Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
COASTAL LOUISIANA
ECOSYSTEM ASSESSMENT & RESTORATION (CLEAR) PROGRAM:
A TOOL TO SUPPORT COASTAL RESTORATION
Final Report To:
Department of Natural Resources
Coastal Restoration Division
Baton Rouge, Louisiana
DNR Interagency Agreement No. 2512-06-02
OCR Interagency Agreement No. 435-600636
Robert R. Twilley
Principal Investigator
Department of Oceanography & Coastal Sciences
Louisiana State University
Baton Rouge, LA 70803
Land Building in the Delta of the Mississippi River: Is it Feasible?
Chapter 10
CLEAR Volume IV
June 2008
Kim, W., D. Mohrig, R. Twilley, C. Paola, and G. Parker. 2008. Land Building in the Delta of
the Mississippi River: Is it Feasible?, Chapter 10. In, R.R. Twilley (ed.), Coastal Louisiana
Ecosystem Assessment & Restoration (CLEAR) Program: A tool to support coastal restoration.
Volume IV. Final Report to Department of Natural Resources, Coastal Restoration Division,
Baton Rouge, LA. Contract No. 2512-06-02.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Land Building in the Delta of the Mississippi River: Is it Feasible?
CLEAR Volume IV, Chapter 10
1
Wonsuck Kim
2
David Mohrig
3
Robert Twilley
4
Chris Paola
1
Gary Parker
1
Department of Civil and Environmental Engineering and Department of Geology
University of Illinois at Urbana-Champaign
Urbana, IL 61801
2
Department of Geological Sciences
University of Texas at Austin
Austin, TX 78713
2
Eco-Hydrology
3
Department of Oceanography & Coastal Sciences
Louisiana State University
Baton Rouge, LA 70803
4
Department of Geology and Geophysics
University of Minnesota
Minneapolis, MN 55455
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Abstract
The delta of the Mississippi River and its associated wetlands are subsiding into the Gulf of
Mexico, causing habitat loss, interfering with human activities, and progressively reducing the
storm surge buffer to New Orleans as the shoreline moves northward. Using a mechanistically
based numerical model, we show that controlled avulsions of the river below New Orleans can
build significant new land and help reverse this deterioration. Using a range of inputs spanning
plausible rates of subsidence, sea-level rise, and sediment supply, the model predicts that ~700 to
1220 km2 of new land could be built over the span of a century.
The problem and a proposed solution: controlled river diversions.
An article in Scientific American in 2001 [Fischetti, 2001] familiarized the public with a disaster
that has been unfolding over the last several decades: the unabated drowning of deltaic wetlands
of the Mississippi River in Louisiana. As documented in e.g. [Morton et al., 2005], this loss has
been proceeding at a rate ~ 44 km2/yr over the last several decades (Fig. 1).
Subsidence is in itself a natural process. Deltas of large rivers carrying fine-grained sediment
subside due to compaction of that sediment under its own weight. In some cases, tectonic and/or
microbial processes also contribute to subsidence. Normally, the land loss associated with
subsidence is countered by sedimentation, either directly from the river during floods and/or
indirectly as river-borne sediment is redistributed by coastal processes. In many deltas, this areal
redistribution is enhanced by the tendency of deltaic river channels to migrate or avulse into low
zones, so creating a shorter path to the sea. Regardless of the precise mechanism, under natural
conditions, sediment deposition tends to balance subsidence and maintain the delta.
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Over the last 4600 years the Mississippi River has undergone avulsions that have resulted in
major depositional lobes (~30,000 km2 in area), each being constructed over a time scale of
~1500 years. At a smaller scale within each major lobe, channel shifts have created minor lobes
(~200 km2 in area) over a time scale of ~175 years [Coleman, 1988].
Two major human-induced changes have altered this balance. First, the present-day river is
now locked into place by bank protection, and is impounded by levees down to near its mouth.
These levees have evolved since the 1700s, and were particularly strengthened after the
disastrous flood of 1927. They have provided needed protection to the citizens of New Orleans
and adjacent floodplain areas within the deltaic plain. At the same time, however, river control
by these public works projects now permit no migration or avulsion of the river, nor do they
allow overbank deposition from the river. As a result, the sediment load of the Mississippi River
is now delivered through the present bird’s foot delta to the shelf-slope break of the Gulf of
Mexico, where it disperses uselessly into the ocean and contributes to anoxia in the Gulf.
The second major change in the overall sediment balance is acceleration of subsidence.
Humans accelerate subsidence by extracting fluids (hydrocarbons and water) from the deltaic
pile, lowering fluid pressures and accelerating compaction. Although some have argued that
current high subsidence rates are natural [e.g. Dokka, 2006], the strong association of land-loss
rate and extraction of subsurface fluids strongly suggests a dominant human influence [Morton et
al., 2005]. The sediment-starved deltaic wetlands behind the levees have been subsiding into the
sea not in geologic time, but in engineering time (Fig. 1). In 1944, the loss rate was 34 km2/yr
[Morton et al., 2005]. By 1967 it had increased to 70 km2/yr, and by 1995 it had declined to the
still-substantial number of 26 km2/yr [Morton et al., 2005]. While New Orleans has always been
at risk of inundation due to hurricane storm surge, and will continue to be so due to its location
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near the sea, Hurricane Katrina (August 29, 2005) underlined the deleterious effects of the loss
of a land buffer between the city and the sea on the mitigation of storm surge.
Can these problems be reversed, at least partially, by means of strategically located river
diversions that act as engineered avulsions?
Arguments against engineered avulsions.
Former Secretary of the Interior Bruce Babbitt has offered an argument against restoring the
Mississippi Delta [Babbitt, 2007].
“I believe the Bush administration should continue to withhold money for coastal restoration
in Louisiana. The projects being served up by the U.S. Army Corps of Engineers are little more
than traditional Louisiana pork.”
“Most of the Mississippi Delta, some 10,000 square miles, lies less than three feet above sea
level. Beset by land subsidence and rising sea levels, much of this vast area will inexorably sink
beneath the waters by the end of this century.”
