More extensive land loss expected on coastal deltas due to rivers jumping course during sea-level rise

Landscape-averaged models (Eqs. 1 and 2) are based on the idea that a river evenly distributes its sediment supply across the subaerial delta plain. Model forecasts can be improved by accounting for potential land loss on inactive delta lobes due to intermittent sediment delivery. Rivers build land on the active delta lobe, but after avulsion, the majority of the sediment supply is diverted elsewhere and some of this land will be lost as sea level rises. Lost land may be rebuilt during the later reoccupation of the delta lobe, but lobes spend most of their time inactive and thus partially drowned (Fig. 2A). To account for drowning on inactive lobes, the total delta-plain area ($A$ in Eq. 1) should be subdivided into the intermittent land area that is gradually lost during lobe inactivity ($A_{\mathit{lost}}$) and the persistent land area that remains dry regardless of lobe state ($A_{\mathit{dry}}$),

Intermittent land area is approximated by a band of terrain along the coastal perimeter ($P$) that progresses landward where lobes are inactive (Fig. 2A). For simplicity, we estimate a characteristic perimeter by the square root of the dry delta-plain area ($P=\sqrt{A_{\mathit{dry}}}$) (13), which should hold for a variety of delta symmetries and basin geometries, and we adopt a characteristic seaward slope $S$ for all lobes. Furthermore, we assume inactive lobes are completely deprived of sediment and do not consider additional sources of sediment (15–17) or erosion (18, 19). Inactive lobes passively drown as the shoreline retreats at a velocity given by the ratio of the rise rate and the seaward slope ($\sigma /S$; Fig. 2B). The intermittent land area is determined by integrating shoreline retreat velocity across the coastal perimeter until a given lobe is reactivated,

where the term in parentheses is the rate of land loss on inactive lobes and the second term is the average time before avulsions reroute the river back to the same lobe, where $f_A$ is the avulsion frequency, $n =\frac{N+1}{2},$ and $N$ is the number of lobes (35).

Previous work shows $f_A$ scales inversely with the time it takes the river to deposit sediment to a thickness comparable to the river channel depth ($H_c$) on the active-lobe delta plain; at this point, flow in the channel is rendered gravitationally unstable, and overbank floods can trigger avulsion (22, 24, 36). Avulsion frequency can be estimated by sediment mass balance, similar to Eq. 1 but here averaged across a delta lobe (Fig. 2C) (37),where the first term on the right-hand-side represents sediment deposition on the delta plain and the second term represents offshore deposition; $z=n\sigma f_A^{-1}$ is the height of sea-level rise before a lobe is reactivated; $D=\frac{\left(H-z\right)}{S}\mathcal{H}\left(\frac{H-z}{S}\right)$ is the nonnegative distance the active lobe builds seaward (where $\mathcal{H}$ is the Heaviside step function); and $H$ and $B$ are the active lobe’s thickness and width of deposition. The lobe length is the avulsion length $L_A$, i.e., the distance up-river from the sea where avulsions occur (Fig. 2C). On lowland deltas, $L_A$ can extend hundreds of kilometers up-river to a distance approximately equal to the backwater length-scale, $L_b\equiv H_c/S$ (23, 38). Together, Eqs. 3–5. provide a means to forecast land loss that resolves intermittent sediment delivery due to switching of the active lobe.

To test the landscape-averaged and lobe-averaged models, we conducted a scaled laboratory experiment. The experiment consisted of a 7-m-long, 14-cm-wide fixed-width flume connected to a 5-m-long, 3-m-wide ocean basin (Fig. 3A and B). Water and sediment were supplied at the upstream end, and sea level was controlled using a programmable standpipe at the downstream end. The basin was initially 4.5 cm deep and free of sediment, and over the course of 105 h, water flow naturally built a river delta. We designed the experiment to replicate realistic backwater-scaled avulsions ($L_A\sim L_b$) over length scales of $\sim 1 \text{m}$, achieved by oscillating input water and sediment supply such that the river experienced persistent backwater effects (39) while maintaining subcritical flow ($Fr<1$), channel depths of $H_c=7.5 \text{mm}$, and gentle bed slopes of $S=0.0042$ ($L_b=H_c/S=1.8 \text{m}$; SI Appendix, Table S1). We systematically raised sea level at four different speeds during four phases of the experiment, lettered A to D (Fig. 3C). Sea-level rise rates were scaled in terms of a dimensionless sea-level rise rate,

