of 17
1
Geophysical Research Letters
Supporting Information for
Origin of a preferential avulsion node on lowland river deltas
A. J. Chadwick
1*
, M. P. Lamb
1
, A.J. Moodie
2
, G. Parker
3
, J.A. Nittrouer
2
1
Division of Geological and Planetary Sciences, California
Institute of Technology
, 1200 E. California Boulevard,
Pasadena, California 91125, USA
,
2
Department of Earth
, Environmental and Planetary
Science, Rice University,
Houston, Texas, USA,
3
Department of Civil and Environmental Engineering and Department of
Geology, University
of Illinois at Urbana
-
Champaign, Urbana, IL, USA
Contents of this file
Text S1 to S
6
Figures
S1 to S6
Table S1
Introduction
This document contains supplementary figures (Figures S1
-
S6), tables (Table S1), and
elaborations on model treatment of the river mouth & delta front (Text S1), model workflow
(Text S2),
flow
variability parameters (Text S3), non
-
dimensionalization (Text S
4), results during
sedimentation of the trunk channel (Text S5), and sensitivity to other input parameters (Text
S6) for the manuscript
Origin of a preferential avulsion node on lowland river deltas
.
2
Text S1.
T
reatment of the river mouth and
delta front
The river mouth and plume set the downstream end of the backwater zone
.
Previous
models
of backwater hydrodynamics
have considered a fixed river mouth position, resulting in
a constant flow
-
width profile and
fixed backwater zone
(Lamb et al., 2012; Chatanantavet et
al., 2012; Chatanantavet et al., 2014). This is a good approximation over the timescales
of
flow
events
considered
by these studies
,
where
the degree of sea
-
level and land
-
surface change is
small compared to the channe
l depth. However, the timescales of river avulsion
are
sufficiently long to drive lobe aggradation and drowning on the order of the channel depth,
which could drive significant river mouth advance and retreat respectively, resulting in
changes to the flow
-
width profile.
In this study we develop a new approach to modeling backwater zones that may
translate over the timescales of river avulsion. The spatio
-
temporal evolution of flow width is
driven by the emergence and submergence of the land surface. At a g
iven time, the location
of the river mouth (
!
"
) is set by the intersection of the
floodplain
profile
#
$
with sea
-
level
%
&
,
!
"
=
!
|
)
*
(
,
)
.
/
.
0
1
.
.
.
.
equation
.
S1
.
where the floodplain elevation is defined as the sum of the bed elevation and channel de
pth,
#
$
=
#
<
+
>
?
.
.
.
.
.
equation
.
S2
The flow
-
width profile is a piecewise function of the channel width, assumed constant and
uniform upstream of the river mouth, and a linearly spreading plume downstream
A
=
B
A
C
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
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.
.
.
.
.
.
.
.
!
<
!
"
A
C
>
?
>
E$
+
A
$FGHI
>
E$
>
?
>
E$
.
.
.
.
.
!
!
"
.
.
.
.
.
.
.
.
equation
.
S3
Where
A
is the flow width,
A
C
is the channel width,
A
$FGHI
is the width of the flare, and
>
E$
=
%
&
#
is the no
-
flow depth. Upstream of the river mouth
(
!
<
!
"
),
flow is confined by the
channel
and the flow width is set by the channel width
(
A
C
)
. Downstream of the mouth
(
!
!
"
)
, the unconfined portion of the flow expands laterally
to form the river plume. In this
setting,
A
is the depth
-
averaged width of
a submerged channel and a linearly expanding flare,
A
$FGHI
=
A
?
+
2
tan
M
(
!
"
!
)
.
.
.
.
.
.
.
.
equation
.
S4
where
M
is the plume spreading angle, here set to fifteen degrees
(
Lamb et al., 2012;
Chatanantavet et al., 2014).
When the river mouth progrades into
an empty basin with flat
topography,
>
?
=
0
and the flow width in equation
S3
reduces to the flare width
(
A
=
A
$FGHI
)
.
When the land is drowned, the river mouth retreats and a portion of the flow
is
confined to the
submerged channel.
The terms
P
Q
P
R
*
and
P
R*
S
P
Q
P
R*
in equation S3 represent the fraction of the no
-
flow depth that is of width
A
C
and width
A
$FGHI
respectively to yield a depth
-
averaged width
of
A
.
