1
Supplementary Information for
More extensive land loss expected on coastal deltas
due to rivers jumping course during sea
-
level rise
A
ustin
J. Chadwick
1
, S
arah
Steele
1
, J
ose
Silvestre
1
, M
ichael
P. Lamb
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA
91125
Corresponding author:
Austin Chadwick
Email:
Austin.chadwick23@gmail.com
This PDF file
includes:
Supplementary Text
Tables S1
to
S
4
2
Supplementary
Text
Details of experimental
setup
.
The laboratory experiment was conducted in the Caltech River
Ocean Facility
,
the same facility
used by Ganti et al. in
previously published
delta experiments
(1,
2)
.
The experimental flume consisted of a 7
-
m
-
long,
14
-
cm
-
wide fixed
-
width river section that
flowed into a 5
-
m
-
long, 3
-
m
-
wide unconfined ocean basin
(Fig. 3a).
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 free of sediment, and over several hours the flow
naturally deposited sediment to form a river d
elta.
The
experiment was designed to feature
backwater
-
scaled avulsions and sea
-
level changes dynamically similar to those on lowland deltas
in nature
(3)
.
To reproduce backwater
-
scaled avulsions, we implemented a variable flood regime
following the example of Ganti et al.
(1, 2)
;
water and sediment input o
scillated between a low
-
flow and high
-
flow discharge, each of which produced significantly different normal flow depths
(Table
S
1). Low
-
and high
-
flow durations were selected to be significantly shorter than the time
required to adjust the channel to norma
l
flow conditions to ensure backwater effects were
persistent
(4)
. Sediment supply was co
-
varied
with water discharge to produce the same riverbed
slope
for both flow events (Table S1). The
riverbed slope was shallow enough to
allow subcritical
flow (
퐹푟
<
1
) and single
-
thread channels.
To achieve
sediment transport at
such gentle slopes,
we used low
-
density sediment: crushed, non
-
cohesive walnut shells (
1300
kg
/
m
!
) with near
-
uniform particle diameter (
0
.
7
mm
).
Flow depth and slope resulted in a backwater length
-
scale of
퐿
"
=
퐻
#
/
푆
=
1
.
8
m
(Table S1).
Over the course of the experiment, we systematically
raised sea
level at four different rise
rates
(
휎
=
0
,
0
.
25
,
1
,
and
4
mm/hr
)
in four phases (
Phase
s
A, B, C,
and
D).
Rise rates were selected to
span
the range of dimensionless rise rates
0
<
휎
∗
<
1
(Table S2
)
which is
common
on
natural deltas
(5, 6)
(Table
S
3
).
Although w
e
did not incorporate subsidence
into the experiment, delta response to uniform subsidence is expected to be mechanically similar
to that of sea
-
level rise
(7
–
9)
. Each phase was allow
ed to continue long enough to allow for many
avulsions (Table S2). Phases were also kept brief enough such that the offshore basin did not
deepen by more than a factor of two; this allowed us to mitigate the
effect
of changing basin
depth
and shoreline aut
oretreat
on
land loss
(6, 10)
, and therefore better isolate the effect of sea
-
level rise rate.
Details of experimental data collection.
Plan
-
view images of the delta were collected every
minute using six overhead cameras mounted above the experimental facility. Photos from each
camera were concatenated to ensure a wide field of view that extended beneath railings in the
facility. The water
was
colored
using a fluorescent green dye that allowed for visual distinction
between subaerial and submerged
land
—
even for shallow (
~
1
cm
) water depths
—
under
ultraviolet light fixtures.
Before starting a flow event we inserted
~
0
.
5
gallons of dye into the end
tank. The flow was then run at a very low discharge (
푄
~
0
.
002
L
/
min
)
with no sediment feed for
~
12
hours of standby, which allowed the dye to disperse evenly throughout the basin without
mobilizing sediment or disturbing the de
lta.
The
imagery was used to map river avulsions and
the
extent of dry and drowned land over the course of the experiment.
Subaerial land
was mapped
within the shoreline, following the boundary of fluorescent green water and the brown sediment
surface. Dro
wned land was mapped between the shoreline and the topset
-
foreset break; during
sea
-
level rise, the shoreline retreat
ed
landward
from the topset
-
foreset break (Fig. 3a
-
b).