where the denominator represents the maximum possible deposition rate averaged across the active-lobe delta plain. Rise rates covered the range $0<{\sigma }^{*}<1$ common to natural deltas (37, 40). To isolate the effect of sea-level rise and river avulsions from other factors that influence land loss, the experiment featured no ocean waves, tides, storms, vegetation, or cohesive sediment (17, 41, 42). Overhead images were collected every minute of the experiment and used to map the area of dry and drowned land and identify avulsions (Fig. 3A and B; Materials and Methods).

The experimental delta evolved through repeated river avulsions (Fig. 3A and B). Without sea-level rise (Phase A; ${\sigma }^{*}=0$; Fig. 3C), there was no land loss; the entire delta plain was persistently dry (Fig. 3D and E). Land area gradually increased at a rate of $\sim 840{ \text{cm}}^2/\text{h}$, in close agreement to models averaged over both the landscape (Eq. 1 in Fig. 3D) and lobe (Eq. 3 in Fig. 3D). Avulsions occurred once per 2 h on average ($f_A=0.5 \left[0.2, 1.0\right]{ \text{h}}^{-1}$, where terms in brackets represent the maximum and minimum; SI Appendix, Table S2) and focused land-building near the active river mouth.

With the onset of sea-level rise (Phase B; ${\sigma }^{*}=0.08$; Fig. 3C), the delta plain lost land in inactive areas cut off from the riverine sediment supply (Fig. 3A and B). Inactive coastlines retreated by $\sim 20$ cm, forming a band of drowned terrain along the coast. The shape of the drowned zone varied over time depending on local slopes (43) and the time since an area was last supplied with riverine sediment, and maintained a roughly constant size of $\sim 0.9{ \text{m}}^2$, which was ∼25% of the entire delta plain (Fig. 3E). Avulsions were more frequent ($f_A=0.9 [0.2, 1.8]{ \text{h}}^{-1}$; SI Appendix, Table S1) and land building outpaced land loss on average, leading to net land-area gain at $\sim 570{ \text{cm}}^2/\text{h}$ (Fig. 3D). The lobe-averaged model accurately captures the amount of land loss (Eq. 4 in Fig. 3E) and the trajectory of the dry land area (Eq. 3 in Fig. 3D). The initial, sudden drop in dry delta-plain area ($43.5-48 \text{h}$) may reflect transient sedimentation of the trunk channel upstream of the delta lobes—a process that can temporarily increase lobe thickness $H$ and delay avulsion (40), thereby reducing dry-land area (Eqs. 3–5).

The lobe-averaged model reproduces the increase in avulsion frequency (Eq. 5), which arises because the river responds to rising sea level by partitioning a greater fraction of sediment to delta-plain deposition ($\sim 13\text{\%}$, as compared to $\sim 8\text{\%}$ during stable sea-level; Fig. 3F), which destabilizes channels more quickly (44). In contrast, the landscape-averaged model predicts no land loss (SI Appendix, Eq. S6 in Fig. 3E), overestimating the dry-land area and more closely following the trajectory of the total delta plain (Eq. 1 in Fig. 3D). This is because the landscape-averaged model does not differentiate between persistent and intermittent land; the area $A$ in Eq. 1 pertains to the delta plain where deposition occurs, and not necessarily the delta plain that is persistently dry. Intermittent land was usually submerged, but was emersed when revisited by the river such that it accreted roughly at pace with persistent land. Thus, the sediment deposited in intermittent zones is a sediment sink not accounted for by landscape-averaged predictions, which leaves less sediment available to build persistently dry land.