“Congress should suspend all coastal funding until the Corps and Louisiana prepare a
comprehensive and realistic land-use plan for the entire delta, applying modern science and fiscal
discipline to determine what can and cannot be salvaged.”
Turner et al. [2006] have argued not that sinking of the delta is inexorable, but that the great
majority of the inorganic sediment in the delta does not come from the river system. They argue
that much of the wetlands in question have been constructed from sediment delivered onshore by
hurricanes, rather than fluvial deposition.
“The accumulation from hurricanes is sufficient to account for all the inorganic sediments in
healthy salt marsh wetlands…” “In particular, hurricanes appear to be the overwhelming
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pathway of new inorganic sediments for coastal wetlands in western Louisiana, because the few
riverine sources bring relatively trivial amounts inorganic sediments into the marsh.” These two
statements provide the background for a third statement by Turner et al. [2007]. “Restoration
programs that restrict the underlying cause-and-effect relationships and remedies to focus on
inorganic sediment source and delivery may be fatally flawed.”
A summary of the arguments for controlled avulsions.
Our response to Babbitt [2007] and Turner et al. [2006; 2007] can be stated simply. a) The river
delta is not a static board sinking into the sea or drowning under sea-level rise, but instead is a
dynamic surface that has been able to maintain its level slightly above sea level for millions of
years, by maintaining a balance between sediment deposition and subsidence. b) That sediment
has over the same millions of years been supplied primarily from the Mississippi River itself.
Several proposals have already been made to reduce wetland loss based on proper river
management of sediment delivery to the inland coast [CPRA, 2007; EFGC, 2006]. The
fundamental assumption in these planning documents is as follows: the use of appropriate
engineering practices to reconnect sediment delivery to the deltaic plain is critical to mitigating
the effects of subsidence and sea-level rise [Day Jr et al., 2007]. What was missing from these
proposals, however, was access to a validated, numerical model describing land building under
various scenarios for subsidence and sea-level rise. Such a model is a necessary tool for
evaluating whether a self-sustaining deltaic landscape that can withstand inundation by
subsidence and sea-level rise over the next 100 years is possible along the Louisiana coast.
In succeeding sections, we justify our response to the assertions of Babbitt [Babbitt, 2007]
and Turner et al. [2006; 2007] in the context of a numerical model that illustrates quantitatively
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the potential for self-sustaining land building through controlled avulsions. We begin by
estimating the supply of sediment available for land building.
Sediment budget.
The Mississippi River system in Louisiana can be divided into five major reaches (Fig. 2): the
Mississippi River in Louisiana above the Old River (OR) Control Structure complex at the
bifurcation of the Atchafalaya River (Mississippi River above OR), the lower Mississippi below
the OR Control Structure complex (Lower Mississippi), the upper Atchafalaya River above the
bifurcation of the Wax Lake Outlet (Upper Atchafalaya), the lower Atchafalaya River below the
bifurcation of the Wax Lake Outlet (Lower Atchafalaya), and the Wax Lake Outlet.
The mean annual suspended sediment discharge of the Mississippi River above OR during
the 19th century has been estimated to be ~400-550 Mt/yr [Allison et al., 2000; Fisk et al., 1954;
Horowitz, 2006; Kolb, 1963]. Since that time, however, over 8000 dams have been built in the
Mississippi River Basin [Graf, 1999]. Many of these dams operate as run-of-the-river
impoundments, and thus do not result in the permanent or semi-permanent sequestration of
sediment. The remaining dams, however, have had a substantial impact on sediment delivery
downstream [Syvitski et al., 2005].
The present best estimate of the mean annual suspended sediment discharge of the
Mississippi River above OR is ~208 Mt/yr, of which 124 Mt/yr flows into the Lower Mississippi
and 84 Mt/yr flows into the Upper Atchafalaya [Allison et al., 2000; Horowitz, 2006]. The
sediment in the Lower Mississippi flows down to the present day bird’s foot delta, which is the
main delta of the system. Of the suspended load in the Upper Atchafalaya, between ~30% to
~45% [Roberts et al., 2003], or around 25-38 Mt/yr flows down the Wax Lake Outlet to form the
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present-day Wax Lake Delta. The remainder, i.e. no more than 59 Mt/yr continues down the
Lower Atchafalaya to form the present-day Atchafalaya Delta.
Of the suspended sediment in the system, about 17% is sand, the rest being mud (silt- and
clay-sized particles) [Allison et al., 2000]. This suspended load is augmented by a sand bedload
transport rate that is roughly 1.6% of the total suspended load [Nittrouer et al., 2008]. At least
the lower portions of the Mississippi system appear to be storing little sediment in the channel on
a long-term basis. Most of the length of the Lower Mississippi River below New Orleans, for
example, is a quasi-bedrock stream with considerable reaches of exposed, consolidated
Pleistocene or early Holocene sediments [Nittrouer et al., 2007].
Subsidence and sea-level rise.
Subsidence in the Mississippi Delta appears to be driven by three factors: a) tectonics (i.e., deep-
seated fault activities and the flow of salt at depth), b) natural subsidence due to compaction and
c) compaction caused by the anthropogenic removal of subsurface fluids (oil, natural gas and
water). The distribution of estimated background subsidence rates over a millennial time spam
[Meckel et al., 2006; Morton et al., 2005] is contrasted with the corresponding distribution over
the last 80 years [Shinkle and Dokka, 2004] in Figure 3. Mean background (geological) rates are
~1.4 mm/yr, whereas mean historical rates over the last 80 years average to ~11 mm/yr (Fig. 3).
