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Supplementary Information for
7
8
Accelerated river avulsion frequency on lowland deltas due to sea
-
level rise
9
10
Austin J. Chadwick
a*
, Michael P. Lamb
a
, Vamsi Ganti
b,c
11
12
a
Division
of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125
13
b
Department of Geography, University of California, Santa Barbara, CA 93106
14
c
Department
of Earth Science, University of California, Santa Barbara, CA 93106
15
16
*
Correspondence may be addressed to Austin J. Chadwick
17
Email:
achadwick@caltech.edu
18
ORCID:
https://orcid.org/0000
-
0002
-
2552
-
0083
19
20
21
This PDF file includes:
22
23
Supplementary text
24
Figures S1 to S2
25
Tables S1 to S3
26
SI References
27
28
29
30
31
32
33
34
35
36
37
38
www.pnas.org/cgi/doi/10.1073/pnas.1912351117
2
S
upplementary text
39
A.
Derivation of numerical model for
backwater
-
scaled avulsions
40
The numerical model
was
built
on
previous work
(1)
that
explored controls on avulsion location on
41
lowland deltas, but did not analyze avulsion frequency and its dependency on relative sea
-
level rise.
The
42
model
consist
ed
of a delta with an imposed number of lobes assumed to form a branching pattern, with
43
only one lobe active at a given time
(Fig. 1C)
(1, 2)
.
E
ach
delta lobe
is modeled
as a
coupled
river and
44
floodplain
in a quasi
-
two
-
dimensional mass balance framework
(3, 4)
,
45
휕휂
휕푡
+
=
1
(
1
!
)
1
"
휕푥
(
S
1
)
46
where
is channel bed elevation relative to sea level,
is time,
is relative
sea
-
level
rise rate,
is
47
downstream distance, and
"
is the volumetric sediment transport capacity at position
. Sediment
is
48
deposited uniformly over
lobe
width
with poros
ity
!
.
At the delta front, fluvial sediment transport
49
transitions
to gravity flow
s
and avalanching,
and deposition drives
foreset
progradation. We approximate
d
50
progradation using a moving
-
boundary formulation
,
with
a foreset of constant
slope
#
set to five times
51
the transport slope
(5, 6)
.
52
We
used the backwater equation to constrain water
mass and momentum under quasi
-
steady
53
flow conditions
(3)
,
54
$
푑푥
=
%
1
&
+
&
1
&
$
$
푑푥
(
S
2
)
55
w
here
$
is
flow depth,
is channel
-
bed slope,
%
=
%
&
is friction slope,
%
is friction coefficient,
퐹푟
is
56
Froude number, and
$
is the width of flow.
We assumed flow width was contained by the channel
57
upstream of the river mouth, and expanded at a constant spreading angle offshore
(3, 7)
, here set to 15
58
degrees
.
Following recen
t work
(1)
, the
location
of the
river mouth
'
was
set by
the intersection of the
59
floodplain profile
%
with sea
level
(
,
60
'
=
|
)
!
(
+
)
.
/
"
(
S
3
)
61
where the floodplain elevation is defined as the sum of the bed elevation and channel
depth
0
,
62
63
3
%
(
)
=
(
)
+
0
(
S
4
)
64
Over time, the floodplain in our model
aggrade
d
in concert with the channel bed, driving river
-
mouth
65
advancement. A mobile river mouth
wa
s
necessary for foreset
progradation to drive topset aggradation
66
(1)
.
67
We routed sediment
in the
river
according to Engelund
-
Hansen
(8)
for total bed
-
material load,
68
"
=
0
<
푅푔
1
%
(
)
3
(
S
5
)
69
where
is submerged specific density of sediment,
is gravity,
is the median grain
-
size of bed
70
material,
is Shields number, and
=
0
.
05
and
=
2
.
5
.
All sediment delivered to the delta front
wa
s
71
captured in the foreset
(6, 9)
.
72
Following
recent work
(1)
,
we approximate
d
deltaic evolution using four separate
quasi
-
two
-
73
dimensional profiles of predefined width
,
represent
ing
four distinct lobes
.
At a given time, one delta lobe
74
was active
(10, 11)
and was governed by
Eqs.
(
S1
S5
)
.
We varied sediment supply at the upstream end
75
with water discharge such that the normal
-
flow bed slope was held constant, and therefore erosion and
76
deposition were not driven by changes in
the ratio of
sediment supply
to
w
ater
discharge
(12)
.
Inactive
77
lobe shapes were unchanged when abandoned
(13)
but were partially drowned in cases due to relative
78
sea
-
level rise.
79
We used an avulsion criterion given by a critical thickness of aggradation, which we refer to as
80
superelevation (
):
81
(
)
(
S
6
)
87
in which
=
0
is the aggradation thickness necessary for avulsion,
0
is the bankfull channel depth
,
82
and
is the avulsion threshold, a dimensionless number that is of order unity
(14
16)
.