More rapid sea-level rise during Phase C (${\sigma }^{*}=0.33$; Fig. 3C) caused more extensive land loss and a long-term reduction in the dry-land area (Fig. 3A–C). Inactive shorelines retreated a distance of $\sim 45 \text{cm}$ before being replenished by the active channel, resulting in a drowned area of $\sim 2.5{ \text{m}}^2$, which was ∼50% of the total delta-plain area (Fig. 3E). Importantly, despite a fourfold increase in rise rate compared to Phase B, land loss increased by only a factor of 2; this is because avulsions were more frequent ($f_A=1.0 \left[0.6, 2.1\right]{ \text{h}}^{-1}$; SI Appendix, Table S2), partially mitigating land loss by limiting the duration of sediment starvation. Similar to Phase B, the lobe-averaged model accurately predicts the observed experimental land loss (Eq. 4 in Fig. 3E) and the trajectory of dry-land area (Eq. 3 in Fig. 3D), whereas the landscape-averaged model overestimates dry-land area by a factor of 2 (Eq. 1 in Fig. 3D) and erroneously predicts negligible land loss (SI Appendix, Eq. S6 in Fig. 3E). By the end of Phase C, the landscape-averaged model predicts that the delta-plain area should have reached its maximum size in the experiment ($\sim 5{ \text{m}}^2$); in reality, over half the delta plain had drowned (Fig. 3D). By taking the ratio of terms in Eq. 5 (SI Appendix, Eqs. S1–S3), we estimate that ∼30% of the sediment supply during Phase C was deposited on the delta plain as a whole (the other 70% was deposited on the delta foreset offshore; Fig. 3F). The ∼30% provided to the delta plain was split roughly evenly between persistently dry land and intermittent land, such that only $\sim 15\text{\%}$ of the total sediment supply contributed to deposition on persistent land. This is only ∼8% greater than the amount partitioned to persistent land during Phase B despite a fourfold increase in sea-level rise rate (Fig. 3F).

During Phase D, sea-level rise was further increased to exceed the maximum possible sedimentation rate (${\sigma }^{*}=1.33>1$; Eq. 6; Fig. 3C). As a result, shorelines retreated even at the active river mouth; land was lost at a rate of $\sim 9000{ \text{cm}}^2/\text{h}$, and the delta plain was completely submerged within 2 h (Fig. 3D). Both models predict land loss in this phase (Fig. 3D and E), but the landscape-averaged model overestimates the amount of dry land at any given time, similar to earlier phases. The lobe-averaged model provides reliable estimates, until the point the entire delta was entirely submerged and avulsions ceased. Nearly the entire sediment supply during this phase was partitioned to the delta plain, consistent with previous studies (37, 45), with the majority (∼65%) being delivered to intermittent land (Fig. 3F)

River deltas receive substantial sediment deposition on terrain that is intermittently exposed subaerially as land but is usually submerged. The lobe-averaged model and experimental results demonstrate how, at a given time, the river builds land on the active lobe, and inactive lobes are drowned to an extent that depends on the time since they were last supplied with sediment (Figs. 3A and B and 4A), creating a zone of intermittent land. After avulsion, deposition shifts to a new lobe where intermittent land is replenished with sediment. At the same time, the previously active lobe commences drowning and the zone of intermittent land on the oldest lobes approaches full submergence (Fig. 4B). This process continues for each avulsion cycle (Fig. 4C), maintaining an approximately constant area of intermittent land delta-wide because the delta is composed of multiple lobes with staggered histories of active sediment supply and inactivity (Fig. 4A–D). In terms of sediment mass balance, deposition in intermittent zones is a sediment sink that leaves less sediment available to nourish persistently dry land (Fig. 3F). In consequence, land loss on deltas is more extensive than predictions from landscape-averaged models (Fig. 3D and E) that have classically relied on the assumption that all available riverine sediment can be deposited on persistently dry land (i.e., $A=A_{\mathit{dry}}$ in Eq. 2).

To assess the effect of deltaic avulsions on land loss globally, we revised previous sediment budgets by evaluating Eq. 2 accounting for deposition in both persistent and intermittent land areas ($A=A_{\mathit{dry}}+A_{\mathit{lost}}$) (Fig. 4E) using available data (SI Appendix). These revised estimates support that deltas are in more danger of land loss than previously anticipated. For example, we find the Mississippi River delta will need more than three times greater sediment supply than previously estimated to sustain its current dry-land area (Fig. 4E), an amount totaling $Q_{s,\mathit{need}}=201 \text{Gt}$. This is more than five times the sediment available by some estimates (46), although others have estimated that greater sediment supply is possible (47). Similarly, we estimate that the Orinoco and Parana deltas in South America need roughly triple the sediment than estimated previously to maintain current dry-land area. In Europe, the Rhone and Rhine-Meuse deltas need more than five times the sediment, and the Danube needs more than 10 times the sediment compared to previous estimates—all far beyond the estimated riverine sediment supply available. Not all deltas are predicted to drown; rivers like the Yellow and Brahmaputra have enough sediment to maintain their current land area. The fate of many other deltas is uncertain (Fig. 4E) because revised estimates require quantification of the avulsion frequency, which is difficult to constrain and is unavailable for many deltas.

