A one-dimensional eco-geomorphic model of marsh response to sea level rise: Wind effects, dynamics of the marsh border and equilibrium

Authors


Abstract

[1] In this study we investigate the long term evolutionary trend of tidal marshes and their possible tendency to approach equilibrium. We account of the dynamic interaction between the marsh and its adjacent environment, allowing for both variations of marsh elevation and displacement of the marsh boundary. We thus consider a 1-D configuration consisting of a tidal channel merging into a 1-D marsh. Starting from some initial configuration of the channel we model how a salt marsh forms at the landward end of the channel and determine the long term evolution of the channel - marsh configuration under different scenarios of relative sea level rise. Previously established results on the morphodynamic evolution of a tidal channel are extended accounting for the effects of vegetation and wind driven sediment resuspension in regions where tidal stresses are too weak to mobilize sediments. Results suggest that, for sufficiently low rates of relative sea level rise the marsh platform may be able to reach an equilibrium elevation, provided wind resuspension is able to maintain a sufficiently large sediment concentration close to the marsh boundary. This is in general agreement with recent results based on zero dimensional modeling. However, we find that the marsh boundary is unstable, as progradation or retreat is generally experienced depending on a delicate balance between tidal transport and transport driven by wind setup. In this sense, actual morphodynamic equilibrium is a rather exceptional and unstable state.

1. Introduction

[2] One of the most alarming consequence of global warming is an accelerated sea level rise [e.g., Rahmstorf, 2007]. In many low-lying coastal areas, increasing vulnerability to inundation is accompanied by a progressive deterioration of natural defences against flooding, i.e., barrier islands and wetlands [Day and Templet, 1989; Seminara et al., 2012]. In order to understand wetland degradation one needs to investigate the variety of mechanisms which contribute to marsh evolution and the way sea level rise may alter them.

[3] Salt marshes are characterized by the presence of halophytic vegetation and of microphytobenthos [e.g., Cronk and Fennessy, 2001], which play a major role in marsh evolution. Vegetation contributes to marsh building through its effects on marsh hydrodynamics. Increasing friction and damping turbulence, marshes oppose the resuspension of sediments, i.e., they function as sinks for suspended sediments [e.g., Le Hir et al., 2007; Mudd et al., 2004; Christiansen et al., 2000]. Vegetation also produces organic sediments through the growth, death and decomposition of aboveground and belowground biomass. Hence, marsh equilibrium results from a delicate interplay among factors affecting the sediment supply to the marsh and factors, such as sea level rise and subsidence, which, by affecting the depth of the marsh, also affect the production of biomass.

[4] A quick glance at the recent history of Venice lagoon, which is one of the main concerns of the paper, suggests that both the above factors have contributed to the degradation process displayed in the last century [Consorzio Venezia Nuova Technital, 2004; Day et al., 1999]. In fact, the amount of mineral sediments supplied to the lagoon has decreased as a result of various anthropogenic actions, most notably the diversion of all the rivers debouching into the lagoon. On the other hand, the construction of inlet jetties has induced inlet deepening, while sea level rise and subsidence (≃2.3 mm/yr) [Carbognin et al., 2004; Canestrelli and Battistin, 2006] have deepened tidal flats [Defina et al., 2007]. Deeper flats have made wind waves more energetic, enhancing sediment resuspension and the loss of sediments from the flats to the adjacent channels. The observed consequence has been the progressive erosion of the salt marsh margins [Cecconi, 2005].

[5] However, examples are known [e.g., Redfield, 1965, 1972] of salt marshes which have been able to adjust their elevation to sea level rise for as long as 4,000 years through deposition of mineral sediments and production of organic matter, such to maintain an elevation of the marsh platform within the intertidal zone close to the mean high tide (MHT) [Krone, 1987]. Observations taken from many estuaries worldwide support the importance of mineral sediment availability in the salt marsh evolutionary process. In the Mississipi/Atchafalaya River, different regions have responded to sea level rise in very different manners despite similar tidal ranges (<1 m) and relative sea level rise rates (≃10 mm/yr) [Cahoon et al., 2000]. Specifically, observations suggest stability and rapid accretion in the extensive marshes surrounding Old Oyster Bayou where the typical mineral suspended sediment concentration is of the order of 70 mg/l [Wang, 1997; Perez et al., 2000], and rapid submergence and erosion in Bayou Chitique characterized by a mean suspended sediment concentration of only 20 mg/l [Wang, 1997]. In some circumstances, such as in the Yangtze River Delta (China), marshes persist and even expand seaward despite high subsidence-generated rates of relative sea level rise (>100 mm/yr) by accreting at rates exceeding 50 mm/yr [Yang, 1999], due to very high mineral sediment input (suspended sediment concentrations of the order of 1000 mg/l).

[6] The variety of salt marsh responses outlined above points at the need to investigate deeply the problem of salt marsh stability. This is a major issue and has attracted the attention of various scientific communities for decades. Fagherazzi et al. [2012]have recently presented a broad overview of numerical models that quantify the formation and evolution of salt marshes under different physical and ecological drivers, especially focusing on spatial models that are mainly two- or three-dimensional. In order to clarify in which context the present model takes place, in the following we will briefly recall the present state of the art on the issue of salt marsh stability.

[7] The first class of models appeared in the literature may be described as zero dimensional models The common characteristic features of these models are: (i) marsh evolution is essentially decoupled from the coevolution of the adjacent flats, modeled as steady sources of minerogenic sediments; (ii) marsh properties are spatially averaged. These models stem from the pioneering works of Randerson [1979] and Krone [1987]. The spatially averaged mass balance envisaged by Krone [1987], can be written in the following general form:

display math

where math formula is the spatially averaged marsh elevation relative to mean sea level; amin is the average net accretion rate driven by the exchange of inorganic (minerogenic) sediments between the marsh and the surrounding environment; aorg is the accretion rate induced by the production of organic sediments on the marsh; scom is the rate of marsh lowering arising from compaction of sediment deposits; srslr is the rate of marsh lowering arising from regional subsidence and sea level rise.

[8] Zero dimensional models have allowed for quali-quantitative predictions of the long term morphological response of marshes to variations of external forcings. The early models [Krone, 1987; Allen, 1990; French, 1993] predicted that the marsh platform would always be able to keep pace with relative sea level rise because an acceleration of the latter would drive a rise in the sedimentation rate. Only the most dramatic eustatic scenarios would result in ecological drowning [French, 1993]. Morris et al. [2002], however, argued that, in accordance with the observations of Allen [1995], marshes are stable against changes in relative mean sea level provided the surface elevation keeps larger than the optimal elevation for primary production. Similar conclusions were drawn by Mudd et al. [2009]. A richer picture emerged from the model of Marani et al. [2007]: various stable and unstable marsh equilibria would be possible depending upon suspended sediment availability, type of vegetation, disturbance of the benthic biofilm and rate of sea level rise. A related zero dimensional model aimed at explaining tidal flat - salt marsh transition in shallow tidal basins was proposed byFagherazzi et al. [2006]. This work was motivated by the observation that extensive tidal flats and salt marshes in shallow basins (notably in Venice lagoon) lie within specific ranges of elevation. Identifying the condition of tidal flat equilibrium with a balance between the average annual erosion and deposition rates, the authors envisage two possible scenarios: (i) if the deposition rate is lower than the maximum erosion rate, then two equilibrium elevations of the flat surface exist, with the larger elevation being stable and the lower being unstable; (ii) for deposition rates higher than the maximum erosion rate no equilibrium elevations of the flat platform would exist and the system would evolve toward the salt marsh configuration. These conclusions have been confirmed in the later works of Fagherazzi et al. [2007] and Defina et al. [2007]: here, the inclusion of the further effect of tidal currents appeared to affect tidal flat equilibrium significantly.

[9] The insight provided by zero dimensional models, along with the awareness of their intrinsic limits, encouraged the development of models where the marsh properties are no longer spatially averaged and/or marsh evolution is no longer decoupled from the morphodynamic evolution of the adjacent environments.

[10] A large body of literature exist describing salt marsh evolution in the context of 1-D modeling [e.g.,Temmerman et al., 2004; Woolnough et al., 1995; Mudd et al., 2004]. In particular, Mudd et al. [2004], considering the evolution of a marsh surface on a transect perpendicular to a tidal creek, show that a marsh dominated by organogenic sedimentation subject to steady sea level rise develops a fairly flat surface and adjusts to changes in sea level less promptly than a marsh dominated by sediment trapping, which develops a gently sloping surface. Van de Koppel et al. [2004]presented a 1-D model that simulates the evolution of the scarp flat-marsh boundary as a function of vegetation biomass and wave characteristics. Quite recently,Mariotti and Fagherazzi [2010]extended the above approach by including wave generation and propagation, tidal oscillations, sediment transport and the feedbacks between vegetation and sediment depositions. In particular, wave propagation was described by the 1-D conservation equation for wave energy, tidal currents were calculated using a quasi-static model and a linear relationship was employed to relate organogenic sedimentation to biomass. Results showed that, for a given sediment supply, the scarp can prograde or retreat depending on the rate of sea level rise. Finally, the problem of coupled evolution of a marsh platform and its channel network has been addressed byD'Alpaos et al. [2007] and Kirwan and Murray [2007]by means of 2-D holistic numerical models, where a simplified treatment of tidal hydrodynamics and the inclusion of few key processes allowed to predict various equilibrium morphologies. Simultaneously,Temmerman et al. [2007] reproduced the patterns of plant colonization and channel formation on the Plaat van Valkenisse tidal flat (Scheldt estuary) by means of numerical simulations using a commercial software to describe the coupling of hydrodynamics, morphodynamics and plant growth.

[11] The performance of some of the above models has been recently analyzed by Kirwan et al. [2010] with the aim to establish what conditions must be fulfilled in order that ecogeomorphic feedbacks may allow coastal wetlands to adapt to projected sea level rise. Results show that models do predict that salt marshes may survive over a large range of sediment concentrations and tidal ranges. With some tuning of the input parameters, models appear to provide reasonable qualitative agreement with observations from estuaries worldwide.

[12] The ultimate aim of the present contribution is to clarify whether the commonly used paradigm of long term morphological equilibrium is appropriate to describe the evolutionary trend of tidal marshes. In order to try and answer this question, we have considered a 1-D configuration consisting of a tidal channel merging into a 1-D marsh: hence, the marsh boundary is allowed to move, unlike in zero-dimensional models where the marsh domain is externally imposed and cannot vary except for changes in marsh elevation.

[13] Furthermore, coupling the morphodynamics of the channel to salt marsh morphodynamics has the advantage not requiring any arbitrary sediment concentration to be imposed at the channel-marsh boundary. Finally, a novel feature of our model, is to allow for the determination of the sediment transport driven by wind setup, a contribution, which, as it will be shown, appears to play a role and has never considered before within the present context.

[14] Note that, due to the 1-D nature of our model, we cannot yet predict morphological features arising from secondary flows generated in the flats surrounding the channel network and from the exchange of sediments between channels and flats. The choice of a 1-D configuration (artificial as it may appear) has, however, a great advantage: simulations can be extended for a very long time, such that the possible tendency of the system to equilibrium can be ascertained, allowing for both variations of marsh elevation and displacement of the marsh boundary.

