We explore the role of plant matter accumulation in the sediment column in determining the response of fluvial-deltas to base-level rise and simple subsidence profiles. Making the assumption that delta building processes operate to preserve the geometry of the delta plain, we model organic sedimentation in terms of the plant matter accumulation and accommodation (space made for sediment deposition) rates. A spatial integration of the organic sedimentation, added to the known river sediment input, leads to a model of delta evolution that estimates the fraction of organic sediments preserved in the delta. The model predicts that the maximum organic fraction occurs when the organic matter accumulation rate matches the accommodation rate, a result consistent with field observations. The model also recovers the upper limit for coal accumulation previously reported in the coal literature. Further, when the model is extended to account for differences in plant matter accumulation between fresh and saline environments (i.e., methanogenesis versus sulfate reduction) we show that an abrupt shift in the location of the fresh-salt boundary can amplify the speed of shoreline retreat.
 The task now is to quantify and incorporate organic sedimentation into a delta growth model. In Figure 1we present a sketch of a cross-section of a river delta. The system is fed by fresh water and sediment from the river system, and evolves on top of the basement under conditions of sea level rise and subsidence. The key geometric feature in this system is the deposited sediment prism. This shape is bounded below by the basement, and from above by the subaerial delta plain and the subaqueous foreset. The delta plain is delimited by the shoreline and the alluvial-basement transition, and the foreset by the shoreline and the delta toe. The movements of these three geomorphic boundaries (alluvial-basement transition, shoreline and delta toe) define the evolution of the delta in cross-section.
 Within the system shown in Figure 1, the sediment volume per unit width in the prism V is determined by the external bulk sediment input from the river system qin, and the rate of accumulation of plant matter within the delta plain. River sediment inputs qin are either trapped in the delta plain, which results in shoreline aggradation, or bypass the delta plain and deposit in the subaqueous foreset, which drives shoreline progradation. The partitioning of sediment between the delta plain and the foreset is controlled by the geometry of the deltaic prism. To first order we assume that all the sediment input from the river is trapped within the deltaic prim, but we recognize that a significant fraction might be transported beyond the delta toe [Allison et al., 1998; Blum and Roberts, 2009]. If we define χ(x, t) to be the thickness of organic as opposed to inorganic sediments at each location, the volume of sediments in the prism, at time t, is then given by
where the second term on the right is an integration over the sediment prism in Figure 1 with x = Uthe location of the alluvial-basement transition andx = W the location of the delta toe.
 In this initial modeling exercise we assume that qin is constant, and that the in situ production of organic matter in the subaqueous foreset (between the shoreline L and the delta toe W) is negligible. We recognize, however, that in real systems the burial of organic sediments in the shelf could be significant [Bianchi, 2011; Sampere et al., 2011], and future (and more elaborate) versions of the model will explore the potential consequences of this process on delta evolution. Under these conditions, it follows from (1) that the rate of change of the prism sediment volume can be written as
Here we recognize that the sediment column has zero height at the alluvial-basement transition and toe, and have definedvorg(x) = ∂χ/∂t as the rate of organic sedimentation at any location x in the delta plain.
 We consider two scenarios: base-level rise and differential subsidence. In the base-level rise case we assume the subsidence, the sum of crustal processes and compaction, to be spatially uniform. In contrast, in the differential subsidence scenario the subsidence rate increases linearly seawards from a pivot location (seeFigure 1). We assume that in a Holocene context this pivot location is located approximately at the landward limit of the onlap, which is ∼100 km upstream from the shoreline [Blum and Törnqvist, 2000]. Thus, the main ingredients in a model for delta evolution are (1) the interplay between the rate of change of the sediment per unit width in the prism , and (2) either the combination of sea level and subsidence expressed as a base-level rise , or the pivot subsidence rate . The base-levelZ and basement slope β are defined as
where Z0 and β0are the base-level and basement slope att = 0. The origin is chosen to be located on the basement, at one characteristic basin length distance landwards from the initial shoreline location, and with x positive in the seaward direction and z positive downward (Figure 2a). As characteristic basin length we choose the distance from the landward limit of the onlap to the shoreline (i.e., [Blum and Törnqvist, 2000]). In this way, in the pivot scenario the origin coincides with the pivot location. For completeness, in Figure 2b, we also show the realization of the concept of the shape preservation under a situation in which, at a point in time, the sediment supply and accumulation cannot sustain a shoreline advance resulting in an abandonment of the foreset and a shoreline retreat.