The fate of river deltas depends upon the frequency of river avulsions (Fig. 4F). In our experiment, river avulsions occurred more frequently as sea-level rise increased (SI Appendix, Table S1), partially mitigating land loss through a negative feedback. As captured in Eq. 4, as rise rate increases, the rate of land loss ($\frac{\sigma }{S}P$) increases, but at the same time more frequent avulsions reduce the amount of time ($\frac{n}{f_A}$) that lobes spend inactive without riverine sediment supply. This change of pace in lobe activity reduces the area of intermittent land and the volume of sediment lost to intermittent land, leading to a relative increase in sediment supply to persistently dry land. As a result, an order-of-magnitude increase in rise rate (${\sigma }^{*}=0.1$ to $1$), like that in our experiment, should incur only a doubling of lost land ($\frac{A_{\mathit{lost}}}{A}=28\text{\%}$ to $60\text{\%}$; Fig. 3E). This negative feedback helps mitigate land loss on deltas, which are also expected to avulse more frequently in the coming century (Fig. 4F) (37). However, there is a limit to the feedback: If rise rate increases beyond ${\sigma }^{*}>1$ (Eq. 6), avulsion frequency ceases to naturally increase and land loss accelerates with rise rate (Eq. 4 in Fig. 3E). Anthropogenic interference is pushing many deltas beyond this limit (5, 37). For example, the Mississippi was built over the last eight millennia with ${\sigma }^{*}\sim 0.3$ (40), but anthropogenic sea-level rise and coastal subsidence may have already forced the Mississippi into ${\sigma }^{*}>1$ conditions (37). Our model suggests that the ${\sigma }^{*}>1$ scenario could render the entire delta plain as intermittent land ($\frac{A_{\mathit{lost}}}{A}=100\text{\%}$; Fig. 3E), implying imminent drowning as far as ∼500 km inland to the avulsion site near Baton Rouge (21, 38) (Fig. 1A and C). Where natural avulsions are not sufficient or prevented by manmade infrastructure, mitigating land loss will require frequent engineered diversions.

Our results indicate that sediment supply is most efficiently leveraged as a land-building resource when engineered diversions commit flow to different delta lobes regularly, maintaining an inland zone of persistent dry land similar to natural avulsions (Fig. 4A–D). Such an approach has already proven successful on the Yellow River, where four engineered diversions in the past 70 y have facilitated long-term land gains (26, 48). For planning future engineered diversions on deltas under accelerating sea-level rise (5), the lobe-averaged model (Eqs. 3–5) provides a framework to optimize sediment resources and mitigate land loss. Without avulsions or diversions, anthropogenic increases to relative sea-level rise ($\sigma$) will result in proportional increases in land loss ($A_{\mathit{lost}}$; Eq. 4). For example, in a scenario where rise rate doubles, diversion frequency ($f_A$) should at least be doubled to prevent further land loss. More frequent diversions provide further benefits by reducing the area of intermittent land in favor of persistently dry land, and effectively maximizing sediment supply to persistent land (Fig. 4F) (SI Appendix). Long-lived, regularly scheduled diversions offer promise for some deltas; on the Mississippi and Orinoco, for instance, the existing dry-land area can be maintained with the available sediment supply if diversions occur at least once per century and once per decade, respectively (Fig. 4F). In contrast, on the Parana, Danube, and Rhine-Meuse, more frequent river diversions are not enough; more sediment must be made available for the existing land area to remain persistently dry (Fig. 4F).

