of 21
1.
Introduction
Permafrost riverbank erosion threatens the homes, infrastructure, and livelihoods of people living in the Arctic
(Hjort et al.,
2018
; Karjalainen et al.,
2019
). Permafrost regions contain 22% of the Earth's landmass (Obu,
2021
;
Obu et al.,
2019
) and ground temperatures are warming rapidly due to climate change (Biskaborn et al.,
2019
;
Isaksen et al.,
2016
). These regions also contain major river systems which can erode their banks up to tens of
meters per year (Rowland et al.,
2019
) (Figure
1
). Hundreds of Alaskan communities experience a combined
risk of bank erosion, permafrost thaw, and flooding (UAF & USACE,
2019
), and it is uncertain how much these
hazards will increase as the Arctic warms. Riverbank erosion has already caused some communities to relo
-
cate entirely (Bronen & Chapin,
2013
; Maldonado et al.,
2013
), but studies disagree whether erosion rates will
increase (Costard et al.,
2014
; Kokelj et al.,
2013
) or decrease (Ielpi et al.,
2023
) in response to climate change.
Accurate mechanistic models of permafrost riverbank erosion are needed to predict bank erosion hazards and
develop mitigation strategies.
Theory has been developed for permafrost riverbank erosion based on the one-dimensional Stefan solu
-
tion (Costard et al.,
2003
; Randriamazaoro et al.,
2007
). In this scenario, the erosion rate is assumed to be
ablation-limited, such that heat transfer and pore-ice melting set the erosion rate, and sediment is assumed to
be immediately entrained following thaw (Figure
2
). It is also possible that bank erosion is limited by sediment
transport or slump blocks (Douglas, Dunne, & Lamb,
2023
; Kanevskiy et al.,
2016
), but our focus here is to eval
-
uate the ablation-limited end member. For ablation-limited erosion, bank erosion rates should depend on river
flow velocity and water temperature because these parameters are the primary control on heat transfer from the
river to the bank (Costard et al.,
2003
). Therefore, since Arctic rivers are experiencing increases in water temper
-
ature and discharge (Brabets & Walvoord,
2009
; Liu et al.,
2005
; Peterson et al.,
2002
), riverbank erosion rates
might significantly increase as the Arctic warms. The theory of Costard et al. (
2003
) for ablation-limited erosion
Abstract
Permafrost thaw is hypothesized to increase riverbank erosion rates, which threatens Arctic
communities and infrastructure. However, existing erosion models have not been tested against controlled flume
experiments with open-channel flow past an erodible, hydraulically rough permafrost bank. We conducted
temperature-controlled flume experiments where turbulent water eroded laterally into riverbanks consisting of
sand and pore ice. The experiments were designed to produce ablation-limited erosion such that any thawed
sediment was quickly transported away from the bank. Bank erosion rates increased linearly with water
temperature, decreased with pore ice content, and were insensitive to changes in bank temperature, consistent
with theory. However, erosion rates were approximately a factor of three greater than expected. The heightened
erosion rates were due to a greater coefficient of heat transfer from the turbulent water to the permafrost bank
caused by bank grain roughness. A revised ablation-limited bank erosion model with a heat transfer coefficient
that includes bank roughness matched our experimental results well. Results indicate that bank erosion along
Arctic rivers can accelerate under scenarios of warming river water temperatures for cases where the cadence of
bank erosion is set by pore-ice melting rather than sediment entrainment.
Plain Language Summary
Many rivers in the Arctic have banks made up of permanently frozen
sand and ice (permafrost) that are susceptible to erosion when they thaw. To understand how bank erosion may
change as the Arctic climate warms, we conducted laboratory experiments using a channel with an erodible
frozen bank. We found that warmer river water and lower bank-ice content produced faster erosion rates. In
contrast, bank erosion was insensitive to the ground temperature. Bank erosion rates were three-fold faster than
predicted by existing theory. We attribute the faster-than-expected erosion to a greater transfer of heat from the
river water due to bank roughness. Our results imply that warming river water may increase riverbank erosion
rates in permafrost regions, threatening communities and infrastructure built along Arctic rivers.