 Taking account of our shape preserving assumption we can constrain the rate of organic sedimentation in (2) by
where P is the organic matter accumulation rate — fully discussed in the previous section—and Athe rate of accommodation due to base-level rise (i.e., ) or differential subsidence (i.e., ). The model in (4) says that, in situations where the organic matter accumulation rate outstrips the rate of accommodation, i.e., P > A, the organic excess—to preserve shape—is either rapidly decomposed via aerobic respiration (i.e., oxidation) or eroded away. Under these conditions, an increase in accommodation rate A would lead to an increase in organic sedimentation vorg, which is consistent with previous formulations and field observations [Morris et al., 2002; Mudd et al., 2009]. On the other hand, if the organic matter accumulation rate cannot keep up with the accommodation rate, i.e., P < A, it is assumed that the shape is preserved by filling in the shortfall with the available inorganic sediment supply. In this case, an increase in accommodation rate A would not lead to an increase in organic sedimentation vorg, and if the inorganic sediment is insufficient there is potentially a retreat in the shoreline location.
 Note that equation (4) implies that spatial changes in the inorganic sedimentation rate vin(x, t) are a function of the space left for deposition by the organic sediments, i.e., max (A-P, 0). For instance, under the base-level rise scenario, if we assume that the organic matter accumulation rateP decreases downstream (as discussed in section 2) and is always less than the accommodation rate , then inorganic sedimentation rate must increase downstream in order to satisfy our shape preservation assumption. In the first instance, however, we assume a constant value for the organic accumulation rate P(an assumption that will be relaxed later). In this way, for the base-level rise case we can use (4) to evaluate the integral on the right hand side of (2) and arrive at
and the differential subsidence case becomes
with initial conditions V = 0, L = 1, and U = 1. For given values of P, qin and A, the coupling of the solution of (5) to the geometric features in Figure 2 will fully realize a model of delta growth. We introduce a list of all the state variables with their dimensions in Table 1.
Table 1. State Variables and Their Dimensions
Dimensions (L, length, T time)
Horizontal distance positive in the downstream direction
Vertical distance positive downward
Volume per unit width of the deltaic prism
River sediment flux into the deltaic system
Alluvial-basement transition location
Delta toe location
Thickness of organic sediment
Rate of change of the volume per unit width of the deltaic prism
 A more general solution approach is achieved by casting (5) into a dimensionless form. Toward this end we use a characteristic basin length , and a time scale using the river sediment input . We identify the following dimensionless variables
And the following dimensionless groups that specify the behavior of the system
With the definitions in (6), and dropping the dsuperscript for convenience of notation, the dimensionless governing equations under base-level rise and differential subsidence become
where , and with initial conditions V = 0, L = 1, and U = 1. The dimensionless base-levelZ and basement slope β are written as follows
We include a list of the dimensionless state variables in Table 2. Using the parameter values discussed in previous section, we constrain the dimensionless groups defined in (6b) as a reference for the following calculations. We note that the fluvial slope ratio is in the range γ ∼ 0.1–0.01, which validates the linear fluvial slope assumption, and that the foreset slope ratio is typically very large (ψ ≫ 1), implying that the sediment stored below the foreset is relatively very small and can be dropped without error from any volume balance calculation. The current dimensionless base-level rise rate in the Mississippi River Delta is ∼ 0. 7 , and the dimensionless organic matter accumulation rate in the range P ∼ 0 – 0.7.
Note that for convenience of notation they share symbol with the state variables with dimensions introduced in Table 1.