2. Formulation

[15] We consider a schematic initial configuration consisting of a tidal channel characterized by a rectangular cross section with constant width, and connected at the upstream end to a water body simulating a tidal sea (Figure 1). We apply the model to a test example given by a tidal channel of initial length L = 20,000 m, initial mean flow depth D0 = 4 m, forcing oscillations at the inlet with amplitude a0 = 0.75 m and tidal period Tt = 12 h. The bottom of the channel is assumed to consist of cohesionless uniform sediments with size ds = 0.1 mm and density ρs = 2650 kg/m3. Note that, as the study of Venice lagoon has motivated our investigation, many parameters of the model have been chosen in order to fit the typical environment of Venice lagoon. Starting from this initial configuration we will model how a salt marsh originates at the landward end of the channel and determine the long term evolution of the channel - marsh configuration under different scenarios of relative sea level rise. Model parameters are reported inTable 1, while a synthesis of the cases considered in the simulations discussed in section 3 is reported in Table 2. We build on previous fairly established results on the morphodynamic evolution of a channel bounded by an inlet (seaward) and merging into the land on the lagoon side. This simplest configuration was investigated theoretically by Lanzoni and Seminara [2002]. The latter model has been extended such to account for the effects of vegetation growth and for wind effects, a major contribution responsible for sediment resuspension in regions where tidal stresses are too weak to mobilize sediments.

Figure 1.

Sketch of the system and notations. The initial configuration consists of a tidal channel of length L connected at the upstream end to a water body simulating a tidal sea, which undergoes tidal oscillations of the free surface with amplitude a0 at the inlet where the average flow depth is D0. Lc is the length of the unvegetated portion of the domain. The terms x and z are the longitudinal and vertical coordinates, respectively. H denotes the water surface elevation relative to the mean sea level. Finally, D and ηare the cross-sectionally averaged flow depth and bottom elevation with respect to the mean sea level.

Table 1. Model Parameters
ParameterValue
L, total domain length20000, 11850 m
D0, initial mean inlet depth4 m
a0, tidal forcing amplitude0.75 m
Tt, tidal period12 h
ds, sediment diameter0.1 mm
ρs, sediment density2650 kg/m3
Bmax, maximum aboveground biomass density1, 3 kg/m2
srslr, relative sea level rise rate0, 2, 10 mm/yr
kN, shoal roughness10 mm
ws, sediment settling speed0.08 m/s
ρa, air density1 kg/m3
ρo, organic material density1200 kg/m3
rw, wind friction coefficient1.5 10−3
k0, free surface roughness50 mm
p, sediment porosity0.4
po, organic sediment porosity0.2
Table 2. Initial Configurations, Wind Directions and Effects, Relative Sea Level Rise Rates (srslr) and Maximum Aboveground Biomass Densities (Bmax) Considered in the Different Simulations
CaseBmax(kg/m2)srslr (mm/yr)Wind DirectionWind Speed
Cases E: Initial Channel Configuration at Equilibrium, No Wind
E110
E230
E412
E532
 
Cases A: Initial Channel Configuration Not at Equilibrium, No Wind
A000
A110
A230
A302
A412
A532
A611
A7110
 
Cases B and C: Initial Channel Configuration Not at Equilibrium, Wind
B112seawardno setup
B212landwardno setup
C112seawardsetup
C212landwardsetup
C311seawardsetup
C411landwardsetup

[16] As discussed in section 4, our approach has various features in common with the recent model of Mariotti and Fagherazzi [2010]: in particular, both models account for the effects of tidal currents, wind waves and vegetation, though the treatment of each of the above effects is somewhat different.

[17] Let us then describe our formulation; it proves convenient, in this respect, to consider the channel domain distinctly from the marsh domain.

2.1. The Channel Domain

[18] The channel domain is defined by the condition: 0 < x < Lc (t), where Lc (t) is the length of the unvegetated portion of the total domain.

2.1.1. Tide Propagation

[19] Tide propagation in the channel domain is investigated following the approach of Lanzoni and Seminara [2002]. In the context of a 1-D model the motion of the fluid phase is governed by the classical continuity and the Saint Venant equations, which express the principles of mass and momentum conservation. Below,t denotes time, D and Uare the cross-sectionally averaged flow depth and flow speed respectively,H is the water surface elevation relative to the mean sea level and g is gravitational acceleration. With the above notations the governing equations read:

display math
display math

with C local flow conductance (estimated by van Rijn's [1984] formula).

[20] Hydrodynamic equations are supplemented by appropriate boundary conditions. At the “sea” boundary we impose prescribed sea level fluctuations while, at the landward end, the boundary condition requires more care. In the initial phase of the morphodynamic evolution, when the bed is still completely submerged throughout the whole domain, the normal component of the velocity at the “land” boundary is simply set to vanish. As the deposition process close to the landward boundary leads to emergence of the bed and salt marsh formation, an appropriate treatment is needed to describe the wet and dry region where the shoreline oscillates back and forth. Here we follow the approach proposed by Defina [2000] wherein the differential equations governing the flow over partially wet elements are reformulated by introducing a subgrid model where bottom irregularities are accounted for statistically, thus allowing for the application of classical numerical schemes without any special modification.

[21] Note that, in the present formulation, the hydrodynamics is coupled to the morphodynamics through the flow depth D = Hη, being ηthe cross-sectionally averaged bottom elevation with respect to the mean sea level.

[22] The above formulation is suitable to model the propagation of tidal currents. However, in shallow waters, sediment resuspension driven by wind acting on the free surface is the major controlling factor for the morphological evolution of the bottom [Carniello et al., 2005; de Swart and Zimmerman, 2009] and determines the mineral sediments supply to the salt marshes. The process of wind induced sediment resuspension occurs through the generation of wind waves. Wave resuspended sediments are then advected by tidal currents as well as by wind driven currents. Let us then analyze each contribution separately.

2.1.2. Wind Generated Waves in Shallow Waters

[23] We suppose wind to blow with a constant intensity throughout each tidal cycle and that every year is characterized by a random succession of wind events whose annual frequency of velocities is reported in Table 3. Furthermore, the wind is assumed to blow constantly or in the seaward or landward direction for the whole duration of each simulation.

Table 3. Annual Frequency of Wind Speed Considered in Our Test Casea
Speed (m/s)Frequency (%)
0–2.557.06
2.5–526.07
5–7.510.24
7.5–103.851
10–12.51.83
12.5–150.662
15–17.50.191
17.5–200.059
20–22.50.022
22.5–250.004

[24] Wind generates waves whose amplitude in shallow waters is spatially dependent and is mainly controlled by the shoal depth, the wind intensity and the available fetch (the length of the region where wind has been able to act). Moreover, a number of additional effects play a role in the process: energy transfer from the wind, energy spreading through non linear interactions, frictional dissipation, breaking. The wind field may also be spatially non uniform, due to the presence of obstacles, such as islands.

[25] To compute wave height in shallow waters many previous works [Umgiesser et al., 2004; Carniello et al., 2005; Fagherazzi et al., 2007; Mariotti and Fagherazzi, 2010] have solved an equation for the conservation of wave energy. Although accounting for various effects, such as whitecapping and depth-induced wave breaking, this approach cannot avoid some approximations. For example, though accounting for processes which are intrinsically nonlinear, such as wave breaking, wave energy is related to wave height employing relationships imported from the classical linear wave theory of monochromatic waves. Here, we calculate the wavefield using the empirical relationships derived byYoung and Verhagen [1996] from field observations performed in shallow environments. For given wind characteristics, the above investigation led to determine the dependence of the significant wave height Hw and wave period Tw on the available fetch xf. Recently, Carniello et al. [2011] have shown that similar relationships for the wave period Tw hold also in Venice lagoon. Denoting by D the flow depth and by Uw the uniform wind speed acting at some conventional distance from the free surface, the dimensionless relationships found by Young and Verhagen [1996] read:

display math
display math

where

display math
display math

[26] The ability of a Stokes wave of known characteristics to entrain sediments is naturally related to its ability to produce turbulence. In the absence of breaking, turbulence occurs only in the Stokes boundary layer adjacent to the erodible bed, where the flow regime may be laminar or turbulent depending on the values attained by two parameters: the Reynolds number RE, defined as (aw U1m/ν) and the relative roughness kN/aw, with ν, kinematic viscosity, aw = TwU1m/(2π) amplitudes of the free stream, U1m = πHw/(Tw sin h(kD)) amplitude of velocity oscillation of the free stream and kN equivalent roughness of the bed.

[27] It may be useful at this stage to get some feeling about the typical hydrodynamic conditions occurring in the flats. Let us consider a typical wind wave with a period Tw ranging about 2 s and a height Hw, say of 0.2 m, propagating on a shoal with depth D, say 1 m and erodible bed with ds of 100 μm. From the classical dispersion relationship ((2π/Tw)2 = gk tan h(kD)), such a wave will have wavelength Lw = 2π/k ≃ 5.2 m, amplitude of velocity oscillation of the free stream U1m ≃ 20 cm/s, amplitude of the free stream oscillation aw ≃ 6.3 cm and Reynolds number RE = 12700.

[28] Estimating the correct value of the equivalent bottom roughness is crucial. As the effective roughness of a tidal flat is determined by the presence of microvegetation and shells lying over the bottom [Amos et al., 2004], below we will assume an equivalent absolute roughness of the flat bottom kN of 10 mm. With aw ≃ 6.3 cm, then the relative roughness attains a value aw/kN ≃ 6.3 and the friction factor diagram of Justesen [1988] suggests that the boundary layer is indeed turbulent.

[29] Finally, the maximum shear stress generated by the wave alone τw can be estimated from the relationship τw = ρfwU1m2/2, with fw wave friction coefficient, which, for turbulent boundary layers, can be calculated using the following predictors [Kamphuis, 1975; Fredsoe and Deigaard, 1992]:

display math
display math
display math

2.1.3. Wave Current Interaction and Sediment Flux in the Channel

[30] We next investigate the distribution of suspended sediments driven by the simultaneous effect of tidal currents and wind waves acting on the free surface in the shoals.

[31] The problem can be tackled taking advantage of the fact that the time scale of the tidal flow, i.e., the tidal period, is orders of magnitude larger than the wave period, which is typically of a few seconds: hence, the tidal flow can be reasonably modeled as a quasi-steady slowly varying sequence of locally and “instantaneously” uniform flows superimposed upon a wavefield which maintains the same intensity throughout the tidal period (an inessential assumption to be eventually relaxed).

[32] In order to evaluate the sediment flux under the combined actions of wind waves and tidal currents we have tested the performances of two well established methods proposed in the literature.