DOUGLAS ET AL.
© 2023. American Geophysical Union.
All Rights Reserved.
Ablation-Limited Erosion Rates of Permafrost Riverbanks
Madison M. Douglas
1
, Kimberly Litwin Miller
1
, Maria N. Schmeer
1
, and Michael P. Lamb
1
1
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, USA
Key Points:
Flume experiments with frozen,
erodible riverbanks were used to test
and revise theory
Erosion rates were affected by
hydrodynamically rough banks that
change the rate of heat transfer
Ablation-limited bank erosion rates
increase with warmer water and lower
pore-ice content and are insensitive to
bank temperature
Supporting Information:
Supporting Information may be found in
the online version of this article.
Correspondence to:
M. M. Douglas,
mmdougla@caltech.edu
Citation:
Douglas, M. M., Miller, K. L.,
Schmeer, M. N., & Lamb, M. P.
(2023). Ablation-limited erosion rates
of permafrost riverbanks.
Journal of
Geophysical Research: Earth Surface
,
128
, e2023JF007098.
https://doi.
org/10.1029/2023JF007098
Received 1 FEB 2023
Accepted 16 JUL 2023
10.1029/2023JF007098
RESEARCH ARTICLE
1 of 21
Journal of Geophysical Research: Earth Surface
DOUGLAS ET AL.
10.1029/2023JF007098
2 of 21
compares well to observed erosion of up to 40 m measured over 1–2 months
for islands in the Lena River (Costard et al.,
2014
), but it over-predicts annual
rates by hundreds of meters of erosion per year if applied over the entire
open-water summer season. This partial disagreement between theory and
observations motivates our investigation of ablation-limited erosion theory
using flume experiments.
There have been few laboratory tests of permafrost bank erosion theory. The
Costard et al. (
2003
) model used an empirical coefficient to parameterize
heat transfer from the river water to the riverbank, based on experiments of
flowing water over ice (Lunardini,
1986
). However, it is unclear if the same
heat transfer coefficient applies to a sediment bank with pore-ice, which is
typical of permafrost floodplains. Other experiments measured erosion of a
small block of frozen sand and ice that was inserted into a smooth-walled
pressurized pipe or duct (Alexander,
2008
; Costard et al.,
2003
; Dupeyrat
et al.,
2011
). They found that higher water temperatures, greater water
discharge, and lower permafrost ice content increased the erosion rate of the
sample, consistent with theory. However, hydraulics are different in a pres
-
surized duct compared to an open channel, and thaw rates in these experi
-
ments may have been affected by any protrusion of the sample into the pipe as
well as the change in roughness from the hydrodynamically smooth pipe wall
to the rough sample. For instance, the size, shape, spacing, and orientation of
roughness elements are known to affect heat transfer by thinning and disrupt
-
ing the thermally diffusive fluid sublayer (Miyake et al.,
2001
; Shishkina &
Wagner,
2011
; Yaglom & Kader,
1974
).
Here, we present results from a permafrost river flume experiment designed
to investigate the erosion rate of a hydraulically rough and erodible frozen
riverbank under open-channel flow. First, we present existing theories for
ablation-limited bank erosion and heat transfer from a turbulent fluid to a
rough wall. Next, we show the experimental methods and results used to test the theories and evaluate the heat
transfer coefficient. Finally, we discuss how the theory applies to natural rivers and the implications for Arctic
riverbank erosion in a warming climate.
2.
Theory for Permafrost Riverbank Erosion
2.1.
Ablation-Limited Erosion Theory
Existing theory for permafrost riverbank erosion typically assumes ablation-limited conditions; that is, the erosion
rate is set by the rate of bank thaw (Costard et al.,
2003
; Dupeyrat et al.,
2011
; Randriamazaoro et al.,
2007
).
This is analogous to the theory developed for the geometry and evolution of subglacial and supraglacial channels,
where the channel geometry is set by heat transfer between the flow and a pure ice boundary (Gulley et al.,
2014
;
Karlstrom et al.,
2013
), but instead uses bank material properties that reflect a mixture of sediment and ice.