What number to use for modeling purposes is open to debate. Data presented in [Morton et
al., 2005] and summarized in Figure 1 indicate a strong temporal correlation between land loss
rates and extraction rates of oil, natural gas and associated formation water. The rate of land loss
is a proxy for a large-scale average of subsidence rate. The correlation between land loss and
fluid extraction demonstrates that the pumping of fluids accelerates subsidence [Morton et al.,
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2005]. The implication is that fluid extraction, which peaked around 1971, may have
overwhelmed natural drivers of subsidence at that time. Between 1971 and 2000, both the fluid
extraction rate and the land loss rate decreased dramatically. Thus the subsidence rate expected
over the next century should be bracketed by 1.4 mm/yr and 11 mm/yr
There is similar variability in estimates of eustatic sea-level rise. If melting of continental ice
sheets were minimal, the rate would be too low to be accurately detectable, and so might as well
be set to zero. Estimates from the Intergovernmental Panel on Climate Change (IPCC) are in the
range 2-4 mm/yr [IPCC, 2007]. Here we use values of 0 mm/yr and 4 mm/yr to bracket the rate
of sea-level rise.
Land building in the Wax Lake Delta.
Any numerical model must be tested by hindcasting before it is used for forecasting. In the
present case, an ideal site offers itself for this purpose: the Wax Lake Delta. The main bird’s foot
delta of the Mississippi River is no longer building out significant amounts of land: instead it is
delivering sediment to the shelf-slope break of the Northern Gulf of Mexico margin, which
functions as an effectively unlimited sink. The Atchafalaya and Wax Lake Deltas are, however,
actively prograding onto the open shelf. They offer direct evidence that the Mississippi River is
capable of building new land, even with its present reduced sediment supply.
Wax Lake Delta provides a near-perfect natural experiment in delta building, and has been
intensively documented by researchers at Louisiana State University since its inception [Roberts
et al., 2003]. The Wax Lake Outlet was constructed in 1941 for the purpose of flood control
[Latimer and Schweitzer, 1951]. Originally, much of the sediment supply to the Wax Lake Outlet
deposited in Wax Lake itself rather than at the coastline. Subaerial development of the delta
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began at the time of the flood of 1973; the delta has been actively building new land in coastal
Louisiana since then.
The evolution of the delta as observed in 1990, 2000, and 2005 is shown in Figure 4. The
Wax Lake Delta, like the Atchafalaya Delta, is evolving as a fan-shaped delta rather than the
shape of the present bird’s foot delta of the Lower Mississippi. Roberts et al. [2003] and
Majersky et al. [1997] estimated the progress of land building in terms of surface area of land
(islands and freshwater channels), and reported the following numbers: 3.8 km2 (1976); 36.3 km2
(1986) ; 19.7 km2 (1981); 47.9 km2 (1989); 57.0 km2 (1992); 84.2 km2 (1994). About 67% of the
Wax Lake Delta deposit is sand, and the remainder is mud [Majersky et al., 1997]. The existence
of a definable deltaic foreset suggests that relatively little sand appears is escaping the delta onto
the muddy shelf.
A suspended sediment supply of 25-38 Mt/yr, which is 17% sand, a sand bedload supply rate
that is 1.6% of the suspended sediment, a deltaic deposit that averages to 67% sand, and the
assumption that negligible sand passes beyond the delta foreset allows a computation of a
sequestration rate of sediment in the Wax Lake Delta ranging from 6.9 to 10.4 Mt/yr. This
corresponds to a 27% sequestration of supplied sediment in the delta, and 0.49 mass units of mud
deposited per unit sand deposited. The sequestration fractions are within the range observed for
other large rivers around the world [Allison et al., 1998; Bobrovitskaya et al., 1996; Goodbred Jr
and Kuehl, 1998].
The land-building model.
The land-building model simulates the evolution of a prograding fan-shaped delta with a topset
and foreset advancing into standing water. The model is an outgrowth of calibrated engineering
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tools for managing the disposal of mine tailings in confined bodies of water [e.g. Parker et al.,
1998]. In the case of one mine, the disposal rate was 21 Mt/yr, and in the other it is 24 Mt/yr, i.e.
numbers on the order of the sediment supply rate to the Wax Lake Delta.
An elaboration of the basic principles of the model can be found in [Parker et al., in press].
The model expresses time-averaged delta evolution: it calculates the characteristics of an
effective channel that amalgamates the major active channels during floods, but the channel is
not specifically located. Instead, it is assumed to migrate, avulse, and flood in such a way as to
maintain overall radial symmetry of the delta as it builds land (Appendix fig. 1). It thus yields the
average downstream bed slope and elevation profiles of the delta, the height of the foreset,
position of the shoreline and delta area as functions of time. The model has the following
features.
• Delta opening angle and vertex position are specified.
• Flood discharge is abstracted into an effective, morphologically active flood flow continued
for a specified fraction of time per year, using the method of [Paola et al., 1992]. This
discharge is assumed to be the bankfull, or channel-formative discharge for the effective
channel.
• The total bed material (sand) transport rate is computed using the Engelund-Hansen equation
[Engelund and Hansen, 1972]. The coefficient has been adjusted based on model runs for the
Wax Lake Delta.
• Sediment mass balance is satisfied rigorously via the Exner equation of sediment
conservation (S4).
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• Sand is deposited in the virtual channel, but the implicit shift of this channel spreads the sand
across the entire delta.
• Mud is carried as wash load in the virtual channel, but can deposit overbank. The zone of
overbank deposition is implicitly spread across the entire delta top.
• For each unit of sand deposited within the delta, a specified unit of mud is deposited with it.
The relevant coefficient is obtained from field data.
• Sand that reaches the downstream end of the delta is deposited on a foreset, the slope of
which is specified from field data. This deposition drives delta progradation.
• The basement over which the delta progrades can be horizontal, or can have a specified slope
that represents an average of the actual bathymetry.
• The basement can be allowed to subside at a prescribed rate. The subsidence is assumed to be
spatially uniform.
• The dimensions of the virtual channel, and in particular its width, are computed via a closure
method using a Chezy flow resistance equation, the above-mentioned bed material load
equation, and a specified, field-verified bankfull Shields number characterizing channel
mobility [Parker et al., 1998].
• The flow in the effective channel is computed using a normal-flow formulation with mean
sea level as the downstream datum. Mean sea level is allowed to rise at a specified rate, and
constant for each model run.
