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Indications of Ongoing Noise‐Tipping of a Bifurcating River
System
Astrid Blom
1
, Clàudia Ylla Arbós
1
, M. Kifayath Chowdhury
1
, Arjen Doelman
2
,
Max Rietkerk
3
, and Ralph M. J. Schielen
1,4
1
Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands,
2
Mathematical
Institute, Leiden University, Leiden, The Netherlands,
3
Copernicus Institute of Sustainable Development, Utrecht
University, Utrecht, The Netherlands,
4
DG Rijkswaterstaat, Ministry of Infrastructure and Water Management, Utrecht,
The Netherlands
Abstract Tipping occurs when a critical point is reached, beyond which a perturbation leads to persistent
system change. Here, we present observational indications demonstrating presently ongoing noise‐tipping of a
real‐world system. Noise in a river system is associated with the changing flow rate. In particular, we consider
the upper Rhine River delta, where flow and sediment fluxes are partitioned over the two downstream branches
(bifurcates) of an important river bifurcation. Field observations show that a sequence of peak flows in the 1990s
resulted in sudden sediment deposition in one bifurcate, triggering a persistent and ongoing change in the flow
partitioning. This has caused the system to move toward an alternative equilibrium state or attractor. An
idealized model confirms that a river bifurcation system under such conditions is prone to tipping, and provides
insight on the onset of tipping.
Plain Language Summary Tipping occurs when reaching a critical point beyond which a
perturbation leads to a significant and often unstoppable change. Real‐life observations of such system tipping,
however, are rare. Here we provide first‐time indications of ongoing tipping of a real‐world system, namely a
major river system. These indications are based on field observations that indicate that a sequence of peak flows
in the 1990s resulted in sudden sediment deposition in one bifurcate, redirecting the flow over the downstream
branches, as well as modeling. These signs of tipping asks for increased attention from water management
authorities, as flood safety and shipping are of large importance.
1. Introduction
Tipping occurs when a perturbation abruptly alters the system state, after which it gradually approaches another
stable equilibrium state or attractor. In other words, tipping is when a system is pushed, or moves, out of the
equilibrium state's domain of attraction (i.e., the set of states from where the dynamical system moves to a stable
equilibrium) into the domain of attraction of another equilibrium (see Box 1).
BOX 1. Mathematics of Tipping
A system crosses a tipping point when the system state leaves the domain of attraction of the original
equilibrium state, and evolves toward an alternative attractor (Rietkerk et al., 2021). Systems that are
near such a critical point may be triggered to tip and undergo a qualitative and often irreversible change.
There are various causes of tipping (Ashwin et al., 2012,2017). Under slowly varying controls, for
instance induced by climate change, the original attractor may become unstable or disappear (typically
in a saddle node bifurcation). As a consequence, its domain of attraction vanishes and the system tips.
This is called bifurcation‐induced tipping (B‐tipping). Noise‐induced tipping (N‐tipping) is when
fluctuation of the controls pushes the system out of the domain of attraction of the original attractor.
Rate‐induced tipping (R‐tipping) occurs when the controls change so rapidly that the system state can no
longer follow the also changing domain of attraction of the equilibrium.
RESEARCH LETTER
10.1029/2024GL111846
Key Points:
•Field observations suggest noise‐
tipping of a river bifurcation system
•Sediment deposition in one bifurcate
resulting from a peak flow sequence in
the 1990s caused a lasting change in
flow partitioning
•Modeling suggests that peak flows can
push a bifurcation system out of an
attractor's stability domain, triggering
noise‐tipping
Supporting Information:
Supporting Information may be found in
the online version of this article.
Correspondence to:
A. Blom,
astrid.blom@tudelft.nl
Citation:
Blom, A., Ylla Arbós, C., Chowdhury, M.
K., Doelman, A., Rietkerk, M., & Schielen,
R. M. J. (2024). Indications of ongoing
noise‐tipping of a bifurcating river system.