W
e
identified avulsions as the establishment of a new channel that captured the majo
rity of flow
through consecutive flow events, accompanied by the partial
or
complete abandonment of the old
channel
, following previous work
(2, 11)
.
Avulsion location and time
were measured as the
location and time when the levee breach in the old channel initiated. Manual identification of
avulsions involved a degree of subjectivity; still, our measurements for avulsion location have an
uncertainty of less than one channel widt
h and much less than the backwater length
-
scale, and
measurements for avulsion time have an uncertainty of roughly one minute
(1)
.
We computed
avulsion length (
퐿
%
) as the distance along the parent channel from the
river mouth to the avulsion
location, and computed avulsion frequency
푓
%
using the definition
푓
%
≡
1
/
푇
%
, where
푇
%
is the time
between the current avulsion event and the previous avulsion event.
3
Details of
lobe
-
averaged
model
implementation
.
For both the experimental and field data, the
lobe
-
averaged model (Eqs. 4
-
6) was
implemented in a
three
-
step approach. First , Eq. (
5
)
wa
s
solved for avulsion frequency (
푓
%
) using an iterative scheme
(5)
and input estimates of
푄
&
,
푐
'
,
퐿
%
,
퐵
,
퐻
,
푆
,
푁
, and
휎
(Tables
S1
-
S2)
.
Second
,
푓
%
wa
s
plugged into Eq
.
(
4
) to estimate land loss
(
퐴
()&*
).
Third
,
퐴
()&*
wa
s
plugged into Eq. (
3
) to
estimate the area of persistently dry land (
퐴
+,-
).
Evaluation of Eqs. (3
-
4) also requires input estimates of total delta
-
plain area (
퐴
)
and perimeter
(
푃
)
—
including both dry and drowned land
;
these values were
computed
using the landscape
-
average
d model (Eq. 1)
for the experiment, and compiled from previous work
(12)
for the field
data
.
The fraction
s
of sediment
deposited on persistently
dry land
(
퐹
+,-
)
,
deposited on
intermittently drowned land
(
퐹
()&*
)
, and farther offshore
(
퐹
)..&
/
),0
)
(Fig.
3e
)
were
estimated by
taking the ratio of terms in Eq. (
5
),
퐹
+,-
=
I
%
!"#
%
J
.
$
1
2
$
3
%
&
4
5
67
8
'
/
:
(
,
(
S1
)
퐹
()&*
=
I
%
)*'+
%
J
.
$
1
2
$
3
%
&
4
5
67
8
'
/
:
(
,
(
S2
)
퐹
)..&
/
),0
=
.
$
46
1
7
,
;
%
&
<
5
8
'
/
:
(
.
(
S3
)
Uncertainty in model predictions arose from
uncertainty in lobe number (
푁
)
;
stochastic
variability in the avulsion threshold (
퐻
) and avulsion length (
퐿
%
)
(6, 13, 14)
;
and gradual changes
in the basin depth (
퐻
"
) as sea
-
level rose and fell
(5, 15)
.
U
ncertainty in Eq. (
5
) was estimated
using the variance formula,
푠
.
$
=
N
O
휕
푓
%
휕푛
푠
#
R
=
+
O
휕
푓
%
휕퐻
푠
7
R
=
+
O
휕
푓
%
휕
퐿
%
푠
2
$
R
=
+
O
휕
푓
%
휕
퐻
"
푠
7
,
R
=
(
S4
)
where
푠
.%
is uncertainty in modeled avulsion frequency, and
푠
7
,
푠
2%
,
and
푠
7
,
represent the
standard deviation in the avulsion threshold (
±
1
.
1
mm
), avulsion length (
±
0
.
28
m
), and basin
depth (
±
2
cm
) observed across the
entire experiment (Table S
2
).
The term
푠
#
represents
uncertainty in
푛
=
>
;
?
=
based on a characteristic number of lobes between four and six (
푁
=
5
±
1
),
consistent with field observations
(16
–
19)
and flume experiments
(15, 20)
.
Next,
the variance
formula was again used to propagate uncertainty in Eqs. (3
-
4) presented in Fig. 3d
-
e,
푠
%
)*'+
=
N
O
휕
퐴
()&*
휕푛
푠
#
R
=
+
O
휕
퐴
()&*
휕
푓
%
푠
.
