of 7
Accelerated river avulsion frequency on lowland deltas
due to sea-level rise
Austin J. Chadwick
a,1
, Michael P. Lamb
a
, and Vamsi Ganti
b,c
a
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125;
b
Department of Geography, University of California,
Santa Barbara, CA 93106; and
c
Department of Earth Science, University of California, Santa Barbara, CA 93106
Edited by Andrea Rinaldo, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, and approved June 3, 2020 (received for review July 17, 2
019)
Sea-level rise, subsidence, and reduced fluvial sediment supply are
causing river deltas to drown worldwide, affecting ecosystems and
billions of people. Abrupt changes in river course, called avulsions,
naturally nourish sinking land with sediment; however, they also
create catastrophic flood hazards. Existing observations and models
conflict on whether the occurrence of avulsions will change due to
relative sea-level rise, hampering the ability to forecast delta re-
sponse to global climate change. Here, we combined theory, numer-
ical modeling, and field observations to develop a mechanistic
framework to predict avulsion frequency on deltas with multiple
self-formed lobes that scale with backwater hydrodynamics. Results
show that avulsion frequency is controlled by the competition be-
tween relative sea-level rise and sediment supply that drives lobe
progradation. We find that most large deltas are experiencing suf-
ficiently low progradation rates such that relative sea-level rise en-
hances aggradation rates
accelerating avulsion frequency and
associated hazards compared to preindustrial conditions. Some del-
tas may face even greater risk; if relative sea-level rise significantly
outpaces sediment supply, then avulsion frequency is maximized,
delta plains drown, and avulsion locations shift inland, posing new
hazards to upstream communities. Results indicate that managed
deltas can support more frequent engineered avulsions to recover
sinking land; however, there is a threshold beyond which coastal
land will be lost, and mitigation efforts should shift upstream.
sea-level rise
|
river deltas
|
river avulsion
C
oastal cities and wetlands are drowning due to global sea-
level rise, accelerated subsidence from fluid extraction, and
reduced fluvial sediment supply (1
3), with significant implica-
tions for the global economy, carbon cycle, and diverse ecosys-
tems (4
6). Most estimates of coastal inundation for the coming
century do not consider the land-building potential of riverine
sedimentation (7
9). Rivers naturally distribute sediment across
deltaic plains through avulsions
catastrophic shifts in the river
course
which occur every 10 to 1,000 years on different deltas
(Fig. 1
A
and
B
) (10, 11). However, it is unknown what sets
avulsion timescales and how avulsion occurrence will change
with increasing rates of relative sea-level rise caused by climate
and land-use changes (12, 13). River avulsions counter land loss
by nourishing wetlands with sediment (14) but also have caused
some of the deadliest floods in human history (10, 15, 16). On
densely populated fluvial systems, dammed reservoirs trap sedi-
ment and engineered levees prevent avulsions, with the un-
intended consequence of heightened land loss (8, 9). Engineered
river diversions are now important parts of future billion-dollar
coastal restoration plans (17). Despite their global importance,
we lack a predictive framework for avulsion reoccurrence on
deltas, which is imperative to mitigate catastrophic flood hazards
and design effective diversions on engineered deltas (8, 9).
Existing observations and models produce conflicting results
as to whether avulsion frequency will increase or decrease with
relative sea-level rise. For example, avulsion frequency on the
Rhine
Meuse delta increased during late Holocene sea-level rise
(19), consistent with observations in fan-delta experiments (20).
In contrast, avulsion frequency decreased on the Mitchell River
delta during Holocene sea-level rise (21). Similarly, numerical
model predictions differ on whether avulsions on the Mississippi
and Trinity Rivers were more or less frequent during the Ho-
locene period (22, 23) despite their geographic proximity (24).
Sequence stratigraphic models (25) and physical experiments
(26) predict sea-level fall causes valley incision preventing avul-
sion, whereas the Goose River delta (27) represents an example
where avulsions persisted during sea-level fall.
Previous work documented that the characteristic avulsion
frequency,
f
A
, scales with the rate that the riverbed aggrades to a
height comparable to its channel depth,
f
A
=
v
a
H
,
[1]
where
v
a
is the in-channel aggradation rate,
H
=
H
p
H
c
is the
aggradation thickness necessary for avulsion, and
H
c
is the bank-
full channel depth (28, 29). The avulsion threshold,
H
p
, is a di-
mensionless number between 0.2
and
1.4 on lowland deltas (30).