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• The model allows for an engineered guide channel of specified width and length to carry the
water and sediment to the vertex of the growing delta. The hydraulics and sand transport in
the guide channels are calculated using the same relations as those for the virtual channel on
the delta.
More detail about the model can be found in the Appendix.
Proof of concept using the Wax Lake Delta.
We first test the model by hindcasting the evolution of the Wax Lake Delta. All the input
parameters are given in Table 1 in the Appendix, along with the data source or justification, both
of which can be found in the Appendix.
Satellite images of the Wax Lake Delta evolution are shown along with corresponding
predictions of the model for the supply rates of suspended sediment estimated above, i.e. a low
value of 25 Mt/yr and a high value of 38 Mt/yr (Fig. 4). Figure 5 shows measured [Majersky et
al., 1997; Roberts et al., 2003; Wellner et al., 2005] and predicted values of the radial distance
from the delta vertex to the shoreline and the surface area (emergent land and freshwater
channels). The two feed rates bracket the data. The land-building model can thus reproduce the
observed land building of the Wax Lake Delta from 1980 to 2005.
The only parameter that was calibrated to obtain the results was the multiplicative factor in
the bed material transport relation, which was set at 2.0. That is, the bed material transport rate
predicted by [Parker et al., 1998] was doubled.
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Application to two engineered avulsions of the Mississippi River below New Orleans.
In order to illustrate the potential for land building in the Mississippi deltaic wetlands below
New Orleans, the model was applied to two adjacent river diversions, one to the southwest into
Barataria Bay and the other to the northeast into Breton Sound. The locations of these engineered
avulsions are chosen based on proposed key strategies in [EFGC, 2006] and [CPRA, 2007]. We
assume that the diversions are implemented via control structures adjacent to the river. These
would serve to activate the diversions only during flood flow. Based on limitations imposed by
the US Army Corps of Engineers to maintain navigation in the main stem, only 45% of the flood
flow would be diverted. This water is split evenly between the two diversions. We assume that
the sediment is similarly split.
In each case, an engineered short guide channel, 5000 m in length is used to stabilize both the
control structure and the vertex of the new delta lobes. Beyond the vertex, the deltas are allowed
to develop freely, just as the Wax Lake Delta has developed since 1973.
We use a base case with a rate of sea-level rise of 2 mm/yr and a subsidence rate of 5 mm/yr
to illustrate typical results. In order to cover the range of uncertainty, variant cases were run with
rates of sea-level rise of 0 and 4 mm/yr, and subsidence rates of 1 and 10 mm/yr, covering the
range discussed above. The input parameters and justifications are given in Appendix.
Figures 6A – 6C shows the results of a run for the base case at 2010, shortly after the
inception of the diversions (Fig. 6A), 50 years later in 2060 (Fig. 6B) and 100 years later in 2110
(Fig. 6C). The model predicts 913 km3 of land building in one century. The results for the base
case and several variant cases are summarized in Table 1.
Figure 6 and Table 1 show that it is indeed possible to build substantial amounts of land over
the time span of a century through engineered avulsions, even in the face of high rates of
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subsidence and sea-level rise. Indeed, the worst case of doubling of both the rate of sea-level rise
and the subsidence rate relative to the base case results in a reduction of the area of land built of
only 24% (Fig. 6D).
Conclusion: the prospects for land-building by means of engineered avulsions.
The continuing growth of the Wax Lake and Atchafalaya Deltas shows that Babbitt’s [2007]
assertion that deltaic drowning is inevitable is fundamentally incorrect. Diverting sediment to the
delta can clearly counteract this sinking and create new land. The assertion of Turner et al.
[2006] that “…hurricanes appear to be the overwhelming pathway of new inorganic sediments
for coastal wetlands in western Louisiana, because the few riverine sources bring relatively
trivial amounts of inorganic sediments into the marsh” is, in our opinion, equally false. The Wax
Lake and Atchafalaya Deltas have been manifestly constructed by sediment derived directly
from distributary channels, not hurricanes, in real time and under close observation.
Both the Wax Lake and the Atchafalaya Deltas are small compared to the main Mississippi
Delta, because they involve relatively small amounts of sediment. The fundamental decision is
not whether delta building is possible, but a) how much sediment we can divert to land building
in the face of competing needs such as navigation, b) how much we reduce the anthropogenic
component of subsidence, and c) how much sea level will rise due to global change. The
scenarios used in our model runs represent realistic, conservative choices for these factors. The
modeling indicates that the main Mississippi Delta cannot be rebuilt to its full pre-settlement size,
but that an area of the order of 1000 km2 of new wetlands could be restored in about a century,
relying mainly on the natural processes that have created and maintained deltas throughout
geologic time. In addition, the model does not yet include the increase in land elevation
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generated by the accumulation of plant organic matter. The role of soil formation may as much
as double the amount of land development during century-long simulations [Neumann and
Macintyre, 1985; Rybczyk and Caboon, 2002].
The physically-based model of deltaic river sedimentation we present here reproduces known
land building with a minimum of tuning. It illustrates that substantial amounts of new land can
be built in the Mississippi Delta by means of engineered avulsions. This new land can be
expected to develop, on its own, new freshwater marshland habitat similar to the habitat
presently being built in the Wax Lake and Atchafalaya Deltas. The models necessary to model
habitat evolution and associated ecological succession are largely available as crude estimates of
ecological change [Twilley et al., 2008]. The next step in our effort is to develop the model into a
form that can be used to a) predict the evolution of the channel structure of the delta, b) provide
input to ecological predictions for delta rehabilitation, and c) create a template for optimizing the
location and scale of controlled avulsions for storm surge suppression, navigation and economic
benefits.
As noted above, the upstream location of any controlled avulsion must be held in place by a
hard-engineered control structure that can control the amount of water diverted. Such structures
are indeed, in the words of Turner et al. [2006] “of considerable cost.” But the cost and
construction methods are well known; the diversion structures would be analogous to the Old
River Control Structure, which currently apportions flow between the Upper Mississippi and
Atchafalaya Rivers upstream of Baton Rouge.