[33] Fredsoe and Deigaard's [1992] Approach.The calculation of the distribution of suspended sediment in a combined wave-current motion would require the solution of the advection-diffusion equation expressing sediment continuity. For the plane case this equation reads:

display math

where z is the vertical coordinate defined in Figure 1, c(x, z, t) is the volumetric concentration of suspended sediments averaged over turbulence, u and w are the mean velocity components in the x and z direction respectively, ws is the particle settling speed and ϵs is an isotropic turbulent diffusivity. The second term on the right hand side can be neglected because the vertical gradients of concentration are typically much larger than the horizontal ones. The advective terms are also negligible, being higher order terms compared to settling and turbulent diffusion, which are processes characterized by timescales much smaller than the tidal period. It may be useful to point out at this stage that the role of advection has been thoroughly analyzed in the past [e.g., Bolla Pittaluga, 2003]. The importance of the so-called settling lag effect (deposition occurring downstream to entrainment) depends on the size of the parameterδsl = U0D0/(wsL0) where U0 is the scale of longitudinal velocity, D0 is the scale of flow depth, ws is the settling speed and L0 is the longitudinal scale (of the order of the tidal wavelength). In the case of the tidal channel considered in our simulations, assuming U0 = 1 m/s, D0 = 4 m, ws = 0.3 mm/s and L0 = 200 km, δsl takes the value 5 10−2. Bolla Pittaluga [2003] showed that results obtained by Lanzoni and Seminara [2002] are practically unaffected by the settling lag effect for values of the parameter δsl around 2 10−3. Below, we will assume a sediment size of 0.1 mm for which, with the Stokes settling speed, the parameter δsl takes the value adopted by Bolla Pittaluga [2003]. We cannot be sure that for finer particles our approximation is still perfectly appropriate, but yet we have decided not to account for sediment advection in order to reduce the computational effort and allow for long term simulations.

[34] Hence equation (11) can be reduced to the form:

display math

[35] The latter equation can be solved for c once a closure relationship for the turbulent diffusivity ϵs in a combined wave current motion is given. Employing the classical analogy between the processes of sediment and water mixing, the sediment diffusivity ϵs can be given the same form as the eddy viscosity. We then follow Fredsoe and Deigaard [1992] and assume that, outside the wave boundary layer the sediment diffusivity has a parabolic form independent of time, hence we write:

display math

with δ boundary layer thickness and ufc friction velocity of the tidal current. In the inner region the sediment diffusivity is given a linear distribution of the form:

display math

where the temporal dependence of the friction velocity ufwc and of the boundary layer thickness δ must be obtained as solutions of the governing equations. Fredsoe and Deigaard [1992] employ a momentum integral approach: they make a reasonable guess on the shape of the velocity profile (a logarithmic shape, corrected for the effect of the wave driven turbulence) and insert it into the momentum equation. For given values of the bottom roughness kN, current speed U, maximum wave speed U1m and wave amplitude aw, the solution of the momentum integral equation allows one to determine the friction velocity ufwc and the boundary layer thickness δ due to both waves and currents as functions of time throughout the wave period.

[36] Having determined the eddy diffusivity, we can finally solve the equation (12) numerically by assuming that c is periodic (c(tz) = c(t + Twz)), and imposing the following boundary conditions: (1) no vertical flux at the water surface; (2) a gradient boundary condition at the bed

display math

with q sediment flux, n unit vector normal to the bed and ce equilibrium concentration of suspended sediments at the reference elevation z = zr + η.

[37] The knowledge of ce is as yet empirically based, even for the simplest case of steady uniform currents. Here we employ van Rijn's [1984] empirical relationship relating ce to the local and instantaneous value of the Shields parameter θ = [ρufwc2/((ρs − ρ)gds)].

[38] Finally, once the vertical distribution of the sediment concentration has been determined, the average sediment flux in a wave period is readily calculated through the following integral:

display math

where uwc (z) is the vertical velocity profile for a combined wave-current motion suggested byFredsoe and Deigaard [1992].

[39] The Simplified Approach of Soulsby [1995, 1997]. A simpler approach to the problem may be pursued by taking advantage of the fact that the thickness of the wave boundary layer is small compared to the flow depth, and hence, as a first approximation, the concentration profile arising from the wave-current interaction can be described by the classical Rouse-distribution:

display math

The equilibrium concentration ce at the reference elevation (z = zr + η) can be evaluated as a function of the Shields parameter under combined waves and currents by means of the van Rijn's [1984] relationship. Rather than estimating the instantaneous value of the shear stress under combined waves and currents solving for the flow field as in Fredsoe and Deigaard [1992], and Soulsby [1995, 1997] proposed an explicit relationship for the average shear stress over a wave period. This turns out to be enhanced beyond the sum of the distinct contributions of wave and current, due to non linear interactions occurring in the boundary layer, and reads:

display math

where τwc is the average shear stress generated by the combined actions of wave and current, τw is the maximum shear stress generated by the wave alone, while τc is the shear stress driven by the current. The maximum shear stress due to the combined action of waves and currents τwcmax is then obtained, assuming a simple linear superposition, as follows:

display math

Note that it is the value of τwcmax which determines the threshold of motion and the reference concentration of the suspended sediments at the bottom.

[40] Finally, the sediment flux advected by the currents throughout a tidal cycle is evaluated as follows,

display math

where uln (z) is the classical logarithmic distribution of the velocity profile of the tidal current and cc is the Rouse distribution (17) for the mean concentration.

[41] Let us now compare the outputs of the above two methods considering a test case: a shoal with average depth D = 1 m subject to the depth and speed oscillations reported in Figure 2 for a typical tidal cycle (Tt = 12 h). Waves are induced by a uniform wind field with speed Uw of 10 m/s and fetch length of 5 km: this determines a velocity oscillation of the free stream at the edge of the Stokes boundary layer with amplitude U1m = 19 cm/s. Finally, we assume the wave propagation aligned with the tidal current.

Figure 2.

(a) Depth and (b) speed oscillations induced by the tide on the shoal. Intensity of the tidal current (solid line) is plotted along with the wind driven amplitude of the velocity oscillation of the free stream (dash-dot line).

[42] Results reported in Figure 3a show that the bottom shear stress obtained by the two approaches exhibits a similar qualitative behavior, though predictions obtained by the approach of Soulsby [1995, 1997] exceed roughly by a factor 1.3 those obtained by the integral momentum method of Fredsoe and Deigaard [1992].

Figure 3.

Test comparison between the time variation of (a) the average value of the bottom shear stress induced by the combined action of waves and tidal currents, (b) the dimensionless sediment flux induced by the combined action of waves and tidal currents and (c) the wind driven dimensionless sediment flux, in a tidal cycle obtained from the integral momentum method of Fredsoe and Deigaard [1992] and the empirical approach of Soulsby [1995, 1997].

[43] Smaller quantitative differences are obtained when evaluating the sediment flux associated with the above shear stress distribution. Figure 3b shows that the simplified model provides estimates for the sediment flux slightly smaller than those obtained from the complete solution of the unsteady wave boundary layer under combined wave and currents. Note that relative differences are smaller than those found for the bottom shear stress; moreover, both approaches predict that the net sediment flux in the tidal cycle is negative (seaward directed) with the complete solution providing a value (−4.19 10−7 m2/s) that is ∼25 % smaller that the value predicted by the simplified solution (−5.64 10−7 m2/s).

[44] Given the limited differences between the results obtained from the two approaches, in section 3.4 we report only the results obtained using Soulsby's [1995, 1997] model.

2.1.4. Wind Driven Currents in the Channel Region

[45] It is well known that the wind stress acting on the free surface of a basin, besides generating wind waves, gives rise to a surface setup i.e., it generates a slope of the free surface in the direction opposite to the wind: as a result, a counter-flow is generated which superimposes on the Couette type flow driven by the wind stress (seeFigure 4). The resulting flow has the direction of the wind close to the free surface and the opposite direction close to the bed. Note that the former effect (a turbulent Couette flow) is the only one accounted for in shallow water models of the type employed by Umgiesser et al. [2004] and Carniello et al. [2005]. In order to account for the latter effect (a quasi-uniform free surface return flow) the knowledge of the actual velocity distribution is needed [Engelund, 1986]. In other words, the model has to be partly 2-D in order to reproduce this feature, which, as it will be shown below, turns out to be significant for our morphodynamic purposes. Decoupling is allowed, as the wind velocity is a small perturbation of the tidal flow (except for a short and irrelevant period close to flow reversal), a feature which allows linearization at least as a first approximation.

Figure 4.

A sketch of the vertical velocity profile of the currents driven by wind stress and wind setup.

[46] The simplest approach to evaluate the vertical velocity profile was proposed by Engelund [1986], who suggested the expression:

display math

where ufsb and ufss are the bottom friction velocity and the surface friction velocity respectively, and k is the von Kármán constant.

[47] In the case of a closed channel with a rectangular cross section, the net flux associated with the wind driven flow field, is bound to vanish, hence the depth averaged velocity must vanish. By imposing the latter condition, the equation (21) leads to the following relationship between the bottom friction velocity and the surface friction velocity:

display math

where fb is the bottom friction factor, given by

display math

Equation (22) allows us to evaluate the bottom friction velocity once the surface friction velocity is given. The latter reads:

display math

where Uw is the wind speed, ρa is air density and rw is the wind friction coefficient which takes typical values ranging about 1.5 10−3.

[48] It may be useful to note that, for a wind speed of 10 m/s, bottom roughness kN = 10 mm and surface roughness k0 = 50 mm [Engelund, 1986] the bottom shear stress induced by the wind setup takes values of about 0.002 N/m2 and the return current has a maximum speed at math formulaof about 3 cm/s. Note that the above value of the shear stress is small compared with the corresponding values associated with the wave-tidal current boundary layer (seeFigure 3a) and too low to induce sediment resuspension. Also the maximum current speed is almost negligible compared with the velocity field associated to the tidal currents (see Figure 2b), however, in the next section it will be shown that, due to its permanent character, the flow speed induced by the wind setup may turn out to be as significant as the speed of the tidal currents in determining the direction and the intensity of the advected sediment flux.

2.1.5. Wind Driven Current and Sediment Flux

[49] A second independent contribution to the sediment flux is due to the effect of the currents driven by wind stress and wind setup. As a first approximation: (i) we neglect interactions between tidally driven and wind driven currents; (ii) moreover, we calculate the flux of sediment advected by wind currents assuming the vertical distribution of concentration negligibly affected by the presence of an additional source of turbulence associated with wind stresses. Again, we test the performance of the two distinct distributions of concentration obtained by the two models described above. In any case, we note that the sediment concentration is greatest close to the bed, where the wind current is directed opposite to the wind direction. Hence, the average sediment flux driven by the wind setup is directed opposite to the wind. Furthermore, as already pointed out, for values of the wind speed of the order of 10 m/s, the return current has speed of the order of few (say 2–3) cm/s, much smaller than the speed of the tidal current (peak values ranging about 0.3 m/s). However, since wind driven currents keep the same direction throughout the whole tidal cycle (provided the wind direction does not vary), the total wind driven return flux in a tidal cycle may be comparable with the net flux advected by tidal currents.

[50] These estimates are confirmed by results of the calculation of the sediment flux advected by the wind currents for the ideal case considered above. They are reported in Figure 3c. Note that, again, the sediment flux calculated using the Rouse distribution of concentration is slightly larger than that calculated by the complete model of Fredsoe and Deigaard [1992]. However, in both cases, the total sediment flux advected by wind induced currents in a tidal cycle (8.21 10−7 m2/s and 6.24 10−7 m2/s respectively) is positive (landward directed) and comparable or even slightly larger than the net sediment flux advected by tidal currents (−5.64 10−7 m2/s and −4.19 10−7 m2/s respectively), suggesting that wind setup can, at least in principle, play a non negligible role in the process of evolution of salt marshes. We will discuss this effect in more detail in section 3.4.