Aggradation rates, in contrast, span orders of magnitude
(
v
a
=
0.5
to
100
mm
=
y; ref. 29). Thus, river avulsions may occur
as frequently as every decade (e.g., the Huanghe; Fig. 1
A
and ref.
31) or as rarely as each millennium (e.g., the Mississippi; Fig. 1
B
and ref. 32), and Eq.
1
is only useful insofar as aggradation rate
can be predicted.
Analytical models are often used to approximate
v
a
through
spatial averaging of sediment mass balance. A common approach
Significance
River deltas host large cities and ecosystems and are sinking
under global sea-level rise and land subsidence from ground-
water and hydrocarbon extraction. Coastal rivers naturally
build land by distributing sediment through abrupt shifts in
river course, known as avulsions, which also have caused some
of deadliest flood disasters in human history. We show that
modern rates of relative sea-level rise should cause avulsions
to occur more frequently on deltas, and in extreme cases
avulsion locations will shift farther inland. Our results provide
a quantitative framework to predict delta response to future
sea-level rise, which is valuable for planning engineered di-
versions to nourish deltas and prevent catastrophic hazards.
Author contributions: A.J.C., M.P.L., and V.G. designed research; A.J.C. and M.P.L. per-
formed research; A.J.C. and M.P.L. analyzed data; and A.J.C., M.P.L., and V.G. wrote
the paper.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
Published under the
PNAS license
.
Data deposition: The data and model code underlying this study are publicly available in
the SEAD Repository (http://doi.org/10.26009/s0FSLKFK
) and GitHub (https://github.com/
achadwick2323/Accelerated-river-avulsion-frequency-on-lowland-deltas-due-to-sea-level-
rise), respectively.
1
To whom correspondence may be addressed. Email: achadwick@caltech.edu.
This article contains supporting information online at
https://www.pnas.org/lookup/suppl/
doi:10.1073/pnas.1912351117/-/DCSupplemental
.
First published July 13, 2020.
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to estimate
v
a
is a radially averaged model, in which the delta apex
is geographically fixed and relative sea-level rise causes the delta
radius to shrink until sediment supply is sufficient to keep delta-
top aggradation at pace with sea level (8, 11, 33
38). At steady
state, the delta land area is given by
A
Δ
=
1
(
1
λ
p
)
Q
s
σ
, where
Q
s
is the
volumetric sediment supply,
σ
is the relative sea-level rise rate, and
λ
p
is the deposit porosity (8, 11). Aggradation rate is given by
v
a
=
1
1
λ
p
()
Q
s
A
Δ
=
σ
.
[2]
For the radially averaged model, aggradation is enhanced during
marine transgression because, with a fixed delta apex and
sediment supply, the delta land area is reduced (33, 39). Eq.
2
shows agreement with steep experimental deltas where the delta
apex was geographically fixed by a canyon-fan transition or flume
inlet (20, 37). In contrast to steep fan deltas, lowland deltas have
an apex that is not fixed but instead is found at a characteristic
distance upstream of the shoreline where avulsions preferentially
occur due to backwater hydrodynamics (22, 31, 40, 41). It is
unknown if lowland deltas will change their area to equilibrate
with sediment supply and sea-level rise as indicated by Eq.
2
,orif
delta response will differ because backwater hydrodynamics set a
constant lobe length (41, 42).
A second analytical approach, termed the channel-averaged
model, is to constrain aggradation rate using sediment mass
balance for a channel. For example, Reitz et al. (43) found
avulsions occurred in a fan-delta experiment at the rate the
sediment supply could fill the channel:
1
(
1
λ
p
)
Q
s
T
A
=
HB
c
L
A
,where
T
A
is the time between avulsions,
B
c
is channel width, and
L
A
is delta lobe length
the distance upstream from the shore-
line where avulsions occur (Fig. 1
C
). Combined with Eq.
1
and 1
=
f
A
T
A
, aggradation rate is given by
v
a
=
1
(
1
λ
p
)
Q
s
L
A
B
c
.
[3]
In Eq.