Because it takes decades to build substantial amounts of new land, levee strengthening must
remain the short-term countermeasure against flooding in New Orleans. In the long term,
however, strategically designed and located controlled avulsions complement “hard” protection
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works by partially reversing land loss in the Mississippi Delta, and thus preventing the shoreline
of the Gulf of Mexico from reaching New Orleans.
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Nittrouer, J. A., et al. (2008), Bedform transport rates for the lowermost Mississippi River, J.
Geophys. Res. Earth Surf., 113(F03004).
Paola, C., et al. (1992), The large-scale dynamics of grain-size variation in alluvial basins; 1,
Theory, Basin Res., 4(2), 73-90.
Parker, G., et al. (1998), Alluvial fans formed by channelized fluvial and sheet flow; I, Theory, J.
Hydraulic Eng., 124(10), 985-995.
Parker, G., et al. (in press), Unraveling the conundrum of river response to rising sea level from
laboratory to field. Part I. Laboratory experiments, Sedimentology, DOI: 10.1111/j.1365-
3091.2008.00961.x.
Roberts, H. H., et al. (2003), An embryonic major delta lobe: A new generation of delta studies
in the Atchafalaya-Wax Lake Delta system, Gulf Coast Assoc. Geol. Soc. Trans., 53, 690-
703.
Rybczyk, J. M., and D. R. Caboon (2002), Estimating the Potential for Submergence for Two
Wetlands in the Mississippi River Delta, Estuaries, 25, 985-998.
Shinkle, K. D., and R. K. Dokka (2004), Rates of vertical displacement at benchmarks in the
lower Mississippi valley and the northern Gulf Coast, 147 pp.
Syvitski, J. P. M., et al. (2005), Impact of humans on the flux of terrestrial sediment to the global
coastal ocean, Science, 308(5720), 376-380.
Turner, R. E., et al. (2006), Wetland sedimentation from hurricanes Katrina and Rita, Science,
314(5798), 449-452.
Turner, R. E., et al. (2007), Hurricane signals in salt marsh sediments: Inorganic sources and soil
volume, Limnol. Oceanography, 52(3), 1231-1238.
Twilley, R. R., et al. (2008), Coastal Louisiana Ecosystem Assessment and Restoration Program:
The Role of Ecosystem Forecasting in Evaluating Restoration Planning in the Mississippi
River Deltaic Plain, paper presented at American Fisheries Society Symposium.
Wellner, R., et al. (2005), Jet-plume depositional bodies: The primary building blocks of Wax
Lake Delta, Gulf Coast Assoc. Geol. Soc. Trans., 55, 867-909.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Table 1. Modeled land building
Sea-level rise rate Subsidence rate Area of land created in a century
mm/yr mm/yr km2
Best case 0 1 1217
2 1 1201
0 5 1002
Base case 2 5 918
4 5 845
2 10 753
Worst case 4 10 701
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. 1. Composite histories of fluid (oil, gas and water) production from oil and gas fields and
wetland loss in southern Louisiana. Adapted from Morton et al. [2005].
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. 2. Five major reaches in the Mississippi River and their estimated suspended sediment loads
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. 3. Distribution of estimated background subsidence rates (gray bars) and that of the historic
subsidence rates over the last 80 years (black bars). Data were binned to each 0.5 mm/yr interval.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. 4. Evolution of the Wax Lake Delta as documented in satellite images from 1990, 2000, and
2005, and modeling results for shoreline positions with the sediment supply rates of 25 (dotted
line) and 38 Mt/yr (solid lines) in the corresponding years.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. 5. Modeling results for A) the surface area and B) radial distance of the shoreline from the
vertex of the Wax Lake Delta since 1980 against the observations [circles [Roberts et al., 2003],
squares [Majersky et al., 1997], triangle [Wellner et al., 2005]].
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. 6. Modeling results of two engineered avulsions, starting from 2110. Figs. 6A, 6B and 6C
illustrate results for the base case (i.e., sea-level rise = 2 mm/yr and subsidence rate = 5 mm/yr)
in years 2010 (initial condition), 2060 and 2110, respectively. Fig. 6D shows the result for the
worst case (i.e., sea-level rise = 4 mm/yr and subsidence rate = 10 mm/yr) in 2110. Note that the
channel geometry is copied from the Wax Lake Delta for illustration.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
APPENDIX
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Formulation
1. Notation and Configuration
An engineered guide channel with specified length LG and width BG is connected to the origin of
B
a fan delta (Fig. S1a). The down-valley coordinate through the straight guide channel is denoted
as x. The fan delta, which has specified open angle θF confined by river valleys and/or coastal
morphology, is assumed to obey axial symmetry in its planform geometry by means of sediment
distribution due to channel lateral migration/avulsion, as well as overbank sediment deposition.
The axial streamwise coordinate in the fan delta from its vertex is denoted as r. Radial distances
to the shoreline and the delta toe are rs and ru, respectively.
The position of the fan-delta vertex is represented as (ηF, r) = (ηro, 0) and/or (ηG, x) = (ηro, LG)
where ηG denotes the bed surface elevation in the guide channel and ηF denotes the bed elevation
in the fan delta (Fig. S1b). The total arc length of the fan BF at downstream distance r is
B
BF = θ F r (S1)
Note that the subscripts “F” and “G” denote the fan-delta and guide channel reaches,
respectively. The positions of the shoreline (the topset-foreset intersection) and the delta toe (the
foreset-bottomset intersection) are denoted respectively as (ηF, r) = (ηrs, rs) and (ηru, ru). The
slope of the guide channel is denoted as SG, the slope of the fluvial topset is denoted as St, and
the slope of the foreset is denoted as Sf. Here the foreset slope is taken to be a specified constant.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Note that the slopes are measured positively down-slope. Z denotes the surface elevation of the
standing body of water, i.e. sea level. The time rate of sea-level changes is denoted as ζ, where
∂Z
ζ= (S2)
∂t
Here constant values of ζ are considered. The fan delta is assumed to be prograding over an
antecedent bed with a slope Sb that is also assumed to be constant. The antecedent bed has
elevation ηb and subsidence rate σ, where
∂ηb
σ=− (S3)
∂t
Here values of σ that are constant in time and space are considered.