2.1.6. 1-D Model of Channel Morphodynamics

[51] Sediment continuity is imposed by requiring that, at each location on the channel surface, the 1-D form ofExner [1925] equation must be satisfied. It reads:

display math

where the term srslr (representing the rate of relative sea level rise) has been introduced because the elevation η is evaluated with respect to the mean sea level (a moving frame). Note that equations (25) and (1)are strictly related, with the major difference that the temporal derivative is here partial rather than ordinary, as the sediment balance is no longer spatially averaged like in zero-order models,amin is equal to math formula, the term aorg is set to vanish as no vegetation is present in the domain and scom has been neglected.

[52] Furthermore p is sediment porosity and qs is the total sediment flux per unit width instantaneously transported by the flow field at a given location. The latter consists of the sum of a bed load component qsb and a suspended load contribution qss. Simulations discussed in section 3 have been performed for different values of the rate of the relative sea level rise (0; 1; 2; 10 mm/yr) as summarized in Table 2. We evaluate the bed load contribution by employing the empirical relationship of Meyer-Peter and Müller [1948] with the local and instantaneous value of the total bottom stress induced by the wave/tidal currents interaction. The flux of suspended sediment is then obtained by summing the tidal current contribution and the wind current contribution as discussed in section 2.1.3 and section 2.1.5 respectively.

[53] In order to complete the formulation of the morphodynamic problem for the channel domain, appropriate boundary conditions must be associated with the sediment continuity equation.

[54] At the sea boundary we assume sediment transport is in equilibrium with the local hydrodynamics during both the flood and the ebb phase, in other words, the sediment flux at the inlet is determined by the local transport capacity of the stream. Note that this assumption has been removed in section 4 in order to investigate the effects of a possible excess or deficit of sediment supply driven by a given sediment concentration imposed at the seaward boundary.

[55] At the land boundary, the exchange of sediments between the channel and the salt marsh is modeled assuming that the sediment flux entering the salt marsh during the flood phase is determined by the transport capacity of the stream at the last channel station (x = Lc (t)). During the ebb phase, the sediment flux leaving the marsh is determined by the residual sediment available to the tidal current after settling and trapping by vegetation has operated over the marsh in the previous phase. As regards settling, we have used Stokes relationship to calculate the settling speed. We have also compared our results with the recent approach of Mudd et al. [2010], whereby the effective settling speed over the marsh can be estimated as a function of flow velocity, biomass and particle diameter: we have found that, in the context of our calculations, this correction leads to a minor correction of the Stokes settling speed (of the order of 1% for 0.1 < B < 3 kg/m2 and 0.1 < U < 0.3 m/s). As regards trapping, our choice has been to ignore it as the conclusion of Mudd et al. [2010] was that “virtually all (>99%) of the increase in accretion rates can be attributed to enhanced settling brought about by reduced kinetic energy in the fertilized canopy”.

[56] Calculations of the residual amount of sediments available in the water column over the marsh at the end of the flood phase have been performed considering a fully developed wave field induced by a wind with constant intensity equal to 10 m/s. This amount turns out to be a small fraction (roughly the 1%) of the volume of sediments entering the marsh in the same period. As trapping, which we have ignored, can only reduce further the residual volume of sediments, and resuspension in the marsh is known to be very small, we felt entitled to assume that the water exiting the marsh carries a negligible sediment load.

[57] The reader will note that our model does not average spatially the hydrodynamic and morphodynamic processes occurring over the salt marsh.

2.2. The Marsh Domain

2.2.1. Tide Propagation

[58] Tide propagation in the marsh is investigated following the same approach [Lanzoni and Seminara, 2002] employed for the channel (see equations (2) and (3)). Note that hydrodynamics does not need any coupling condition at the channel/marsh boundary to be imposed. The governing equations are solved in the entire domain (channel and marsh), with the only difference that frictional effects are enhanced due to the presence of vegetation in the marsh domain.

[59] The first attempt to model the above effect has been proposed by Mudd et al. [2004] elaborating the results of Nepf [1999]. More recently Mudd et al. [2010] combined field measured macrophyte characteristics with the laboratory results of Tanino and Nepf [2008], finding a new relationship relating drag forces to the physical characteristics of emergent vegetation.

[60] Here we follow the primitive formulation of Mudd et al. [2004], we assume that flow conductance in the presence of vegetation is equal to:

display math

where C is the local bed flow conductance in absence of vegetation, cD is the bulk plant drag coefficient, δs [m] is the stem diameter and αp [m−1] the projected plant area per unit volume. The latter quantities are known functions of biomass, measured in the field for Spartina [Morris and Haskin, 1990]:

display math

where we denote by Bag [kg/m2] the spatial density of biomass associated with aboveground halophytic vegetation. Furthermore, following Mudd et al. [2004], who took advantage of the findings of Nepf [1999], cD can be expressed as a decreasing function of biomass:

display math

again with Bag in kg/m2.

[61] Through the latter relationships, the marsh hydrodynamics becomes dependent on the growth of vegetation. The next step is then to model the latter process.

2.2.2. Growth of Halophytic Vegetation

[62] Prediction of the growth and decay of vegetation is a subject which has received great impetus in the last decade [Day et al., 1999; Morris et al., 2002; Mudd et al., 2009].

[63] In particular, it has been firmly established that the above ground production of vegetation depends on several factors, most notably the elevation of marsh platform relative to mean sea level (which controls flood frequency and soil salinity [Morris, 1995]) and evapotranspiration (which also affects soil salinity). Furthermore, Morris et al. [2002] point out that there is an optimum marsh elevation for Spartina productivity which varies regionally as well as with tidal range.

[64] A different, though related, approach has been recently employed by Marani et al. [2007], who rely on the logistic model of Levins [1969]. In the Spartina dominated case, both the biomass reproduction rate and the mortality rate were assumed to vary linearly with elevation.

[65] The procedure here adopted is the following. First, following Morris [2006] the annual peak of spatial density of biomass associated with aboveground halophytic vegetation of Spartina alterniflora BP [kg/m2] is assumed to be a function of the depth of the marsh platform below MHT, D + a0, as follows:

display math

where, a, b, and c are site specific dimensional coefficients (a = 8.23 kg/m3, b = −9.85 kg/m4 and c = −0.724 kg/m2). According to equation (29) the upper and lower limits of marsh elevation for macrophyte survival correspond to depth below mean high tide of 10 cm and 70 cm, respectively, with a maximum biomass density at depths between 40 cm and 60 cm below mean high tide.

[66] Note that the peak aboveground biomass is bounded, Bmax being the maximum biomass spatial density of the particular species characterizing the salt marsh. Simulations have been performed for different values of the maximum biomass density (1; 3 kg/m2).

[67] Note that the parabolic relationship (29) was originally proposed by Morris et al. [2002] for biomass productivity rather than biomass density. Later, Morris [2006] stated that “productivity and standing biomass density are used here interchangeably as they are roughly proportional.” However, the two formulations are not equivalent when employed in the context of morphological models. Indeed, qualitative and questionable differences in the output of the present model would arise using the original approach. While, to our knowledge, this issue has not been pointed out before, it is the latter formulation which is invariably used in existing models [Mudd et al., 2004; Morris, 2006; Kirwan and Murray, 2007; Mariotti and Fagherazzi, 2010].

[68] Furthermore, since vegetation varies through the seasons peaking in the summer months, Mudd et al. [2009] propose the following relationship for the temporal dependence of the aboveground biomass Bag:

display math

where t is time, P is the period of the cycle, assumed to be 1 yr, and tp is the time of the year when aboveground biomass is at its peak. Note that, the minimum aboveground biomass has been assumed to vanish.

[69] Finally, the spatial density of below-ground biomassBbg is strictly related to the value of the aboveground biomass. As in Mudd et al. [2009] we approximate the ratio root/shoot of Spartina as a linearly decreasing function of the depth below MHT, D + a0, as follows:

display math

with D + a0 expressed in meters.

[70] Rather than allowing for the prediction of the effective bottom friction, the knowledge of the amount of biomass available in the marsh is the first ingredient preliminarily needed in order to quantify the production of organic sediments. It will be discussed in the next section.

2.2.3. 1-D Model of Salt Marsh Morphodynamics

[71] The evolution equation of the marsh surface can be expressed in the form (1)with the major difference that the temporal derivative is here partial rather than ordinary, as the sediment balance is no longer spatially averaged like in zero-order models.

[72] In the following we will describe how the accretion rates aorg and amin have been treated within the present framework. Note that we neglect the effects of autocompaction.

[73] Production of Organic Sediment. The production of organic matter is a complex issue as a number of mechanisms are involved [Day et al., 1999].

[74] An old proposal [Randerson, 1979] suggests that the marsh accretion rate associated with the production of organic matter (both above- and below-ground) is as proportional to the spatial density of biomass associated with aboveground halophytic vegetation through a given constantγ. Following Marani et al. [2007] the latter relationship can be employed using the annually averaged values of above ground biomass and assuming γ = 2.5 10−3 m3 kg−1 yr−1. A complete model for the production of organic matter has been recently proposed by Mudd et al. [2009]. It accounts explicitly for both above and below ground organic processes including root growth and decay of organic carbon and was based on a combination of 20 years of biomass data, various in situ experiments and sediment cores.

[75] These reasons lead us to implement Mudd et al.'s [2009] approach in our morphodynamic model. Differences in the results obtained from the application of the two approaches (Randerson vs Mudd) will be discussed in section 3.2.

[76] The rate of marsh accretion associated with the production of organic matter aorg is evaluated following Mudd et al. [2009] as follows:

display math

with

display math
display math

where Mt is the local mortality rate of the total biomass per unit area, a function of both time and location on the marsh, while ρo = 1200 kg/m3 and po = 0.2 [Mudd et al., 2009] are the density and porosity of the organic material respectively, finally mlab is the mass of the labile fraction of organic material per unit area, which may be calculated integrating equation (33) in time, being alab = math formula, or more directly as

display math

[77] Indeed, following Mudd et al. [2009], we consider that part of the organic sediment is subject to decomposition. In particular, χlab is the fraction of deposited organic matter that decays in time (labile) and klab is its decay rate. The values of both χlab and klab may show some variability. Here, we employ the average values suggested by Mudd et al. [2009], namely χlab = 0.842 and klab = 0.2 yr−1.

[78] The mortality rate of the total biomass per unit area Mt is the sum of the two contributions Mag and Mbg associated to the above ground and below ground biomass, respectively.

[79] Mag can be calculated according to the following equation of biomass dynamics:

display math

where Gagis the growth rate of aboveground biomass. Based on a long-term record of biomass dynamics from the North Inlet estuary in South Carolina,Mudd et al. [2009] assume that Gag can be approximated as follows:

display math

[80] Here ts = 0.15 yrs is the phase shift between the maximum growth rate and the maximum biomass and Gp is the peak growth rate, which is related to the peak biomass BP by the following relationship

display math

with image a coefficient equal to 5.037 yrs−1.