3
,
B
c
can be substituted by an effective delta lobe width,
B
,
to approximate lateral distribution of sediment across a delta
lobe, for example due to bifurcations (35, 44
48). While the
channel-averaged model considers sedimentation within a dis-
crete channel or lobe, it does not account for backwater hydro-
dynamics, multiple lobes, or sediment partitioning between lobe
aggradation and progradation.
More complex two-dimensional (2D) morphodynamic models
designed to study delta bifurcations and channelization (14, 27,
49
53) have yet to be run systematically to explore backwater-
scaled avulsions. While simpler one-dimensional morphody-
namic models that include backwater hydrodynamics exist, these
models are tuned to specific case studies (22, 23) and yielded
opposite trends for avulsion frequency response to relative sea-
level rise. A 2D reduced-complexity model found that avulsion
TRUNK
L
A
D
H
H
b
avulsion
node
shoreline
S
TRUNK
D
H
H
b
z
avulsion
node
shoreline
L
A
σ
DELTA
SEA
drowned
Mississippi River delta
100 km
TRUNK
avulsion
node
modern
channel
historic
channels
Huanghe delta
avulsion
node
40 km
SEA
TRUNK
DELTA
2
3
4
SEA
1
DELTA
avulsion node
B
B
c
TRUNK CHANNEL
L
A
(5)
AB
C
DE
Fig. 1.
Natural and modeled deltas. (
A
) Huanghe delta, China and (
B
) Mississippi River delta, United States, showing modern (solid line) and abandoned
(dotted line) channel pathways and avulsion node (yellow star) (Google Earth). The thin dashed line in
B
indicates approximate shoreline before human
management (18). (
C
) Model conceptualization, showing channels of width
B
c
that occupy lobes of width
B
(shaded regions) and length
L
A
(yellow line).
Shaded regions are deposits created during avulsion cycles 1 through 4. Once the channel avulses from the active lobe 4 to the space 5, all lateral accom
-
modation is filled, and the avulsion node moves downstream in tandem with progradation. (
D
) Model conceptualization in cross-section for constant relative
sea level and (
E
) relative sea-level rise rate of
σ
, showing lobe length
L
A
, lobe-progradation distance
D
, topset slope
S
, basin depth
H
b
, relative sea-level rise
magnitude
z
, and aggradation height
H
in a single avulsion cycle on the active lobe (gray shaded region).
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frequency was insensitive to sea-level rise for small rise rates due
to progradation (54). However, this model did not include
backwater hydrodynamics or a model spin-up phase. A spin-up
phase was shown to eliminate bias in predicted avulsion locations
associated with the assumed initial topography, by reworking the
initial topography for at least one avulsion cycle per delta
lobe (55).
Recent models for backwater-scaled avulsions have included
quasi-2D nonuniform flow, lobe aggradation and progradation,
multiple cycles of lobe growth and abandonment, and a spin-up
phase (55, 56). These models provided new insight into the
controls on avulsion location on deltas (55) and were validated
against field observations from the Huanghe delta (56); however,
they have yet to be used to explore how relative sea-level rise
affects avulsion frequency. Here we built on the model of
Chadwick et al. (55) to develop a generic framework to predict
the response of avulsion frequency on deltas to different rates of
relative sea-level rise and fall. We also derived an analytical
approximation to the model that can be directly compared to the
commonly used radially averaged and channel-averaged ap-
proximations.
Backwater-Scaled-Avulsion Modeling and Field Data.
The numerical
morphodynamic model for avulsions consisted of a delta com-
posed of multiple lobes that were assumed to form a branching
pattern (Fig. 1
C
) (30, 55) (
SI Appendix
, section A
). We repre-
sented each lobe by a separate long profile and a shared trunk
channel, and only one lobe was active at a given time. The
evolution of the active lobe was governed by quasi-2D non-
uniform hydrodynamics (22), sediment transport (57), and sed-
iment mass balance of the lobe topset and foreset (58). Avulsions
occurred when and where the active lobe first aggraded to a
specified height
H
relative to the lowest neighboring lobe (Eq.
1
)
(22, 28, 41, 54). Subsequently, the active channel was rerouted
downstream and the lowest-elevation abandoned lobe became
the new active lobe (
SI Appendix
, section A
). Inactive lobes were
unchanged, except for inundation due to relative sea-level rise,
approximating a river-dominated delta with negligible reworking
from waves or currents (54, 56). Every model run began with a
spin-up phase in which the river occupied each lobe at least once,
thereby eliminating the bias of the imposed initial conditions
(55) (
SI Appendix
, section B).