This scheme leads to dynamic backwater profiles that advance with river mou
th
progradation
(i.e., increasing
!
"
)
and back
-
step during shoreline retreat
(i.e., decreasing
!
"
),
and conveniently facilitiates numerical stability in our simulations by producing a gradient in
width
(
TA
/
T!
)
.
that is everywhere differentiable.
A
t the river mouth
lateral flow expansion
drives
an abrupt deceleration of flow,
resulting in a mound of sediment that
accumulate
s
and steepen
s
. At sufficiently steep slopes,
3
fluvial sediment transport gives way to gravity flows and avalanching
to form
a delta front, or
foreset. We model the development of delta fronts in terms of a threshold slope condition
following Hotchkiss and Parker (1991). A delt
a front develops at position
!
V
if the bed ste
epens
to the threshold slope
W
G
associated with grav
ity flows and sediment avalanching.
!
V
=
!
|
X
Y
X
Z
.
.
.
.
.
.
.
.
equation
.
S5
Once a delta front initiates, t
he slope of the
front is fixed at
W
G
and deposition drives
progradation of the new delta front and delta toe according to shock
-
capturing co
nditions
(Kos
tic & Parker, 2003; Parker et al., 2004;
Kim et al., 2006;
Parker et al., 2008
a; Chatanantavet et
al., 2014).
Previous work has focused on conditions where the delta front is sufficiently far
downstream such that the water velocity
is approximately zero
.
Over longer timescales,
however, we find that lobe
-
switching over antecedent topography can drive the creation of
shallow foreset wedges
farther upstream
,
with significant flow velocities
at their toes
. The
foreset wedge
s pose
an order of magnitude discont
inuity in bed slope that violates the
gradually
-
varied flow assumption in the backwater equation
(Parker, 2004). Across the shallow
foreset wedge
, we reason the water surface is more accurately described by the Borda
-
Carnot
relationship for flow encounteri
ng a sudden expansion (Sturm, 2010).
\
%
+
]
^
2
_
`
a
=
\
%
+
]
^
2
_
`
b
+
1
2
_
(
]
a
]
b
)
^
.
.
.
.
.
.
.
.
.
equation
.
S6
where
%
is the water surface elevation,
]
is flow velocity,
_
is acceleration
due to gravity, and
subscripts
d
and
T
denote conditions upstream and downstream of the shallow foreset
respectively. We can rearrange to find that this describes a “jump” in the water surface at the
lip of the foreset.
%
>
a
=
f
g
a
^
(
1
g
h
)
.
.
.
.
.
.
.
.
.
.
equation
.
S7
where
%
is the increase in w
ater surface elevation from the upstream to downstream end and
g
h
is the ratio of the upstream to downstream cross
-
sectional areas of the flow. For the low
Froude number scenarios considered in our study, water surface deflections are limited to 1
-
5% of
the flow depth measured upstream of the foreset. Nevertheless, this treatment is
important for avoiding erroneous and unstable application of the backwater formula to
reaches with a steep, thin delta front. Downstream of the foreset, the flow can again be
adequately described in terms of quasi
-
steady, gradually varied flow.
Text S2. Model workflow
Our simplified
modeling scenario consists of an
initial
planar delta surface
with a topset
slope equal to the normal flow bankfull transport slope
(
W
C
)
,
a delta foreset slope equal to five
times the topset slope
(
W
G
=
5
W
C
)
(Borland, 1971; Hotchkiss & Parker, 1991)
, and a horizontal
basin floor.
We vary
water discharge at the upstream end, and co
-
vary
sediment
supply
such
that all discharges have the same
equilibrium transport slope,
simulating
an alluvial river
profile that is always at
transport capacity and isolating
backwater effects from long
-
term
adjustment in riverbed s
lope due to changes in sediment
-
supply and wa
ter
-
discharge ratios
(Dade & Friend,
1998; Paola, 2000; Parker et al., 2004; Church, 2006; Ganti et al., 2016b
). For
each timestep, the river mouth is id
entified according to S14
and any developing shock fronts
4
are detected using equation S5
. The backwater equation is s
olved using an upwind
predictor
-
corrector scheme applied to equation 2,
except across foreset wedges where equation
S7 is
locally applied. Next, sediment is routed
upstream of the foreset wedge with
equation 3
utilizing a “ghost node” at the upstream end (Parker 2004). After ro
uting water and sediment,
the delta foreset and
river
bed
profile
#
<
for the next timestep are
calculated using a finite
-
difference approximation of
equation 1 in a moving
-
boundary formulation
that is explicit,
centered in space, and forward in time
(Kost
ic & Parker, 2003; Parker et al., 2004)
.