Geophysical Research Letters,51,
e2024GL111846. https://doi.org/10.1029/
2024GL111846
Received 6 AUG 2024
Accepted 6 NOV 2024
Author Contributions:
Conceptualization: Astrid Blom,
Arjen Doelman, Max Rietkerk, Ralph
M. J. Schielen
Data curation: Clàudia Ylla Arbós,
M. Kifayath Chowdhury
Formal analysis: Astrid Blom,
Clàudia Ylla Arbós,
M. Kifayath Chowdhury, Arjen Doelman,
Max Rietkerk, Ralph M. J. Schielen
Funding acquisition: Astrid Blom
Investigation: Astrid Blom, Ralph
M. J. Schielen
Methodology: Astrid Blom
Project administration: Astrid Blom
Software: Ralph M. J. Schielen
Validation: Astrid Blom, Ralph
M. J. Schielen
Visualization: Clàudia Ylla Arbós,
M. Kifayath Chowdhury
© 2024. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.
BLOM ET AL. 1 of 9
Research on system tipping has mainly focused on ecosystems (Carrier‐Belleau et al., 2022; Meng et al., 2020;
Rietkerk et al., 2004,2021; C. Wang et al., 2016), climate (Alley et al., 2003; Ashwin et al., 2012; Bakke
et al., 2009; Klose et al., 2020; Lenton et al., 2008,2019; Lohmann & Ditlevsen, 2021), finance (Gatfaoui & De
Peretti, 2019), biology (He et al., 2012; Song et al., 2021), and water systems (Notebaert et al., 2018; Phil-
lips, 2018). Tipping in river systems has been particularly associated with peak flow events, sea level rise,
avulsions, and anthropogenic impact on the sediment supply (Notebaert et al., 2018; Phillips, 2018). Extremely
enhanced sediment supply to a fluvial system due to a giant rockslide (Lavé et al., 2023), earthquake (Sai-
kia, 2020), or dam break (Barrera Crespo et al., 2024) can also trigger approach to a new equilibrium state.
Despite the recent focus on tipping, there has been limited direct evidence of tipping in real‐world systems.
Evidence is generally restricted to ecosystems with limited extent and complexity such as shallow lakes (Scheffer
et al., 2001). Tipping in larger and more complex Earth system components has been suggested (Lenton
et al., 2008): it has been argued that the western Greenland ice sheet (Boers & Rypdal, 2021), the Amazon rain
forest (Boulton et al., 2022), and the Atlantic meridional overturning circulation (Boers, 2021; Ditlevsen &
Ditlevsen, 2023; Van Westen et al., 2024) are approaching a tipping point, indirectly inferred from early warning
signals (Scheffer et al., 2009; Van Westen et al., 2024). However, no conclusive evidence exists to confirm
tipping events in these systems, either past or present.
Here we focus on the bifurcation region in the upper delta of the Rhine River (Figure 1), where branches have
been narrowed and channelized in the past and, resultingly, are no longer able to change their planform and width
in response to changes of the controls (Chowdhury et al., 2023; Ylla Arbós et al., 2021,2023). At a bifurcation, the
Writing – original draft: Astrid Blom
Writing – review & editing: Clàudia Ylla
Arbós, M. Kifayath Chowdhury,
Arjen Doelman, Max Rietkerk, Ralph
M. J. Schielen
Figure 1. The system and its noisy forcing. (a) Rhine River basin with inset showing our domain of interest with river
bifurcation Pannerdense Kop. (b) 123‐year hydrograph at the German‐Dutch border (Lobith) illustrating that the flow forcing
has a noisy character. (c) Probability density function for the Lobith discharge.
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river splits into two bifurcates that each transport a fraction of the upstream flow, and gravel and sand fluxes.
Recent research by Chowdhury et al. (2023) has illustrated that two to three successive peak flow events have led
to abrupt change in flow partitioning among the two bifurcates. Our research question is: “To what extent can we
link the abrupt response to the successive peak flows to system tipping, specifically noise‐induced tipping?” (see
Box 1). In a river system, noise is associated with the forcings of flow rate, gravel flux, and sand flux, which all
vary with time (Arkesteijn et al., 2019,2021; Blom, Arkesteijn, et al., 2017), see Figure 1b. Our assessment is
based on field observations and application of a idealized river bifurcation model.
2. River and Observations
The Rhine River originates in the Swiss Alps, flows through six countries, and discharges into the North Sea.