Following Randriamazaoro et al. (
2007
), we derive the position of the thawing bank (
y
=
s
; m) and the bank
temperature (
T
; °C) at a given time for the 1-D case (Figure
2a
). The control volume consists of a thawing portion
of a frozen riverbank with thickness
ds
(m) (Figure
2a
). Following the conservation of heat,
푝,푏
푑푇
푑푡




=
푑푠
+
=
,
(1)
where
q
f
(J/m
2
/s) is the latent heat flux into the bank,
q
w
(J/m
2
/s) is the heat flux from the flowing river water to
the bank,
q
r
(J/m
2
/s) is the sensible heat flux conducted from the control volume to the frozen bank,
ρ
b
(kg/m
3
)
and
c
p
,
b
(J/kg/°C) are the bulk density and specific heat of the bank material, and
y
is the coordinate normal to
the bank. Equation
1
assumes a 1-D heat budget where the only heat source is water flowing past the bank. This
assumption is supported by field observations that flowing water cuts deep thermoerosional niches and creates
characteristic overhangs in permafrost riverbanks, implying that the heat flux from the air is a relatively minor
Figure 1.
(a) Field photo of eroding permafrost sand and gravel riverbank
along the Yukon River near Beaver, AK. The exposed bank is approximately
3 m tall. (b) Field photo of eroding permafrost silt and peat riverbank along the
Koyukuk River near Huslia, AK. The exposed bank is approximately 1.5 m tall.
21699011, 2023, 8, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2023JF007098 by California Inst of Technology, Wiley Online Library on [03/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Journal of Geophysical Research: Earth Surface
DOUGLAS ET AL.
10.1029/2023JF007098
3 of 21
component of bank erosion (Walker & Hudson,
2003
). Bank material properties are assumed to be spatially and
temporally homogeneous, so that
ρ
b
and
c
p
,
b
are constants.
A thawing bank should be at the melting temperature, such that
T
|
y
=
s
= T
f
, where
T
f
(°C) is the temperature of
fusion of pore ice; thus, in Equation
1
,
푑푇
푑푡



=
=0
.
In addition, the heat flux due to fusion is
푞푞
푓푓
=
휌휌
푏푏
퐿퐿
푓푓
푑푑푑푑
푑푑푑푑
,
(2)
where
L
f
(J/kg) is the latent heat of fusion of the frozen bank. Substituting these expressions into Equation
1
and
rearranging results in
휌휌
푏푏
퐿퐿
푓푓
푑푑푑푑
푑푑푑푑
=
푞푞
푤푤
푞푞
푟푟
.
(3)
To evaluate
q
r
in Equation
3
, heat flow is modeled by conduction within the frozen bank (i.e., where
y
s
),
such
that
Figure 2.
1-D model for permafrost bank erosion (
E = ds/dt
, m/s) by ablation. (a) Schematic 1-D cross-section showing a
temperature profile (
T
) into a bank (
y
-direction) with the river flowing into the page. The erosion model considers heat fluxes
from the flowing water to the bank (
q
w
, J/m
2
/s) in a control volume of width
ds
(m). Heat flux from the river depends on
water flow velocity (
U
, m/s), temperature (
T
w
, °C), density (
ρ
w
, kg/m
3
), specific heat capacity (
c
p
,
w
, J/kg/°C), and the bank
coefficient of friction (
C
f
,
b
, dimensionless). The permafrost bank has a constant thermal conductivity (
κ
b
, W/m/°C), specific
heat capacity (
c
p
,
b
, J/kg/°C), latent heat of fusion (
L
f
, J/kg), and bulk density (
ρ
b
, kg/m
3
). The bank temperature in the control
volume is at the temperature of fusion (
T
f
, °C) and decreases to the background temperature (
T
0
, °C) over a distance
δ
(m)
driven by conduction (
q
r
, J/m
2
/s). (b) Cartoon cross-section of the bank showing how roughness affects heat transfer from
a fully turbulent fluid (flowing out of the page) to a hydraulically rough wall. The bank has median grain size
D
50
(m) and
volumetric ice content
λ
p
(m
3
/m
3
). Far from the wall, heat transfer is dominated by heat advection in turbulent eddies, while
heat transfer near the bank occurs by molecular diffusion through a thin sublayer. Roughness elements cause more rapid heat
transfer to the bank by thinning or protruding through the diffusive sublayer.