$
R
=
(
S5
)
푠
%
!"#
=
푠
%
)*'+
(
S6
)
w
here
푠
%
)*'+
and
푠
%
!"#
represent uncertainty
in modeled
intermittent land area and persistently dry
land area, respectively.
Details of landscape
-
averaged model implementation
.
For field data, l
andscape
-
averaged
model predictions
were
compiled
from previous work
(12)
(Table S3).
For the experiment,
landscape
-
averaged model predictions were calculated by solving Eq. (1) for the total
delta
-
plain
area (
퐴
)
numerically using finite differences
.
Landscape
-
averaged land loss (
퐴
()&*
)
shown in Fig.
3e
was estimated using a scaling analysis of Eq. (1)
:
t
o first order, the area of lost land (
퐴
()&*
) is
equal to the rate of land loss t
imes the duration of relative sea
-
level rise (
푇
,@&0
), i.e.
퐴
()&*
~
−
+%
+*
푇
,@&0
. Combining this with the landscape
-
averaged model (Eqs. 1
-
2) and rearranging gives
4
퐴
()&*
=
I
8
'
,
.//!
3
8
'
:
(
7
,
J
푇
,@&0
,
(
S6
)
w
here
푇
,@&0
is
estimated by the experimental phase duration (Table S2).
T
hus, landscape
-
averaged models predict that the area of land loss is directly proportional to the deficit between
the sediment available (
푄
&
) and the sediment needed to vertically accr
ete the entire delta plain at
pace with sea
level (
푄
&
,
#00+
; Eq. 2).
Details of
revising
sediment
estimates
for field data
.
The sediment supply a delta
needs
to
sustain its current area is given by Eq. (2).
Past
landscape
-
averaged estimates have evaluated
Eq. (2) using modern measurements of dry
-
land area for
퐴
;
this implies that all available sediment
can be deposited on persistently dry land (i.e.,
퐴
=
퐴
+,
-
)
.
We revised these estimates by
accounting for sediment deposition
i
n
both the area of persistent
ly
dry land (
퐴
+,-
) and the area of
intermittently drowned land (
퐴
()&*
)
by
plugging in
퐴
=
퐴
+,-
+
퐴
()&*
in Eq. (2).
T
he additional term
퐴
()&*
was computed using the lobe
-
averaged model (Eq.
4
).
Importantly,
퐴
()&*
depends on the
frequency of river avulsion and diversion
(Eq. 5)
;
to assess delta sustainability under different
diversions scenarios,
w
e estimated the
needed
diversion frequency
(
푓
%
,
#00+
)
to sustain current
dry
-
land area with
out changing
the available sediment
supply
(i.e., the
border between shaded
and unshaded r
egions in
Fig. 4b)
. This was done
by
combining Eq. (
4
-
5
) and Eq. (
2
)
under the
condition that
there is
exactly
enough sediment to maintain the delta area (
푄
&
=
푄
&
,
#00+
), and
rearranging for
avulsion frequency
,
which gives
푓
%
,
#00+
=
푛
I
B
C
푃
J
I
8
'
B
−
퐴
+,-
J
3
?
.
(
S7
)
To plot the boundary shown in Fig. 4b we
calculated Eq. (
S7
)
for the scenario of 1 m sea
-
level
rise in 100 years (
휎
=
1
cm
/
yr
)
us
ing
characteristic values for field data
(
퐴
+,-
=
1
e
4
km
=
,
푆
=
7
.
4
e
-
5
,
푛
=
2
.
5
;
Table S3) and estimated delta perimeter using
푃
=
Z
퐴
+,-
.
5
Table S
1
.
Variable flood regime of the experiment
Low flow
High flow
Water discharge
[
liters
/
min
]
14.4
20.4
Sediment supply
[
g
/
min
]
30.4
69.4
Normal
-
flow depth,
퐻
!
[
mm
]
7.5
11.7
Flow duration
[
min
]
22
8
Normal
-
flow transport slope,
푆
[
−
]
0.0042
0.0042
Backwater length
-
scale,
퐿
"
=
퐻
!
/
푆
[m]
1.8
−
Froude number,
퐹푟
[
-
]
0.5
9
0.4
3
6
Table S
2
.