After crossing the German‐Dutch border, it bifurcates into two branches at the Pannerdense Kop bifurcation,
which is our bifurcation of interest (Figure 1). The domain of interest has been characterized by channel bed
incision, which is due to channelization measures conducted over the 18th to 20th centuries (Chowdhury
et al., 2023; Ylla Arbós et al., 2021). Recent observations by Chowdhury et al. (2023) have illustrated that two
to three successive peak flow events, in 1993, 1995 and possibly 1998, have led to abrupt sediment deposition
and bed level increase (and hence flow depth decrease) at the upstream end of the Pannerden Canal bifurcate
(Figure 2a). In other words, the gravel and sand supply exceeded the sediment transport capacity at that
location.
The swift sequence of peak flows is succeeded by a notable reduction in the channel bed incision rate in both
bifurcates. The peak flows appear to have resulted in the downstream displacement of relatively coarse
sediment and its deposition over the upstream parts of the two bifurcates, which explains the significant
temporal bed surface coarsening over the upstream parts of the bifurcates over this period (Figure 3). A
coarser bed surface erodes more slowly than a finer one, as coarser sediment is less mobile and characterized
by a larger equilibrium slope (Blom, Arkesteijn, et al., 2017; Blom et al., 2016). This explains the decrease in
the channel bed incision rate in the bifurcates.
The abrupt change in bed level in response to the rapid succession of peak flows is accompanied by an abrupt
change in the flow partitioning between the bifurcates. Since this sequence of peak flows, the Waal discharge
has gradually increased at the expense of the Pannerden Canal discharge (Figure 2b). This continues to present
day, and is associated with the Waal channel incision rate being larger than the one of the Pannerden Canal
(Figure 2a).
Measured data illustrate that the bed surface sediment has coarsened with time (Figure 3). As the bed surface
grain size distribution largely depends on the one of the sediment flux (Blom, Arkesteijn, et al., 2017; Blom
et al., 2016), the latter has gradually coarsened with time, as well. This is associated with the downstream
migration of the Rhine River's gravel‐sand transition, which is a natural phenomenon and seems to have been
accelerated by relatively coarse sediment nourishments in the German Rhine (Blom, Chavarrías, et al., 2017;
Chowdhury et al., 2023; Ylla Arbós et al., 2021). The system incision and coarsening are expected to
continue, as the system will respond to climate change over the next century (Ylla Arbós et al., 2023).
The observations depicted in Figure 2illustrate the ongoing tipping process at the Pannerdense Kop bifurcation.
3. Model and Simulations
There exists a number of models describing the stability of river bifurcations (see Box 2). These idealized models
are valuable for understanding qualitative behavior such as equilibrium solutions and potential tipping, yet they
are less suitable for precise quantitative predictions due to conceptual simplifications. These models show that
there exists one solution to the equilibrium state of a one‐channel system with non‐erodible banks (Blom,
Arkesteijn, et al., 2017; Blom et al., 2016; Howard, 1980), yet three to five solutions for a river bifurcation system
(Bolla Pittaluga et al., 2003,2015; Ragno, 2023; Ragno et al., 2023; Schielen & Blom, 2018).
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BOX 2. Bifurcation Stability Models
The Z. B. Wang et al. (1995) model was the first to address a river
bifurcation as a stability problem, and to show that three equilib-
rium states exist (i.e., two active bifurcates or either one of the
bifurcates active). Simplifying assumptions are: unisize sediment;
each bifurcate adjusting uniformly; no effects of the lateral slope
upstream of the bifurcation; and a nodal point relation partitioning
the sediment flux over the bifurcates. Subsequently lateral bed slope
effects were accounted for (Bolla Pittaluga et al., 2003,2015;
Ragno et al., 2023), as well as mixed‐size sediment (Ragno
et al., 2023; Schielen & Blom, 2018). All these models are highly
idealized and exclude several physical mechanisms, such as the
temporal variation of the flow rate, helical flow due to the offtake
angle or Bulle effect (Bulle, 1926; Dutta et al., 2017), and flow
separation in the offtake channel (Van der Mark & Mossel-
man, 2013). Stability analyses reveal that bifurcation systems are
characterized by at least three or five equilibrium states. Under
conditions with multiple stable equilibrium states, the initial con-
ditions govern which stable equilibrium the system approaches
(Paudel et al., 2022; Schielen & Blom, 2018).