[81] The mortality rate Mag is then calculated by feeding equation (37) and the temporal derivative of (30) into (36), while Mbg is readily known once recognized that the ratio of aboveground to belowground mortality scales with the ratio of aboveground to belowground biomass.

[82] Note that we also assume that deposition affects the whole bottom surface, i.e., we ignore the fraction of marsh bottom occupied by vegetation, an assumption which leads to slight underestimation of marsh accretion.

[83] Marsh Accretion Driven by Settling of Minerogenic Sediments. The presence of vegetation over the marsh prevents the occurrence of bed load transport as well as of sediment resuspension [e.g., Mudd et al., 2010]. Hence, in order to estimate the rate of marsh accretion amin (or ∂ηmin/∂t) driven by deposition of minerogenic sediments, we are left with the problem of evaluating the suspended sediment flux, i.e., the local and instantaneous value of the suspended sediment concentration. This would strictly require the solution of an advection-diffusion equation coupled with appropriate boundary conditions at the marsh borders and a closure relationship for the eddy diffusivity in the presence of vegetation. However, as a first approximation, it is sufficient to employ a simpler approach where turbulent diffusion is neglected noting that the presence of vegetation has a strong damping effect on the turbulence of the stream. A pure settling-advection equation for the mean concentrationc averaged over turbulence is then obtained. It reads:

display math

with u mean local speed (averaged over turbulence) and ws settling speed of sediment particles. With the further assumptions that the local speed u can be approximated by its depth averaged value U and that the settling speed ws can be estimated based on the classical relationships available for isolated particles, the latter equation is readily amenable to analytical treatment. Using the approach described in the Appendix, the following analytical solution is found for the distribution of concentration of mineral sediments over the salt marsh domain seen by an observer leaving the initial cross section of the marsh x = x0 at time t = t0 and moving with the depth averaged velocity U(t0):

display math

where f(z) is the Rouse distribution for the sediment concentration and the quantity c0 is its amplitude given by matching the marsh solution with the channel solution at the initial cross section of the salt marsh at time t = t0. In order to implement the solution (40) we only need to know the quantity c0 as well as the depth averaged flow speed at the initial cross section in the marsh U(t0) as a function of time throughout the flood phase.

[84] Finally, the rate of marsh accretion associated with settling of inorganic sediments amin is readily calculated as the spatial and temporal distribution of concentration is known from (40):

display math

Note that the above scheme is justified as the flow field in the marsh is slowly varying both in time and in space or, more clearly, variations of the flow field occur on temporal and spatial scales much larger than the scale of particle settling. In fact, the settling speed of sediments of size 0.1 mm is roughly equal to 8 mm/s. With values of the depth and flow speed of the order of 1 m and 0.2 m/s respectively, it follows that the time required for a particle at the surface to settle down to the bottom is of the order of 100 s (much smaller than the tidal period), while the horizontal displacement of the particle is of the order of 20 m (much smaller than a typical marsh size). Furthermore, the above estimate suggests that the water exiting the marsh during the ebb phase is practically devoid of minerogenic sediments.

[85] The one dimensional equations governing mass and momentum conservations, coupled with the equation of mass conservation of the solid phase are solved numerical by means of a classical [MacCormack, 1969] scheme in the entire domain. Note that biomass distribution, organic accretion and the position of the channel-marsh boundary are updated at the end of each tidal cycle.

3. Results on the Long Term Evolution of Salt Marshes

[86] We apply the framework discussed above to a thought experiment where we start considering the simplest configuration, a system tidal inlet-tidal channel (characteristic parameters reported inTable 1) communicating with a tidal sea at the inlet, no marsh, no wind acting on the free surface and vanishing rate of sea level rise. We will then progressively enrich this configuration by varying the forcing and the boundary conditions. We focus on the issue of morphodynamic equilibrium, that we discuss in the light of previously available knowledge and present results. This seems of particular importance as the paradigm of equilibrium is widely employed in the literature.

3.1. Results for the Tide Dominated Case in the Absence of Vegetation

[87] The Simplest Configuration by Lanzoni and Seminara [2002]. The simplest configuration described above was analyzed by Lanzoni and Seminara [2002], who showed that, provided sediment supply at the inlet equals transport capacity, then the channel eventually reaches morphological equilibrium with a shore forming at the inner end of the channel and the net sediment flux in a tidal cycle vanishing everywhere. More recently, Seminara et al. [2010] have argued, based on a theoretical solution of the problem, that an even stronger constraint is required for equilibrium: namely, that the sediment flux must vanish (i.e., the Shields stress must be critical) everywhere throughout the channel at each instant. We called this static equilibrium. We attach a special importance to the latter paper as it overcomes the delicate issue of reaching equilibrium numerically: indeed, as equilibrium is approached, the sediment flux progressively decreases, hence the morphodynamic evolution slows down, which makes computations more expensive. Moreover, numerical calculations in the wet and dry region employ approximate schemes which have never been rigorously substantiated. The analytical solution of Seminara et al. [2010] bypasses these problems, at least for the simplest configuration, and provides a conclusive proof that, under those conditions, rigorous equilibrium conditions do indeed exist. These seem to agree reasonably with some field observations where [Friedrichs and Aubrey, 1996] it is suggested that some real tidal channels do operate close to static equilibrium. The latter analysis predicts the following relationship between the length of a non-converging channelLeq and the inlet depth D0eq at equilibrium

display math

where Ucr is the critical velocity for sediment motion and ω is the frequency of the forcing tide.

[88] Sediment Supply at the Inlet in Excess to Transport Capacity. Having established the above starting point, let us next modify the boundary condition at the inlet. We have seen that equilibrium was attained assuming that, throughout the evolution process, the flood currents entering the inlet carried a sediment load at capacity: hence, at equilibrium, no sediment flux may be exchanged through the inlet. However, in the real world, the net sediment flux entering the inlet is forced by the sea and depends on inlet geometry and sea conditions: in particular, during coastal storms, the flood currents may be overloaded by sediments resuspended in the swash zone around the inlet. In other words, under these conditions, the flood currents carry a suspended load that cannot be transported by the tidal flow if the channel is at equilibrium: as a result, the sediment settles and alters the equilibrium bed profile which, in turn, affects the hydrodynamics and the channel starts to evolve further. Since no sediment flux is allowed to leave the channel at the landward end, it is clear that equilibrium is impossible if a non vanishing net sediment flux is forced to enter the inlet: in the absence of further effects, the fate of such a tidal channel would be to fill up.

[89] Nonvanishing Rate of Sea Level Rise, rslr ≠ 0. Let us now assume that the rate of sea level rise does not vanish. Recalling the sediment continuity equation (25), averaging over a tidal cycle and integrating in the longitudinal direction, it turns out that morphodynamic equilibrium (〈η〉 = 0 everywhere with 〈〉 temporal average over a cycle) requires that the sediment flux per unit width averaged over a tidal cycle (〈qs〉) must be equal to rslr (Lcx). Hence, at equilibrium, the net sediment flux per unit width at the inlet 〈qs〉|x=0 must balance exactly the quantity (rslr Lc). Needless to say, there is no a priori reason why this constraint should be met, inasmuch as the controlling factors of sea level rise and sediment supply at the inlet are totally independent. We then conclude that, again, exact equilibrium can hardly be established and the channel will slowly either degrade or fill up.

3.2. Results for the Tide Dominated Case: The Role of Vegetation

[90] Let us now allow vegetation to grow on the shore adjacent to the landward end of the channel, still considering tide dominated conditions, i.e., no wind waves.

[91] Randerson [1979] versus Mudd et al. [2009]. Let us first perform a comparison between results obtained employing Randerson's [1979] and Mudd et al.'s [2009] approaches for the estimate of the production of organic sediments. It turns out (Figure 5) that, in absence of sea level rise, the maximum salt marsh elevation increases slightly more rapidly in the latter case. On the contrary, in the presence of a moderate sea level rise, (Bmax = 1 kg/m2), starting from the same bottom configuration, at the end of a 100 year simulation, Mudd et al.'s [2009] approach predicts a marsh elevation which is slightly lower than that predicted using Randerson's [1979] relationship. However, differences appear to be fairly small and purely quantitative hence, below, we will report only results obtained using the more complete Mudd et al.'s [2009] approach.

Figure 5.

(a) Bed profiles obtained in the tide dominated case using Randerson's [1979] approach for the production of organic sediments, after a simulation of 100 year in cases A1, A4 and A7. The symbol η is the bottom elevation and xis the longitudinal coordinate aligned with the channel axis and oriented landward, with origin at the channel inlet. (b) A zoomed view of the channel-marsh transition showing vegetation: the elevation of the dotted lines with respect to bottom elevation is proportional to the local biomass density. (c and d) The results obtained with the same conditions usingMudd et al.'s [2009] approach.

[92] Channel Initially at Equilibrium, No Sea Level Rise. Let us next monitor the evolution of the system, starting from an initial equilibrium state of the channel such that marsh accretion can only be driven by the deposition of organic matter. Results of our simulations for this case are given in Figure 6.

Figure 6.

Temporal evolution of the elevation of the salt marsh platform starting from an initial configuration of the channel at equilibrium, for different values of Bmax and rates of relative sea level rise. (a) E1: Bmax = 1 kg/m2, srslr = 0 mm/yr. (b) E2: Bmax = 3 kg/m2, srslr = 0 mm/yr. (c) E4: Bmax = 1 kg/m2, srslr = 2 mm/yr. (d) E5: Bmax = 3 kg/m2, srslr = 2 mm/yr. The elevation of the green lines with respect to the bottom elevation is proportional to the local biomass density. In the initial state the channel was in morphodynamic equilibrium with no marsh.

[93] Since the channel is initially at equilibrium, it is unable to transport sediments, hence the bed elevation in the channel does not vary except in the narrow dry and wet region close to the marsh margin where the growth of vegetation leads to organic accretion. Accretion also occurs in the rest of the marsh, hence the marsh profile raises toward mean high tide, until the flow depth in the marsh has decreased sufficiently to reach the lower limit of biomass productivity and the vegetation disappears (Figures 6a and 6b). The final state is indeed an equilibrium state, with a fairly steep scarp and no marsh!

[94] Channel Initially in Equilibrium, Moderate Sea Level Rise. Next, assume to switch on sea level rise as soon as we allow vegetation to grow. The picture now changes (Figures 6c and 6d): as a result of sea level rise, the channel bed (in equilibrium) sinks, the flow speed in the channel then decreases, hence the transport capacity of the stream decreases further and the channel bed keeps inactive. Some variations occur in the bed profile of the channel close to the marsh margin. The marsh accretes non uniformly and the marsh margin moves depending on the balance between marsh accretion rate and rate of sea level rise. For moderate biomass productivity (Figure 6c), the effect of sea level rise prevails, the marsh retreats, the channel lengthens and steepens near the marsh margin. Steepening arises from the fact that marsh accretion near the margin is non uniform, it increases from zero at the upper limiting depth for biomass productivity to some maximum at the optimal depth. For high biomass productivity (Figure 6d) the marsh no longer retreats, but it does not prograde either: indeed, if the channel is unable to transport sediments, there is no mechanism to allow sediment deposition near the margin, hence the channel continues to deepen and the profile close to the margin steepens, while the marsh progressively attains a flat and stable profile. Hence, in this case, the final state is a stable equilibrium state for the marsh, but not for the channel, which continues to deepen everywhere as a consequence of the unbalanced effect of sea level rise.