Horizontal distance positive in the seaward direction
Vertical distance positive downward
Volume per unit width of the deltaic prism
Alluvial-basement transition location
Delta toe location
Rate of change of the volume per unit width of the deltaic prism
Rate of inorganic sedimentation
Rate of inorganic sedimentation that exceeds the accommodation rate and is trapped within the delta plain
Dimensionless rate of organic sedimentation
Fluvial plain slope
Rate of pivot subsidence
Subaqueous foreset slope
Rate of base-level rise
Rate of accommodation
Rate of organic matter accumulation
Rate of organic matter accumulation in fresh water environments
Length of the fresh water environments
Rate of organic matter accumulation in saline environments
Length of saline environments
Fraction of the delta plain with saline ecosystems
Time of conversion from fresh to saline environments
5. Model Solution
 At a given time t > 0, if the right hand side of (7) and the current volume of the delta deposit Vold are known, the value of the volume V at a small increment of time Δt beyond time t can be calculated with a simple Euler scheme
where is given by (7). The integral on the right hand side of (7b) is estimated numerically using a simple trapezoidal rule
where N + 1 is the number of equally spaced grid points.
 In order to define the delta profile evolution and to generate the information to complete subsequent time steps calculations we need a means of extracting the positions of the alluvial-basement transitionU(t) and the shoreline L(t) from this updated value of the total volume, V. In Tables 3 and 4 we include a list of the relevant calculation steps to achieve this; all the expressions are derived from direct geometric arguments.
Table 3. Geometric Relationships for the Base-Level Rise Scenarioa
The value of Z is obtained from equation (2a).
L = Lold + ΔL
Alluvial-basement transition position
Table 4. Geometric Relationships for the Pivot Subsidence Scenarioa
The value of β is obtained from equation (2b).
L = Lold + ΔL
Alluvial-basement transition position
 The calculation requires three input parameters: the slope ratio γ, the organic matter accumulation rate P, and either the base-level rise rate or the pivot subsidence rate . In operation, following the update of the volume V from (9) we first calculate the shoreline change increment within a time step ΔLand then the shoreline and alluvial-basement transition positionsL and U. If the shoreline moves seawards (i.e., ΔL ≥ 0), we follow the calculations on the left column in Tables 3 and 4. At some point in the calculation, however, the sediment supply and the organic matter accumulation cannot sustain a shoreline advance causing a shoreline retreat indicated by an estimated value ΔL < 0. In this case we switch to the expressions in the right column in Tables 3 and 4. In connecting to previous models, we note that if the organic matter accumulation rate is zeroed out, i.e., P = 0, the model presented in Table 3 recovers the previous model developed by Kim and Muto .
 In Figure 3we present the typical model behavior under constant base-level rise (Table3). We choose a base-level rise rate = 0.8, a the slope ratio γ = 0.01, and an organic matter accumulation rate P = 0.4. Initially the total rate of sediment input exceeds the total accommodation rate in the delta plain (L − U), and, as depicted in Figure 3, the shoreline trajectory moves seawards ΔL > 0. As time increases the delta length (L − U) increases monotonically and at some point in time cannot be maintained by the sediment input, which leads to shoreline retreat, i.e., ΔL < 0. Both the alluvial-basement transition and the shoreline reach a constant landwards speed ΔU/Δt = ΔL/Δt = − and the delta length reaches a steady value (L − U) = 1/((1 − γ) − min( , P)).
 In Figure 4 we present the typical model behavior under pivot subsidence (Table 4). We use two subsidence rates: (solid lines) and (dashed lines). The slope ratio is γ = 0.01, and the organic matter accumulation rate is P = 0.4. Initially, similarly to the base-level rise scenario, the total rate of sediment input exceeds the total accommodation rate , and, as depicted in Figure 4, the shoreline trajectory moves seawards ΔL > 0. At later stages of delta growth, however, the shoreline can have two possible behaviors: (1) it can monotonically approach a steady location (solid line), or (2), it can overshoot and retreat before reaching the steady location (dashed line). The shoreline reaches the steady location and the delta length reaches a steady value when the sediment input rate balances accommodation rate: at this point the alluvial-basement transition is fixed at the pivot location (seeFigure 5).
 We note that the behaviors depicted in Figures 3 and 4 apply for any set of parameter values γ, P, , and as long as the accommodation rate is positive and finite (i.e., > 0 or ).