2. Governing Relations
The time-averaged Exner equation of bed sediment continuity takes the following respective
forms on the fan-delta and guide channel regions
⎛ ∂rqr ∂qθ ⎞
(1 − λ )r ⎛ ∂η
⎜
⎞
+ σ ⎟ = −Ι f (1 + Λ )⎜
F
+ ⎟ (S4a)
⎝ ∂t ⎝ ∂r ∂θ ⎠
p
⎠
⎛ ∂q ⎞
(1 − λ )⎛ ∂η
⎜
⎞
+ σ ⎟ = −Ι f (1 + Λ )⎜ x ⎟
G
(S4b)
⎝ ∂t ⎝ ∂x ⎠
p
⎠
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
where λp denotes the porosity of the sediment deposit, t denotes time, Ιf denotes an intermittency
of channel-forming flood flows, Λ denotes the number of units of wash load deposited in the fan
per unit bed material load deposited, qx denotes the volume transport per unit width of bed
material load in down-valley direction, and (qr, qθ) denotes the volume transport per unit width of
bed material load in the (r, θ) direction. Equation (S4a) is integrated in θ from -θF/2 to θF/2,
where θF denotes the fan angle, under the following constraints: a) the fan is approximated as
axially symmetric in the angular direction and b) channels migrate/avulse across the entire fan.
Equation (S4b) is similarly integrated in the transverse direction y from 0 to BG. The respective
B
results are
⎛ ∂η ⎛ ∂Q ⎞
(1 − λ )θ ⎞
r ⎜ F + σ ⎟ = −Ι f (1 + Λ )⎜ tF ⎟ (S5a)
⎝ ∂t ⎝ ∂r ⎠
p F
⎠
⎛ ∂ηG ⎛ ∂Q ⎞
(1 − λ )B ⎜
⎞
+ σ ⎟ = −Ι f (1 + Λ )⎜ tG ⎟ (S5b)
⎝ ∂t ⎝ ∂x ⎠
p G
⎠
where
θF / 2 BG
QtF = ∫ rqr dθ, QtG = ∫ q x dy (S6a, b)
−θ F / 2 0
The mobility of bed material load (sand) can be quantified in terms of total bed material relation
of Engelund and Hansen [1972], applied during floods when the flow assumed to have attained
the appropriate channel-forming Shields number for sand-bed streams, τ∗form :
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
α EH τ∗form
Qt = Qw St (S7)
R Cf
In the above relation, αEH denotes a dimensionless sediment transport coefficient, R denotes the
submerged specific gravity of the sediment, Qw denotes water discharge and Cf denotes a Chezy
friction coefficient. Parker et al. [1998] have found that the value τ∗form = 1.86 is an appropriate
approximation for many sand-bed streams.
The boundary condition at the shoreline position, r = rs, is given as
ηF rs
=Z (S8a)
where ηF rs
denotes the bed elevation at the shoreline. A continuity condition on sediment
surface elevation is imposed at x = LG, where r = ro = 0:
ηF r = ro
= ηG x = LG
(S8b)
Equations (S5a, b) must be solved subject to a) an initial condition, here specified as a bed with a
specified constant slope, b) a specified upstream sediment feed rate Qt at x = 0, c) a continuity
condition QtF = QtG at r = 0 and d) a shock condition at the foreset as specified below.
3. Shock Condition across the Foreset
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
The Exner equation (S5a) may be integrated across the foreset with respect to r and θ to yield the
following shock relation
⎛ ∂η θ F / 2 ru ∂rq
(1 − λ )∫ θF / 2
− θ F / 2 ∫rs
r ⎜ F + σ ⎟drdθ = −Ι f (1 + Λ )∫
ru ⎞
− θ F / 2 ∫rs ∂r
r
drdθ (S9)
⎝ ∂t
p
⎠
The bed profile across the foreset is given as
ηF = ηF rs
− S f ( r − rs ) (S10)
Taking the time derivative of (S10) yields
∂η F ∂η F ∂η F
= − S s rs + S f rs , S s = −
& & (S11a, b)
∂t ∂t rs ∂r rs
where Ss is the river slope at the shoreline. Combining (S11) and (S9), it is found that
⎛ ∂η ⎞ θ F / 2 ru ∂rq
(1 − λ )∫
p
θF / 2
−θ F / 2 ∫
ru
r⎜ F
rs ⎜ ∂t
+ (S f − S s )rs + σ ⎟drdθ = −Ι f (1 + Λ )∫
&
⎟ − θ F / 2 ∫rs ∂r
r
drdθ (S12)
⎝ rs ⎠
Note that the delta toe location ru is a function of θ over the sloping basement, and thus an
additional relation is required in order to complete the integration in (S12). After some geometry,
this relation is found to be
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
−1
⎛ η F r − ηb r ⎞⎛ ⎞
ru = ⎜ s o ⎟⎜1 − cos θ S b ⎟ (S13)
⎜ Sf ⎟⎜ Sf ⎟
⎝ ⎠⎝ ⎠
The shock condition, which is obtained by substituting (S13) to (S12), takes the following final
form:
⎞ ⎡⎛ ηF rs − ηb ro ⎤
2
⎛ ⎞
(1 − λ p ) 1 ⎜ ∂ηF + (S f − S s )rs + σ
& ⎟ ⎢⎜ + rs ⎟ Θ − θ r 2 ⎥ = Ι (1 + Λ )Q (S14a)
2 ⎜ ∂t ⎟ ⎢⎜ ⎟ ⎥
F s f tFs
⎝ ⎠ ⎣⎝ Sf ⎠ ⎦
rs
where
θF / 2 ⎛ ⎞
Θ=∫ ⎜1 − cos θ S b ⎟ dθ (S14b)
−θ F / 2 ⎜ Sf ⎟
⎝ ⎠
and
θF / 2
QtFs = ∫ rqrs dθ, qrs = qr (S14c)
−θ F / 2 rs
It is assumed that all the sediment delivered to the shoreline is deposited on the foreset, so that qr
= 0 at r = ru (i.e., no sediment is delivered beyond the delta toe).