We adopt the Schielen and Blom (2018) model of a gravel‐sand bifurcation
system to assess whether sudden sediment deposition or associated abrupt
decrease in flow depth in one bifurcate can lead to system tipping. The
Supporting Information S1 provides more detailed information on the model
and model parameters. We compute the model equilibrium solutions, and
determine whether they are stable (nodes) or unstable (saddles). The phase
plots in Figure 4consider cases with five equilibrium states, three of them
being stable and two of them being unstable. Each stable equilibrium has a
domain of attraction, which is the collection of states from where trajectories
converge to the specific equilibrium. The two red areas in Figure 4each
indicate an attraction domain of an equilibrium governed by a different active
branch each, and the blue area is the domain of attraction of an equilibrium
with two active branches.
Figure 4addresses a situation where the system state initially is not far from,
and gradually approaches, a stable equilibrium of two active bifurcates. We
examine the case of abrupt change described in the previous section.
Following the observations, we assess a situation where the peak flow suc-
cession leads to an abrupt increase in bed level and associated decrease in
flow depth in bifurcate 2. The phase plot in Figure 4a represents a case where
the domain of attraction is sufficiently large for the flow depth decrease in
bifurcate 2 not to push the system to another domain of attraction. Thus, for a
sufficiently large domain of attraction, the abrupt decrease in flow depth in bifurcate 2 leads to a somewhat
adjusted path toward the stable equilibrium state but does not lead to tipping.
We argue that the temporal coarsening of the bed surface sediment (Figure 3) and sediment flux reduce the
domain of attraction with time. This can be explained as follows. The partitioning of wash load (particle sizes
generally considered smaller than 63 μm) over bifurcates typically correlates with the partitioning of the water
discharge, while the partitioning of coarser sediment is more strongly related to the relative width of the bifurcates
Figure 2. Observations of bed level and flow partitioning. (a) Bed level
relative to Dutch reference level NAP at the upstream ends of the Waal and
Pannerden Canal bifurcates (i.e., data have been averaged over a 3 km
window right downstream of the bifurcation). (b) Ratio of Waal discharge to
the upstream channel's water discharge (termed Lobith discharge). Subfigure
(b) distinguishes two water discharge ranges: the lower range represents
closed weir conditions for the three weirs in the Nederrijn‐Lek branch
(Figure 1a), which leads to backwater effects reaching up to the
Pannnerdense Kop bifurcation and, resultingly, the Waal bifurcate taking a
relative large share of the Lobith discharge at the expense of the Pannerden
Canal. This weir‐induced backwater effect is absent for the higher flow
range as the weirs are lifted. Water discharge data are derived from Ott
current meter until 1999 and since 2000 ADCP measurements (Chowdhury
et al., 2023).
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(Schielen & Blom, 2018). This is because wash load is more uniformly
distributed over the cross‐sectional area, while the transport of coarser sedi-
ment occurs close to the bed. This implies that the coarser the size fraction, the
smaller the dependence on the discharge partitioning and the larger the
dependence on the ratio of the width of the bifurcates. In the Schielen and
Blom (2018) model, such a reduced dependence on the flow partitioning is
expressed by a smaller value of the gravel‐related nodal point coefficient, kg.
This yields a smaller domain of attraction of the stable equilibrium state. We
refer to the Supporting Information S1 for further details on the nodal point
relations and associated nodal point coefficients.
Figure 4b illustrates a phase plot for a case with similar stability properties as
Figure 4a but a smaller domain of attraction of the equilibrium state (resulting
from a reduced value of the gravel‐related nodal point coefficient, kg). Here
the abrupt decrease in flow depth in bifurcate 2 does push the system state into
another domain of attraction. Thus, a small domain of attraction allows the
abrupt decrease in flow depth in bifurcate 2 to cause tipping (Figure 4b).