6. Calculating Organic Fraction
 A simple and worthwhile extension of the model presented above is to calculate the organic fraction Cf, defined as the ratio between the organic and total (i.e., organic and inorganic) sediment volume in the sediment column. We define the average organic fraction in the time interval (t1, t2) at a given location x in the delta plain in terms of the organic vorg and inorganic vin sedimentation rates as follows
The organic sedimentation vorg is described in equation (4). The inorganic sedimentation vin first fills the fraction of the accommodation A not occupied by organics, and the excess (if any) is distributed between the delta plain and the subaqueous foreset. Since the accommodation rate Ais constant in time for the base-level riseA = and differential subsidence scenarios, we can write (11) as follows
where (vin)dpis the excess of inorganic sediments retained in the delta plain. We exclude the excess that accumulate in the foreset; an approach consistent with how measurements are made in the field, which only consider the organic-rich sections of any given core [Bohacs and Suter, 1997; Diessel et al., 2000]. During shoreline advance ΔL ≤ 0, (vin)dp is obtained through geometric construction, and during shoreline retreat ΔL ≤ 0 the entire inorganic sediment supply is included in the accommodation term A. We can then write it as follows
7. Comparison With Coal and Peat Data
 Coal geologists have observed that the fundamental control of the organic fraction in the sediment column Cf is the ratio between accommodation rate A (space made for sediment accumulation) and organic matter accumulation rate P. [Bohacs and Suter, 1997; Diessel et al., 2000].
 Here we assume Pto be a constant value, an assumption that will be relaxed in the next section. In the base-level rise scenario, the accommodation rateA = is also a constant. Thus, substituting A = and P into equation (12) we obtain a constant organic fraction Cf along the delta plain. In Figure 6 we plot the organic fraction as a function of the ratio /P for the input parameters indicated. A key observation, independent of the parameter values chosen, is the occurrence of a maximum organic fraction when the accommodation rate matches the organic matter accumulation rate, i.e., /P = 1. This result matches the widely made observation in the coal literature of maximum peat fractions in systems where organic matter accumulation rate matches the accommodation rate, i.e., /P ∼ 1 [Bohacs and Suter, 1997; Diessel et al., 2000]. We also include the range for coal accumulation (i.e., Cf > 0.75), and note that the model recovers within a narrow range the upper limit previously obtained by Bohacs and Suter  (i.e., /P = 1.18). The lower limit for coal accumulation, however, is lower than what is typically observed in the field [Bohacs and Suter, 1997; Diessel et al., 2000] and more sensitive to the input parameters. Future work will study the sensitivity of the coal accumulation range in more detail.
 Under differential subsidence, both the accommodation rate and the organic fraction Cf are functions of location on the delta plain x. In Figures 7 and 8 we plot the organic fraction as a function of . In Figure 7 we vary the ratio and fix the location at x = 1, whereas in Figure 8 we vary the location x and fix the ratio . Again, the key observations are: (1) the occurrence of the maximum organic fraction when the accommodation rate matches the organic matter accumulation rate A/P = 1 [Bohacs and Suter, 1997; Diessel et al., 2000], and (2) the upper limit for coal accumulation that emerges from the model matches within a narrow range the result obtained by Bohacs and Suter .
 We view these results as a validation of our modeling approach, in particular our assumption of fluvial shape preservation that constrains the organic sedimentation to be the minimum of the organic matter accumulation rate P and accommodation rate A (equation (4)).
8. The Role of the Fresh-Salt Boundary Dynamic in Delta Evolution
 In the initial derivation and application of our model we have assumed a constant value for the organic matter accumulation rate P. We now generalize this assumption by accounting for the potentially important effects of different organic matter accumulation rates between saline and fresh environments (see discussion in section 2). On deltaic coasts, several factors control the extent of the fresh water region such as the fresh water supply, precipitation intensity, base-level fluctuations, and topography, but numerical modeling and field studies in the Ebro delta point out the fresh water inputs from the river as the main control [Ibañez et al., 1997; Sierra et al., 2004]. A reduction in the river water supply leads to saltwater intrusion, which accelerates the decomposition of organic soils due to the shift in anaerobic respiration from methanogesis to sulfate reduction [Capone and Kiene, 1988; Portnoy and Giblin, 1997; Portnoy, 1999; Ibañez et al., 2010]. Human activities such as channelization play a key role in controlling the fresh water supply [Day et al., 1997; Williams et al., 1999; Sierra et al., 2004]. Additionally, changes in water supply on centennial to millennial time scales can be caused naturally by climate change or an abandonment of the river channel. Paleoflood chronologies from the Mississippi river have been used to conclude that minor changes in climate can produce very high changes in water discharge [Knox, 1993].