4. Condition for Movement of the Delta Toe
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Elevation continuity must be satisfied at the foreset-basement break i.e., the delta toe position. In
the present simplified model, this is imposed along the centerline of a fan delta, i.e., θ = 0 as
ηb ru , θ = 0
= ηF rs
− S f ( ru − rs ) θ =0 (S15)
Evaluating the time derivative in the same way as was done for the shock condition, the
following relation between the delta toe and shoreline migration rates is found;
∂ηb
∂t
− S b ru
& θ=0
=
∂η F
∂t
& (
− S s rs − S f ru
& θ=0
− rs
& ) (S16)
ru ,θ = 0 rs ,θ = 0
5. Moving Boundary Formulation
The following transformation is made on the fluvial zone of the fan delta reach, i.e. 0 < r < rs:
r
tˆ = t , r =
ˆ (S17a, b)
rs
It follows that
∂ ∂ r ∂
& ∂ 1 ∂
= −r s
ˆ , = (S18a, b)
∂t ∂tˆ rs ∂r ∂r rs ∂r
ˆ ˆ
so that the Exner equation (S5a) becomes
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
⎛ ∂η r ∂ηF ⎞ Ι f (1 + Λ ) ∂QtF
(1 − λ )θ &
r r⎜ F − r s
ˆ⎜ ˆ + σ⎟ = −
⎟ (S19)
⎝ ∂t rs ∂r ∂r
p F s
ˆ ˆ ⎠ rs ˆ
The shock condition (S14) then transforms to
1 ⎡ ∂ηF 2Ι f (1 + Λ )QtFs ⎤
rs = ⎢− −σ+
&
Sf ⎢ ∂tˆ
⎣ r =1
ˆ (
(1 − λ p ) ΨΘ − θ F rs ⎥
2
⎥
⎦ ) (S20a)
where
2
⎛ η F r − ηb ⎞
Ψ =⎜ + rs ⎟
ro
s
(S20b)
⎜ Sf ⎟
⎝ ⎠
The condition for the delta toe (S16) then transforms to
1 ⎛ ∂η ⎞
&
ru = ⎜ S f rs + F
⎜ & + σ⎟
⎟ (S21)
θ=0
S f − Sb ⎝ ∂tˆ r =1
ˆ ⎠
Input Variables
The land-building model was calibrated and verified against the observed evolution of the Wax
Lake (WL) Delta since 1980. It was then applied to two adjacent diversions of the Mississippi
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
River, one to the southwest into Barataria Bay (BB) and the other to the northeast into Breton
Sound (BS), both for 100 years starting in 2010. The input parameters and justifications are
given in Table 1S.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Table 1S. Model input parameters.
Parameter Units Wax Lake Delta Barataria Bay Breton Sound
Channel-formative flood
m3/s 4,800 25,600 25,600
discharge1
Mean annual sediment
Mt/yr 25.6, 38.4 126 126
discharge2
3
Fraction sand 1 0.183 0.183 0.183
Flood intermittency4 1 0.35 0.34 0.34
Total fraction of water diverted5 1 1 0.45 0.45
Number of diversion sites (water 1 2 2
1
split equally)6 (from WL) (One is BB) (The other is BS)
Unit wash load deposited per
1 0.49 1 1
unit sand7
8
Bed material grain size mm 0.10 0.21 0.21
Sediment specific gravity9 1 2.65 2.65 2.65
Porosity of deposit10 1 0.6 0.6 0.6
Dimensionless Chezy resistance
1 20 20 20
coefficient11
One-time settling thickness into
m 0 0.65 0.65
prodelta12
Elevation of top of foreset above
m 0 0 0
mean sea level13
Initial elevation of bottom of
m -1.5 -0.1 -0.1
foreset14
Initial length of delta from
m 4300 2000 2000
vertex15
16 -5 -5
Initial slope of delta 1 3.0×10 1.5×10 1.5×10-5
17 -4 -4
Subaqueous basement slope 1 1.8×10 2.0×10 5.2×10-5
18
Slope of foreset face 1 0.002 0.005 0.005
Delta spread angle19 deg 120 180 180
Rate of sea-level rise20 mm/yr 2 0, 2, 4 0, 2, 4
Subsidence rate21 mm/yr 5 1, 5, 10 1, 5, 10
Initial slope of guide channel22 1 3.0×10-5 1.5×10-5 1.5×10-5
Width of guide channel23 m 300 5,000 5,000
Length of guide channel24 m 25,000 300 300
Date of model commencement25 1 1980 2010 2010
Channel-forming Shields
1 1.86 1.86 1.86
number26
Multiplicative factor in sediment
1 2 2 2
transport relation27
1
WL: Based on effective flood discharge in the Upper Atchafalaya River with the assumption that 41% of the
floodwater enters the Wax Lake Outlet based on gage records at Atchafalaya Simmesport and Wax Lake Outlet
Calumet [Wright and Parker, 2005]; BB and BS: Effective flood discharge in the Lower Mississippi River [Wright
and Parker, 2005].
2
Values include suspended mud and sand, including sand bedload. The annual load is carried only during floods.
The two values for WL are based on the assumption that 30% (low) or 45% (high) [Roberts et al., 2003] of the flood
flow of the Upper Atchafalaya enters the Wax Lake Outlet. The numbers for BB and BS correspond to the values for
the Lower Mississippi estimated in the text [Allison et al., 2000; Horowitz, 2006; Nittrouer et al., 2008], from where
the floodwater is assumed to be diverted.
3
17% of the suspended load is sand [Allison et al., 2000], the sand bedload is equal to 1.6% of the suspended load
[Nittrouer et al., 2008].
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
4
The value in Wright and Parker [2005] for the Upper Atchafalaya is applied to WL; the corresponding value for
the Lower Mississippi is applied to the flow diverted to BB and BS.
