes [Wolinsky et al., 2010], leads to decelerating progradation, which can be seen in the numerical delta (Figure 3a; in this case due to a sloping shelf). While capture ratios of ≈30% are typical of deltas with significant wave energy [Kim et al., 2009; Wolinsky et al., 2010], the numerical and experimental deltas achieve f ≈ 90% in the absence of marine waves or currents, suggesting that 40% capture scenarios for Mississippi river diversions [Kim et al., 2009] may be overly pessimistic.

[7] While none of the deltas maintains a constant shape during their evolution (Figure 1), the curves of L_{shore} vs. A_{land} lie remarkably close to the square root curve expected for isometric growth of a semi-circle (Figure 3b). This is particularly interesting because erosional shorelines are the prototypical fractal [Mandelbrot, 1967]. However, given the fixed ruler length (pixel size), if the delta shorelines were fractal we would expect to resolve increasingly intricate roughness as the delta grows (i.e., the curves should lie distinctly in the grey-shaded “excess length” region of Figure 3b). This is not the case—time series of relative shoreline roughness (Figure 3c) are statistically indistinguishable from the constant shape null model of isometric growth (except for a slight smoothing trend in the numerical delta).

[8] What can explain this phenomenon? The rocky coast of Britain, Mandelbrot's original fractal, is essentially the sea-level contour of mostly erosional topography. Contours of erosional (nested valley) topography are known to be fractal [Turcotte, 1997], with dimensions D ≈ 1.20 ± 0.05 quite similar to the coast of Britain, D ≈ 1.25. For the deltas we studied, static box counting of shoreline snapshots gives D = 1.03 ± 0.04, suggesting that shorelines of growing deltas are not fractal (since mathematically D > 1 corresponds to an “infinitely rough” fractal curve, while D = 1 corresponds to a “smooth” curve with only finite roughness [Mandelbrot, 1967; Turcotte, 1997]). Combined with previous observations on the non-fractality of beach coasts [Murray and Barton, 2007], where wave-driven alongshore transport is a significant process [Jerolmack and Swenson, 2007], this suggests that constructional coasts made of mobile sediment may be generally nonfractal. We note that allometric growth and a fractal boundary are conceptually distinct, i.e., isometric growth of a fractal curve should be possible in principle (although if measured with a fixed ruler size, isotropic expansion of a fractal would still yield allometric length-area scaling).

4. Wetted Area Growth

[9] The story becomes more complex when we consider evolution of wetted areas (Figure 4). As subaqueous deposition builds up and land emerges, the deltas are initially almost entirely wet (Figure 4a), but with continued growth A_{wet} rapidly decreases until it reaches a constant fraction of A_{land}, after which the wet fraction exhibits fluctuations (associated with avulsions [e.g., Martin et al., 2009; Seybold et al., 2009]), but remains statistically constant (Figure 4a). However this does not reflect statistically isometric growth of the channel pattern, because the constant wet fraction is accompanied by steady growth in the relative wetted length (Figure 4b).

[10] This scaling pattern is consistent with channel growth by bifurcation around subaqueous mouthbars during delta progradation [Edmonds and Slingerland, 2007], where channel splitting creates wetted length faster than would isometric channel extension. Indeed, curves of L_{wet} vs. A_{wet} all lie significantly below the isometric growth line (Figure 4c), indicating creation of “excess” channel length during delta growth (i.e., fractal growth), and static box counting of channel-edge snapshots gives D = 1.31 ± 0.07. Furthermore, the Mossy and numerical deltas are remarkably consistent with the simple distributary fractal tree structure expected from successive mouthbar-driven bifurcations [Edmonds and Slingerland, 2007] (Figure 4c), and all deltas show significant roughening of channel banks through time (Figure 4d).

5. Discussion

[11] These results suggest that in growing deltas the channel pattern exhibits fractal growth, due to bifurcation and successive channel splitting, while the shoreline and land area exhibit statistically isometric growth. A simple explanation of this phenomenon could be compensational stacking [Straub et al., 2009] during delta growth: although deposition is focused around channels, as channel avulse to shorter paths they tend to fill in shoreline asperities, smoothing the shoreline. Detailed patterns of delta evolution (auxiliary material) suggest overbank deposition also plays a key role in filling interdistributary bays, although the distinction between overbank flows and weakly channelized avulsions is ambiguous.

[12] Our results suggest that first order delta allometry can be reduced to four scaling laws that can be used to predict delta growth: (1) A_{land} = at/(1 + t/T), where the initial growth rate a and deceleration time T (see Appendix A) are primarily controlled by sediment supply and aggradation rate, respectively (Figure 3a); (2) L_{shore} ∼ A_{land}^{b}, where b ≈ 1/2 (Figures 3b and 3c, where b ≈ 0.50 ± 0.08); (3) A_{wet} ≈ FA_{land}, where the mean wet fraction F stabilizes after an initial transient (Figure 4a, where F ≈ 35% ± 20%); and (4) L_{wet} ∼ A_{wet}^{c}, where c > 1/2 (Figures 4c and 4d, where c ≈ 1.7 ± 1.0).