Dimensional analysis revealed that model behavior depen-
ded primarily on the ratio of rates of relative sea-level rise and
sediment supply to the delta. We termed this dimensionless
parameter the normalized relative sea-level rise rate,
σ
p
=
σ
Q
s
=
nL
b
B
(
1
λ
p
)
,where
n
=(
N
+
1
)
=
2,
N
is the number of delta
lobes,
L
b
=
H
c
=
S
is the backwater length scale, and
S
is channel
bed slope (
SI Appendix
, sections A and C
). Delta lobe pro-
gradation occurs for
σ
p
<
1, and when
σ
p
>
1 the shoreline re-
treats and the delta lobe drowns. Delta response also depends
on the offshore basin depth
(
H
b
)
relative to the channel depth
(Fig. 1
D
),thelobewidth
(
B
)
relative to the channel width, the
avulsion threshold
(
H
p
=
H
=
H
c
)
, and six additional di-
mensionless parameters that describe river flow hydraulics and
sediment transport (
SI Appendix
, section B
). For each modeled
delta, we computed the average time between avulsions
(
T
A
)
over 13 avulsion cycles and calculated
f
A
1
=
T
A
under dif-
ferent scenarios of normalized relative sea-level rise rate
(
σ
p
)
,
with all other parameters held constant at values representative
of typical lowland deltas (
SI Appendix
, section B and Tables S1
and S2
).
We also derived an independent analytical model for avulsion
frequency by setting lobes to a fixed length scaled by
L
b
(22, 40)
and averaging sediment mass balance over an avulsion timescale
for a river channel of constant slope (
SI Appendix
, section D
).
These assumptions circumvented the need to solve the nonlinear
equations for water and sediment transport. The result is
f
A
=
1
1
λ
p
()
Q
s
L
A
D
()
BH
+
DB H
b
+
z
+
DS
=
2
()
if
D
0
1
1
λ
p
()
Q
s
L
A
BH
if
D
<
0
[4]
where
D
=(
H
z
)
=
S
is the lobe-progradation distance and
z
=
n
σ
f
1
A
is the magnitude of relative sea-level rise during an
interavulsion period (Fig. 1
D
and
E
and
SI Appendix
, section
C). The lobe length,
L
A
, typically varies between 0.5
L
b
and 2
L
b
for large, coastal rivers (30, 55), and in Eq.
4
it is a specified
constant, unlike in the numerical model where
L
A
is a model
outcome that emerges due to backwater hydrodynamics (55).
For
D
0, the first and second terms in the denominator account
for sediment partitioned to the lobe topset and foreset, respec-
tively. For
D
<
0, all sediment is sequestered in the topset. We
used the analytical solution as a comparison to the numerical
model and to explore the effect of relative sea-level rise on
avulsion frequency with covarying basin depth, avulsion thresh-
old, lobe length, and lobe width.
We compared the numerical and analytical models to a
compilation of data from 15 natural deltas spanning a wide range
of known avulsion frequencies
f
A
=
0.5
to
140
ky
1
()
and lobe
lengths (
L
A
=
30
to
490
km;
SI Appendix
, Tables S1 and S2
; refs.
21, 23, 27, and 29). Due to infrequent historical avulsions,
documented avulsions pertain to the Holocene period, with the
exception of the Huanghe where historic natural avulsions were
documented between 1889 and 1930 prior to major engineering
(Fig. 1
A
) (31). Consistent with our model assumptions, these
deltas feature a single major channel and lobe lengths that scale
with the backwater length (22, 45). For each delta, we computed
the average relative sea-level rise rate during the period of
avulsions as the sum of eustatic sea-level rise rate
(
σ
eu
)
and
coastal subsidence rate
(
σ
subs
)
, that is,
σ
=
σ
eu
+
σ
subs
(4, 59
63).