5, 6
All the floodwater in the Wax Lake Outlet goes to WL. Only 45% of the floodwater in the Lower Mississippi is
assumed to be diverted; this is split equally between BB and BS. Such a diversion leaves 14,150 m3/s or 500,000
ft3/s in the main-stem Mississippi River during floods to allow for navigation.
7
The value of 0.49 for WL is explained in the text. Because WL faces the open shelf, but BB and BS are in more
protected zones, a higher value, i.e. a reasonable estimate of 1 is used there, based on personal communications from
H. Roberts, Louisiana State University (emeritus) USA and T. Tornqvist, Tulane University, USA.
8
WL: [Dumars, 2002]; BB and BS: Value in the Lower Mississippi near the diversions [Nittrouer et al., 2008].
9
The value for quartz has been used.
10
From [Meckel et al., 2006; Meckel et al., 2007] for bay mud and sand bars.
11
Estimate for large, low-slope sand-bed streams based on Parker et al. [in press]
12
It is assumed here that the prodelta can have a layer of high-porosity mud, which is rapidly compacted on a one-
time basis as the delta foreset propagates over it. In the case of BB and BS, the layer is assumed to be 1.45 m thick
[Morton et al., 2005] with a porosity of 0.78 [Meckel et al., 2006], which compacts on a one-time basis to produce
the indicated subsidence.
13
Initial sea level is set at zero.
14
WL: [Roberts et al., 2003]; BS and BB: Estimated using the National Land Cover Data (NLCD) 2001 Elevation
and Bathymetry Data.
15
WL: As of 1980, [Majersky et al., 1997]; BS and BB: Arbitrarily chosen short distances to start the model.
16
Estimated using relations of Parker et al. [in press].
17
WL: [Roberts et al., 2003]; BS and BB: NLCD 2001 Elevation and Bathymetry Data.
18
WL: [Wellner et al., 2005]; BS and BB: Estimated based on the dip section of the Mississippi River Delta in
Coleman [1988] and Gould [1970].
19
WL: From satellite images; BB and BS: Maximum space-filling value.
20
WL: Adopted middle value; BS and BB: Low value, middle value and high value, as justified in the main text.
21
WL: Adopted middle value; BS and BB: Low value, middle value and high value, as justified in the main text.
22
WL, BB and BS: Estimated using relations of Parker et al. [in press].
23
WL: Measured from Wellner et al. [2005]; BS and BB: Assumed values for a short channel intended to maintain
pathway from diversion structure on Mississippi River to the delta vertex.
24
WL: 25 km reach of the Wax Lake Outlet; BS and BB: Assumed values for a short channel intended to maintain a
pathway from diversion structure on Mississippi River to the delta vertex.
25
WL: Date by which the delta length had reached the initial length that is given in the 15th row; BB and BS:
Assumed for illustrative purposes.
26
Taken from Parker et al. [in press], based on data for large, low-slope sand-bed rivers.
27
Value evaluated by calibration to the Wax Lake Delta.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
References
Allison, M. A., et al. (2000), Development and reworking of a seasonal flood deposit on the
inner continental shelf off the Atchafalaya River, Cont. Shelf Res., 20(16), 2267-2294.
Coleman, J. M. (1988), Dynamic changes and processes in the Mississippi River delta, Geol. Soc.
Am. Bull., 100(7), 999-1015.
Dumars, A. J. (2002), Distributary Mouth Bar Formation and Channel Bifurcation in the Wax
Lake Delta, Atchafalaya Bay, Louisiana, 88 pp, Louisiana State University.
Engelund, F., and E. Hansen (1972), A Monograph on Sediment Transport, 3rd ed., 62 pp.,
Technisk Forlag, Copenhagen, Denmark.
Gould, H. R. (1970), The Mississippi Delta complex, in Deltaic sedimentation: Modern and
ancient, edited by J. P. Morgan, pp. 3-30, SEPM Special Publication.
Horowitz, A. J. (2006), The effect of the "Great Flood of 1993" on subsequent suspended
sediment concentrations and fluxes in the Mississippi River Basin, USA, IAHS-AISH
Publ.(306), 110-119.
Majersky, S., et al. (1997), Facies Development in the Wax Lake Outlet Delta: Present and
Future Trends, Basin Res. Inst. Bull., 7, 50-66.
Meckel, T. A., et al. (2006), Current subsidence rates due to compaction of Holocene sediments
in southern Louisiana, Geophys. Res. Lett., 33(11).
Meckel, T. A., et al. (2007), Sediment compaction rates and subsidence in deltaic plains:
Numerical constraints and stratigraphic influences, Basin Res., 19(1), 19-31.
Morton, R. A., et al. (2005), Historical subsidence and wetland loss in the Mississippi Delta plain,
Gulf Coast Assoc. Geol. Soc. Trans., 55, 555-571.
Nittrouer, J. A., et al. (2008), Bedform transport rates for the lowermost Mississippi River, J.
Geophys. Res. Earth Surf., 113(F03004).
Parker, G., et al. (1998), Alluvial fans formed by channelized fluvial and sheet flow; I, Theory, J.
Hydraulic Eng., 124(10), 985-995.
Parker, G., et al. (in press), Unraveling the conundrum of river response to rising sea level from
laboratory to field. Part II. The Fly-Strickland River System, Papua New Guinea,
Sedimentology, DOI: 10.1111/j.1365-3091.2008.00962.x.
Roberts, H. H., et al. (2003), An embryonic major delta lobe: A new generation of delta studies
in the Atchafalaya-Wax Lake Delta system, Gulf Coast Assoc. Geol. Soc. Trans., 53, 690-
703.
Wellner, R., et al. (2005), Jet-plume depositional bodies: The primary building blocks of Wax
Lake Delta, Gulf Coast Assoc. Geol. Soc. Trans., 55, 867-909.
Wright, S., and G. Parker (2005), Modeling downstream fining in sand-bed rivers. I: formulation,
J.Hydraulic Res., 43(6), 613-620.
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Hydrogeomorphology CLEAR Vol IV, Chapter 10 June 2008
Fig. S1. Definition sketches for the definition of variables: A) planform and B) downstream
section.
40