 To illustrate how our model can be generalized to account for spatial variations in organic matter accumulation controlled by changes in fresh-water inputs, we consider a scenario of a delta initially in a purely freshwater environment that at some timet* converts to a delta that contains both a fresh region of length f and a saline region of length s.This transition is taken to be instantaneous, i.e., short compared to the time scale of delta response. This example does not intend to model any system in particular, but aims to explore the importance of the fresh-salt transition on delta evolution. If we denote organic matter accumulation rate in the fresh region byPf and in the saline region by Ps, a modified form of the governing equation in (7) can be written as
which on specification of an appropriate relationship between f, s, L and U can be solved by a simple extension of the approach detailed in Tables 3 and 4. For the case under consideration these relationships are
where k is the fraction of the delta that is eventually under saline conditions.
 Under a steady base-level rise, an early shift of the fresh-salt transition modifies the time and the location at which the shoreline retreat begins (Figure 9a). When the delta is always fresh (k = 0), the shoreline advances farther and for a longer time period than a combined fresh/saline delta (k = 0.5, 1). In contrast, a shift in a more advanced stage of delta growth leads to an abrupt increase of the shoreline retreat speed (Figure 9b).
 Similarly, under differential subsidence the fresh-salt transition shift can also lead to an abrupt increase of the shoreline retreat speed before reaching a steady location (Figure 10). Therefore, Figures 9 and 10suggest that the extension of the fresh water region can potentially have a strong effect on shoreline dynamics. The higher organic matter accumulation in fresh water environments compared to saline environments implies that a reduction in fresh water inputs can lead to a rapid shoreline retreat. Episodes of punctuated shoreline retreat observed in the Gulf of Mexico, however, are usually interpreted as an increase in the rate of base-level rise [Rodriguez et al., 2010], a reduction in river sediment input [Milliken et al., 2008], or topography complexity [Rodriguez et al., 2004, 2005]. We present the motion of the fresh-salt boundary as a potential candidate to explain (or amplify) this back stepping shoreline behavior.
 We present a simple geometric model that for the first time captures the basic interplay of organic and clastic deposition in deltas under base-level rise and differential subsidence. The model reproduces a central observation from coal geology, that organic fraction is maximized when the organic matter accumulation and accommodation rates are just balanced. The model also recovers the upper limit for potential coal accumulation predicted byBohacs and Suter . Moreover, the model shows that the imbalance in organic sedimentation between fresh and saline environments can significantly alter delta evolution. In particular, a landwards shift of the fresh-salt transition caused by a reduction in fresh water inputs can lead to a punctuated shoreline retreat.
 A number of modifications will significantly improve the utility of the model presented here. In particular we plan to include a more complete mechanistic description of the movement of the fresh-salt boundary, which is currently assumed to be an input of the model. Additionally, we need to incorporate field and experimental data to better constrain the model parameters of organic matter accumulationPf and Ps. These improvements will lead to a consistent mathematical framework for the characterization of the accretion/oxidation of organic soil in deltas. Such a tool can be used to advance existing modeling efforts pertaining to land building in the Mississippi River Delta [Kim et al., 2009]. Additionally, we intend to use it to better understand the relative importance of variations of the fresh-salt boundary compared to variations on allogenic controls such as base-level or river sediment input. Future versions of the model will also aim to address current concerns related to the increase of nutrient supply into the wetland ecosystems of the Mississippi River Delta [Perez et al., 2011]. Among other important effects, an increase in nutrient concentration in river water can enhance decomposition and aboveground versus below-ground productivity [Perez et al., 2011]. This can result in an increase in vulnerability of wetland vegetation, and thus, a decrease in long-term plant matter accumulation in the sediment column.
 We are grateful to the reviews from Kevin Bohacs, Torbjörn Törnqvist, Matt Kirwan, and Alexander Densmore which have helped to improve the clarity of the manuscript. This work was supported by the STC program of the National Science Foundation via the National Center for Earth-surface Dynamics under the agreement EAR-0120914.