We update
the floodplain elevation profile using equation S2, assuming the channel depth profile is equal
to the flow depth profile (equation 2) under bankfull discharge conditions. The updated
floodplain profile is
used to determine the superelevation using equation 5, and also
determines the river mouth for the next timestep via Equation S1. We repeat this numerical
scheme, stepping through time until the avulsion threshold is exceeded somewhere along the
long
-
prof
ile according to equation 4. At this point an avulsion occurs, and avulsion length is
measured by the stream
-
wise distance along the parent channel between the avulsion
location and the river mouth.
We find that s
table
numerical simulation requires an es
pecially
small timestep when
large floods erode the bed near the river mouth and drive progradation of the delta front.
For
computational expendience, we employ
a discharge
-
dependent CFL condition based on rates
of change of bed topography near the river m
outh, maintaining numerical stability during
high flows and vastly speeding up model simulation during low flow periods
where smaller
timesteps are not necessary
. This is in contrast to previous users of this technique, who have
employed a constant river d
ischarge (Kostic & Parker
, 2003
) or abandoned the moving
-
boundary framework during high flow events (Chatanantavet et al. 2012).
Because the
floodplain profile is set by the bankfull
-
water
-
surface profile (equation S15), changes in the
bankfull flow depth
over time violate equation 1. However, we note found that changes in the
bankfull flow depth over time were so small that the error in mass
-
balance incurred is
acceptable
it is less than the ~3% truncation error introduced by our numerical scheme,
which
is a common value for similar morphodynamic models (Parker, 2004)
and should not
significantly affected modeled avulsions.
In our simulations
we have imposed four delta lobes
, represented by four one
-
dimensional stream
-
wise long profiles in parallel
. A
t a given time, a single lobe is actively
routing flow, and the other three lobes are abandoned. When the active lobe experiences
avulsion, flow finds a new path downstream of the avulsion location along one of the
abandoned lobes, and the flow path upstre
am remains unchanged,
#
<
,
EIk
(
!
)
=
l
MIN
p
#
<
,
G<GEb&EIb
q
(
!
)
,
#
<
,
G<GEb&EIb
^
(
!
)
,
#
<
,
G<GEb&EIb
r
(
!
)
s
.
.
.
.
.
!
>
!
h
.
#
<
(
!
)
.
.
.
.
.
.
.
.
.
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.
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.
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.
.
.
.
.
.
.
.
.
.
.
.
!
!
h
.
e
q
.
S8
where
!
is distance downstream,
!
h
is the avulsion location,
#
<
,
EIk
is the new riverbed profile
after avulsion,
#
<
is the riverbed profile before avulsion, and
#
<
,
G<GEb&EIb
q
,
#
<
,
G<GEb&EIb
^
,
and
.
#
<
,
G<GEb&EI
b
r
are the three abandoned lobe long profiles. The
MIN
operator here selects the
abandoned profile that has the minimum mean elevation,
#
̅
<
, downstream of the avulsion
node,
#
̅
<
=
1
!
"
!
h
y
#
<
(
!
)
,
z
,
{
T!
.
.
.
.
.
.
.
.
.
.
equation
.
S
9
5
where
!
"
is the downstream coordinate of the river mouth. For example, if
#
<
,
G<GEb&EIb
^
(
!
)
yields a lower value of
#
̅
<
than both
#
<
,
G<GEb&EIb
q
(
!
)
and
#
<
,
G<GEb&EIb
r
(
!
)
yield, then
#
<
,
G<GEb&EIb
^
(
!
)
is selected as the path dow
nstream of the avulsion location. This simple
selection scheme mirrors the tendency of river deltas to fill in topographic lows when avulsing
(Straub et al., 2009).
Text S3.
Flow
variability parameters
In this study, we explore how deltaic avulsion patt
erns respond to upstream
flow
regimes and downstream changes in relative
-
sea
-
level through the systematic variation of
these variables on a base case of the model that is characteristic of large, low
-
sloping deltas.