We used available data to constrain reasonable values for other
model inputs
H
p
=
0.5,
L
A
=
L
b
,
H
b
=
2
H
c
()
and to test model
results against avulsion frequencies for six deltas where all model
inputs could be constrained (
SI Appendix
, Table S2
). We also
computed modern expectations for avulsion frequency using
relative sea-level rise rates for each delta measured by tide
gauges over the 20th century (64). All calculations used historical
measurements to estimate river sediment loads prior to damming
(65). To facilitate comparison to natural deltas, the model and
data were normalized through dimensional analysis (
SI Appen-
dix
, sections A and D and Table S2
). Results were cast in terms
of normalized avulsion frequency,
f
p
A
=
f
A
Q
s
=
H
c
B
c
L
b
(
1
λ
p
)
, which ap-
proaches unity when the sediment supply is entirely sequestered
in the delta lobe topset, rather than the foreset.
Results
Results from backwater-scaled-avulsion modeling show that
avulsion frequency responds nonlinearly to relative sea-level rise
rate, falling within three regimes depending on the competition
between relative sea-level rise and sediment supply as described
by
σ
p
(Fig. 2). In the regime 10
4
<
σ
p
K
10
1
,termedprogradation-
dominated, sediment supply outpaces the accommodation space
created on the delta top due to relative sea-level rise, causing
lobe progradation (Fig. 1
E
and
SI Appendix
, Fig. S1
). Lobe
progradation causes channel aggradation because the channel
adjusts to maintain a transport slope as the river mouth and
avulsion node advance seaward (31). Because avulsion frequency
scales with aggradation rate (Eq.
1
),
f
A
is insensitive to relative
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sea-level rise in this regime (Fig. 2), similar to previous reduced-
complexity model results (54).
Results indicate that channel aggradation increases with rel-
ative sea-level rise when the rates of relative sea level rise and
sediment supply are similar 10
1
K
σ
p
<
10
0
()
(Fig. 2 and
SI
Appendix
, Fig. S1
). In this regime, termed rise-dominated, rela-
tive sea-level rise enhances channel aggradation because more
sediment is partitioned into the lobe topset relative to the
foreset, in a process similar to the radially averaged model (8, 37,
38, 58), but here manifesting at the scale of delta lobes. Conse-
quently, avulsion frequency accelerates as relative sea-level rise
rate increases (Fig. 2).
For
σ
p
>
10
0
, relative sea-level rise outpaces the rate at which the
sediment supply can aggrade the reach downstream of the avulsion
node, causing coastal land to drown and the avulsion node to shift
upstream. In this regime, termed supply-limited, aggradation rate
and rise rate are decoupled. Instead, aggradation rate is limited by
the sediment supply per unit lobe area (i.e., the product of lobe
length
L
A
and width
B
), and avulsion frequency reaches a maximum
value that is insensitive to rise rate (Fig. 2).
Independently varying parameters in the analytical model
revealed that relative sea-level rise universally increases avulsion
frequency in the rise-dominated regime 0.1
K
σ
p
<
1
()
by creating
accommodation space at a rate commensurate with the sediment
supply (Fig. 3). Smaller basin depths also increase avulsion fre-
quency (Fig. 3
A
), but only for low rise rates. Small basin depths
facilitate rapid progradation when
σ
p
K
0.1, driving high aggra-
dation rates on the topset to maintain a transport slope. Results
show that avulsion frequency increases with decreasing avulsion
threshold and lobe length (Fig. 3
B
and
C
) because less sediment
is needed on the delta top to cause an avulsion. Avulsion fre-
quency decreases as lobe width increases (Fig. 3
D
), consistent
with the recent finding that wider distributary networks are more
resilient to environmental change (47). When rapid relative sea-
level rise causes marine transgression
D
<
0
()
, the sediment
supplied to the delta lobe is entirely sequestered on the lobe top,
and
f
A
is maximized to a value controlled by lobe length
(
L
A
)
,
width
(
B
)
, and aggradation thickness necessary for an avulsion
(
H
) (Eq.
4
). For cases with shallow basins and
L
A
=
L
b
>>
1, such
as steep fan deltas (31), avulsion frequency is insensitive to
σ
p
because the large topset area causes a transition from progradation-
dominated to supply-limited conditions at relatively low rise rates
(Fig. 3
C
). For
σ
p
J
2.5, the avulsion node drowns before in-channel
sedimentation can reach the avulsion threshold.