Phases of the experiment
Phase A
Phase B
Phase C
Phase D
EXPERIMENTAL DESIGN
Dimensionless sea
-
level rise rate,
휎
∗
[
−
]
0
0.08
0.33
1.33
Sea
-
level rise rate,
휎
[
mm
/
hr
]
0
0.25
1
4
Flood
-
averaged s
ediment supply
[
g
/
min
]
40.8
40.8
40.8
40.8
Run time
[
hr
]
0
-
43.5
43.5
-
82
82
-
101
101
-
105
Duration,
[
hr
]
43.5
38.5
19
4
Number of low flows
[
−
]
87
77
38
8
Number of high flows
[
−
]
87
77
38
8
MEASURED DURING
EXPERIMENT
Number of avulsions
[
−
]
10
22
16
2
Average avulsion frequency,
푓
$
[
hr
%
&
]
0.5
0.9
1.0
2
Average avulsion length,
퐿
$
[
m
]
1.3
±0.28
Approximate lobe thickness,
퐻
[
mm
]
2.3
±1.1
Approximate number of lobes,
푁
[
−
]
5
±
1
Approximate lobe width,
퐵
[
m
]
0.2
Solids fraction of sediment deposit,
푐
'
[
−
]
0.7
Lobe width (
퐵
)
was estimated based on
width of the active channel, as flood deposition was
negligible during the experiment. L
obe thickness (
퐻
)
was
measured in
a
companion experiment
(2)
.
For
simplicity we adopted a characteristic number of lobes in the range
푁
=
5
±
1
, consistent
with field observations
(16
–
19)
and flume
experiments
(15, 20)
. This range is consistent with our
experiment, based on estimating the number of lobes by the ratio of the delta width (~1 m)
to
the
lobe width (~0.2
m)
.
The solids fraction
of the sediment deposit was calculated as
푐
'
=
(
휌
"D(E
−
휌
FG*0,
)
/
(
휌
HG,*
−
휌
FG*0,
)
using direct measurement
s
of
the deposit’s
bulk density (
휌
"D(E
=
1210
kg
/
m
!
),
the water
density (
휌
FG*0,
=
1000
kg
/
m
!
), and
the sediment
particle density (
휌
HG,*
=
1300
kg
/
m
!
)
.
7
Table S3.
Field data used in this study.
흈
[mm/yr]
푳
푨
[km]
푳
풃
[km]
풇
푨
[1/kyr]
푸
풔
[Mt/yr]
푯
풄
[m]
푯
[m]
푩
풄
[km]
푩
[km]
푯
풃
[m]
푵
[
-
]
푻
풄
[kyr]
흈
∗
[
-
]
푨
풅풓풚
[km
2
]
Parana
3
210
295
0.6
79
11.8
8.2
1.3
50.8
40
5
±
1
3.6
2.3
1.4e5
Danube
0.2
95
125
0.5
67
6.3
5.0
1.3
50
50
5
±
1
0.9
0.1
2.5e4
Nile
4.5
210
254
—
120
16.2
—
0.2
9.6
120
5
±
1
0.5
0.4
1.0e5
Mississippi
2.3
490
480
0.8
400
21
12.5
0.7
26
80
5
±
1
1
0.3
3.6e5
Rhine
-
Meuse
1.6
51
45.5
0.7
3.1
5
2.3
0.7
28
18
5
±
1
3.3
2.6
3.3e3
Magdalena
2.9
67
63.2
—
220
6
—
1.1
44
200
5
±
1
0.1
0.1
6.3e3
Orinoco
2.6
78
133.3
1
150
8
2.1
2
80
110
5
±
1
0.9
0.7
2.8e4
Amazon
2.9
404
400
—
1200
12
—
3
120
50
5
±
1
0.8
0.5
2.5e5
Rhone
2.8
—
183.5
0.7
31
7.3
2.9
0.4
15.1
70
5
±
1
1
1
5.3e4
Yellow
1.7
31
35
142.9
1100
3.5
0.7
0.5
20
30
5
±
1
3.5e
-
3
4e
-
3
1.9e3
Brahmaputra
11.4
—
70
2
540
7
10
3.3
132
80
5
±
1
0.2
0.8
7.7e3
Relative sea
-
level rise rates (
휎
) are reported by Chadwick et al.