Flow variability is parameterized in terms of a distribution of
flow
e
vents with varying
frequency, magnitude, and duration. Table
S
1 provides field examples of the relevant
parameters, and summarizes the range of parameter space explored in this study.
The water discharge of alluvial rivers can fluctuate across a range of
timescales, but
generally stage height will fill the channel banks on a recurrence interval of ~2 years (Wolman
& Miller, 1960). Many have argued that the morphodynamics of alluvial rivers can be well
-
approximated using an intermittency factor and a consta
nt discharge that is equal to this
bankfull condition, on the grounds that the bankfull flood represents a balance of frequency
and magnitude that has the maximum impact on alluvial form (Wolman & Mi
ller, 1960;
Andrews, 1980; Parker et al., 2007
). However,
this approximation should break down for rivers
in their backwater zone, where the downstream boundary enhances deposition during lower
-
than
-
bankfull flows and drives erosion during larger floods (Lamb et al., 2012). We explicitly
model variable flows
usi
ng a log
-
normal distribution of stage height upstream in the normal
-
flow reach. A log
-
normal distribution of stage height sufficiently describes flow in many river
systems measured on a monthly
-
mean basis (Stedinger et al., 1993
; Leboutillier & Waylen,
199
3; Lague et al., 2005
), and is uniquely defined by a bankfull
-
exceedence probability and a
coefficient of variation (Figure S1). The bankfull exceedence probability
f
<$
describes the
frequency of overbank flows relative to all
possible flows, and can ra
nge from zero to unity. On
many low
-
gradient alluvial rivers, monthly
-
averaged flows will exceed bankfull between 1
-
10% of the time, corresponding roughly to a 1
-
2 year recurrence flood (Langbein & Leopold,
1964). The coefficient of variation (CV) describe
s the magnitude of low flows and high flows
relative to the average flow, and is defined by the standard deviation of the stage height
divided by the mean. Among the lowland deltas considered in table S1, the coefficient of
variation ranges from 0.18
-
0.91.
In our numerical model, we discretize the distribution into twenty logarithmically spaced bins
that span from low flow (less than bankfull) to high flow (greater than bankfull) conditions.
Over time, each bin is randomly sampled at a defined event timesc
ale,
}
~
. In our scaled
framework the normalized event timescale (
}
~
=
}
~
/
}
ÄÅÇ
)
describes how long flow events
persist relative to
channel adjustment timescale,
the time required for those flows to transport
enough sediment to aggrade the backwater
reach by one channel depth. Based on previous
work (Chatanantavet et al. 2014), we expect flow regimes to maintain bed disequilibrium and
a persistent backwater zone when the normalized event timescale is much less than unity. This
condition is satisfied
for many deltaic rivers, where we calculate
}
~
=
ÉÑ
S
Ö
ÉÑ
S
É
.
If
}
~
>
É
,
6
individual flow events may persist long enough to mute backwater effects through
aggradation or degradation of the river profile to quasi
-
un
iform flow conditions.
Text S
4
.
Non
-
dimensionalization
Our simulations explore how deltaic avulsion patterns respond to river
flow regime
,
relative sea
-
level rise, and
initial topography
by systematically varying the discharge and sea
-
level parameters. We non
-
dimensionalize the model to
develop a framework that can be
applied to a wide range of river conditions,
reduce the number of model inputs, and identify
key controls on model behavior. Channel
-
bed elevation
(
Ü
á
)
is scaled in terms of bankfull
channel depth in the normal
-
flow reach
, flow width
(
à
)
is scaled in terms of the channel width
in the normal
-
flow reach, and stream
-
wise distance
(
â
)
is
scaled in terms of the backwater
length
-
scale,
#
<
=
#
<
>
C
.
.
.
.
.
.
.
.
.
.
equation
.
S10
A
=
A
A
C
.
.
.
.
.
.
.
.
.
.
equation
.
S11
.
!
=
!
ä
<
=
!
ã
>
C
W
C
å
.
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S12
where
#
is dimensionless channel
-
bed elevation,
A
is dimensionless flow width,
!
is
dimensionless distance downstream, and
>
C
,
A
C
,
and
W
C
are the channel depth, width,
and
slope in the normal
-
flow reach upstream
. We also scale time
(
ç
)
in terms of the time required
to fill a backwater reach with the sediment supply. Here, we modify the bed
-
adjustment
timescale to apply to a sinuous channel that exchanges sediment with its nearby floodplain,
making the simplifyin
g assumption that floodplain width
(
A
$
), channel sinuosity
(
Ω
)
, deposit
porosity
(
è
ê
)
, and the ratio of wash load to bed
-
material load
(
Λ
)
are constant and uniform,
.