Model results also indicate that avulsions can occur during
relative sea-level fall, so long as the channel is aggradational
(Fig. 3
E
) (27, 66). Basin depth is the fundamental control on
avulsion frequency during sea-level fall because topset aggrada-
tion is driven primarily by lobe progradation, similar to the case
of slow relative sea-level rise.
Our analytical model incorporates the crucial component of a
backwater-scaled avulsion node that fixes the lobe length, con-
sistent with the numerical model results and natural lowland
deltas (Fig. 2). The backwater-scaled avulsion node moves sea-
ward or landward in tandem with shoreline progradation or re-
treat, consistent with field observations (31, 56). Consequently,
lobe length remains constant, and the competition between
shoreline progradation, relative sea-level rise, and sediment
supply controls avulsion frequency. The radially averaged model,
in contrast, fixes the avulsion location such that shoreline pro-
gradation and retreat cause adjustment of delta top area, which
scales inversely with the aggradation rate and avulsion frequency
(Eq.
2
and Fig. 2 and
SI Appendix
, Fig. S2
). The radially averaged
model and our backwater-scaled-avulsion models produce
qualitatively similar trends only in the rise-dominated regime
10
1
K
σ
p
<
10
0
()
; our model predicts more frequent avulsions in
this regime due to additional topset aggradation from pro-
gradation that is not included in the radially averaged model.
The channel-averaged model predicts
f
p
A
=
2 regardless of rise
rate because all sediment is sequestered in the channel (Fig. 2
and
SI Appendix
,Fig.S2
). Replacing channel width,
B
c
,inEq.
3
with
lobe width,
B
, yields predictions similar to our backwater-scaled-
avulsion model, but only in the supply-limited regime
(
σ
p
1
)
.For
σ
p
<
1, our model predicts fewer avulsions because it partitions
sediment to the delta foreset, which diminishes channel aggradation
rates.
Data for avulsions during the Holocene period for the Dan-
ube, Mississippi, Rhine
Meuse, Orinoco, and Paraná deltas and
the preindustrial historical avulsions on the Huanghe (
SI Ap-
pendix
, Tables S1 and S2
) fall within the prediction envelope of
the analytical backwater-scaled-avulsion model (Fig. 2). The high
sediment load of the Huanghe yielded
σ
p
0.004, placing it in
the progradation-dominated regime, where avulsion frequency is
insensitive to relative sea-level rise. Avulsions on the Danube,
Mississippi, and Orinoco are predicted to have been in the rise-
dominated regime, in which avulsion frequency increases with
relative sea-level rise rate (
σ
p
0.1,
0.3, and 0.77, respectively).
The Paraná and Rhine
Meuse were in the supply-limited regime
(
σ
p
2.3 and 2.6, respectively), where avulsion frequency is
Fig. 2.
Normalized avulsion frequency,
f
*
A
, versus normalized relative-sea
level rise rate,
σ
*
. Black circles and error bars show the median, minimum,
and maximum from 13 avulsions that occurred for each backwater-scaled
numerical model run. The gray solid line is the backwater-scaled analytical
model (Eq.
4
), and dashed lines are radially averaged model (Eq.
2
), channel-
averaged model (Eq.
3
), and channel-averaged model incorporating a lobe
width (Eq.
3
with
B
c
B
) with other variables set to constant values typical of
large lowland deltas (
H
*
=
0.5,
L
A
=
L
b
,
H
b
=
2
H
c
,
N
=
4, and
B
=
40
B
c
;
SI
Appendix
, Table S2
). White diamonds are data for Holocene avulsions on
natural deltas and preindustrial historical avulsions on the Huanghe, where
all parameters are constrained, and gray shaded regions are envelopes of
analytical model solutions for input parameters corresponding to each
natural delta (
SI Appendix
, section B and Table S2
).
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sensitive to sediment supply and delta size, and not to relative
sea-level rise rate. While some model input parameters were
unavailable for the other nine deltas in our compilation (
SI
Appendix
, Table S2
), they still can be placed within the model
parameter space (Fig. 3). Model comparison suggests that the
Nile, Magdalena, Amazon, Rhone, Brahmaputra, and Trinity
deltas likely resided in the rise-dominated regime (Fig. 3
A
D
),
whereas the Goose and Mitchell deltas were in the progradation-
dominated regime, such that avulsions occurred despite sea-level
fall (Fig. 3
E
), consistent with observations (21, 27). We expect
that accelerated relative sea-level rise over the past century (
SI
Appendix
, Table S3
) is causing many of these deltas to transition
into the rise-dominated or supply-limited regimes (Fig. 3), which
will result in either more frequent avulsions or avulsions that
occur farther upstream.