(5)
and reflect the sum of eustatic
sea
-
level change
(21)
and coastal subsidence
(22
–
26)
estimated over the time that avulsion
s
occurred. Avulsions occurred during the late Holocene period (last
7
ky
)
, with the exception of the
Yellow
where pre
-
industrial historical avulsions are documented
(13)
. Avulsion lengths (
퐿
%
)
and
backwater length
-
scales (
퐿
"
) are reported in
(13, 27)
. Avulsion frequency (
푓
%
), channel depth (
퐻
:
),
and channel width (
퐵
:
) are reported in
(28)
. Basin depths (
퐻
"
) are reported in
(29)
. Sediment
supplies (
푄
&
) are reported in
(30)
, and are converted here to volumetric rates using a sediment
density of
2650
kg
/
m
!
and 40% porosity
(
푐
'
=
0
.
6
). Channel filling timescales are
estimated as
푇
:
=
퐻
:
퐵
:
퐿
"
푐
'
/
푄
&
following
(4, 31)
.
Data for the Danube are
reported in
(32)
.
Deltas were
assumed to be composed of four
to six
lobes (
푁
=
5
±
1
) with width of forty times the channel
width (
퐵
=
40
퐵
:
), which are reasonable estimates
(18, 19, 33, 34)
.
Depositional thickness of the
delta lobe at avulsion
(
퐻
)
is
reported by Chadwick et al.
(5)
and
dimensionless sea
-
level rise (
휎
∗
)
is calculated using Eq. (
5
)
.
Area of persistent dry land (
퐴
+,-
)
is reported
by Giosan et al.
(12)
, or
where unavailable was estimated by
?
=
휋
퐿
"
=
following Ganti et al.
(1)
. Area of intermittently drowned
land (
퐴
()&*
) was calculated using Eq. (4).
Empty table entries indicate data were not available.
8
Table S
4
.
Land loss
forecasts
for field data that account for avulsions and delta
lobes
Intermittent
land area,
푨
풍풐풔풕
[km
2
]
Fraction of land
lost,
푨
풍풐풔풕
/
푨
[%]
Sediment needed
푸
풔
,
풏풆풆풅
[
Mt
]
Predicted avulsion
frequency
[1/kyr]
Parana
4.5
e
5
±
7.5e4
76
±
35
9.4
e
5
±
1.2e5
0.6
Danube
4.9e4 ± 8.2e3
66
±
25
8.5
e
4
±
1.3e4
1.8
Mississippi
1.0
e
5
±
1.7e4
22
±
5
2.3
e
5
±
2.7e4
1.6
Rhine
-
Meuse
2.3
e
4
±
3.8e3
87
±
53
4.1
e
4
±
6.0e3
0.6
Orinoco
8.4
e
4
±
1.4e4
75
±
33
1.8
e
5
±
2.2e4
7.2
Rhone
2.5
e
5
±
4.2e4
83
±
44
4.8
e
5
±
6.6e4
—
Yellow
1.6
e
2
±
2.7e1
8
±
1
5.4
e
3
±
4.3e1
189.3 ± 6.4
Brahmaputra
1.3
e
4
±
2.2e3
63
±
22
3.3
e
4
±
3.5e3
—
Intermittent land area
(
퐴
()&*
) was calculated using Eq. (4)
.
Fraction of land lost (
%
)*'+
%
=
%
)*'+
%
!"#
;
%
)*'+
)
was calculated
using the predicted intermittent land area and the dry land area from Table S3.
Needed sediment
(
푄
&
,
#00+
)
was calculated using Eq. (2)
accounting for deposition in both
persistent and intermittent land areas (
퐴
=
퐴
+,-
+
퐴
()&*
; Eq. 3
)
.
Predicted
avulsion frequency was
calculated using Eq. (5).
Calculations were performed for a scenario of 1
-
m of sea
-
level rise over
100 years following Giosan et al. (2014)
, and
w
here applicable show
±
1
standard deviation in
uncertainty propagated from an estimated lobe number of
푁
=
5
±
1
using Eq. S5
.
Empty table
entries indicate
field
data
necessary to perform calculations were not available
(Table S3)
.
9
SI
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