3.3. Results for the Tide Dominated Case: The Role of Initial Conditions

[95] The evolution of the system depends also on the initial conditions: indeed, if the channel is not in equilibrium initially, it can transport sediments which may be deposited close to the marsh margin allowing for the possible growth of vegetation and marsh progradation. We have investigated this possibility, repeating the simulations starting from a non equilibrium initial state. This was generated numerically by letting an initially flat bed profile in a channel closed at the landward end evolve into a sloping profile deepened at the inlet and aggraded landward. After roughly 20,000 tidal cycles (grossly 27 year), the bed starts to emerge and a shore develops in the region close to the landward end. This configuration, which was not yet in equilibrium, has then been used as initial bottom profile for simulations where we have allowed for vegetation growth and sea level rise. Results of these simulations are reported in Figure 7.

Figure 7.

Temporal evolution of the elevation of the salt marsh platform for different values of Bmax and rates of relative sea level rise. (a) A1: Bmax = 1 kg/m2, srslr = 0 mm/yr. (b) A2: Bmax = 3 kg/m2, srslr = 0 mm/yr. (c) A4: Bmax = 1 kg/m2, srslr = 2 mm/yr. (d) A5: Bmax = 3 kg/m2, srslr = 2 mm/yr. The elevation of the green lines with respect to the bottom elevation is proportional to the local biomass density.

[96] Vanishing Sea Level Rise. For vanishing rate of sea level rise, the Figures 7a and 7b show that the marsh invariably progrades with a rate increasing as the biomass productivity increases. This result differs from that observed in the corresponding simulation starting from initial channel equilibrium. Why? The explanation is simple. In the present case, the stream transports sediments; moreover, as soon as vegetation grows the flow decelerates close to the marsh margin, hence the transport capacity of the stream decays in the landward direction leading to sediment deposition. The channel bed then aggrades progressively reaching elevations such that vegetation is able to grow. The marsh then advances seaward leaving behind (Figure 7b) a flat marsh region which has reached the upper limit for biomass productivity, hence it does not accrete any longer. Note that, for a vanishing rate of sea level rise, the maximum elevation reached by the salt marsh for a maximum biomass density of 1 kg/m2 is the same as for a maximum biomass density of 3 kg/m2: indeed, the elevation at which marshes stabilize ranges about 0.65 m relative to mean sea level, corresponding to a depth below high tide equal to 10 cm, i.e., the lower limiting value of depth below MHT for macrophyte survival. At the end of the simulation, the channel still transports some sediments, hence it is not yet in exact equilibrium after 1000 years. This is also confirmed by a glance at Figure 8c showing the evolution of the ratio channel length to inlet depth: at the end of simulations A1 and A2, the latter ratio is still significantly higher than the theoretical value predicted by equation (42) which, for the present simulations, ranges about 2500.

Figure 8.

Comparison between the time evolution of (a) the position of the tidal channel/salt marsh border, (b) the maximum salt marsh elevation and (c) the ratio between the channel length and the inlet depth, for the simulations A1, A2, A4 and A5. Note that the theoretical value of Lmarsh/D0 corresponding to equilibrium has been obtained from equation (42) with Ucr = 0.27 m/s, ω = 1.45 10−4 s−1 and a0 = 0.75 m.

[97] It is reasonable to expect that, eventually, equilibrium would be reached: the marsh evolution would then follow the trend already discussed in the previous section (Figures 6a and 6b). We have checked this by continuing one of the above simulations until channel equilibrium was indeed reached: results of this simulation are reported in Figure 9 and confirm our expectations.

Figure 9.

The temporal evolution of the elevation of the salt marsh platform for the configuration A1 of Figure 7, (Bmax = 1 kg/m2, srslr = 0 mm/yr) has been extended until channel equilibrium was achieved. The elevation of the green lines with respect to the bottom elevation is proportional to the local biomass density.

Figure 10.

(a) Spatial distribution of the relative accretion rate along the marsh at the end of the 960th year in cases A1, A2 and A5. Note that, as shown in (b), at the end of the 960th year, the minerogenic component of the marsh accretion rate vanishes. (b) Temporal evolution of the relative mineral accretion rate in x = Lc.

[98] The Effect of a Nonvanishing Sea Level Rise. If we now allow for a relative sea level rise, the picture changes. For a moderate biomass productivity (Bmax = 1 kg/m2) the marsh turns out to be unable to keep up with even a small or moderate relative sea level rise: basically, its border retreats and the vegetation disappears completely (Figure 7c). With a rate of relative sea level rise of 2 mm/yr this process is completed after roughly 125 years. Note that the trend exhibited by this simulation is qualitatively similar to that displayed by the corresponding simulation described in Figure 6c. For a high biomass productivity (Bmax = 3 kg/m2), initially the marsh progrades seaward at a rate slower than in the case of vanishing rate of sea level rise (Figure 7d); after roughly 550 yrs the salt marsh border slightly retreats and then keeps fixed up to 1000 yrs. Furthermore, initially the salt marsh platform rises at a rate slower than in the case of vanishing rate of sea level rise; after roughly 200 yrs, the marsh platform stabilizes at an elevation, relative to mean high tide, obviously lower than the elevation it would have reached in the absence of sea level rise. This result is again qualitatively similar to that displayed by the corresponding simulation described by Figure 6d. This is not surprising as Figure 8c shows that the final state of the channel in simulation A4 is quite closed to equilibrium, hence the trend is eventually close to that shown in Figure 6d. Incidentally, the reader should not be surprised to see that the channel at the end of simulation A1, which is still far from equilibrium, is much shorter than the channel at the end of simulation A5 which is close to equilibrium: indeed, in the latter case, the channel has deepened as a result of sea level rise, hence the final ratio length to inlet flow depth in simulation A5 is smaller than in simulation A1.

[99] Finally, Figure 10a shows that the marsh rate of relative accretion does not vanish everywhere on the marsh at the end of the 960 year simulations A1, A2 and A5.

3.4. The Morphological Effects of Wind

[100] Finally, we investigate the important role played by sediment resuspension in the channel, driven by a wind acting either landward or seaward. We pursue this goal by performing a third series of simulations for the same set of geometrical and hydrodynamic parameters employed in the tide dominated case (see section 3.3). In each test, we assume the presence of a single species characterized by maximum biomass density Bmax equal to 1 kg/m2. The temporal sequence of wind fields acting on the free surface was a random realization of a process characterized by the annual frequency distribution of wind speeds reported in Table 3. Again, in order to illustrate the role of various effects we will progressively include additional features to the problem. In all cases herein we allow for sea level rise and vegetation growth.

[101] No Setup. We first consider a wind blowing in the seaward direction but neglect the effect of wind setup, i.e., we restrict our attention to the effect of the wind driven wave boundary layer.

[102] The bottom profile in the shoaling region of the channel close to the marsh boundary is characterized by water depths which increase seaward. Accounting for the sheltering effect of marsh vegetation one finds that the spatial distribution of bottom stress associated wind waves evolving in the shoaling region has the following trend: proceeding from the marsh seaward, the stress initially increases, reaches a peak at some distance from the marsh border and then gradually decreases. Indeed, on one hand, as the channel deepens far from the marsh, τw necessarily decreases being inversely proportional to local depth (see section 2.1.2); on the other hand, τw must also decrease in the shallow region close to the margin, as the significant wave height Hw decreases there due to the sheltering effect of vegetation. A similar trend is exhibited by sediment transport which increases landward up to a peak and then decreases.

[103] Let us then compare the salt marsh configuration reached at the end of a 100 yr simulation in the present case with the corresponding configuration obtained in the tide dominated case for a rate of relative sea level rise equal to 2 mm/yr. A variety of novel features are brought up by the presence of wind. They are illustrated in Figure 11. (i) First, the region of the shoal located between the marsh border and the area close to the stress peak is subject to sediment deposition while erosion occurs when proceeding further seaward: as a result, the marsh progrades; (ii) secondly, salt marsh deposition is more intense than in the tidally dominated case as wind introduces an additional input of mineral sediments at the salt marsh boundary; (iii) thirdly, due to this additional depositional effect, the profile of the salt marsh changes significantly, becoming flatter than in the tidally dominated case.

Figure 11.

(a, b) Comparison is performed between the numerical solution for the longitudinal bottom profiles attained after 100 year, in cases A4, B1, C1, B2, and C2. The symbol η is the bottom elevation and x is the longitudinal coordinate aligned with the channel axis and oriented landward, with origin at the inlet. (c, d) A zoomed view of the landward reach of the channel with vegetation. The distance of the dotted lines from the bottom elevation is proportional to the local biomass density.

[104] Note that (i) is a consequence of the flood dominant character of the tide propagation in the channel region close to the marsh border. In fact, under these conditions, the net sediment flux associated with the tidal currents is directed landward and the Exner equation suggests that the region of the shoal located between the marsh border and the area close to the stress peak is subject to deposition while erosion occurs when proceeding further seaward. This process induces a seaward marsh progradation. If the shoal were ebb dominated the net flux of sediments transported by the tidal currents would be directed seaward leading to landward migration of the marsh margin. This is an important feature that will require attention when the role of tidal flats bounding the channel will be investigated, as they are known to let the tidal currents become ebb dominated in the channel.

[105] Suspended sediment is now present at the boundary between the marsh and the channel, a feature absent under tide dominated conditions with the channel at equilibrium. These sediments are advected by tidal currents over the marsh platform during the flood phase, causing marsh aggradation and eventually the formation of sort of levees close to the marsh border, arising from the fact that sediment deposition is maximum at the salt marsh edge and decays gradually in the landward direction.

[106] Note that the wind direction plays a purely quantitative role (compare Figure 11a with Figure 11c) as landward directed winds do not feel the sheltering effect of vegetation and their fetch is consequently larger than the fetch of seaward directed winds.

[107] The Effect of Wind Setup. Wind setup enhances the above processes. Note that, when the wind blows in the seaward direction, the seaward setup of the free surface simply implies a landward set down, as there cannot be any setup at the sea boundary, where the free surface elevation is imposed Wong and Moses-Hall [1998]. In this case, the sediment flux associated with the wind currents is directed toward the marsh. As a consequence, adding wind setup effects in our simulation, with wind blowing in the seaward direction, we increase the total sediment flux during the flood phase and reduce it during the ebb phase. In other words the morphodynamic effect of flood dominance, most notably marsh progradation, is enhanced (see Figure 11a).

[108] The opposite scenario occurs if the wind blows in the landward direction. In this case, the sediment flux associated with the wind setup is directed seaward: as a result, the effect of wind setup is erosive in the region adjacent to the salt marsh. Whether marsh progradation or retreat will actually prevail depends on the balance between tidal transport and transport driven by wind setup. In the case described in Figure 11b tidal transport prevails but the effect of setup is to let the marsh prograde less fast than in the no setup case.