.
ç
=
ç
í
C
=
ç
>
C
A
$
ä
<
Ω
ì
î
ï
ñ
1
è
ê
ó
(
1
+
Λ
)
.
.
.
.
.
.
.
.
.
.
.
equation
.
S13
where
ç
is dimens
ionless time,
í
C
is the reach
-
filling timescale, and
ì
î
ï
is the time
-
averaged
sediment supply per unit width. It should be noted that
previous authors have hypothesized
that backwater effects may drive downstream fining trends (Nittrouer et al., 2011; Ni
ttrouer et
al., 2012; Venditti & Church, 2014
; Maselli et al., 2018
)
and downstream reductions in
floodplain and channel
-
belt width (Fernandes et al., 2016) in some systems, which could alter
the bed
-
adjustment timescale. For example,
narrow
floodplain
s
will aggrade faster than wide
floodplains for the same amount of sediment
-
flux convergence (Equation 1
)
,
and therefore
backwater
reaches with narrow
floodplains
adjust more quickly after flood events
and
may
avulse more frequently compared to wider floodpl
ains upstream.
We non
-
dimensionalize equations 1
-
3 in the main text by inserting equations S10
-
S13
and simplifying,
ò
#
ò
ç
+
ô
=
.
1
ì
ö
î
ò
A
ì
î
ò
!
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S14
.
.
ò
>
ò
!
=
W
W
$
1
f
g
^
+
f
g
^
1
f
g
^
>
A
T
A
T
!
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S15
.
.
7
õ
$
ì
î
=
ú
(
ù
)
E
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S16
.
where
>
=
>
/
>
C
is the dimensionless flow depth,
W
=
W
/
(
>
C
/
ä
<
)
is the normalized bed
slope,
W
$
=
f
g
^
õ
$
/
W
C
is the normalized friction slope,
ì
î
is the Einstein number representing
dime
nsionless bed
-
material transport (Einstein, 1950; Parker, 1979) and
ì
ö
î
is the time
-
averaged Einstein number.
We note that
#
in equation
S14
is
defined as elevation relative to
sea level, not relative to the basin floor (as it is defined in some previo
us work, e.g. Baumanis &
Kim, 2018). We prefer this reference frame because it condenses relative sea
-
level rise into the
single parameter
ô
,
illustrating that sea
-
level rise and subsidence have the same effect on
sediment mass
-
balance.
While
fg
and
ù
are commonly defined in terms of gravity (
_
)
and
grain
-
size
(
û
)
, we can reduce the number of model inputs by casting them in terms of their
counterparts in the normal
-
flow reach upstream during bankfull conditions.
f
g
^
=
]
^
_>
=
f
g
<$
^
õ
$
,
<$
õ
$
ã
1
A
å
^
ã
1
>
å
r
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S17
ù
=
õ
$
]
^
ü_û
=
ù
<$
>
õ
$
õ
$
,
<$
\
fg
f
g
<$
`
^
.
.
.
.
.
.
.
.
.
.
.
equation
.
S18
where the subscript
†°
denotes bankfull conditions in the normal
-
flow reach
.
Another
important parameter is the dimensionless relative sea
-
level rise (or basin subsidence) rate,
ô
=
ô
ä
<
ì
ö
î
A
$
Ω
A
(
1
è
ê
)
(
1
+
Λ
)
.
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S19
where
ô
describes the balance of accommodation space created by relative sea
-
level rise
ove
r the active floodplain, as compared to the sediment supply to the backwater reach. When
ô
1
,
sediment supply far outpaces the rate of sea
-
level rise and we expect lobe growth and
avulsion similar to steady sea
-
level scenarios. As
ô
approaches unity, w
e expect that sea level
will cause intermittent or permanent drowning of delta lobes and potentially affect the
location of avulsions. This parameterization is similar to the “A/S” ratio concept” of Muto and
Steel (1997, 2002) and similar theories for radi
ally averaged deltas (Galloway, 198
9
; Paola et al.,
2011; Liang et al., 2016) but is here applied to avulsion cycles and discrete deltaic lobes.