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Journal of Geophysical Research: Earth Surface
DOUGLAS ET AL.
10.1029/2023JF007098
4 of 21
−Δ∼


=
휕휕휕휕
휕휕휕휕
(4)
with
q
as the heat flux (J/m
2
/s) within the frozen bank. Heat conduction occurs from
y
=
s
to
y
=
s
+
δ
, and beyond
y
=
s
+
δ
the bank temperature is set to a constant background value
T
0
(°C) (Costard et al.,
2003
). Integrating
Equation
4
from
y
=
s
to
y
=
s
+
δ
and using the chain rule results in the following equation:
휌휌
푏푏
푐푐
푝푝푝푏푏
푠푠
+
훿훿
푠푠
휕휕휕휕
휕휕휕휕
휕휕휕휕
휕휕휕휕
푑푑휕휕
=
푠푠
+
훿훿
푠푠
휕휕휕휕
휕휕휕휕
푑푑
휕휕푑
(5)
As the bank erodes,
δ
(m) is assumed to remain constant, so the thermal gradient within the bank translates in the
y
-direction at the rate of bank erosion; thus,
dy/dt
=
ds/dt
. Then, Equation
5
can be solved and rearranged using
the boundary conditions
T
(
t
,
y
=
s
+
δ
)
= T
0
,
T
(
t
,
y
=
s
) =
T
f
,
q
(
t
,
y
=
s
+
δ
) =
0
, and
q
(
t
,
y
=
s
) =
q
r
, to find,
푞푞
푟푟
=
휌휌
푏푏
푐푐
푝푝푝푏푏
푑푑푑푑
푑푑푑푑
(
푇푇
푓푓
푇푇
0
)
.
(6)
The latent heat of fusion in Equations
2
and
3
for a saturated sand-ice mixture (Dupeyrat et al.,
2011
) depends on
the mass fraction of ice in the bank (
f
ice
; kg ice/kg frozen bank) and the latent heat of fusion of ice (
L
f
,ice
; J/kg):
퐿퐿
푓푓
=
푓푓
∼≤−
퐿퐿
푓푓푓
∼≤−
.
(7)
The specific heat of the bank (
c
p
,
b
) is calculated as a sum of the specific heat of ice (
c
p
,ice
; J/kg/°C) and the specific
heat of quartz sand (
c
p
,
s
; J/kg/°C), weighted by the mass fraction of ice:
푐푐
푝푝푝푝푝
=
푓푓
ice
푐푐
푝푝푝
ice
+
(
1−
푓푓
ice
)
푐푐
푝푝푝푝푝
.
(8)
Typically, the latent heat of fusion for ice is orders of magnitude greater than its specific heat, so we expect that
the phase change and not the permafrost temperature should set the rate of pore-ice thaw.
The heat transfer from a turbulent fluid to a wall depends on fluid velocity,
U
(m/s), and an empirical coefficient
that describes the efficiency of heat transfer (Nield & Bejan,
2017
). Thus,
q
w
in Equation
3
can be written as,
푞푞
푤푤
=
퐶퐶
휌휌
푤푤
푐푐
푝푝푝푤푤
푈푈
(
푇푇
푤푤
푇푇
푏푏
)
(9)
where
C
h
(dimensionless) is the heat transfer coefficient,
ρ
w
(kg/m
3
) is water density,
c
p
,
w
(J/kg/°C) is the specific
heat of water,
T
b
(°C) is the bank temperature, and
T
w
(°C) is the water temperature. In the transient solution,
T
b
may change in response to
q
w
, but the ablation-limited solution given by Equation
3
requires
T
b
=
T
f
.