[13] These scaling laws provide a basis for predicting how the internal geometry of a restored delta lobe will change as the lobe grows. While the deltas considered here exhibited linear growth, (1) can be used to predict decelerating growth due to relative sea level rise [e.g., Wolinsky et al., 2010]. Isometric shoreline growth (2) implies that newly created land is “compact”, with minimal exposure to coastal erosion. Preliminary analysis of strongly cohesive delta simulations [Edmonds and Slingerland, 2010] suggests that birdfoot-style growth [Seybold et al., 2009] can result in linear shoreline-length scaling L_{shore} ∼ A_{land} (i.e., the birdfoot extensions lengthen while maintaining constant width; similar to green-dashed curve in Figure 4c). Since a birdfoot geometry would have increased exposure to coastal erosion, this suggests the importance of getting the sediment mix (cohesive vs. non) right in diversions aimed at delta restoration. Additional work is needed to understand the influence of vegetation and avulsions on the mean and variability of wetted fraction (3). Finally, channel edge length (4) is critical for ecological prediction since channel-edge zones provide critical spawning habitat and in salt marshes are primary productivity maxima [Mitsch and Gosselink, 2003].

[14] Further investigation will be required to determine if these growth laws apply in other types of deltas. In degrading deltas, enhanced compaction in organic-rich overbank areas [Törnqvist et al., 2008] may result in more fractal shorelines as non-channel areas subside below sea level, while in wave-dominated deltas diffusive dampening of the channel-bifurcation instability [Jerolmack and Swenson, 2007] could result in non-fractal channel patterns.

[15] While we have focused on morphological allometry in growing deltas, we have left unexplored the possibility of metabolic allometry, i.e., scaling of energy or material consumption rates with size [West et al., 1999] (for a suitably defined “metabolic rate”). While this possibility has been explored in terrestrial tributary channel systems [Banavar et al., 1999], results were inconclusive [Haff, 2000], and the analogous problem for coastal distributary channel systems may be a fruitful area for future research. Now that methods to predict deltaic land building are available [Syvitski et al., 2009; Kim et al., 2009; Wolinsky et al., 2010], additional work connecting geomorphic allometry to ecologic allometry [Hood, 2007] in deltaic systems is a natural step toward establishing the science of delta restoration.

Appendix A:: Methods Details

[16] We solve the box model using A_{land} [0] = 0, H [0] = H_{0}, and dH/dt = R, giving A_{land} [t] = at/(1 + t/T), where a = (f/c)Q_{s}/H_{0} and T = H_{0}/R can be estimated by least squares fits to A_{land} [t] data. For the Mossy delta, the box model fit gives emergence time 1929, growth rate a ≈ 0.3 km^{2}/yr, and R ≈ 0. For the experimental delta, the box model fit gives a ≈ 2.3 × 10^{−2} m^{2}/hr and R ≈ 0. For known Q_{s} = 6.1 × 10^{−4} m^{3}/hr, experimental deposit measurements of H ≈ 46 mm and c ≈ 50% give f ≈ 87%. For the numerical delta, the box model fit gives a ≈ 3.9 km^{2}/yr, R ≈ 0.2 m/yr, and H_{0} ≈ 1.3 m. For known Q_{s} = 1.4 × 10^{6} m^{3}/yr and c = 25%, this gives f ≈ 91%. In the numerical delta, rapid rates are due to continuous flood-flow simulation, and aggradation arises from the shelf slope (37.5 cm/km), which drives deposit thickening during progradation into deeper water.

[17] For a fractal tree we compute L_{wet} and A_{wet} assuming a binary tree where at bifurcation order k we have n_{k} = 2^{k} branches of length ℓ_{k} = ℓ_{0}f_{L}^{k} and width w_{k} = w_{0}f_{W}^{k}. We take f_{L} ≈ 0.58 and f_{W} ≈ 0.71, consistent with modern river-dominated deltas [Edmonds and Slingerland, 2007]. From the geometric series F_{k} = r^{k} = (r^{k+1} − 1)/(r − 1) we obtain formulas for net channel network wetted length L_{k} = 2n_{k}ℓ_{k} and area A_{k} = n_{k}w_{k}ℓ_{k} in a tree of order k. The curve in Figure 4c uses continuous extrapolation of discrete formulas with k = [−1, 3].

Acknowledgments

[18] This work was supported by the STC program of the National Science Foundation via the National Center for Earth-surface Dynamics under the agreement Number EAR-0120914. We thank Ahammed Anwar Chengala for help mapping DB07 wetted areas.