Inserting
equation S10
into
equation 4, we find the equation for normalized avulsion
setup,
#
>
>
?
.
.
.
.
.
.
.
.
.
.
.
equation
.
S20
where
#
=
Δ
#
/
>
C
is normalized superelevation and
>
?
=
>
?
/
>
C
is the dimensionless
channel depth. T
he avulsion threshold
>
is equal to fifty percent of the channel depth
.
(
>
=
0
.
5
)
for all
our
simulations
presented here
,
representing
a value that is consistent with field
and laboratory observations
(Mohrig et al., 2000; Ganti et al., 2016b). However,
field evidence
suggests that the avulsion threshold is systematically reduced under flashier discharge
regimes in the range
of H* = 0.2
-
1 (Ganti
et al.,
2014).
At each model timestep, normalized
superelevation
#
is calculated by inserting equation S10 into equation 5,
#
(
!
)
=
l
#
$
(
!
)
#
$
,
G<GEb&EIb
(
!
)
.
.
.
.
for
.
!
!
"
,
G<GEb&EIb
#
$
(
!
)
%
VIG
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
for
.
!
>
!
"
,
G<GEb&EIb
.
.
.
.
equation
.
S21
8
where
#
$
=
#
$
/
>
C
is dimensionless floodplain elevation
,
!
"
=
!
"
/
ä
<
is dimensionless river
mouth location,
%
VIG
=
%
VIG
/
>
C
is dimensionless
sea
-
level elevation, and
subscript
“abandoned” indicates quantities on the abandoned delta lobe with lowest elevation.
The
dimensionless avulsion length (
ä
h
)
is the ratio of the avulsion length (
ä
h
)
to the backwater
length
-
scale (
ä
<
),
ä
h
=
ä
h
ä
<
=
ä
h
>
C
/
W
C
.
.
.
.
.
.
.
.
.
.
.
.
equation
.
S22
For lowland deltas with a preferential avulsion length set by the backwater length, we expect
that
ä
h
~
1
. We ran our simulations for a total of 13 avulsion cycles, which we found sufficient
to capture trends in avulsion location between our
different model runs. Running the model
for many more avulsion cycles yields similar results but is computationally expensive given the
broad parameter space considered in our study.
Text S5. Results for trunk
-
filling avulsion cycles
Most simulated avulsion cycles feature focused deposition within one backwater
length
-
scale of the river mouth (Figure 2b
-
c, e
-
f). However, we also observed occasional
avulsion cycles with significant deposition farther upstream in the trunk channel. These
trunk
-
filling avulsion cycles occur when the initial
floodplain
profile
#
$
is significantly lower than all
other inactive lobes, for example during avulsion cycle 4 (Figure S2). Consequently, the active
lobe begins construction with substantially lower
superelevation compared to other avulsion
cycles, requiring greater aggradation along the entire river long
-
profile before reaching the
avulsion threshold.
Trunk
-
filling avulsion cycles occurred periodically in all our simulations, usually during
cycle n
umber 4, 7, 10, and 13. In simulations with a preferential avulsion node, we observed
that trunk
-
filling avulsion cycles were also associated with downstream translation of the
avulsion node (Figure S2b). The avulsion node translated downstream with major
shoreline
progradation, as a result of greater aggradation of the river long
-
profile with a constant
transport slope. Similar behavior involving a periodic shift in the avulsion node has been
documented for the Yellow River in China (Ganti et al., 2014).
During avulsion cycles 8 and 11 in the variable
-
discharge case, avulsions occurred far
upstream of the backwater zone, but without a prominent peak in superelevation. These
avulsions were similar to those in the constant
-
discharge case and were due to a t
ransient
period of nearly uniform deposition rate as the trunk channel adjusted to a new profile
immediately following trunk
-
filling avulsion cycles. Occasional avulsions far upstream of a
backwater
-
mediated node have been interpreted from Mississippi Rive
r deposits
(Chamberlain et al., 2018).
Text S6. Model sensitivity to other parameters
In the main text, we present three conditions that can produce a backwater
-
scaled
avulsion node in our model: 1) a uniformly downstream
-
sloping initial condition, 2)
flow
9
variability, and 3) rapid sea
-
level rise. During the course of our analyses, we also explored how,
and to what extent, changing other model parameters impacts our main results. Here we
discuss the avulsion threshold parameter
>
and the number of imp
osed delta lobes
ß
which,
after the important roles of
flow
variability, the initial condition, and sea
-
level rise, we found to
have a notable effect on model behavior.