The final expression is found from substituting Equation
6
for
q
r
in Equation
3
, Equation
9
for
q
w
in Equation
3
,
and using Equations
7
and
8
to account for the fraction of pore ice (
퐴퐴퐴퐴
ice
) in the latent heat of fusion and heat
capacity. Solving for the bank erosion rate
퐴퐴퐴퐴
푑푑푑푑
푑푑푑푑
for the 1-D ablation-limited case results in Randriamazaoro
et al. (
2007
):
퐸퐸
퐶퐶
휌휌
푤푤
푐푐
푝푝푝푤푤
푈푈
(
푇푇
푤푤
푇푇
푓푓
)
휌휌
푏푏
(
푓푓
ice
퐿퐿
푓푓푝
ice
+
(
푓푓
ice
푐푐
푝푝푝
ice
+
(
1−
푓푓
ice
)
푐푐
푝푝푝푝푝
)(
푇푇
푓푓
푇푇
0
))
.
(10)
2.2.
Heat Flux Parameterizations
Applying Equation
10
requires specifying the heat transfer coefficient
C
h
. Different empirical relations have been
proposed for
C
h
. Costard et al. (
2003
) and Dupeyrat et al. (
2011
) calibrated
C
h
based on a series of frozen flume
experiments to evaluate the rate of heat transfer from the water to a frozen bank. Both coefficients were calcu
-
lated as a function of the thermal conductivity of water (
κ
w
; J/m/s/°C), the Prandtl number (Pr), and the Reynolds
number (Re), using flow depth as the characteristic length scale:
퐶퐶
=
퐴퐴퐴퐴
푤푤
Pr
훼훼
Re
훽훽
(
휌휌
푤푤
푐푐
푝푝푝푤푤
푈푈푈푈
)
.
(11)
The Prandtl number (Pr
= ρc
p
,
w
ν
/
κ
w
) represents the dimensionless ratio of momentum diffusivity over thermal
diffusivity and depends on the fluid kinematic viscosity (
ν
; m
2
/s). The Reynolds number is the non-dimensional
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Journal of Geophysical Research: Earth Surface
DOUGLAS ET AL.
10.1029/2023JF007098
5 of 21
ratio of fluid inertial forces over viscous forces, with Re =
UH/ν
. For application to natural rivers, Costard
et al. (
2003
) used values of
A
= 0.0078,
α
= 0.3333, and
β
= 0.9270 constrained from flume experiments of water
flowing over pure ice (Costard et al.,
2003
; Lunardini,
1986
). During these experiments, the ablating ice devel
-
oped scallops on the scale of tens of centimeters, so heat transfer may have been influenced by form drag from
the scallops (Lunardini,
1986
). In this case,
β
∼ 1, so
C
h
is mostly independent of flow velocity (in Equation
11
,
Re/
UH
∼ 1/
ν
).
Yaglom and Kader (
1974
) proposed an alternative formulation for
C
h
that explicitly considers how wall roughness
affects heat transfer. They used the Reynolds analogy and asymptotic mapping of the advective and diffusive
thermal sublayers analogous to the derivation of the log law (Figure
2b
). Their formulation has been extensively
used in sea-ice models (Malyarenko et al.,
2020
) but has not been applied previously to permafrost riverbanks.
Assumptions in their theory include a negligible longitudinal pressure gradient and homogeneous wall rough
-
ness. They used linear interpolation to find a solution that includes hydrodynamically rough flow (roughness
Reynolds number
퐴퐴
Re
푘푘
푠푠
=
푘푘
푠푠
푢푢
휈휈
> 100, with
퐴퐴퐴퐴
푈푈
퐶퐶
푓푓푓푓푓
) as well as hydrodynamically smooth flow. These
are reasonable assumptions for permafrost rivers, which are fully typically turbulent with hydraulically rough
banks due to coarse sand and gravel grains and morphological roughness elements such as slump blocks and
vegetation (Figure
1
). Their final expression when integrated over the flow field is Yaglom and Kader (
1974
);
their Equations 22 and 23:
퐶퐶
=
퐶퐶
푓푓푓푓푓
훼훼
ln
휂휂
1
+
훽훽
1
+
훽훽
푡푡
.