The temporal decrease of the domain of attraction due to recent system
coarsening would explain why the 1993, 1995 and 1998 peak flow succession
led to abrupt system response and subsequent gradual approach to a new equilibrium state, while there is no sign
that previous sequences of peak flows (for instance, in 1924 and 1926) led to a similar response. Apparently, at
that time the sediment flux was still sufficiently fine and the domain of attraction sufficiently large to prevent
tipping from happening.
Figure 3. Observations of median bed surface grain size over Bovenrijn‐
Waal branches.
Figure 4. Phase plots illustrating possible development of a river bifurcation. (a) A relatively large domain of attraction of a
stable equilibrium state with two open bifurcates. (b) A relatively small domain of attraction of a stable equilibrium state with
two open bifurcates. Phase plots are predicted using the Schielen and Blom (2018) model. The nodal point coefficient for
sand, k
s, is 1, while the nodal point coefficient for gravel, kg, equals 3.5 (subfigure a) and 2.4 (subfigure b). We refer to the
Supporting Information S1 for further details on the nodal point relations and other model parameters. Black arrows indicate
“abrupt change”, and yellow arrows indicate “gradual change.” Note that here the phase plots are symmetric as we consider
bifurcates with mostly equal properties, but typically phase plots would not be symmetric.
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4. Conclusions
Over the past decades the bed surface sediment in a channelized bifurcation system has gradually coarsened. This
seems to have slowly decreased the domain of attraction of the original stable equilibrium or attractor to the extent
that tipping became possible. This enabled a peak flow sequence to initiate tipping, and the system has gradually
evolved toward a different attractor since. The latter gradual change is reflected by one bifurcate (the Waal
bifurcate) receiving an ever increasing share of the water discharge of the upstream branch, which is accompanied
by it incising at a higher rate than the other bifurcate.
The reduced domain of attraction seems to have increased the probability for the system to be pushed out of the
domain of attraction through the noisy controls for flow and sediment fluxes. Noise‐induced tipping (N‐tipping) is
thus the probable tipping type.
An idealized model illustrates that there are a few stability regions in the parameter space, and that a reduced
domain of attraction of the original attractor can allow a peak flow sequence to perturb the river system into a
different attractor's stability region.
Tipping is a bidirectional process: the process may be reversed by system control change or interventions. This
requires insight on the consequences of control changes or interventions that may (a) manipulate the system
trajectory or (b) enlarge the domain of attraction of the stable equilibrium that the system originally approached.
The current analysis assumes a fixed channel network. River bifurcations are typically part of a river delta, and
many deltas worldwide consist of a fixed channel network as a result of past and modern channelization works.
Examples are the Mississippi River, Yangtze River, Pearl River, Yellow River, Po River, and Nile River, which
are increasingly governed by fixed channels.
Recent research has illustrated that complex systems can evade tipping if they are governed by sufficient degrees
of freedom regarding their response to control change (Rietkerk et al., 2021). The degrees of freedom of channel
response of the considered system were reduced over the past few centuries, as river planform and channel width
have been fixed. As a result, system response is limited to adjusting the main channel slope and the bed surface
grain size distribution. Consequently, the extensive channelization measures of the past may have enabled the
present tipping process of the Rhine River bifurcation.
One bifurcate attracting a gradually increasing share of the water discharge at the expense of the other bifurcate
affects flood risk and shipping, and, as such, is relevant to water management authorities. This is because the
Dutch flood protection system (i.e., a network of levees) is designed according to a specific flow partitioning at
bifurcations. Also shipping is affected by a changing flow partitioning, as a decreasing share of the water
discharge in a bifurcate reduces vessel draught, which has economic consequences.
Appendix A: Field Observations
Our analysis is based on field observations (Appendix A) and application of a idealized river bifurcation model
(Appendix B).
We refer to Chowdhury et al. (2023) and Ylla Arbós et al. (2021) for information on measurement techniques,
data collection, and data treatment regarding bed level and surface texture. There have been several temporal
changes in measurement techniques (e.g., single‐beam to multibeam echo sounders for bed level). In addition, as
the Rhine River spans across borders, techniques also vary spatially. Such temporal and spatial changes in
measurement techniques increase the uncertainties in the measured data.