In previous simulations the avulsion threshold was
set
to
>
=
0
.
5
.
To relax this
assumption, we varied the avulsion threshold across a range comparable to modern deltas
(
0
.
2
>
1
, Ganti et al., 2014) under constant
-
discharge conditions and under variable
-
discharge conditions, with all other parameters set to the base ca
se (Figure S6). Similar to the
scenario of
>
=
0
.
5
, a preferential avulsion node emerges only in simulations with
flow
variability
. When avulsions occurred in constant
-
discharge simulations, a reach of ~2
backwater lengths was within 10% of the avulsion th
reshold, indicating no dominant avulsion
location regardless of the value of
>
. In both constant
-
and variable
-
discharge scenarios,
increasing the avulsion threshold leads to an increase in the observed avulsion lengths. This is
because, at higher values
of
>
, lobes prograde farther seaward of the inactive
-
lobe shoreline,
where avulsions are unlikely to occur. Thus, the avulsion threshold influences the location of
the avulsion node in our model, but it does not control the occurrence of a preferential
avulsion node itself, which still depends on
flow
variability, initial conditions, or sea
-
level rise.
The avulsion threshold was set to a constant value during all simulations for simplicity.
However,
avulsion
threshold may not be a constant in reality
a
nd may in fact depend on
flow
variability at different sites
,
as argued by Ganti et al. (2014). Nonetheless, the sensitivity
analysis
shows
that changing the threshold does not change our main conclusion that
flow
variability is necessary for emergence of
a persistent node
.
Varying
>
under variable
-
discharge conditions
only shifted avulsion lengths between 0.5
ä
<
to 2
ä
<
(Figure S5b), which
was
minor compared to spread in avulsion lengths resulting from a constant discharge (Figure
S5a)
, and also within
the scatte
r
of
backwater
-
mediated
avulsion
lengths
observed
in
the field
(
Figure 1a).
In previous simulations, we also imposed a fixed number of 4 delta lobes. This was an
arbitrary but
reasonable choice based on field observations (Pang & Si, 1979; Robe
rts, 1997;
Coleman et al., 1998;
Chu et al., 2006) and flume experiments (Reitz et al., 2010; Carlson et al.,
2018). We found that changing the number of delta lobes alters the timing of
an avulsion
node’s behavior
, but does
not affect the occurrence of the avulsion node.
For example, the
shift from preferential to non
-
preferential avulsions in the constant
-
discharge scenario occurs
after 4 avulsion cycles in our model (Figure 2A) because we impose four delta lobes, and so it
takes 4 avulsions for the delta to bury its initial conditions. On
a
delta with
ß
discrete lobes,
this same behavior occurs after burying the initial conditions, but it requires
ß
avulsion cycles.
Other examples include outlier avulsions far upstream of
the backwater zone, and
downstream translation of the avulsion node, both of which occur every
ß
1
avulsion cycles
in our simulations. Both outlier avulsions and translation of the avulsion node are a
consequence of trunk
-
filling avulsion cycles in our mo
del (Text S5), which occur when all
lateral space has been filled and the new lobe is forced to prograde farther seaward than
previous lobes, corresponding to every
ß
1
avulsion cycles after the delta has buried its
initial conditions. Thus, simulated del
tas with 4 lobes filled their trunk channels every 3
avulsion cycles
(
usually cycles 4, 7, 10, and 13
)
and experienced a shift in the avulsion node
and then an outlier
avulsion
. A delta with fewer lobes has less lateral space to fill before
advancing seawa
rd, and therefore experiences more frequent filling of its trunk channel, a
more mobile avulsion node, and more common outlying avulsion sites.
10
Figure S1
.
a)
Exceedence probability of normal
-
flow depth normalized by bankfull depth for
the Mississippi an
d Huanghe
(Ganti et al., 2014)
, illustrating how bankfull exceedence
probability (
f
<$
) and the coefficient of variation (
õ®
) were estimated in Table 1. Steeper trends
of exceedence probability correspond to lower values of
õ®
. b) Schematic time
-
series of
modelled normal
-
flow depth in after non
-
dimensionalization, showing how input flow depth
is determined by randomly sampling a log
-
normal distribution for fixed flow events of
duration
í
I
.