(12)
C
h
depends on the bank coefficient of friction (
C
f
,
b
; dimensionless), the relative roughness element height
(
퐴퐴퐴퐴
1
=
푘푘
푠푠
퐻퐻
; dimensionless), the von Kármán constant (
κ
= 0.41), and empirical constants from the law of the
wall (
α
= 2.12;
β
1
= 0.5). For hydraulically rough flows (Re
ks
> 100),
β
t
=
β
r
, with
=
Re

1
Pr
2
3
2

.
For smooth to transitional flow (Re
ks
≤ 100),
β
t
=
β
r
(Re
ks
/100) +
β
s
(1 −
Re
ks
/100), where
β
s
= 12.5Pr
2/3
− 6. Next,
we describe the experimental approach and methods to test the bank erosion model (Equation
10
) and the two
different relations for the heat transfer coefficient (Equations
11
and
12
).
3.
Methods
3.1.
Experimental Goals and Approach
The goal of our frozen flume experiments was to evaluate the relations for the heat transfer coefficient for condi
-
tions similar to permafrost rivers. We simulated ablation-limited permafrost riverbank erosion, where permafrost
was directly in contact with the flowing river water. The experiments were not intended to be scale models of
any specific river, but we did consider important dimensionless numbers so that the experiments had similar
thermal and hydraulic states as natural permafrost rivers. The experiments were conducted under fully turbulent
(Re ∼ 10
5
) and subcritical (Froude number < 1) flow with hydraulically rough bed (
퐴퐴
Re
푘푘
푠푠
> 100) and an eroding
bank in the transitional roughness regime (Table
1
). We also scaled our flume experiments to the thermal regimes
of natural permafrost rivers. We used the Biot number to compare heat transfer to the bank versus conduction
within the bank (Bi =
C
h
ρ
w
c
p
,
w
UH/κ
b
), where
κ
b
is the bank thermal conductivity (W/m/°C), and the Stefan
number to compare heat transfer to the bank (St =
c
p
,
w
(
T
w
T
f
)/
H
). Using calculated values for bank thermal
conductivity and best-fit model results for
C
h
(see Section
4.2
), we find Bi ∼ 0.10 and St ∼ 130–660. For compar
-
ison, we estimated similar values (Bi ∼ 0.40 and St ∼ 8–80) from field measurements of the gravel-bedded Atigun
River, Alaska, using data from Scott (
1978
).
We conducted five experiments to vary water temperature, bank temperature, and mass fraction of water ice while
holding the other variables approximately constant. The effect of water temperature was evaluated by comparing
Experiments 1–3; bank temperature was evaluated by comparing Experiments 3 and 4; and pore-ice fraction was
evaluated by comparing Experiments 2 and 5 (Table
2
).
3.2.
Experimental Methods
The experiments were designed to simulate a straight half-width channel by using one fixed hydrodynamically
smooth wall and one erodible permafrost bank in the Caltech Earth Surface Dynamics Laboratory (Figure
3
).
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We placed the channel along the smooth wall, rather than in the middle of the flume, to suppress meandering
or braiding. The smooth plexiglass wall had minimal friction relative to the rough bed and sediment bank, and
therefore the half-width experiment can be considered representative of a full-width channel with two sediment
banks that is twice as wide (Pitlick et al.,
2013
).
The flume was 0.75 wide and 7.60 m long, ending in a reservoir filled with chilled water. We evaluated bank
erosion within a test section 0.70 m long centered in a 1.8 m reach bound by the clear plexiglass wall of the
flume on river right and an erodible bank consisting of a frozen mixture of sand and ice on river left. The
bankfull channel was initially set to 0.056 depth and 0.10 m width for each experiment. During the experiment,
the channel width increased due to bank erosion to a final value of about 0.3 m. We increased water discharge
to maintain a constant water depth as bank erosion progressed and the channel widened and used a backwater
model (see Text S3 in Supporting Information
S1
) to account for variations in channel hydrodynamics through
-
out the experiment.
We controlled water temperature and bank temperature using a Mokon AL Iceman chiller. The chiller recirculated
a 30% mixture of glycol in water through flexible pipes and mats arranged in the end tank and end barrels to chill
the water, and in the channel under the bed material and on the river-left flume wall to freeze the bank material
(Figure
3
). The experiments were not conducted in a climate-controlled room, so air temperature was variable.