Discussion and Conclusions
Our analytical and numerical models for backwater-scaled
avulsions show that avulsion frequency on lowland deltas de-
pends on the dominant cause of channel aggradation. To first
order, avulsion frequency scales with the sediment supply to the
delta and the delta top area [i.e.,
f
A
Q
s
=
H
c
B
c
L
b
(
1
λ
p
)
or
f
p
A
=
constant], similar to the channel-averaged avulsion model
(Fig. 2). However, our results indicate significant deviation from
this scaling relation depending on whether channel aggradation
is driven by progradation, relative sea-level rise, or limited by
sediment supply (Figs. 2 and 3); these processes in turn de-
termine whether or not avulsion frequency is sensitive to changes in
relative sea-level rise rate. Our a
nalytical model, despite its sim-
plicity, accurately predicts avulsion frequencies observed in the
physics-based numerical model (F
ig. 2). Minor differences in mass
balancearosebecauseweassumed
L
A
=
L
b
in the analytical model,
whereas
L
A
in the numerical model emerged autogenically and
varied between 0.5
L
b
and
2
L
b
. Consistency between numerical and
analytical model results (Fig. 2) suggests that the sediment parti-
tioning between the delta topset and foreset exerts the primary
control on avulsion frequency. Thus, beyond accounting for a
backwater-scaled topset, modeling hydrodynamics and transient bed
adjustment are not necessary to predict
f
A
to first order.
The parameter
σ
p
distinguishes between avulsion-frequency
regimes because it compares the relative sea-level rise rate to
the aggradation rate attained when sediment supplied to the
delta lobe is entirely sequestered on the backwater-scaled topset
(Fig. 4
A
D
). Deltas with lobe lengths set by the backwater
length
(
L
A
=
L
b
1
)
feature a transition between the rise-
dominated and supply-limited regimes at
σ
p
=
1, at which point
lobe aggradation can only barely keep pace with sea-level rise.
Similarly,
σ
p
0.1 marks a transition between the rise-dominated
and progradation-dominated regimes. Our model indicates deltas
may also transition between regimes as a result of increasing basin
depth
(
H
b
)
and lobe length
(
L
A
)
associated with shoreline autore-
treat (Fig. 3
A
and
C
)(33,39).
Our results have important implications for delta flood haz-
ards and morphodynamics under changing
σ
p
and basin depth.
Basin depth plays a significant role in the progradation-dominated
regime
σ
p
K
0.1
()
(Figs. 3
A
and 4) because less sediment is required
for a given amount of progradation to occur in shallow basins
(67
69). Thus, lowland deltas building into shallow epicontinental
seas likely reside in the progradation-dominated regime and their
avulsion frequency may be insensitive to relative sea-level rise. This
model result can explain the puzzling observation of decreased
avulsion frequency during Holocene sea-level rise on the Mitchell
River delta (21), which builds into a shallow basin
H
b
=
15
m
()
compared to the channel depth
(
H
c
=
7
m
)
. We reason that pro-
gradation rates may have decreased on the Mitchell delta as its
basin deepened over the Holocene, causing less-frequent avulsions.
In contrast, on deltas with deep offshore basins, such as shelf-edge
deltas during sea-level lowstands (70), progradation is slow and so
avulsion frequency should be more responsive to sea-level change.
A
B
C
D
E
Fig. 3.
Analytical model results for backwater-scaled avulsions. Contours of
normalized avulsion frequency,
f
*
A
, as a function of normalized relative sea-
level rise rate
σ
*
and (
A
) basin depth, (
B
) avulsion threshold, (
C
) lobe length,
and (
D
) lobe width, with other variables set to constant values typical of
large lowland deltas (
H
*
=
0.5,
L
A
=
L
b
,
H
b
=
2
H
c
,
N
=
4,
B
=
40
B
c
;
SI Appen-
dix
, Table S2
). (
E
) Solutions for falling sea level and varying basin depth. Gray
zones indicate where the analytical model is not applicable due to shifting
avulsion location or channel incision (
SI Appendix
, section D
). White dia-
monds are data for preindustrial historical avulsions on the Huanghe and
Holocene avulsions for the other deltas (filled markers;
SI Appendix
, Tables
S1 and S2
), and modern expectations based on rise rates documented over
the 20th century (unfilled markers;
SI Appendix
, Table S3
).