Where existing dry land cannot be saved, smaller-scale diversions can also improve the outlook for prioritized coastal zones. Plans for the Mississippi (11, 12, 34) call for focused diversions within smaller portions of the delta compared to what has historically been replenished. Reducing the size of the management area effectively reduces sediment supply requirements for maintaining land persistently dry therein (Fig. 4F; SI Appendix). But this plan comes with the trade-off that intermittent land outside the management area will become permanently lost if it is not supplied with sediment. Planned land loss can be cost-effective depending on the delta (34, 49) by creating new wetland ecosystems that capture biogenic and offshore sediment in residual channels (15–17) and providing storm dissipation for prioritized zones inland (11, 27). Partial diversions can also be effective, as demonstrated by the Atchafalaya diversion of the Mississippi River that has persisted since 1963 (Fig. 1A) (30–32). Partial diversions increase the number of simultaneous active lobes, which can produce local land gains (e.g., the Wax Lake subdelta; Lobe 6 in Fig. 1C) (12, 32), but can also exacerbate land loss in other areas with reduced sediment supply (e.g., the Birdfoot subdelta; Lobe 5 in Fig. 1C) (50, 51). Even for smaller-scale and partial diversion strategies, committed decisions need to be made as to where and how much land should be prioritized for management through diversions.

Once a management zone is adopted, efforts must commit to this zone with regular, long-lived diversions in order to fully leverage sediment resources. Temporary opening of flood byways provides only short-term land gains (52, 53); if these areas are left to drown in later decades then the sediment resources diverted there represent a sunk cost. With the same sediment supply, larger management areas can be maintained using more frequent diversions because this minimizes the zone of intermittent land (Fig. 4A–D), but frequent diversions also introduce engineering costs, relocation efforts, and ecosystem pressures (11, 25, 26). Economic analysis suggests that the benefits of frequent diversions can outweigh the costs, even when considering the active lobe alone (25). Diversions will never be frequent enough to fully eliminate intermittent land; doing so would require an infinite diversion frequency (i.e., $A_{\mathit{lost}}\to 0$ when $f_A\to \infty$ in Eq. 5). With this in mind, management plans need to decide upon a target, acceptable amount of land loss ($A_{\mathit{lost}}$) and implement diversions at a scheduled frequency ($f_A$) that meets this target. Diversions need to reoccur indefinitely—just like natural avulsions—or else inactive zones will drown and sediment previously delivered there will be a wasted resource. Furthermore, diversions should be placed to minimize deposition on intermittent land near the coast, prioritizing deposition in inland areas that can be maintained indefinitely. Doing so will make efficient use of riverine sediment as a natural resource to restore sinking land, and strike a balance between coastal sustainability and river stability (27) that will be crucial for the survival of coastal communities in the coming century.

In the laboratory experiment, overhead cameras captured plan-view imagery of the delta every minute. The water was dyed fluorescent green, which allowed visual distinction between subaerial and submerged land under ultraviolet light fixtures. We manually mapped the extent of subaerial and submerged land (Fig. 3A) and identified avulsions where and when the flow established a new channel that captured the majority of flow (Fig. 3B) (44, 54). For each avulsion, we measured the avulsion length ($L_A$) and avulsion frequency ($f_A\equiv 1/T_A$, where $T_A$ is the time since the previous avulsion). Further details regarding the experimental setup can be found in Ganti et al. (54) and the SI Appendix.

For both the experiment and field data, the lobe-averaged model (Eqs. 3–5) was implemented by solving Eq. 5 for $f_A$, using an iterative scheme and input estimates of $Q_s, c_0, L_A, B, H, S, N$, and $\sigma$ (37), and plugging the result into Eqs. 4 and 5 to determine $A_{\mathit{dry}}$ and $A_{\mathit{lost}}$ (SI Appendix, Tables S2–S4). For simplicity we adopted a characteristic number of lobes $N=5\pm 1$, consistent with field observations (20, 21, 29, 55) and experiments (56–58), and propagated uncertainty associated with lobe number to land-loss predictions (SI Appendix). Landscape-averaged model predictions were reported by Giosan et al. (10) for field data, and for the experiment were determined by numerically solving Eq. 1 for $A$ and performing a scaling analysis to estimate $A_{\mathit{lost}}$ (SI Appendix). The fraction of sediment partitioned to persistent dry land, intermittent land, and farther offshore (Fig. 3F) was estimated by taking the ratio of terms in Eq. 5 (SI Appendix).

We thank Vamsi Ganti for useful discussions and acknowledge NSF Grant EAR 1427262 and the Resnick Sustainability Institute at the California Institute of Technology for support.