[109] Long Term Evolution. Having clarified the physical mechanisms, let us finally examine the effect of wind on the long term temporal evolution of the marsh. This can be ascertained from Figure 12 for the case of maximum biomass density Bmax = 1 kg/m2, seaward or landward winds and two different values of the rate of relative sea level rise (1–2 mm/yr).

Figure 12.

Temporal evolution of the salt marsh bottom and vegetation distribution over the marsh. Simulations have been performed in cases: (a) A4, (b) C1, (c) C2, (d) A6, (e) C3, and (f) C4. The distance of the green lines from the bottom elevation is proportional to the local biomass density.

[110] In all cases, the salt marsh is eventually found to retreat and gradually disappear. However, various differences are brought up by the presence of wind. The position occurring in the shoal region close to the marsh margin leads to initial marsh progradation and slows down the subsequent marsh retreat compared to the no wind case. For a rate of sea level rise of 2 mm/yr the marsh disappears after 320 yrs (Figures 12a–12c). For a rate of sea level rise of 1 mm/yr the salt marsh manages to survive for 500 yrs in the no wind case and for 750 yrs in the case of a seaward blowing wind. For landward blowing winds marsh is even more resilient to sea level rise (Figures 12d–12f). Note that, in the latter case, although the sediment flux associated with wind setup is directed seaward, sediment deposition in the marsh region adjacent to the channel is more intense than in the case of seaward blowing wind. This is due to the fact that the fetch is greater when wind blows toward the marsh than in the opposite case. Anyhow, after 1450 yrs, the vegetated region disappears even in the case of a landward blowing wind. This final fate originates from two facts: on one hand a moderate organic production alone is insufficient to compensate for the effects of an albeit fairly small rate of relative sea level rise; on the other hand, the amplitude of wind waves and wind resuspension decreases as the channel deepens, hence the amount of mineral sediments reaching the marsh decreases and ceases to be able to add a significant contribution to marsh accretion. This mechanism is essentially operating in Venice Lagoon, where flats are deepening and salt marshes retreating or even disappearing.

[111] Finally, note that results do not depend on the choice of the location of the landward boundary of the marsh. Indeed, until vegetation is present, the rate of salt marsh accretion at the landward boundary obtained in our simulations appears to be only determined by organic production, which depends on the maximum biomass density, as well as by the rate of relative sea level rise. On the contrary neither wind intensity nor wind direction affect the marsh accretion rate at the landward boundary: indeed, the boundary is far enough from the marsh margin for the process of deposition of mineral sediments to be completed before reaching the landward end.

4. Discussion

[112] Let us now ascertain how the picture emerged from our simulations compares with previous results, starting from those obtained by Kirwan et al. [2010]. The main outcome of the latter work is the prediction of a threshold value of the rate of relative sea level rise, depending on sediment concentration and tidal range, above which marshes are replaced by subtidal environments. Predicted values are reported to exhibit a satisfactory, albeit qualitative, agreement with observations from several estuaries worldwide.

[113] We have then performed simulations where we have tried to interpret the role of the ‘averaged sediment concentration’ employed in zero order models. To this aim, we have evaluated the concentration of suspended sediment generated by the effect of wind resuspension, at the channel-marsh boundary (total domain lengthL = 20 km), for given characteristics of the forcing tide (a0 = 0.5 m) and checked how a marsh characterized by a maximum biomass density Bmax = 1 kg/m2 responded to different persistent wind conditions (Uw = 5, 10, 15 m/s) for different rates of relative sea level rise (srslr= 2, 6, 10, 15, 20, 25 mm/yr). These simulations allowed us to determine the threshold rate of relative sea level rise above which the marsh platform is unable to reach an equilibrium elevation: this threshold depends on the sediment concentration experienced at the channel-marsh boundary for given characteristics of the forcing tide.

[114] A comparison of the latter results with those found by Kirwan et al. [2010] for marshes subject to a 1 m tidal range is reported in Figure 13 where the solid line identifies the mean threshold rate predicted by [Kirwan et al., 2010], while the points represent the output of our simulations. Circles denote configurations where the marsh platform was able to reach an equilibrium elevation, while crosses denote cases where the marsh proved unable to keep up with sea level rise and disappeared.

Figure 13.

Predicted threshold rate of relative sea level rise above which marshes are replaced by subtidal environments. The solid line represents the mean threshold rate (±1 SE) predicted by Kirwan et al. [2010]. Circles denote configurations where the marsh platform was able to reach an equilibrium elevation, while crosses denote cases where the marsh proved unable to keep up with sea level rise and disappeared. The morphological evolution of the marsh in cases denoted by CA and CB is reported in Figure 14.

[115] It turns out that our predictions are in good agreement with the predictions of Kirwan et al. [2010]. However, an important difference exists. In fact, although not all the models used in Kirwan et al. [2010] are zero dimensional, the concept of marsh stability in Kirwan et al. [2010]was associated with the ability of the marsh surface to keep pace with sea level rise. Our concept of marsh equilibrium requires an additional constraint, namely that the channel-marsh boundary does neither prograde nor retreat. In other words, we include the morphological evolution of the environment adjacent to the marsh among the ingredients contributing to establish morphodynamic equilibrium. The latter feature, which cannot obviously emerge from zero-dimensional models, is indeed captured by our 1-D simulations. InFigure 14 we provide two examples of our simulations which led to equilibrium states in the sense of Kirwan et al. [2010]: this figure shows that in both cases the marsh boundary is not in equilibrium as it migrates seaward.

Figure 14.

The morphological evolution predicted by our simulations for the two configurations denoted by CA and CB in Figure 13, which are equilibrium configurations in the sense of Kirwan et al. [2010], show that the marsh boundary migrates seaward in both cases, with a rate depending on the concentration experienced at the marsh boundary: in this sense the marsh is not in equilibrium.

[116] The latter findings are not surprising in the light of the analysis of Mariotti and Fagherazzi [2010], who showed that, for a given rate of sea level rise and non vanishing sediment concentration at the sea boundary, a unique value of the latter exists such that the marsh boundary does not undergo any horizontal displacement. However, small perturbations of this ‘equilibrium’ condition induce migration of the marsh boundary in a direction which depends on the values of the rate of sea level rise and sediment concentration at the inlet. Note that our approach differs from that followed by Mariotti and Fagherazzi [2010]in the following respects: (i) rather than solving the full 1-D equations of mass and momentum conservation for the fluid phase,Mariotti and Fagherazzi [2010]employ a quasi-static model for tide propagation; (ii) wind waves are modeled solving the 1-D form of the wave energy conservation equation at steady state; (iii) the morphodynamic evolution of the bottom is computed by solving the 1-D form of the Exner equation coupled with the 1-D form of the advection-diffusion equation for sediment concentration; (iv) rather than followingMudd et al.'s [2009] approach, Mariotti and Fagherazzi [2010] adopted the linear relationship of Randerson [1979] for the calculation of the organogenic sedimentation rate; (v) Mariotti and Fagherazzi [2010] accounted for the effect of erosion of cohesive marsh scarps by wave attack; (vi) finally, the ability of wind acting on the free surface to generate wind currents driven by wind shear and the effect of wind currents on sediment advection is ignored in Mariotti and Fagherazzi [2010].

[117] It is of interest to note that our model does also predict the formation of a scarp. In order to show this, we have performed a series of further simulations starting from a shorter channel length (500 m). Two sets of simulations were performed, corresponding to different scenarios for the sediment availability at the inlet (0.5, 0.1 g/l), using values of wind speed (uniformly distributed between 0 and 20 m/s), storm duration (12 h every 10 days of calm), tidal amplitude (a0 = 1 m) and maximum biomass density over the marsh (Bmax = 2 kg/m2) as in Mariotti and Fagherazzi [2010]. The simulations started with an initial sloping bottom (slope 1/130). During the flood phase of the storm period, the sediment concentration at the seaward boundary was set equal to 0.5 g/l; otherwise, concentration was determined by the local transport capacity. Figure 15a shows that sediments start to accumulate at the landward side and bottom erodes at the marsh boundary. When the accreting area is close to MHT, the sediments form a terrace. Thereafter, the marsh profile progrades undergoing a rigid translation, i.e., without variations in shape. The presence of a steep profile between salt marsh and tidal flat suggests the tendency to form a scarp. In the second set of simulations (Figure 15b), we reproduced salt marsh retreat. The initial configuration was set equal to the configuration reached after 200 years in the previous simulation (see Figure 15a). Moreover, the sediment concentration at the seaward boundary was set equal to a fairly low value (0.1 g/l), such that a net sediment flux in a tidal cycle would exit the domain. Results show that erosion lowers the tidal flat and increases the slope of the platform, leading to the formation of a scarp. Our model generates a shoal profile (Figure 15a) which is flatter and more shallow compared with that obtained by Mariotti and Fagherazzi [2010]; however, the qualitative response of the flat marsh system to variations of inlet concentration agrees with that displayed by Mariotti and Fagherazzi's [2010] results. This suggests that the erosion of the marsh scarp produced by breaking waves, an effect neglected here and empirically accounted for in the model of Mariotti and Fagherazzi [2010], may undoubtedly play some role, but is not the only mechanism responsible for scarp formation.

Figure 15.

(a) Evolution of the bed profile starting from initial sloping bottom with an imposed sediment concentration (0.5 g/l) at the seaward boundary. (b) Evolution of the bed profile starting from a fully developed salt marsh (200 yrs profile of Figure 15a), imposing a sediment concentration equal to 0.1 g/l at the seaward boundary.

5. Concluding Remarks

[118] The main message we wish to deliver through this paper is the need to recognize that marsh equilibrium must be viewed in relation to the morphological equilibrium of adjacent environments. Let us summarize our conclusions.

[119] In the simple 1-D configuration investigated herein, marsh equilibrium can only be achieved provided the adjacent channel is also in equilibrium.

[120] If channel equilibrium is static (no sediment supply at the inlet), then no sediment is fed to the marsh which can only reach equilibrium provided it experiences an accretion rate associated with organic production able to balance the rate of sea level rise. With no sea level rise, the final state is an unvegetated marsh with a very sharp scarp at the margin. With moderate sea level rise, a highly productive marsh may reach a final vegetated equilibrium, with a sharp margin separating the marsh from a continuously sinking channel.

[121] If the channel is not in equilibrium initially, it undergoes a morphodynamic evolution which interacts with that of the marsh: indeed, a channel not in equilibrium is ‘active’, hence it allows for some sediment supply to the marsh. The final fate of the system, however, is not different from the previous one, albeit the timescale of the evolution needed to reach equilibrium may be much larger.

[122] If the external supply of sediments at the inlet differs from the transport capacity of the stream, the equilibrium of the system becomes even more uncertain. In the absence of a marsh, channel equilibrium cannot be attained unless the net sediment flux per unit width entering the channel balances exactly the sinking effect of sea level rise, a fairly unlikely event. When a marsh is present, equilibrium in the present sense is only achieved for some precise value of the concentration at the inlet. This equilibrium would however be unstable, as small perturbations of the inlet concentration would lead to an albeit slow marsh progradation or retreat.