We constructed the frozen, erodible bank and floodplain in layers to make a uniform mixture of sand
(
D
16
= 0.16198,
D
50
= 0.26132, and
D
84
= 0.36361 mm; Figure S1 in Supporting Information
S1
) and pore ice.
Experiment
Bed
slope
(m/m)
Water
temperature
(°C)
Bank
temperature
(°C)
Fraction
ice (wt%)
Bulk density
(g/cm
3
)
Measurement time
interval (stage 1)
(min:sec)
Bank erosion
rate (mm/s)
Experiment 1
0.0156
1.9 ± 0.1
−5.8 ± 0.7
33.0 ± 0.5
1.54 ± 0.04
40:06
0.075 ± 0.032
Experiment 2
0.0144
6.9 ± 1.5
−4.1 ± 0.8
23.6 ± 1.1
1.57 ± 0.06
19:32
0.16 ± 0.07
Experiment 3
0.0249
8.8 ± 0.6
−4.4 ± 0.7
27.7 ± 5.7
1.71 ± 0.09
6:52
0.26 ± 0.06
Experiment 4
0.0149
6.3 ± 1.0
−7.1 ± 0.6
21.4 ± 0.5
1.65 ± 0.36
15:00
0.19 ± 0.06
Experiment 5
0.0205
6.2 ± 0.2
−8.2 ± 0.9
31.3 ± 2.9
2.05 ± 0.16
5:46
0.23 ± 0.03
Table 2
Frozen Bank Properties for Each Experiment, With Variability Reported as 1 Standard Deviation
Variable
Symbol
Units
Values
Bank median grain size
D
50,bank
m
0.00026132
Bank 84th percentile grain size
D
84,bank
m
0.00036361
Bed median grain size
D
50,bed
m
0.019
Bed 84th percentile grain size
D
84,bed
m
0.021
Water discharge
Q
w
m
3
/s
0.00221–0.00756
Channel depth
H
m
0.056
Channel width
B
m
∼0.10–0.30
Average water flow velocity
U
m/s
∼0.6–0.7
Water Reynolds number
Re
Dimensionless
∼3.04 × 10
5
Bank roughness Reynolds number
퐴퐴
Re
푘푘
푠푠
Dimensionless
∼90
Water Froude number
Fr
Dimensionless
∼0.83
Prandtl number
Pr
Dimensionless
10
Stefan number
St
Dimensionless
∼130–660
Biot number
Bi
Dimensionless
∼0.10
Note
. Water discharge and channel width were increased throughout each experiment as the channel widened.
Table 1
Experimental Hydraulic Conditions for All Frozen Banks
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Journal of Geophysical Research: Earth Surface
DOUGLAS ET AL.
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7 of 21
We used a 0.1 m wide mold along the length of the sandy bank to form the initial channel on river right. We
filled the region between the mold and the river-left flume wall with the sand-water mixture and placed it on
top of a graded gravel bed. We built the bank material by laying ∼1.5 cm thick layers of saturated sand, graded
each layer to parallel the bed slope, and then covered it with insulation to freeze overnight. Temperature sensors
Figure 3.
Frozen channel experimental setup in Caltech Earth Surface Dynamics Laboratory. (a) Top-down cartoon of the flume setup. Glycol was cooled by the
chiller and circulated through a set of flexible pipes and mats to freeze the bank and cool the water in the end tank. Water was circulated by the pump from the end tank
through the flow diffuser and into the experiment headbox, where it flowed past gravel and sand. Overflow of the stand-pipe went into external barrels. (b) Side-looking
cartoon of the flume test section. Glycol mats line the side and base of the flume, and the exterior of the flume was covered by insulation. An array of temperature
sensors was frozen into the eroding, sandy bank, and we recorded 10-s timelapse imagery using down- and side-looking cameras. (c) Photograph during a flume
experiment. The glycol mats and temperature sensors are visible protruding up past the bank. The instrument cart ran on rails (visible in the lower right foreground) and
carried a laser to measure topography as well as a sonar to measure water slope throughout the experiments.
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