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Chadwick et al.
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Our model also shows that high progradation rates can drive
channel aggradation and avulsion even during sea-level fall
(Fig. 3
E
), which is typically thought to cause channel incision and
prevent avulsion. This finding complicates sequence-stratigraphic
interpretations of deltas building into shallow seas, such as the
Cretaceous Interior Seaway (25), where unconformities interpreted
as sequence boundaries reflecting sea-level fall could instead form
intrinsically due to flood variability and avulsions (30, 71).
Our model indicates avulsion hazards should become more
common and occur farther inland under modern rates of relative
sea-level rise, as compared to over the Holocene. (Fig. 3 and
SI
Appendix
,TableS3
). Many deltas on continental shelves resided in
the rise-dominated regime 0.1
K
σ
p
<
1
()
during the Holocene
period, and so modern acceleration of rise rates is expected to
cause more frequent avulsions (Fig. 3). More frequent avulsions
will increase the potential for catastrophic flood hazards (15) but
also present an opportunity to mimic natural processes on man-
aged deltas by using engineered diversions to build land (8, 9, 17,
31). Results also point to distinct hazards associated with the
supply-limited regime
σ
p
>
1
()
. Unlike radially averaged models
with a geographically fixed avulsion node (Fig. 4
F
), the avulsion
node in our model moves upstream when
σ
p
>
1 in order to keep
pace with shoreline retreat and maintain a constant, backwater-
scaled lobe length (Fig. 4
E
). Notably, under modern rise rates, the
Mississippi and Rhone are predicted to transition to supply-
limited conditions, similar to the Rhine
Meuse and Paraná
(Fig. 3). As a result, we expect coastal land will drown, and shifting
avulsion locations will introduce new flood hazards upstream of
historic avulsion sites, such as at the Old River Control Structure
on the Mississippi River (72). While the Huanghe has experienced
a drastic increase in
σ
p
in the past century due to subsurface fluid
extraction (64), our model indicates negligible change to avulsion
frequency because aggradation is still predominantly set by pro-
gradation. Climate and land-use changes over the coming century
are likely to further increase the rate of global sea-level rise, ac-
celerate coastal subsidence, and reduce river sediment supply (4,
73), all of which can further amplify future avulsion hazards.
Overall our results indicate lowland deltas will not passively
drown but instead will respond to relative sea-level rise through
more frequent cycles of sedimentation and river avulsion
(Fig. 4). While this response acts to mitigate land loss in coastal
wetlands, it also heightens flood hazards. Engineered diversions
that mimic rivers
natural tendency of more frequent avulsions
during relative sea-level rise may offset land loss, but our results
indicate there is a threshold beyond which sediment supply
cannot keep pace with increasing rise rates, and avulsion hazards
will shift upstream.
Materials and Methods
The numerical morphodynamic model solves the coupled equations for flow
hydraulics, sediment transport, and topographic evolution (
SI Appendix
,
sections A
C). The model is quasi-2D and assumes a specified number of
lobes in a branching pattern. Only one lobe is active at a time, and the flow
switches between lobes when aggradation reaches a critical value that
triggers an avulsion, following ref. 55. The analytical model averages sedi-
ment mass balance over an avulsion cycle for a lobe that is assumed to build
to a fixed length
L
A
set by backwater hydrodynamics (22, 40) with a constant
riverbed slope (
SI Appendix
, section D
). These assumptions allow for the
prediction of avulsion frequency without solving nonlinear equations for
water and sediment transport. Model parameters and avulsion frequencies
for natural deltas were compiled from previous work (
SI Appendix
, Tables
S1
S3).
ACKNOWLEDGMENTS.
We thank Andrew Moodie, Gary Parker, Jeffrey
Nittrouer, Hongbo Ma, and Brad Murray for useful discussions and three
reviewers for constructive comments. We acknowledge NSF Grant EAR
1427262 and the Resnick Sustainability Institute at the California Institute
of Technology for support.
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