We triggered an
83
avulsion when and where the floodplain elevation of the active lobe exceeded the floodplain elevation of
84
the lowest
-
elevation abandoned lobe (
%
,
#5#36(376
)
, evaluated at the same distance downstream from the
85
trunk channe
l:
86
(
)
=
I
%
(
)
%
,
#5#36(376
(
)
for
'
,
#5#36(376
%
(
)
87#
for
>
'
,
#5#36(376
(
S
7
)
88
4
where
'
,
#5#36(376
is the stream
-
wise coordinate of the abandoned
-
lobe shoreline. Seaward of the
89
abandoned lobe, superelevation is measured relative to sea level (
87#
)
(1, 17)
.
Extreme
floods may also
90
affect
the timing of any one
avulsion
(15, 18)
,
but
these factors were neglected
following previous work
(1,
91
19, 20)
.
For simplicity we
ignored
the river reach laterally spanning lobes
,
because lobes are much longer
92
than they are wide
(21, 22)
.
93
After avulsion, the river was rerouted to the lowest abandoned lobe by joining the bed profile of
94
the active channel upstream of the avulsion site with the bed profile of the new flow p
ath downstream,
95
37$
(
)
=
I
MIN
S
#5#36(376
9
(
)
,
#5#36(376
&
(
)
,
#5#36(376
1
(
)
U
>
:
(
)
:
(
S
8
)
96
where
is distance downstream,
:
is the avulsion location,
37$
is the new riverbed profile after
97
avulsion,
is the riverbed profile before avulsion, and
#5#36(376
9
,
#5#36(376
&
,
and
#5#3
6(376
1
are the
98
three abandoned
-
lobe long profiles. The
MIN
operator here selects the abandoned profile that has the
99
minimum mean elevation,
̅
, downstream of the avulsion node,
100
̅
=
1
'
:
X
(
)
+
#
+
$
푑푥
(
S
9
)
101
w
here
'
is the downstream coordinate of the river mouth. For example, if
#5#36(376
&
(
)
yield
ed
a lower
102
value of
̅
than both
#5#36(376
9
(
)
and
#5#36(376
1
(
)
yield, then
#5#36(376
&
(
)
wa
s
selected as the path
103
downstream of the avulsion location.
This
procedure
mimics
the tendency of rivers to select steeper
104
paths, fill in topographic lows
(10, 23)
, and to reoccupy previously abandoned channels
(24)
.
After
105
establishing the new flow path, lobe co
nstruction (
Eqs.
S1
S5
)
and avulsion setup (
Eqs.
S6
and
S7
)
106
began anew.
107
To enable applicability across a wide range of river conditions, the model was
normalized
using
108
the channel dimensions and
the
characteristic
aggradation rate
of the backwater zone
,
^
#
=
9
(
9
;
<
%
)
=
&
>
'
?
(
.
109
Normalizing
Eqs.
(
S1
S9
)
yields
110
+
=
1
`
"
"
(
S
10
)
111
'
=
|
)
!
(
+
)
.
/
"
(
S
11
)
112
5
%
(
)
=
(
)
+
1
(
S
12
)
113
$
=
%
1
&
+
&
1
&
$
$
(
S
13
)
114
%
"
=
(
)
3
(
S
14
)
115
(
S
15
)
116
(
)
=
I
%
(
)
%
,
#5#36(376
(
)
for
'
,
#5#36(376
%
(
)
87#
for
>
'
,
#5#36(376
(
S
16
)
117
37$
(
)
=
I
MIN
S
#5#36(376
9
(
)
,
#5#36(376
&
(
)
,
#5#36(376
1
(
)
U
>
:
(
)
:
(
S
17
)
118
̅
=
1
'
:
X
(
)
+
#
+
$
(
S
18
)
119
where
=
/
5
is
normalized
distance downstream,
=
9
(
9
;
<
%
)
"
@
'
>
'
?
(
/
=
&
is
normalized
time
and
8
is
120
volumetric sediment supply
averaged over many flood cycles
,
=
/
0
is
normalized
lobe
width,
$
=
121
$
/
0
is the
normalized
depth
of flow,
$
=
$
/
0
is
normalized
width of flow,
=
/
(
0
/
5
)
is the
122
normalized bed slope,
%
=
&
%
/
(
0
/
5
)
is the normalized friction slope,
"
is the Einstein number
123
representing dimensionless bed
-
material transport
(25, 26)
and
`
"
is the time
-
averaged Einstein number.
124
All elevation variables
were
normalized
by the
channel depth (e.g.,
87#
=
87#
/
0
,
=
/
0
).
125
126
B
.
Model implementation
127
For comparison of results among
the numerical backwater
-
scaled
-
avulsion model (Eq
s
.
S1
-
S9
),
128
the analytical backwater
-
scaled
-
avulsion model
(Eq. 4),
the radially averaged model (Eq. 2), and the
129
channel
-
averaged model
(Eq. 3)
, we varied the
normalized relative sea
-
level rise rates
for
constant
130
values for the other
input
parameters that were representative of lowland delta
s (Table S2)
.
Parameter
131
valu
es correspond to the case of a delta
that
builds into a basin that is twice as deep as the channel
132
(
5
=
2
0
) and experiences avulsions when the active lobe has aggraded to a height of half the channel
133