[123] Our conclusions are then that, under conditions commonly experienced in the real world, where sediment supplied from the sea depends on the coastal climate, actual morphodynamic equilibrium is a rather exceptional and unstable state. This may appear not very reassuring for the reader. However, it must be appreciated that the timescale of the natural evolution process being very large (of the order of centuries), in the absence of strong anthropogenic (or climatic) effects, variations undergone by these systems are typically very slow, as shown by our simulations. A slowly evolving system may then be easily perceived as a system in equilibrium.

[124] A number of approximations limit the validity of our model. Numerical modeling of tidal flow in dry and wet regions has been performed employing an approximate scheme which has poor theoretical substantiation. Sediment transport in the channel was assumed to be equal to the local and instantaneous transport capacity, an approximation which is equivalent to neglecting advective effects compared with settling and turbulent diffusion in the advection diffusion equation for suspended load. As discussed in the paper, the validity of this approximation decreases as the sediment size is reduced. Sediment cohesion was ignored. Sediment trapping in the marsh was not explicitly accounted for. The characteristics of wind generated waves were estimated based on field observations on fairly uniform shallow lakes which were then extended to the present slowly varying geometry. The calculation of sediment resuspension by waves and currents acting simultaneously was also based on linearized treatment, ignoring breaking, a feature fairly common in shallow lagoons. Removing or improving the above approximations may undoubtedly affect the quantitative output of the model but can hardly modify the general features and the delicate nature of marsh equilibrium emerged from the present analysis. Finally, the 1-D nature of our model does not allow to fully clarify the role of tidal flats, which are delicate transitional environments between channels and marshes, which affect the hydrodynamics and morphodynamics of both channels and marshes. Developing some 2-D extension of the present model, sufficiently simplified to make long term simulations feasible but still able to capture the main physical mechanisms operating in each environment, is the next challenge for research.

Appendix A

[125] In order to obtain a formal derivation of the solution (40) one must note that, given a typical flow depth in the marsh Dm0 (say the mean flow depth in a tidal cycle), the settling process occurs on a temporal scale (Dm0/ws) which is typically much smaller than the timescale of the forcing tide ω−1: indeed, with Dm0O(1 m) and ws ∼ O((0.5–1)10−2 m/s) it follows that Dm0/ws ∼ O((1–2)102s) whereas ω−1 ∼ O(7 · 103s). Similarly, settling occurs on a spatial scale (Um0Dm0/ws) (with Um0 a typical speed in the marsh) which is much smaller than the spatial scale L0 of the tidal wave propagating over the marsh domain. The above facts can be restated in mathematical terms introducing ‘fast’ spatial and temporal variables X and T:

display math

which describe the settling process as opposed to the ‘slow’ variables ξ and τ:

display math

which describe the ‘slow’ variation of the forcing tide. Moreover, noting that the flow depth D varies on the slow spatial and temporal scales, it is convenient to rescale the vertical coordinate z by D and write:

display math

[126] With the above definitions the concentration c may be expressed as a function c(Z, X, T; ξ, τ) where the slow variables ξ and τ play the roles of parameters. The advective equation (39) may then be set in the following dimensionless form:

display math

where math formula and math formula are dimensionless longitudinal speed and flow depth scaled by Um0 and Dm0 respectively, parametrically dependent on the slow variables. Moreover, we have denoted by O(σ, α) higher order quantities which are not shown here for the sake of simplicity. They are proportional to the small parameters σ and α, which read:

display math

The equation (A4) must be solved with the initial condition

display math

where f is Rouse distribution for the sediment concentration and the quantity C0(τ) is given by matching the marsh solution with the channel solution. Since the coefficients math formula and math formula in (A4) are independent of the fast variables X and T, they can be treated as constants. The solution of (A4) is readily obtained for an observer moving with the depth averaged dimensionless velocity math formula. For such an observer the concentration varies with T and Z, i.e., we can write:

display math

Moreover, the advection equation for this observer takes the simpler form:

display math

which is readily solved by inspection and reads:

display math

[127] Note that this solution has various features. (i) It obviously satisfies the initial condition. (ii) An observer moving with the depth averaged speed math formula sees the concentration decrease from f[Z] at X = 0 to math formula at math formula (Figure A1). (iii) As the vertical coordinate Z is defined in the range 0 < Z < 1, the above solution applies in the range math formula: hence, settling is completed at math formula (Figure A1). (iv) When back transformed into dimensional form the solution (A9) attains the form (40).

Figure A1.

Vertical distribution of the dimensionless concentration c/ce at different points on the marsh (X = 0, 0.5 math formula, math formula). The terms math formula and math formula are dimensionless longitudinal speed and flow depth scaled by a typical flow speed and depth in the marsh, respectively.

Notation
math formula

Empirical coefficients (equation (29)).

a0 [L]

Amplitude of the forcing tide.

alab [LT−1]

Accretion rate associated with the production of labile (decaying) organic sediments on the marsh.

amin [LT−1]

Average net accretion rate driven by the exchange of inorganic (minerogenic) sediments between the marsh and the surrounding environment.

aorg [LT−1]

Accretion rate induced by the production of organic sediments on the marsh.

aref [LT−1]

Accretion rate associated with the production of refractory (not decaying) organic sediments on the marsh.

aw [L]

Amplitude of the oscillations of fluid displacement due to the wind waves.

Bag [ML−2]

Aboveground biomass density.

Bbg [ML−2]

Belowground biomass density.

BP [ML−2]

Annual peak of spatial density of biomass associated with aboveground halophytic vegetation.

Bmax [M L−2]

Maximum value of aboveground biomass density.

C [/]

Local flow conductance.

c [/]

Volumetric concentration of suspended sediments averaged over turbulence.

CD [/]

Bulk plant drag coefficient.

cc [/]

Vertical Rouse distribution for the mean concentration.

ce [/]

Equilibrium concentration of suspended sediments at the reference elevation.

Cv [/]

Local bed flow conductance in presence of vegetation.

C0f (Z) [/]

Vertical distribution of the sediment concentration at the initial cross section of the salt marsh at time t.

D [L]

Cross-sectionally averaged flow depth.

math formula

Dimensionless depth over the marsh platform

D0 [L]

Initial mean flow depth.

Dm0 [L]

Depth scale in the marsh domain

D0eq [L]

Equilibrium inlet depth.

ds [L]

Sediment diameter.

fb [/]

Bottom friction factor.

fw [/]

Wave friction coefficient.

g [LT−2]

Gravitational acceleration.

Gag [M L−2T−1]

Growth rate of aboveground biomass.

Gp [M L−2T−1]

Peak growth rate.

k [/]

Von Kármán constant.

kN [L]

Bottom roughness of the shoal.

k0 [L]

Free surface roughness.

klab [T−1]

Decay rate of the labile organic matter.

H [L]

Water surface elevation relative to the mean sea level.

Hw [L]

Wind waves height.

Lc [L]

Length of the unvegetated portion of the total domain.

Lmarsh [L]

Marsh length.

L [L]

Total domain length.

L0 [L]

Tidal wavelength.

Leq [L]

Equilibrium channel length.

Mt [M L−2T−1]

Local mortality rate of the total biomass per unit area.

Mag [M L−2T−1]

Local mortality rate associated to the above ground biomass.

mbg [M L−2T−1]

Local mortality rate associated to the below ground biomass

mlab [M L−2]

Mass of the labile fraction of organic material per unit area

n [/]

Unit vector normal to the bed

p [/]

Sediment porosity

po [/]

Organic sediment porosity

P[T]

Period of the biomass cycle, assumed to be 1 yr.

qs [L2T−1]

The total sediment flux per unit width instantaneously transported by the flow field at a given location

qsb [L2T−1]

Bed load component of the total sediment flux

qss [L2T−1]

Suspended load component of the total sediment flux

q [L2T−1]

Sediment flux

RE [/]

Reynolds number

rw [/]

Wind friction coefficient

scom [LT−1]

Rate of marsh lowering arising from compaction of sediment deposits

srslr [LT−1]

Rate of marsh lowering arising from regional subsidence and sea level rise.

t [T]

Time

Tt [T]

Tidal period

tp [T]

Time of the year when aboveground biomass is at its peak

ts [T]

Phase shift between the maximum growth rate and the maximum biomass

T [T]

Fast temporal variable

Tw [T]

Wind wave period

ufsb [LT−1]

Bottom friction velocity

Ufss [LT−1]

Surface friction velocity

U [LT−1]

Cross-sectionally averaged flow speed

U0 [LT−1]

Scale of the longitudinal velocity in the channel

Um0 [LT−1]

Scale of the longitudinal velocity in the marsh

Uw [LT−1]

Uniform wind speed acting at some conventional distance from the free surface

U1m [LT−1]

Amplitude of the fluid velocity oscillations due to the wind waves

ufc [LT−1]

Friction velocity of the tidal current.

ufwc [LT−1]

Friction velocity of the wave-tidal currents

uwc [LT−1]

Vertical velocity profile for a combined wave-current motion

uln [LT−1]

Logarithmic distribution of the velocity profile of the tidal current

math formula

Dimensionless velocity over the marsh platform

ws [LT−1]

Particle settling speed

x [L]

Longitudinal coordinate

X [/]

Fast spatial variable

xf [L]

Fetch

z [L]

Vertical coordinate

Z [/]

Dimensionless vertical coordinate

zr [L]

Reference elevation with respect to the bottom for the equilibrium concentration

α [/]

Dimensionless parameter in advection equation

αp [L−1]

Projected plant area per unit volume

χlab [/]

Fraction of deposited labile organic matter

δsl [/]

Dimensionless parameter measuring the ‘settling lag effect’ in the channel

δs [L]

Stem diameter

δ [L]

Boundary layer thickness due to the wave currents interaction

ϵs [L2T−1]

Isotropic turbulent diffusivity.

γ [L3T−1 M−1]

Randerson's constant

η [L]

Cross sectionally averaged bottom elevation with respect to the mean sea level.

math formula

Spatially averaged marsh elevation relative to mean sea level

ν [L2T−1]

Kinematic viscosity

image[T−1]

Peak growth rate coefficient

ρ [M L−3]

Water density

ρs [M L−3]

Sediment density

ρa [M L−3]

Air density

ρo [M L−3]

Density of the organic material

σ [/]

Dimensionless parameter in advection equation

τ [/]

Slow temporal variable

τw [M L−1T−2]

Maximum shear stress generated by the wavefield

τwc [M L−1T−2]

Average shear stress generated by the combined actions of wave and current

τw [M L−1T−2]

Maximum shear stress generated by waves

τc [M L−1T−2]

Shear stress driven by the current

τwcmax [M L−1T−2]

Maximum shear stress due to the combined action of waves and currents

θ [/]

Shields parameter

ω[T−1]

Angular frequency of the forcing tide.

ξ [/]

Spatial slow variable

Acknowledgments

[128] The first author holds a Researcher position at the University of Genova. Funding provided by Thetis SpA (Venezia, Italy) to establish this position is gratefully acknowledged.

Ancillary