Field observations comprise the following: (a) a 123‐year hydrograph at the German‐Dutch border; (b) flow
partitioning based on water discharge in the branches; (c) channel bed level; and (d) median bed surface grain size.
Ad 1. The 123‐year hydrograph at the German‐Dutch border (Lobith) consists of water discharge data between 1
January 1901 and 31 December 2023. Water discharge (Figure 1) is derived from a stage‐discharge relationship,
which has been calibrated using ADCP measurements (Toonen, 2015). The stage‐discharge relationship is
regularly updated to account for local temporal change in bed level and hysteresis effects.
Ad 2. The flow partitioning between the two bifurcates (Figure 2b) is not based on stage‐discharge relationships
(data at gauging stations). This is because the associated uncertainty is large, and the Waal gauging station is not
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representative of the Waal branch due to its slightly upstream position. Instead, water discharge data are derived
from Ott current meter until 1999 and ADCP measurements since 2000 (Chowdhury et al., 2023). Considered
cross‐sections were approximately 1 km downstream of the Pannerden bifurcation for the Waal and Pannerden
Canal. Ott current meter data are associated with a larger inaccuracy (smaller than 20%) than ADCP data (smaller
than 10%). ADCP transects were traversed repeatedly at least 10 times, within at least 5% agreement to the mean
discharge value. Resulting values are averaged over a hydrologic year (as an example, the 1990 data point reflects
the period between 1 October 1989 and 30 September 1990).
Ad 3. Bed level data were averaged over the cross‐sectional profile between the groyne tips, and is relative to the
Dutch reference level for mean sea level (NAP). Details on data processing are described by Ylla Arbós
et al. (2021). Figure 2a shows data that are representative of the upstream ends of the Waal and Pannerden Canal
bifurcates: the data have been averaged over a 3 km window right downstream of the bifurcation, and the moving
average window does not extend across the bifurcation.
Ad 4. Data on surface median grain size over the Bovenrijn‐Waal branches (Figure 3) stem from grain size
sampling campaigns that have varied in space and time (Ylla Arbós et al., 2021). Three samples (spaced 65 m)
were taken per cross section, and we consider cross‐section averaged values of bed surface grain size. The data
have been smoothed using the Loess method. For smoothing, we used a span equal to 20% of the data points,
except for the limited‐extent 2008 and 2016 data series where the span was equal to 90%.
Appendix B: Mathematical Modeling
We adopt the Schielen and Blom (2018) model of a gravel‐sand bifurcation system to assess whether sudden
sediment deposition or associated abrupt decrease in flow depth in one bifurcate can lead to system tipping. It
consists of conservation equations for mass (flow, gravel and sand) and momentum (flow), and nodal point re-
lations that prescribe the partitioning of the gravel and sand fluxes over the bifurcates. The model predicts the
temporal adjustment of branch‐averaged bed level and surface gravel and sand content in the two bifurcates under
constant water discharge, sediment supply, base level, and channel width. The Supporting Information S1 pro-
vides more detailed information on the model and model parameters.
Data Availability Statement
Data on bed level and bed surface grain size for Bovenrijn and Waal branches (1926–2018) are available from the
4TU.ResearchData repository (Ylla Arbós, 2021). Bed level and bed surface grain size along the Pannerden Canal
branch, and the water discharge data for the Rhine branches are available at the 4TU.ResearchData repository
(Chowdhury et al., 2022). The code to reproduce Figure 4is hosted at the 4TU.ResearchData repository (Schielen
et al., 2024).
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Acknowledgments
AB and RS contributed equally and
share first authorship. The authors thank
Rijkswaterstaat for providing the field
data, Kieran Dunne for proofreading the
manuscript, and Eric Barefoot and an
anonymous reviewer for helpful
feedback. The research of A.D. and M.
R. is supported by the ERC‐Synergy
project RESILIENCE (101071417) and
NWO project “Resilience in complex
systems through adaptive pattern
formation” (OCENW.M20.169).
Contributions by K.C. and C.Y.A. are
part of the research program
Rivers2Morrow, which is financed by
the Dutch Ministry of Infrastructure and
Water Management.
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